On Numbers and God

You...

2010-12-02T02:56:00.002-06:00

I wrote this a while ago, and nearly forgot about it. Found it in an old forum I no longer post in:

Like every day, I've spent most of today doing math. Sheet upon sheet black with ink, dusty green chalkboards covered in arcane symbols; definitions and theorems that are far too elaborate to explain. For the first time in a while I remembered: I cannot convince myself that any of this is real. There are no infinite collections or compact spaces in the real world. There is no finite axiomatization for life. This is all meaningless symbols on paper.

You, however, are real. Your breath, your eyes, the taste of your lips, are ever so beautifully, painfully real. You have meaning-- you mean everything. But reality and meaning...

They scare me. Far, far too much. There is no reason, no logic behind it, but I am afraid. So I will stay in my fantasy world. I will miss you terribly, but infinity is easier to deal with than love.

10 things your barista should want you to know.

2010-10-05T22:01:00.001-05:00

This post has been removed.

It was really bad... It was mostly a place for me to complain about something that really grates me (the "ghetto latte"), and to talk about something that people don't seem to think about: unseen costs. The biggest cost for a business is operating costs, not the cost of the product itself. "Doing it yourself" costs a lot more than it may seem at first. Sometime soon(tm), I'll write something about the idea of unseen costs directly. In a less obnoxious and boring way.

Cheers.

Hungarian math education

2010-08-19T01:11:00.001-05:00

Since going to Hungary, I've been wondering why exactly the math education in Hungary is so great; there hasn't been a concerted effort to "improve curriculum" or any formal attempt to make the system so great, but Hungarian math education is fantastic, at least from high school on. The Hungarian math circle got started as something of a spontaneous cultural phenomenon, but I think there are some deeper cultural reasons that it sprouted.

Today I was thinking about Hungary. The things I miss, as well as the things I found annoying. One of the annoying things is the Hungarian mentality. In part because of 800 years of sidelining and oppression from almost all of their neighbors, and in part because of the depression which came from Soviet influence, Hungarians are very reserved, and wear a facade of depression. Along with this, the Hungarians picked up from the Habsburgs a German practicality. As a result of culturally enforced depression, and culturally enforced practicality, open display of excitement and passion for something are frowned upon. If you don't believe me, spend a week or two in Budapest, and watch for how easy foreigners are to spot (hint: they're the loud people who laugh in public), and watch how silent and serious children are.

Hungarian mathematicians, as opposed to most other Hungarians I met, are very excited, passionate people. I think a small group of young students who were interested in math, and couldn't have given a damn what other people thought of them were very public about their passion for math, and this became a sort of counter-culture movement in post-war Hungary. Youth who wanted to open up found this community as a natural place to revolt against the sullen Hungarian attitude. As with most "revolutionary" cultural movements, this group pushed the boundaries. A lot of modern methods and ideas in combinatorics and set theory came out of this group when they were still pretty young.

The math culture in Hungary has perpetuated itself quite well. Partly, this is the natural result of passion being imparted to the students by the instructors, but I think it's largely a continuation of the revolt against the Hungarian mentality: math remains culturally acceptable, but at the same time disillusioned youth can express themselves freely in a culture which continues to uphold an image of stoic depression.

(Or maybe I'm being too hard on Hungarians... Bocsanat, Magyar!)

LaTeX

2010-07-06T02:55:00.001-05:00

It seems the LaTeX engine I use on this blog is broken... I'll be replacing it soon, assuming it doesn't fix itself first.

Structuralist philosophy and methodology in mathematics

2010-07-06T02:53:00.002-05:00

There is a philosophy of mathematics (or rather a collection of related philosophies) called structuralism. In brief, a structuralist believes that mathematical "objects" are positions in a structure, rather than existent objects. This is a rather incohesive and shallow ramble about structuralist methodology and philosophy in mathematics.

Since I hold a rather formalist, and somewhat classical platonic view of mathematics-- which is to say, I do not believe mathematical notions exist in the real world in any real capacity, but rather in some external abstract universe-- I intend to talk about brands of structuralism which do not in any way invoke "reality". Perhaps mostly because of my biases against "the real world", I cannot rightly fathom philosophies of mathematics which invoke the real world in any way.

Anyway, it seems that structuralism as a methodology pre-dates structuralism as a philosophy. What is a "structuralist methodology"? It is the approach which emphasises structures over systems. To use some language from logic, a structural methodology approaches theories, instead of models. A simple example is the tacit tendency to forget the difference between isomorphic groups: $\mathbb{Z}/3\mathbb{Z}$ is "the same group" as $\mathbb{Z}_3$. From a purely set-theoretic or material point of view, this is not correct: the first group has cosets of $3\mathbb{Z}$ in $\mathbb{Z}$ as group elements, and the second has $\{1,2,3\}$ as its underlying set. But the two groups are isomorphic, which means that they act the same as groups The tendency to forget the (quite irrelevant) difference between the two groups is the heart of structuralist methodology.

The Bourbaki group was one of the first to emphasize "high abstraction". Their methods are truly similar in spirit to the modern category-centric structural approach. While I have never read Bourbaki, all the information I can find leads me to believe that the set-theoretic foundation is a result of 2 things: when Bourbaki started, set theory was the only thing to work with (categories had not yet been invented), and "Bourbaki is relentlessly linear in its exposition". With this linearity in mind, changing to a categorical perspective late in the game was out of the question.

The structural approach permeates mathematics, particularly in algebraic areas, and in almost all contemporary approaches to foundations. Isomorphic structures are taken as identical; in category theory (and especially in higher category theory), there is a real push to eliminate notions which do not interact sensibly with equivalence-- equivalence is a weaker notion than isomorphism, but it is still considered a "good enough" notion of equality.

With the ubiquity of structural methodologies in mind, it should be no surprise that a closely related philosophy should spring up. I'm only surprised that it took so long (at least 40 years from the start of Bourbaki) to really pin down "structuralism". A structuralist philosophy takes this methodology as a philosophical starting point: it is not simply productive to study mathematical ideas from a structural viewpoint, but mathematical objects are structures. 3, for example, is not a specific set (e.g. {{}.{{}}.{{}.{{}}}} or {{{{}}}}), but rather a convenient short-hand for any object satisfying a "3-like" position in a structure. This seems "obvious" to me, since any structure which satisfies the Peano axioms will have natural number arithmetic. My formalist tendencies are at work here; the notion of an intended model seems somewhat foreign to me. There are many, many ways to construct the reals within ZFC; if they all act like reals, then what is the "correct" model? All statements true in a specifc model, but not in others are not part of real analysis; the "correct interpretation" is one where real numbers are taken as sui generis objects.

Finally, a change of topic. There seems to be a deep relationship between structuralism and phenomenology, which seems under-explored. Levinas, for one, makes a big deal of "existence without existents"; that is, being without thing-ness. This is exactly the idea of structuralism: we are studying mathematical notions without reference to a specific object to which the notion applies.

With You

2010-06-21T02:10:00.003-05:00

I originally wrote this in Ghent (or Brussels?) in early January, but I've had trouble tying it all together. I'm still not competely satisfied with it, but I am content, so here goes.

Edit: Ugh. There are a few lines which just won't work... the first stanza is so seperate from the rest of the poem; I am sorely tempted to cut it out entirely, but I have very difficult-to-explain reasons for wanting to keep it.
Edit again: Yeah. I think the first stanza is the biggest problem. Until/unless I find a way to express what the first stanza is supposed to express, consider the poem to start with "Through days and day-dreams".
Edit again (again): I've removed the first stanza from this post.
****

Through days and day-dreams, I walk alone,
along ancient ways on cobblestones.

Yet in all my wandering, there is one thought
which fills my heart:
I would like,at last,
to be warm at Home.
In Bed.

A few thoughts on the officiating in the 2010 World Cup...

2010-06-18T21:18:00.002-05:00

Edit: Sorry about the lack of linkage... I'm too lazy to dig back up all my sources

I'll try not to bore you too much with sports, but I'm super-stoked for the World Cup, and as always, I have some skepticism about some of the officiating.

I was excited last week when the officials were completely, always right. There were a number of times (in each game) when I said "are you kidding?" and then when watching the replay, I realized that the referee had made the correct call, and in a hard-to-see play. My hat's off to them.

Unfortunately, this hasn't kept up. A few days ago (I've forgotten which game), a goal was scored by a clearly off-sides striker. Why wasn't it called? The assistant referee was 3 or 4 yards up-field from the last defender. This is possibly pardonable in a club match, but not at the international level; especially in the World Cup finals.

Then today, we see an inconsistent, booking-happy referee in the Serbia vs Germany game. I'm really skeptical about most of the cautions he made. Regardless, he was inconsistent the whole match, and clearly was approaching his role with a very different mentality in the second half.

And of course, it'll be a while until people shut up about the USA-Slovenia draw... From the first moment I was questioning the referee-- he was inconsistent, missed some clear infringements, and called some spurious "fouls" where there was little contact, and a clean challenge. There were a number of nearly identical little pushes from behind throughout the match. One earned a caution, 2 (I think) earned the proper free kick they deserve, and at least 2 earned an absent-minded turn of the head.

The first booking had me laughing. Findley's yellow for a "handball" (clearly unintentional and off of a hand in a natural position-- viz, clearly not an infringement according to the Laws of the Game) had me groaning. The third booking was fair. The late-game caution near midfield was ever-so-slightly iffy. Had the referee not lost the benefit of the doubt with sub-par decision, I would hardly even bring it up, but his performance all game was questionable enough that I don't feel bad questioning it.

The booking on Josy's break-away was hardly cautionable-- the only reason I see for a booking was stopping a "clear goal-scoring opportunity", so if there's a booking, the guilty player should be sent off.

Finally, we get to the decision that Americans will be moaning about for months to come-- the spurious infringement during Donovan's (would-be goal-scoring) free-kick. Firstly: the announcers were wrong, it was not off-sides, the referee explicitly said it was for a foul. He did, however refuse to say what sort of foul, and who committed it.* Pictures of the play show what appears to be no less than 4 Slovenian defenders fouling American players, and I've seen only one picture which shows what may be an American foul-- Bocanegra appears to have his arms around a defender (Pecnik). In that picture, it looks to me like Bocanegra is falling, and (instinctively) lifting his arms to grab something. Watching the replay only confirms that pictures can never capture what's going on before set-pieces-- a pair of the apparent Slovenian infringements clearly were not, and I can only find 2 certain infringements: Radosavljevic bear-hugging Bradley and Cesar holding back DeMerit. Also, Bocanegra (as I suspected) is laid out at the beginning of the play (He's the guy rolling around when Edu shoots). I can't tell how legitimate his fall is, but considering Pecnik doesn't come down with him, he is clearly not holding Pecnik.

In any case, no less than 2 fouls in the action of the play are missed by the referee, and we have no idea what it was he did see.

*Two notes: first, it is officially (according to something I read on the internet...) a foul against Edu. Considering he hardly touches the only defender who's near him, the only possible call is offsides. Which the referee denied. And which he clearly wasn't.

Second, apparently this is coyness is within the referee's rights. Even in the post-game report, referees are not required to say (i)who committed an infringement, or (ii)what the infringement was, unless the player is booked.
During the game, fine, but in the post-game report? That is unacceptable, precisely for situations just like this. No one knows what the call was, and apparently, this includes the esteemed Mister Coulibaly. Fans need to know what it was that he saw.

Unlike many (American) fans, I won't say that Coulibaly should be investigated for any sort of gross misconduct-- he was hardly fair to the Slovenians, and this is not the first match where his decisions have been loudly questioned-- and I won't say that FIFA should rectify anything. I also (unlike many fans the world over) won't say that new technology should be used in-game to rectify wrong decisions-- part of the beauty of soccer is the pace, and I'd prefer FIFA avoid bad decisions by being more vigilant about using referees with a history of very controversial decisions (3 of the 5 Africa Cups he's refereed in!), than by slowing the game down. But I will say this:

FIFA needs to force referees to say who committed an infringement, and what it was, at least in a post-game report-- allowing the referee to archive these after re-watching the game, so as not to slow down the game.
FIFA needs to make public statements whenever a referee consistently makes controversial calls, and whenever a referee makes a highly controversial call. Either defending the referee by saying something along the lines of "You may not agree with his interpretation of the Laws, but we consider them acceptable", or admitting that the referee was wrong. A clear procedure for such situations (for petitioning for a statement, and for the subsequent review.)
In the case that the referee is wrong, FIFA needs to administer punative actions. If a player acts in an unacceptable manner, they are typically fined, and often suspended for longer than the one-game suspension that comes with a send-off. Similar measures should be instituted for referees whose decisions are not within acceptable interpretation of the Laws.

Edit (About 20 minutes after posting): It seems FIFA is in fact reviewing Coulibaly's performance, and will likely be unassigned from future matches in the tournament. This is a good first step, but I'll be waiting for a statement from the tournament organizers... I'm also interested in hearing from FIFA regarding the censorship claims

Art and Meaning

2010-06-07T22:00:00.000-05:00

A recent Abstruse Goose reminded me of Preface to Dorian Gray. Of course, people deride critics all the time. Especially modern art critics. I tend to defend art critics even though I don't understand them, because I know they are working from a different framework, a different background and dialogue than most people (including myself), and that this framework is not necessarily wrong.

But this Abstruse Goose reminded me of something that I don't like about... well, most people who discuss art, and this thing is what reminded me of Dorian Gray: Trying to find meaning in art is to murder it. The criticism and analysis of art should concern itself only with the aesthetics of the art-- the emotional complex that the art inspires. Yes, a piece of art can have a moral, or a meaning, but if the piece does not stand without that moral, then it is a failure as a work of art. The meaning should be extracted and the work discarded.

If something can be said, it can be said directly. "But," you say, "what about metaphors and paraboles?" Yes, they have their place, but they are not art, and should not be treated as such. Pilgrim's Regress is not a good piece because it tells the tale of someone coming to God; it is a good piece because it captures the beauty, the sehnsucht that helped to define Lewis's faith. Without capturing this beauty, it would be nothing. Speaker for the Dead is not a good piece because it questions the neo-colonialism inherent in anthropological methodology; it is a good piece because it captures so many emotional struggles.

I mentioned two novels, because novels most often do have a meaning; but the meaning is not what makes them good art. Aesthetic considerations are what make or break a novel. On the other hand, most visual art, and most good poetry do not have a meaning, and trying to suck meaning out a poem or picture is to fail to see the piece as art. This is unforgivable.

You may say that understanding the meaning helps to appreciate the aesthetics. Take my example of Pilgrim's Regress: is the aesthetic appeal not greater appreciated by understanding the God and the path to salvation? Yes, it surely is, but this understanding is assumed by the work, not expressed by the work. Someone approaching the work from a framework similar to Lewis's own will be better able to appreciate the work. But this says nothing about meaning. Understanding the background of a work, understanding the dialogue that the work is a part of and the "etymology" (if you will forgive the abuse of terminology) of the work will give someone a better appreciation for the aesthetics of the work. But to say that this is achieved by studying the meaning of the work is to commit cum hoc ergo propter hoc. Just because understanding the background helps both to understand the meaning and the aesthetics does not mean that understanding the meaning helps you to understand the aesthetics.

By the way, this is why you will rarely (never?) see me engage in a semantic analysis of a poem; I will study it syntactically, and I will study the way images are invoked, but I will not attempt to understand the author's intent, or understand the meaning of a poem. If the intent is anything but to create something of beauty, then I don't care.

Another poem.

2010-05-22T22:29:00.003-05:00

Can't think of a title:

A man. A woman.
Sitting on the steps
of a Church in London.

A stifling silence.
Seconds fade. The man sighs,
"So what are we to do?
Will you be my tragic muse?"

A mirthless smile.
A soft sad kiss on the shoulder.
And a heavy, hesitant stance.

A man. Sitting
silently on the steps
of a churchyard, thinking.

Categories of paths; functors and natural transformations...

2010-05-22T19:48:00.003-05:00

Somehow blogger screwed up, and a post I started (but never finished) a long time ago is "older" than another post of mine. It's about path categories, and higher categories... Hopefully worth a read?

Mu/Wu...

2010-05-06T15:45:00.004-05:00

Mu is a Japanese word. I'll let you look it up there, since it's easier.

Programmers tend to like to re-imagine koans and Buddhist stories; so here's a classic question re-envisioned.

Prelude> Does the dog have the Haskell nature?

Couldn't match expected type `Human' against inferred type `Dog'.
In the first argument of `Haskell nature?', namely `dog'
In the expression "Does the dog have the Haskell nature?"

Cheers.

Categories of paths; functors and natural transformations thereon

2010-04-19T21:58:00.009-05:00

Remerber when I said I'd post something about natural transformations and path categories? Well, here it is.

I talked briefly about paths in that post. Let's talk about them a little more. We want a way to talk about a path from a to b in an arbitrary topological space. In $\mathbb{R}$, this is easy enough: take any portion of a curve which starts at a and ends at b. While that's easy to understand, it's a bit unwieldy to work with directly. But we do know that a curve is some sort of a function (think back to 8th grade algebra). So, let's decide this curve is a function. And let's say $f(0)=a$, and $f(1)=b$, and for every $c\in [0,1]$, $f(c)$ lies on the curve we just talked about.

There are a two important things to notice here:

We now know the domain of our function: the unit interval (let's call it $I$).
This curve should be continuous, or else we have to jump... and what kind of a path is that?

There's something great about these two facts: Where did I mention that our path is a path on $\mathbb{R}$? Nowhere. This means that we can replace $\mathbb{R}$ with any topological space.
So: A path on a topological space $X$ is a continuous function $p:I\rightarrow X$, and we call $p(0)$ the starting point, and $p(1)$ the ending point.

I mentioned categories in the titles, so you may be wondering about now where the category comes from. Let our objects be the points of $X$, and let a morphism from $a$ to $b$ be a path with end-points $a$ and $b$. Is this actually a category? let $id_a$ be constant function $\forall c\in I, p(c)=a$, and let the composition be convolution: $q\circ p = p*q$, where $p*q (c) = p(2c)$ if c<1/2 and $q(2(c-1))$ otherwise.

We run into one big problem here: composition isn't associative, and composition with the identity isn't quite right... but they're both close. You hit all the same points in the right order, but the "speeds" aren't quite right.

We can rectify this with homotopies, but if we use just any homotopy, we'll get pretty boring spaces... so we do what topologists always do to their homotopies when they need to restrict them: fix the end points. So, two paths equivalent up to homotopy with fixed endpoints are now the same.

This gives us a topological space as a category. If this is a category, we should be able to get continuous functions as functors. Yep!
If $F:X\rightarrow Y$, a path $p$ will be sent to the path $F\circ p$. (Exercise: Check that this is indeed a functor, with our wishy washy paths.)

I'm not sure if I've talked about natural transformations. A natural transformation intuitively is a way to transform one functor into another, at the objects.
For two functors $F,G:C\irghtarrow D$, a natural transformation $\eta:F\rightarrow G$ is a collection of morphisms $\eta_x : Fx \rightarrow Gx$, one for each object of $C$. These morphisms need to interact properly with $F$ and $G$. Namely, if $f:x\rightarrow y$ in $C$, then
$\eta_y\circ Ff = Gf\circ\eta_x$. In other words-- starting from $Fx$, following $Ff$ and then a component takes you to the same place as following a component, and then following $Gf$. This must happen everywhere.

One way to understand what's going on is this: There are 3 worlds, $C$, and two other worlds living inside of $D$: $F$-world, and $G$-world. The components of a natural transformation allow us to travel from $F$-world to $G$-world, and if $\eta$ is truly a natural transformation, than we can travel from $F$-world to $G$-world, and then around $G$-world, or you can make the same trip in $F$-world, and then cross over to $G$-world, and either way, you end up in the same place.

What do natural transformations look like when our categories are these path-spaces? Let's see if we can't figure out what is going on. First, let $F,G:X\rightarrow Y$, where X and Y are topological spaces. Now let's look at a path p with endpoints a and b. This will give us two paths in Y, $Fp:Fa\rightarrow Fb$, and $Gp:Ga\rightarrow Gb$. We need a way to turn the first path into the second... How do we do this? Take a homotopy $H:X\times I\rightarrow Y$, where $H_0=F,\; H_1=G$. If we take $H\circ (p\times id)$ (I.e., we take our path, and then apply the homotopy), we get a homotopy from $Fp$ to $Gp$. If we further restrict ourselves to $H_0$, we get... a path $Fa\rightarrow Ga$, and likewise, if we look at $H_1$ we get a path $Fb\rightarrow Gb$. So if $H$ is truly a homotopy, it defines a natural transformation from $F$ to $G$. Likewise, if we have such a natural transformation, we can define a homotopy.

So natural transformations are homotopies. I'm going to stop here, but a fun remark: natural transformations turn Cat what's called a 2-category. So our "2-dimensional" homotopy space (i.e., paths as morphisms and homotopies as natural transformations) turn Top into a 2-category. We can keep going: homotopies between homotopies make 3-morphisms, homotopies between homotopies between ... form n-morphisms, and suddenly we have some notion of $\infty$-category. Moreover, homotopies (and paths) are invertible; which means we actually have an $\infty$-groupoid.

And hopefully that helps motivate some of (higher) category theory. Cheers.

(Disclaimer: There may be gross inaccuracies in this post... please let me know if you find any)

Interesting notes on Dedekind

2010-04-10T11:04:00.002-05:00

I'm reading through Dedekind's The Nature and Meaning of Numbers(as translated by W. Beman), an early treatment of set theory. I find the following convention interesting:

A system [set] $A$ is said to be part of a system $S$ when every element of $A$ is also element of $S$. Since this relation between a system $A$ and a system $S$ will occur continually in what follows, we shall express it briefly by the symbol $A\subset S$. The inverse symbol $S\superset A$, by which the same fact might be expressed, for simplicity and clearness I shall wholly avoid, but for lack of a better word, I shall sometimes say $S$ is whole of $A$ [$S$ contains $A$], by which I mean to express that among elements of $S$ are found all the elements of $A$. Since further every element $s$ of a system $S$by (2) can be regarded as a system, we can hereafter emply the notation $s\subset S$.

(The bold is mine, and the symbol used by Dedekind is not, in fact, $\subset$, but the same symbol is used throughout. The [...] is also my own clarification.)

The question is, of course: is he confusing the two notions $A\subset S$ and $s\in S$, or is he just abusing notation? Considering the context, I doubt the latter, so it would appear he is confusing the two notions. On the other hand, his reasoning seems to be clear throughout, and points (1) and (2) (the text of which I will not force upon you), seem to suggest that he well understands the difference between the idea of "system" and "thing" (as he puts it), that I find the first alternative likewise hard to accept. Although perhaps not, as (2) may provide the source of his confusion. He says "For uniformity of expression it is also advantageous to include the special case where a system $S$ consists of a single (one and only one) element $a$, i.e., the thing $a$ is element of $S$, but every thing different from $a$ is not an element of $S$." This seems to suggest that that the confusion is not elementhood versus subsethood (forgive the abuse of the English language...), but rather, $a$ and $\{a\}$. Either way, it's fascinating, and it seems that this confusion (if that's what it is), does not pop up in the rest of the text.

Another interesting point is that I see the first (that I know of) use of a few common words and notations. A few that come to mind:

The word identity to mean what is commonly meant in the mathematical community. That is
The simples transformation of [function from] a system is that by which each element of its elements is transformed into itself; it will be called the identical transformation of the system.
. for composition: "This transformation [the composition] can be denoted briefly by the symbol $\psi .\phi$ or $\psi\phi$." This same paragraph has the first proof I've seen that sets with functions forms a category... albeit, not in those words, and as his set theory is naive, it is not technically correct (as it is not even a consistent system!)

There also appears to be (at least) two flawed proofs.
He shows first that $f(A)\subset f(B)\Rightarrow A\subset B$, and from this concludes that $f(\cap_{i\in I} A_i) = \cap_{i\in I} f(A_i)$. (You can show the first is false by taking some $s\in S\setminus B$, and mapping it into $f(B)$. Then $A=\{s\}$ is a counterexample. Actually, the second is quite correct, assuming the first statement...) Although again, I'm not quite certain: This theorem appears where he is discussing bijections (which he calls "similar transformations"), which might lead one to believe he means only in the case of bijective functions, but in every other theorem in the section he is clear to point out if the function is supposed to be bijective. Further, it appears every function is a priori surjective up until the next section.

While I'm picking apart such a crucial text, I might as well continue complaining: the translation is also infuriating at times as it translates phrases such as dann ist $A$ and dann gibt es as "then is $A$" and "then is there", rather than the more natural "then $A$ is" and "then there is". I do like Miltonic inversion, but this is hardly poetic writing...

On algebra

2010-04-04T01:25:00.004-05:00

Here, someone posted a question about "what algebraists do."

I think the question is interesting, and I like my response (man, does my voice sound good... or something), so I'm posting it here. As the discussion progresses, I'll continue to update this post.

*****
(forcesofodin)
Seems like most of the math majors at my school call themselves algebraist. I really am unsure what an algebraist does. It seems like they're the mathematical equivalent of biologists, observing, categorizing, all the while linking categorizations and members thereof together in new (sometimes surprising ways). But having a name and label for everything (it's been done with finite groups I believe) seems to uninteresting a goal for so many people to be algebraist. Indeed, over categorization and labeling breeds repugnant amounts of technical terms. I know some people like to name-drop with technical terms, but to me it seems more beneficial working to not use the technical terms, to be able to explain to those without the background. Even our major tools, the morphisms, are just ways of categorizing new groups/rings into a variety of already encountered sub types of rings/groups. Such a goal would be wholly useless for someone doing analysis on differential equations.

*****

(me)
In the grand scheme of things, what mathematicians do is categorize and describe increasingly abstract structures. This isn't a pursuit unique to algebraists. The Poincare conjecture was part of a classification movement which is similar in spirit to the classification of finite groups: What manifolds are diffeomorphic to R^n? To S^n? Through the 20th century, you saw the same push for this classification as you did for FSGs. You also see similar attempts to classify things in graph theory-- there are two "forbidden" minors for planarity, but there are 33 (I think?) for the projective plane, and hundreds for other spaces; graph theorists are spending a good deal of time categorizing embeddability.
When you look at category theorists (who I consider to be algebraists...) you see that they aren't categorizing (uh... sorry) anything any more than anyone else-- in fact higher dimensional category theorists are just starting to really figure out what exactly it is they're trying to talk about; they don't have a whole lot of time to worry about how to taxonomize these things.

Regarding term-dropping: Think of terms like Hausdorff, regular, normal, and compact in topology (and continuous, uniformly continuous in analysis); these are all convenient shorthands that say "the object we are looking at satisfies some extra properties." These extra properties give us information about the structure we are looking at. Would you really rather I say "Let G be a topological space for which every open cover has a finite subcover" every time I talk about compact spaces, or would you rather I say "Let G be compact" and move on to what I'm trying to say?
Yes, there are people who like to drop big words to feel good about themselves, but the point of these abstract, esoteric definitions is not to be esoteric or precocious-- the point is to get past the things we see over and over again, and move onto what we're trying to talk about. The words, like any word, are a way for us to communicate information efficiently. Because mathematicians work with new structures all the time, we have to also be in the business of creating language. Since we are trying to describe structures for which there has never been a need for words, by using other structures for which there has never been a need for words, anything we tried to say would very quickly become unruly if we didn't have a quick way of saying it.

My favorite example recently is from ETCS (a structuralist set theory): the axioms for it can be very conveniently stated "The category of sets is a well-pointed topos with a natural number object satisfying the axiom of choice." If you know what a well-pointed topos is, what a natural number object is, and what the axiom of choice is, then you understand the axiom system. Compare that to ZFC-- while the ZFC axioms might be easier to pick apart (explaining the whole axiom system for ETCS in words that "any" mathematician could understand immediately would take... a while), a number of mathematicians are familiar with all 3 of the things needed to understand that axiom (at least, as familiar as they are with the formalism of ZFC) from other areas, so this sentence conveys a good deal of information-- so long as you have the language. It allows someone talking about ETCS to move past the definition, and get to "real" mathematics quicker.

Anyway, onto your question "what does an algebraist do?"
That's a difficult question, in large part because "algebraist" is a much vaguer term than "analyst" or "topologist". A category theorist could be called an algebraist, someone doing finite group theory will be using very different methods than someone doing infinite group theory, and they work with completely different structures than someone doing ring theory or galois theory.
So the question becomes: what about a pursuit makes it "algebraic"? I would say the focus is on some notion of transformation. An action is "algebraic" if it involves pushing some object through a transformation to see what happens. An algebraist studies the way these transformations interact with each other. Turning it around "algebraic [insert mathematical field here]" is the study of a given class of objects (those of the mathematical field we are "algebra-izing") by studying how the objects move under these transformations.

So, I would say an algebraist studies transformations. This is my principal reason for calling category theorist algebraists: when it comes down to it, they aren't studying categories, they are really studying functors and natural transformations-- ways that categories can be transformed.

Also, you say

But having a name and label for everything (it's been done with finite groups I believe) seems to uninteresting a goal for so many people to be algebraist.

Interestingly, there was a discussion about the classification of finite simple groups on the FOM mailing list, in which someone said John Conway was "pessimistic" about the classification: he meant that Conway was pretty sure the classification was complete. So mathematicians spend a good deal of time classifying things, but really, the goal isn't to classify things, it is to understand the structures that we see. The classification is a (possibly unfortunate, possibly fortunate) side effect.

Cheers,
Cory

*****

I failed to respond to the following statement in the above, and I'm feeling rather philosophical (and not particularly sleepy... and also, apparently, verbose) today, so I'll say something about this.

Even our major tools, the morphisms, are just ways of categorizing new groups/rings into a variety of already encountered sub types of rings/groups. Such a goal would be wholly useless for someone doing analysis on differential equations.

that's not all what morphisms are; A morphism from an object A to an object B is a way of saying that you have a relation between to objects-- it means you can say something about B by looking at A (or just as often, you can say something about A by looking at B). The beautiful thing about morphisms is that they show up everywhere: functions are morphisms of sets, homomorphisms are morphisms of (algebraic) structures, continuous functions are morphisms of (topological) spaces, paths are morphisms of points (in a topological space), homotopies are morphisms of continuous functions, proofs are morphisms of propositions, functors are morphisms of categories, natural transformations are morphisms of functors (in more than one way)... the list goes on; an example which is close to home at the moment is morphisms of graphs: a k-coloring of G is a morphism from G to the k-clique*.

In fact, whenever you have a transitive, reflexive relation, you have morphisms, and vice versa. The idea of morphism has very much permeated all of math. Even if it's not (explicitly) being used in an algebraic sense, this categorical language is becoming more and more common, because it very nicely captures something all of matehmaticians do: apply a certain type of function to our objects. What type of function? One that preserves the "interesting" structures of our object. I find it hard to believe that such a general and pliable notion is useless for any mathematician.

*There are some really great results that prove the colorability of whole classes of graphs, simply by making use of composition of morphisms, and apparently graph homomorphisms are being used to precisely and neatly say things that could only be said using rather messy and approximate arguments before.

*****
(forcesofodin)
fair enough, I wish you had taught me algebra.

*****
(pseudonym)
There's so much more to algebra than groups, rings and fields! In broad terms an algebra is just a pair $(X,\Omega)$, where $X$ is a set and $\Omega$ is a set of operations of finite arity on , in which a number of additional rules may hold governing the actions of the operations. The additional structure that can be placed on a general algebra, such as demanding that certain identities hold in the application of sequences of operators (e.g. associativity etc.) make the concept of an algebra very flexible in what it can be used to model. Along with the familiar objects mentioned above algebras have applications in order theory (lattices), logic (boolean algebras with operators, cylindric algebras etc.), theoretical computer scientists can even use algebra to describe the way computer programs work (Kleene algebras), and there is plenty more besides these examples.

With regards the terminology, on an undergrad course it can seem like its just there for its own sake. You prove a lot of stuff that seems like busywork. But this is just because even relatively advanced undergrad/beginning grad courses are really only introductions. They're trying to give you an overview of the tools that are available but they rarely have time to motivate them by going into the problems from which the definitions emerged.

*****
(me)

fair enough, I wish you had taught me algebra.

No you don't, I really don't have the background in algebra I should considering the amount of time I spend raving about it...
(If only I spent that time doing it...)

Thanks for your post, pseudonym, that's a really important point; it also explains why "algebraic combinatorics" focuses so much on lattice theory. (At least, if my description of "algebaric ___" is correct in general.)

Also,

But this is just because even relatively advanced undergrad/beginning grad courses are really only introductions. They're trying to give you an overview of the tools that are available but they rarely have time to motivate them by going into the problems from which the definitions emerged.

This is definitely the hardest part of math education, and is also one of the biggest problems (although there may not be a good solution to it.) The step from solving exercises to original math is really more of a leap, and one with which I am currently floundering. Were it somehow possible to introduce these motivating situations sooner, I think this leap would be easier to make, as students would get to see why we do it that way instead of some other way.

I think this shows up in topology a lot; the definition is signficiantly more abstract than anything most students have seen in analysis at that point, and some understanding seems to get lost along the way. The number of questions on math overflow revolving around "Why is topology definted this way" is some interesting evidence for this.

*****

(pseudonym)
When I look back at my undergrad days I can see how several of the tutors tried to work motivation and exposition into their problem sets, but at the time a lot of it went over my head. I was fairly good at solving problems but I was a long way from seeing them in a wider context. I think the problem is that often the motivating issues are too complex to get across to people who haven't aquired the mathematical maturity gained from a few years of grappling with terminology and educational 'toy' problems.

*****

(jason.chase)
You all sound very intimidating. I am just about to leave my world of problem sets for this wider, terrifying world. I don't know whether reading this is inspirational of scary.

*****

(me)
Hmm... that very well could be the problem... And I do know that my instructors seem to have gotten better at communicating motivation over the past couple of years... perhaps I've just gotten better at understanding it.

You all sound very intimidating. I am just about to leave my world of problem sets for this wider, terrifying world. I don't know whether reading this is inspirational of scary.

Heh. :)
I can promise it is much more requarding and enjoyable once you start trying to break out of problem sets-- pursuing an idea (even a fruitless one!) is very exciting, and gives you a much deeper understanding of the thing you're working with than any problem set can. Suddenly seeing a connection (such as noticing a surprising structure show up "in the wild") is a wonderful feeling that is very difficult to get across with problem sets. (Although I certainly have had this happen while working on a problem set.)

Of course, problem sets will always be important-- I never expect to understand a book until I work the problems, and never expect to understand a lecture or paper without working out the proofs on my own, even when they are "trivial"-- so you'll be able to comfortably hide inside a cozy problem set for a bit whenever you get too afraid of the wilderness.

*****

What I meant by overuse of terminology is when a fellow mathematics student throws in technical terms specific to their expertise that they know I don't know, instead of trying to offer possibly longer explanations in terms that are appropriate to my background. In my experience it is the algebra whiz kids that are the most likely to do this, but perhaps it's only a mistake of not realizing that they at one point didn't know these words. I wish I could remove this statement altogether though as it's a gross generalization fueled by finitely many cases of personal frustration.

With regards the terminology, on an undergrad course it can seem like its just there for its own sake. You prove a lot of stuff that seems like busywork. But this is just because even relatively advanced undergrad/beginning grad courses are really only introductions. They're trying to give you an overview of the tools that are available but they rarely have time to motivate them by going into the problems from which the definitions emerged.

Yes this is an excellent point, and a topic that should be explored in its own thread (but not on the algebra forum of course). It's interesting to look back at high school books, and early undergrad books at the problems to see how they were really setting you up for later material. Like integral convergence questions in my calc book use for the exponent the power p, as a primer to showing the difference between convergence in the different Lp spaces. That's a bad example perhaps, but you know what I mean.

I think a good professor will tell the students why something will be important later. The downfall to this, is that it can lead to students ignoring other "less relevant" parts of the course material. But if only I knew how important Taylor's theorem was when I was learning integral calculus as a freshman. Something I know consider to be the most important tool in applied mathematics is something I used to think was busy work to fill the end of the semester.

Of course, problem sets will always be important-- I never expect to understand a book until I work the problems, and never expect to understand a lecture or paper without working out the proofs on my own

This is an excellent point as well. In the transition to theoretical mathematics I foolishly began overlooking the importance of "drill work". However, in studying for the GRE math subject test I've seen an amazing improvement in my problem solving skills as a whole, that are no doubt a result of repeated drill work. Tools I knew about but in practice never thought to use are now actively surfacing in my consciousness , and I feel so much more empowered.

Above all the foundations of your knowledge base need to be practiced over and over again as you progress (i.e. algebra, geometry, trig. , calculus calculus calculus). A building is only ever as strong as its foundation, and an A grade almost never implies true mastery. I can't tell you how many kids who get A's in algebra can't apply the same tricks in the calculus setting or beyond.

even when they are "trivial"-- so you'll be able to comfortably hide inside a cozy problem set for a bit whenever you get too afraid of the wilderness. :D

This can help build confidence and help alleviate some of the fear of mathematics, it's important for the student to realize 'hey, I CAN do this stuff'. Fear of mathematics is such a powerfully negative force for some people. In tutoring calculus I have seen near brilliant people fail to answer the simplest of questions, only because of the fear and preconceptions of calculus. If I had asked the same questions without calculus floating in the air, they would have thought I was belittling them. So in learning mathematics an air of confidence (but not over confidence or self importance) is powerful and necessary. Maybe I should really say an understanding of one's own potential. I have a saying I made up about this:

Knowledge is only useful if you know you have it
But only a fool thinks himself otherwise
So praise not what you think you know
And embrace only the potential to grow

The fundamental group functor part 2

2010-03-01T15:38:00.005-06:00

So... in the previous post I promised to finish what I was saying about the fundamental group functor... So far I've sketched the proof that this is, indeed, a functor. I would show the proof in detail, but it's long, tedious and not very informative-- the point is, it's a map from Top$_*$ to Grp which preserves morphisms. There are much more interesting things a functor can preserve. Namely, it can preserve products and coproducts.

So what's a product? As a motivating example, look at Set. When we talk about the product of two sets, we clearly mean the cartesian product. Since we're interested in category theory at the moment, we don't really want to talk about members of the product, we want to talk about maps to and from the product.

It turns out there are two really nice maps $pr_1:A\times B\rightarrow A$ and $pr_2: A\times B\rightarrow B$, the projections onto $A$ and $B$ respectively. It turns out that they have a really nice universal property: Given an object $V$, and two maps $f:V\rightarrow A$, $g:V\rightarrow B$, we can "factor" $f$ and $g$ through $A\times B$ in a unique way. This means that we have a unique map $h:V\rightarrow A\times B$ such that $f=pr_1\circ h$ and $g=pr_2\circ h$. At first it may be a bit surprising that this map (sometimes called $f\times g$) is unique. But really, our two projections forget everything about one side of our product, so the function needs to act "independently" on $A$ and $B$, and there's really only one way to get this to interact properly with the projections.

Something that's more surprising is this: the product is unique up to unique isomorphism. This means that if there is a "different" product (Why not try $B\times A$?), there is a single, canonical isomorphism between the two objects-- just factor the projections from one product through the other. This map is unique, and it damned-well better be an isomorphism. (To see that it is, factor the projections back the other way, wave your hands about and say something about "the identity morphism".)

Ok. So, by analogy with Set, we (sort of) get what a product is. What about coproducts? A nice thing about category theory is that whenever you see a word that starts with "co", you can figure out what it means in 3 easy steps:

remove the "co" from the word.
Draw the diagram that represents the word you just found.
Turn around all the arrows.

So, this means the coproduct, $A\coprod B$, should have two maps $i_1:A\rightarrow A\coprod B$ and $i_2:B\rightarrow A\coprod B$ (called the imbeddings) such that for any pair of maps $f:A\rightarrow V$ and $g:B\rightarrow V$, we have a unique morphism $h:A\coprod B\rightarrow V$ such that $h\circ i_1 = f$ and $h\circ i_2 = g$. For some reason this always seems a little harder to follow. Let's work it out in Set. Let's look at $f$ and $g$ as in the definition. We want some map (call it $f*g$ because I can't think of what the actual notation is) that goes from somewhere to $V$ such that $f*g \circ i_1 = f$ and the same with $g$. We want $i_1$ and $i_2$ to do almost nothing... what happens if we take $A\coprod B$ as the disjoint union? (Hence the notation...) What is the imbedding? It's the "move me from $A$ to $A\coprod B$" function. And what could $f*g$ possibly be? Well obviously, it's the function which sends $a\in A\mapsto f(a)$ and $b\in B\mapsto g(b)$.

Ok. Great. We know what the product and coproduct need to look like (at least when we only care about the product of two objects). What exactly are they in Top$_*$? It turns out they are the wedge products-- take the disjoint union (familiar?) and glue the two spaces together at their base-points. This means that we have two completely unrelated spaces (modulo open sets containing the basepoint.)

This idea of a coproduct being the result of "smashing together" two objects without making them at all related is basically consistent throughout basically every category. In fact, in the category of groups, it's the free product, which is the "freely generated" product of the two groups-- For groups $G$ and $H$, this means the set of all words on $G\cup H$, where things reduce in the "obvious" way and no other way... I'm going to pretend like this makes sense to you, since (as you've surely learned by now) I have yet to decide what level of audience I'm writing for.

So, taking this back to the fundamental group functor: for two (pointed) spaces $(S,s)$ and $(T,t)$, we would like $\pi_1(S\times T, (s,t)) = \pi_1(S,s)\times \pi_1(T,t)$ and $\pi_1(S\coprod T, (s,t)) = \pi_1(S,s)\coprod \pi_1(T,t)$.

Guess what? I'm going to cop out of actually proving this! (Are you surprised? You should be used to this by now...)

However, I will at least wave my hands around a bit and give you a feel for why it's true. First let's look at products. As an example, look at the torus-- $S^1\times S^1$. Draw a path on this. We want to be able to push this path down to a path which only lives in one copy of $S^1$ in each component. (I.e., a path which stays on $(S^1\times\{0\})\cup(\{0\}\times S^1)$. ) We can do this by pushing (in a continuous fashion-- i.e., homotopically) all points of our path onto one of our two reference circles.

For coproducts: it's a little more obvious in some sense--- any path is going to stay in one of our two spaces for a while, and then cross over to the other. The homotopy group we get here "reduces" in the obvious way, and no way else-- i.e. it's the free product.

Ok. there. I've fulfilled my promise. Expect a more detailed and less obnoxiously hand-wavy post about natural transformations and path categories soon (TM)

The Zahir and Asterion

2010-02-22T11:35:00.002-06:00

Sorry... I know I promised to finish that last post about 2 weeks ago... I'll get around to that soon. In the mean time, I've just started reading The Zahir by Borges, which I somehow haven't read. I could have sworn I had read the whole of The Aleph, but I digress. I stumbled upon the following passage (I don't know who the translator is):

Until the end of June I distracted myself by composing a tale of fantasy. The tale contains two or three enigmatic circumlocutions: “water of the sword”, it says, instead of blood, and “bed of the serpent”, for gold, and is written in the first person. The narrator is an ascetic who has renounced all commerce with mankind and lives on a moor. (The name of the place is Gnitaheidr.) Because of the simplicity and innocence of his life, he is judged by some to be an angel; that is a charitable sort of exaggeration, because no one is free of sin. He himself (to take the example nearest at hand) has cut his father’s throat, though it is true that his father was a famous wizard who had used his magic to usurp an infinite treasure for himself.
Protecting this treasure from mad human greed is the mission to which the he has devoted his life; day and night he stands guard over it. Soon, perhaps too soon, that watchfulness will come to an end: the stars have told him that the sword that will cut him off forever has already been forged. (Gram is the name of the sword.) In an increasingly tortured style, the narrator praises the luster and flexibility of his body; one paragraph offhandedly mentions “scales”; another says that the treasure he watches over is of red rings and gleaming gold. At the end, we realize that the ascetic is the serpent Fafnir and the treasure on which the creature lies coiled is the gold of the Nibelungen. The appearance of Sigurd abruptly ends the story.

This sounds rather amusingly like... The House of Asterion which was published in the same collection. This is one of the things I really like about Borges: He makes very subtle references to other works of his. Can anyone think of any other specific examples of this?

The fundamental group functor

2010-01-30T14:25:00.015-06:00

This is something I've always (read: since I learned about it less than 6 months ago) found pretty neat. There's nothing terribly original here-- everything can be found in any algebraic topology book, and in most general topology books, but I don't think categorical language makes its way in there all the time...

The point of this "little" post is to point out that the operation taking a (pointed) topological space $(X,x_0)$ to it's fundamental group, $\pi_1(X,x_0)$ is a functor which preserves products and coproducts... (Did that sentence have a point? Sorry... I'm done.)

First, as a technical point: we need to work int he category of pointed spaces: Top$_*$. (A pointed topological space is just a pair $(X,x_0)$ where $x_0\in X$. The morphisms are continuous functions $f:(X,x_0)\rightarrow (Y,y_0)$ such that $f(x_0)=y_0$. The idea is we are distinguishing a point, just as we do to get the fundamental group.) The reason for this is that it gives a nice way of distinguishing between base points (for our fundamental group) in different path-components-- every selection of base point gives us a new space-- Some are isomorphic. This allows us the avoid the technical nightmare of what to do with non-path-connected spaces. (I.e., we don't get a functor if we're only working in Top) There's another reason for this: Wedge products give Top$_*$ a sensible notion of coproduct-- or at least, one which is actually preserved by the functor.

So, first of all, what does it mean for us to have a functor? A functor is a map between categores which preserves identities and composition of morphisms. In other words, for categories $C$ and $D$, $F:C\rightarrow D$ is a functor if $F(id_c)=id_{F(c)}$ for every object $c\in C$, and for every pair of morphisms
\[c_0\stackrel{f}{\rightarrow}c_1\stackrel{g}{\rightarrow}c_2\]
In C, we have that $F(g)\circ F(f) = F(g\circ f)$.

Given a function $f:(X,x_0)\rightarrow(Y,y_0)$, $f$ induces a homomorphism $f_* : \pi_1(X,x_0)\rightarrow \pi_1(Y,y_0)$-- Any path in $X$, when fed through $f$ becomes a path in $Y$. Since the map preserves basepoints, a loop at $x_0$ becomes a loop at $y_0$-- seeing that this is compatible with homotopy isn't too difficult.

To say that $\pi_1(-)$ is a functor means that $f_*(\pi_1(X,x_0)) = \pi_1(\operatorname{Im} f,y_0)$ and that $(id_X)_* = id_{\pi_1(X)}$ (Sorry, commutative diagrams are not working so hot in this $\LaTeX$ package... I'll need to do something about that.) A quick diagram chase shows that this is the case.

Now is where things finally get interesting... and... I'm tired, and will finish this later today.

A few words on Balaam's Error

2010-01-19T15:37:00.002-06:00

I'm not sure why I'm writing this down now, but: I don't agree that "Balaam's error" has anything to do with money. Based on his actions, monetary reward seems to be a small concern for him. His error comes from this: He is afraid to contradict the Moabites. He is too polite, too unwilling to offend.
Sometimes things need to be said which are offensive-- causing offense is rarely good in its own right, but offensive things are important. Balaam was too afraid (either socially, or for his life) to tell the Moabites something offense, "God says 'no!'"

We should learn from this.

Balaam

2009-12-12T04:44:00.003-06:00

I wrote this sometime last year, but apparently forgot about it.
***

I tread on with the Moabites, seeking their praise, silver and jewels. "I cannot contradict the ineffable," but I march on, afraid to contradict these messengers.

But still my ass, wiser than I, stubbornly refuses to move. Twice she's flogged, and twice she stands up and walks. Again she's flogged, but finally she speaks her mind:
"Why do you treat me so? Have I not carried you from your home?"
And thus I lie: "No."

Nuclear energy

2009-12-04T05:54:00.000-06:00

is safe and clean. Ask anyone who knows anything about it.

Mathematical insight...

2009-11-20T11:07:00.002-06:00

This is from a reply I posted here to a question about gaining mathematical insight.

* Nothing is "obvious".
Try to be extremely formal with all of your proofs. Make sure your steps all follow immediately from previous steps, definitions or theorems. Spend some time proving the "really basic" properties that follow immediately from applying the definition. Also, ask yourself what sort of objects satisfy certain properties, and which don't. Eg. For complete metric spaces, come up with a "canonical" example of a complete metric space, a "canonical" incomplete metric space, and a degenerate example of each. For example, the discrete metric is complete (if you know about metric spaces, you may want to prove this), but it really doesn't match our intuition for what a complete metric space "should be."

On that note, try to understand what the intuition for a property or object is-- what does it "mean" for a set to be a group under an operation? Also, try to keep track of where intuition departs from math-- For example, we like to think of topological spaces geometrically, but there are some very non-geometric topological spaces.

* Rewrite the same thing as many different ways as you can.
For example, if the problem asks a question about a normal subgroup, you should be thinking of all the characterizations of normality-- It's the kernel of a homomorphism, it's invariant under conjugatian (which really is the same as its left and right cosets are the same), if a and b are in the same coset of N, then a-b is in N.

* When working on a proof, pay attention to everywhere you use your assumptions.

* After writing a proof, make sure the result seems to make sense.
Does it match up with intuition? If not, figure out why. If the problem is with your intuition, try to figure out what you are assuming to be true, and make a note of it.

Are any basic examples of the structure a counter-example to your "theorem"? Does each step follow from the last? Are you sure?
(I have a friend who has written 3 or 4 wrong proofs this semester, and every time, he realized it was wrong based on these checks, although normally I had to pick out the false step for him :D )

* Learn to look for counter-examples.
If you're asked to prove something wrong, look at some basic examples of the structure you're looking at. Does the statement hold for them? If so, can you see what properties make it work? If so, try to come up with an example where that property doesn't hold. Does the statement fail now? Rinse and repeat.

* Rewrite your assumptions. Rewrite them in different words. Rewrite them with the definitions of any terms you are uncomfortable with.

* Look for connections.

* Rewrite any objects you're looking at in terms of other objects. The complement of an open set is closed. The complement of a closed set is open. A connected space has proper (non-empty) clopen sets. g is in the Center of G means gh=hg for any h.

* State the obvious. Often. And then state it again.

* Ask stupid questions. Then answer them.
Is R complete? Why is a polynomial continuous? Is Z abelian? Finitely generated? What about Z^n? What does Abelian mean anyway?

* Don't be afraid to ask someone else stupid questions.

* Don't be discouraged when you sit for hours without understanding what to do; let the gears keep grinding.
Put on some music and rock out while you think. Rewrite the assumptions. Try to do something. When you get stuck, try to figure out why that doesn't work. Does it get you anywhere at all?

* Don't be afraid to go do something else for an hour or 2 and then come back to work on a problem.
This is when some of the best insights happen-- go make some tea, read a book, watch a movie, get coffee with a friend, do something. Then come back and start again. Sometimes it'll be hard to get back in the zone-- redo some easier problems: Try to reword your argument or try to find a cleaner argument.

* Work on a simpler problem.
Need to separate two compact sets? Don't! separate a compact set from a point. Can you use this same argument again? Will a similar argument work for two sets?

* Work on a more general problem.
Don't show that n is divisible by 3, show that all numbers of a certain form are divisible by 3. Then show that n has that form.

Hope these give you something useful to think about.

The size of the list of things to learn...

2009-10-21T14:22:00.004-05:00

$\displaystyle 2^{2^{.^{.^{.^{2^{\aleph_\omega}}}}}}$

Cruel Irony.

2009-10-14T14:14:00.002-05:00

Worth watching. I won't say you should never buy Monster again, or anything... actually I will, but mostly because Monster is awful-- if the sugar is over-saturated, there is too much.

Also in the Cease and Desist letter, the following quote' "VERMONSTER in connection with beer will undoubtedly create a likelihood and/or dilute the distinctive quality of Hansen's MONSTER marks." Self-fulfilling prophecy, much?

Art

2009-10-11T23:53:00.003-05:00

I've finally found a definition of art that I think I agree with... Came to me as I woke up this morning.

****
Art (as a verb) is a creative or transformative process undertaken primarily as an appeal to some aesthetic, in order to induce a sense of "aesthetic euphoria" in those who experience the resulting object.

An object created (or transformed) in this way (this is, with this aesthetic goal as a primary objective) is a work of art.

The broad category of all artistic processes is art-- Everything which is done primarily as an appeal to an aesthetic. Any category of process which is primarily undertaken for aesthetic appeal is an artistic discipline.

Anything which has an aesthetic appeal, but was not designed with the aesthetic appeal as the primary objective is craft.
****

This definition is pretty loose (yet mathematically precise; I won't apologize for who I am), but it seems to explicitly exclude "useful" objects from the category of art... This isn't entirely true. An object which is useful, but was designed with its aesthetic appeal as a primary objective is still art: Something can be both craft and art.

I also am not trying to be derogatory towards craft: many great artists are primarily craftsmen, and a lot of craft is more aesthetically appealing than a lot of art. Further, what separates a good craftsman from a great craftsman, is that a great craftsman elevates the artistic value of his creation to an equal footing with it's utility-- without sacrificing function for form.

It's also, I assume, a very modernist definition... So my poetry and my artistic ideals are 70+ years behind the times; C'est la vie.

The one thing I'm struggling with is how kitsch fits into this. I would like to say kitsch is not art, but I don't think this definition excludes it.
On the other hand, I tend to refer to kitsch as "the unart" in the same way that zombies are undead. So it makes sense that kitsch will fit the definition of art; now how does in fit the definition of non-art?

My poetry...

2009-10-04T13:48:00.001-05:00

Is so formulaically modernist...

Rhapsody in Blue

2009-10-04T13:42:00.004-05:00

(For Sasha)

"I", she said to me, looking up from a cup of tea.
"I should have been a rhapsodist.
But, my dear, I have no skill for words,
no aptitude for meter."
With poetic eloquence she explains,
"I wrote when I was younger."
But never since.

"I was born with a talent of gold, you see.
But I've earned nothing more."
I know, my friend, you are terrible with money,
and have gotten rather poorer.
But a talent is a hefty sum,
and can always be made
to last a little longer.

Testing LaTeX

2009-09-30T08:28:00.014-05:00

$\displaystyle|\mathcal{F}|^k\leq \prod_{i\in I} |\mathcal{F}_i|^k$
That's a corollary to Shearer's Lemma, by the way.
(I haven't told you what $I$, $\mathcal{F}$ and $\mathcal{F}_i$ are; oh well)

Anyway. This is courtesy of Watch Math. It's pretty simple, in fact. And it'll make the math on this site prettier.

I may (read: probably won't) get around to rewriting all my math in $\color{white}\LaTeX$.

Edit: Hmm... it seems the LaTeX sometimes takes a little while to load properly. Please be patient.

Introduction to Logical Languages (2)

2009-09-16T07:16:00.003-05:00

In my last post (earlier today), we defined a logical language. But we ended wondering how to give meaning to this language. Since we are looking at mathematical logic, we want a mathematical structure to talk about-- every logical statement fits inside of some logical structure: A group G, ZFC, N, the theory of groups, etc.
So, what is a structure and how does this relate to a logical language? A structure is just a set which has some additional material attached; since we have a language we're not using, we might as well attach it to the set.

For a set A, and a language L, an L-Structure is a non-empty set A, (called the universe of structure for A) such that:

For each constant symbol c, we have a c^A in A.
For each k-ary function symbol f, there is an f^A:A^k->A.
For each k-ary relation symbol R, there is an R^A in A^k

These x^A are called interpretations of x. You can think of an L-structure relating to L as a meaning for L. They just drop these symbols into this universe structured around A, and give them a home. But we still don't actually have a way to get from L to its "meaning" in A! Namely, our structure does not contain any information about variables.

A quick recap: we've taken a set A, and equipped it with a structure by dropping symbols from a logical language into it-- notice that constants are just members of A; this is a good thing. And k-ary functions (relations) are functions (relations) which take k elements of A; also a good thing. But we still haven't done anything with our variables So, what kind of variables do we have if we're looking at a set A? We have elements of A. So, let s be a function s:V->A, and let's call it an evaluation map [remember, V is the set of variables in our language]. So, we have a function which gives each variable a value. Good!

But right now, our function and our L-structure are kind of disjoint. We need a function which can take the value of our variables and push those through the interpretations of our functions and relations. So!
For s, let s':T->A with the following properties:

s'(x)=s(x) [when x is a variable]
s'(c)=c^A [when x is a constant]
s' preserves function application: s'(f(t_1,...,t_k))=f^A(s'(t_1),...,s'(t_k)).

These properties mean that s' preserves the structure we've defined on A (so s' is a homomorphism.) If the structure weren't preserved, then we'd run into the problem that if you evaluated f first, you might get something different than if you evaluated the arguments first. Also, you may be wondering why relations aren't included in this: remember that T is the set of terms, and relations aren't terms.

So far, we've defined a logical language, and we've given our language a set to play in, and given our terms a meaning. Pulling back out, when our terms mean something, then we can ask whether a statement about our terms is true. So, truth:
For an L-structure X on A, and an evaluation map s:V->A (X is easier to write than a new font), a statement a is called true (with evaluation s)-- in symbols X|- a [s] if:

When a: t=u, then X|- a [s] iff s'(t)=s'(u)
When a: R(t_1,...,t_k), then X|- a [s] if and only if R^X(s'(t_1),...s'(t_2)) holds.
Similarly for non atomic formulas.

Basically all we are saying is "The statement is true in a structure if and only if it makes sense to call it true in that structure." We've just jumped through some hurdles so that we can say "it makes sense" in a rigorous way.

I hope this was easier to follow than the lectures it came from (to be fair, I've left out some useful material about homomorphisms, which was almost as abstract as the evaluation maps)

Introduction to Logical Languages (1)

2009-09-16T06:12:00.009-05:00

This is mostly to clarify in my mind the material from the first 2 lectures of the (mathematical) logic class I'm taking. Hopefully it will also help someone else.

So. We want to be able to look at logic from a formal, rigorous perspective-- which means we need to define logic formally. The guiding question when defining it should be: What does a logic look like? We want variables and all the fun logical connectives, and we want them to mean something, and we want all the meanings (eventually) to boil down to the question "Is this statement true?".

So, let's make each of those happen, one at a time;
First, we need some symbols. We will call S our alphabet-- the set of symbols. This may be tediously formal, but it must be done. Since we want to create a logic, we need the logical symbols: So S contains the connectives and quantifiers, (I'm not going to list them) and just as importantly: variables. These are our logical symbols. Since this is mathematical logic, we need some mathy non-logical symbols; these will be constant symbols, relation symbols and function symbols (these will also "look like" variables, but we want to distinguish between them for reasons we will see shortly). We will also consider '=' to be a logical symbol-- I know it's a relation, but it's a special one, and we want to keep it special.

So, we have a bunch of symbols; that doesn't do us much good, so let's let S* be the set of finite sequences from S. Namely, the empty sequence (I'll use _|_) is in S*, S is a subset of S*, and if a and be are in S, ab (the concatenation of a and b) is in S*.
We want to define a language, L, as a subset of S*, but while any subset is a language, we don't want just any subset: it has to be something that we can "read" in a meaningful way. We have to define our language inductively, and piece by piece. I'll let you know when we get there.

Let the set of L-terms (or just terms) be the smallest T such that:

V (the set of variables) is in T.
C (the set of constant symbols) is in T.
Whenever t_1,...t_k are in T, and f is a k-ary function symbol, f(t_1,...,t_k) is in T.

Notice that this definition is recursive: any t_i in the last condition may have the form f'(t_1',...,t_k').
A term is an object which we can equip with some non-logical value, which may need some sort of context (a value for a variable). But a term does us no good on its own, if we are interested in logical values. So, an atomic formula is:

if t, u are terms, then "t=u" is an atomic formula.
if t_1,...,t_k are terms, and R is a k-ary relation symbol, then R(t_1,...,t_k) is an atomic formula

So, like any set of "atoms", atomic formulas are the building blocks for formulas.
The set of L-formulas (or just formulas) is defined inductively:
Base case: Atomic formulas are formulas.
Closure: The set of formulas is closed under logical connectives, and quantifiers.
(What this means is that if you have two formulas, p and q, you can connect them using a logical connective, and you can quantify them, and the sequence of symbols you create will also be a formula.)

So, we now have something that looks like the logic we know. But there is a problem: if x and y are variables, "x=y" is a valid formula. But what are x and y?
In that formula, both x and y are free variables. For a given formula the free variables are :

(for s, t terms) FV(s=t)= var(s)Union var(t) [var(t) means the variabls in t]
(for R a k-ary relation) FV(R(t_1,...,t_k))= Union(t_i)(over 1≤i≤k)
(for ~ a logical connective, a,b formulas) FV(a~b)= FV(a)UnionFV(b)
(for x a var, b a formula) FV(forall x, b)= FV(there exists x, b)= FV(b)\{x}

(Sorry for the formatting...) All this says is that anytime a variable is introduced, it is "free" until it has been quantified.
So finally, we have: A sentence is a formula with no free variables, and our logical language, L, is the set of all sentences from S*.

Cool! We have a logical language. Looking back at our list of things we want, we've got connectives, and we've got quantifiers and just a little bit more. Unfortunately, this means we're not quite done: this language is meaningless. We haven't once said what these symbols actually mean. And we will learn how to do that next time.

Random Conversation

2009-09-08T13:26:00.001-05:00

(Laughter has been removed)

MC: hmm. I'm demoting you
CK: ?
MC: you are no longer Master Commander of Hypothetical Operations.
CK: But if you don't demote me, think about how great everything would be!
MC: Your title is now Chief Executive of Jello Affairs.
CK: But!!! I'm not jiggly enough!
MC: hmm... actually, I don't know if I like that title... Jello Executive, Fruity Faction.
CK :I do, however, know that every conditional with a false antecedent is true... and I am responsible enough to only imagine badass scenarios.
MC: mostly so it abbreviates to JEFF
CK: If I had not been demoted, you would be richer than google.
MC: a googol dollars!!!
CK: If I were currently MC of HO, then yes.
MC: hmm
CK: (Hurray vacuous truths!!!) Allowing mathematicians to make vacuous promises since 1000BC
MC: right, could have doesn't actually imply causal effect, does it?
CK: Well, it's that If x Then y is always logically true when x is false. Because the implication is only broken when x is true and y is false. I had a prof who was in the habit of using vacuously true cases for the base case of an induction. Like, a statement about edges in a graph; his base case would have no edges...
CK: Why did I get demoted, by the way?
MC: glitch in the payroll system.
CK: Ah. Well; can't be helped.
MC: actually, we introduced the glitch after the fact
CK: I can't blame anyone, can I.
MC: there was a glitch in the name placards and they came out wrong.
CK: Well, if the name placard says so, it must be so.
MC: so we demoted/promoted people accordingly. It only made sense.
CK: Of course.
MC: we didn't want to waste the money we spent printing them.
CK: A company needs principles if it's to run smoothly. Principals? I don't know which.
MC: well, it needs both; who else is going to turn the hamster-wheel-power-generator?
CK: Right. This is why you can promote, and I'm only the JEFF. Best conversation ever, by the way.
MC: no, they only let me demote. I don't have authority to promote. They only give that authority to the janitor's secretary.
CK: I see. That seems sensible.
MC: I'm not sure this conversation would make any sense were I to read through it after forgetting the fact itself.
CK: You forget that it makes no sense now.

The mathematician on the street...

2009-08-22T12:07:00.003-05:00

That last post was a bit of frustration about an ongoing discussion of AC/CH on the FOM mailing list. Not everything about the discussion has been quite as frustrating as the whole discussion-- namely, some fantastic quotes have come from it. Here are some of my favorite in posting order (I think they slowly get less and less technical...):

"All that we have here in this quasi-paradox is confirmation that reals are not a perfect model of dart throwing and vice versa." -Thomas Lord

"It would be useful to provide some rationale why the continuum having cardinality aleph_1 leads to more unusual results than, say, the Banach-Tarski paradox. Furthermore, I would like to know why you think these results should lead us to reject the continuum hypothesis but not the axiom of choice. Finally, I would be interested to know what has led you to conclude that most 'mainstream mathematicians' find your arguments convincing." -Lasse Rempe

"They are not claiming to have an argument formalizable in ZFC; they are merely claiming that mathematicians have overreacted to the results of Banach-Tarski, Godel, and Cohen by throwing out too much of their intuition about assigning measures to subsets of R^n." -Joe Shipman

"If you think it's an interesting question to investigate plausible extensions of ZFC that settle CH, then you're already a dyed-in-the-wool f.o.m.er." -Tim Chow

"But in my experience, if you pick a random mathematician who is not already interested in f.o.m., there's at least a 50% chance that you'll have to remind them of the definition of a well-ordering of the reals and of its relationship to the axiom of choice." -Tim Chow

"So you do not accept AC in the same way you accept the other ZF axioms? That's fine, but it's not the position of the mathematician in the street." -Joe Shipman

" I don't think the mathematician in the street will respond, 'Gee, since I accept AC as gospel, I am forced to blame these pathologies entirely on CH!'" -Tim Chow

"For starters, [the mathematician on the street] is unlikely to be able even to list the axioms of ZF, but he or she will know AC explicitly, precisely because it is known to have some strange consequences." -Tim Chow

(Included because of how wrong it is): "I think we are in danger of forgetting that not only do most mathematicians-in-the-street not believe AC, most of them have no intuitions about it and cannot state it even roughly, let alone have any idea how to use it." -T Foster

"Someone who does not know such basic material cannot be called 'a mathematician' (neither in the street nor anywhere else)." -Arnon Avron

"I am reminded of a time in graduate school [...] when I delivered my self of the opinion that cardinal trichotomy was, intuitively, OBVIOUSLY true and that the Well-Ordering Theorem was, intuitively, OBVIOUSY very fishy." -Allen Hazen

"This certainly circumvents the use of AC, but I submit that it is somewhat contrary to the mathematical practice of *not* equipping structures with non-canonical stuff that is extraneous to their essence. You could define a vector space as something that comes equipped with a basis, or a manifold as something that comes equipped with an embedding in R^n, or a group as something that comes equipped with a homomorphism to an automorphism group of something, etc." -Tim Chow

"mathematicians tend to replace the use of existential statements by the introduction of skolem functions. This is such a common procedure that they do not even notice that they are using AC when they do so." -Arnon Avron

"In particular we agree that these street mathematicians (one pictures them performing Hilbert's Nullstellensatz while passers-by drop coins in their hat) are enumerating witnesses to countability rather than countable sets." -Vaughn Pratt

"Depriving the street mathematician of her witnesses is like depriving a boxer of his fists. [...] Why should she care that foundationalists make things harder by killing off her witnesses?" -Vaughn Pratt

A little rant about AC...

2009-08-16T05:00:00.002-05:00

So. Before we get started. Math is based on logic. The most important result of this fact is that there must be some axioms-- a starting point for the logical "gears". So, there must be a few things which are accepted as true, without argument, in order to "prove" anything. The really nice thing about this is you can use different axiom sets for different purposes; for example, the Peano Axioms are the axioms for number theory; any set theory capable of producing arithmetic will have axioms that imply the Peano Axioms.

The problem with this forced reliance on axioms is that mathematical truth is based to some small extent on human intuition. And human intuition of mathematical concepts is notoriously fickle.

One point that is still a hot topic amongst foundational researchers, and amongst those who spend their time discussing the philosophy of math, is called the Axiom of Choice. The "normal" statement (which gives the axiom its name) is a bit technical, but there's a completely equivalent statement: A non-empty Cartesian product of non-empty sets is non-empty. In other words, if we have a bunch of non-empty sets, and we take the set of all tuples (ordered lists) of these sets, we have a non-empty set. An example: X={1,2}, Y={1,3} Z={3}, X×Y×Z={(1,1,3),(1,3,3),(2,1,3),(2,3,3)}. (For the more interested reader, the first statement of AC in the wikipedia article mentions a choice function. Any point in the cartesian product encodes such a choice function. If we have a non-empty cartesian product, we have a choice function.)

The debate is mostly about whether or not this statement is intuitively true-- can we say it can be placed with the "obvious" axioms? It seems to make sense to do so, but it leads to a lot of counterintuitive results. The most famous such result is the Banach-Tarski "paradox", which says it is possible to take apart a sphere to create 2 spheres whose sizes are each equal to that of the first. Let's repeat that: Start with one sphere of a certain volume. Split it into 2 in a very clever way. Now you have two spheres, each with volume equal to the first. The "clever way" of splitting the sphere requires the axiom of choice (in a way I'm not sure I have the background to understand.)

The problem, of course, is that counter-intuitive starts at infinity, not at AC. We can split the set of all even numbers into two copies without choice (pull out 2,6,10,... then divide those by two and add 1, and simply divide the rest by 2), so why is a sphere less intuitive? How is it intuitive that there are as many rationals as there are integers? How is it intuitive that you have 0 probability of selecting an algebraic number from the reals, despite the fact that they are dense?

I don't mind people rejecting choice for certain work: constructive logic is incredibly useful for CS, but it explicitly contradicts choice. What I mind is people bringing up "counter-intuitive" results that are no more counter-intuitive than results that have long been taken for granted, because we're so used to seeing them.

My last sentence reminds me of another problem with the whole discussion: Human intuition is so fickle! The results I mentioned are not considered counter-intuitive to most working mathematicians, because the results are so fundamental. In addition, in the same breath that they say "AC leads to counter-intuitive results", they talk about how certain people haven't built up an intuition for these sorts of foundational results. Perhaps none of us have built up an intuition for certain results?

Copycenter

2009-06-30T11:26:00.003-05:00

So, I guess I should point out that unless noted otherwise, or the work is not by me, everything on this blag is licensed under the CC-by license. In other words, do anything you want with it as long as you attribute me (Cory Knapp) in any distribution or modification in a way that does not imply that I endorse your use of the work, without explicit permission to do so, more here:

I'd like to get into a discussion about copyrights here, but I just don't care enough... Let me just say, I prefer copycenter to copyleft, and I prefer copyleft to copyright... I think open source (and the non-software equivalents) is the right thing to do in a "give to charity" sort of way, not in a "don't kill" sort of way...

Remind me...

2009-06-29T22:34:00.001-05:00

To write a story about Feynman diagrams. And a poem about candles. I'll know what you mean.

War! Huh! What is it good for?

2009-06-08T14:12:00.002-05:00

Actually... Absolutely nothing. So, I hear people bring up the economic benefits of war every so often, mostly regarding how WWII "got us out of the depression." before I talk about how little sense this makes, I'd like to point out that the US had mostly recovered from the depression before 1940. Last I checked, the US didn't get involved until December 1941, so time disagrees with this theory.

Now, let's talk about wars. Specifically, let's talk about WWII. People talk about how American industry was mobilized for the war. This is true, but there's a subtle fallacy at play here. Jobs, in and of themselves, do not add to the economic vitality of a nation. If they did, we could have everyone working rolling rocks up hills, and letting them fall down again, and we'd have a booming economy. What does add to economic vitality is the creation of capital. Capital is a good which can be used to make more goods. In other words, a booming economy is an economy which is one which is increasing its capacity to produce.

So, the construction of the factories and machinery to create the weapons of war was economically healthy because it created capital, but it ends here. All of those factories went to work building supplies for the war. What we then have is capital-- raw goods (mostly ore and oil) and processed goods (metal alloys, gasoline, rubber) -- being turned into finished products. All well and good, but these products are leaving the economy: To go be used (and destroyed) in a war.

What, then do we have? Capital that is not being used to create more capital. Capital which is being used exclusively to push products out of the economy. We were wasting capital.

And this capital waste rears its ugly head in the shortages and rationing. Capital was leaving the economy at an enormous rate. The market response is hyperinflation: If goods are rare (as they will be if all capital is leaving the economy), goods are expensive. This also happened in Europe after WWI, because all of their monetary capital was going to reparations (Germany, Austria), or repaying loans (Allied powers). (To be fair, Weimar monetary policy didn't help the hyperinflation.)
This hyperinflation can be curbed by fixing prices, but as we saw in the 70s (and every other time prices have been fixed), this leads to shortages. The rationing in the US was a response to the shortages. These shortages were caused because the capital was leaving the country.

In case you hadn't caught this, rationing is not a sign of a healthy economy. It is quite the opposite.

Next. The Marshall Plan. Total US aid in Europe was over 1.2 Billion. A healthy economy does not need billions of dollars pumped into it. Money from the US even came after the European economy had started turn around: After a few years of mass shortages. So a few years after the war, the economy, while recovering, was still in shambles. Should war have brought about a boom?

Finally, take a look at all the poorest nations in the world. They all have one thing in common: They have been war-torn for at least 10 years. In almost all of these conflicts, the war started for socia-political reasons, and after a short time, much of the fighting became centered around mines. Why? Because the combatants ran out of money, and need a way to finance the war. But if a war is good for the economy, they shouldn't need half-working mines to fund their wars.

Poem

2009-05-22T21:58:00.006-05:00

It's been a while since, I've given you a poem, eh?

***

Multitudes, multidudes in the valley of derision!
But the glory of the Lord is near in the valley of decisions,
and a wicked wind blows through the sea of visions
and revisions before the taking of toast and tea.

In His house we come and go,
talking, oh, of Michaelangelo.

Remember when David danced in his ephod?
Well, I stood by and scoffed:
The king of Israel naked (with the slave girls!)
for all the world to see!

And centuries later:
"Your mind is not far from the kingdom of Heaven."
My mind is close! O my soul, rejoice!
But, O soul, alack!
It seems you've fled the winter of despair,
and won't be coming back.

These pedals on a wet black bough,
they'll fall off soon, any day now--
drifting to the ground
like nameless faces, wandering through the crowd;
While my mind, lofty, in the clouds,
crashes into a mountain, and comes tumbling down.

***
There will be another one shortly.

Summer goals

2009-05-07T19:15:00.003-05:00

Here's a tentative list of things I want to get done this summer... As with every break, I hardly expect to get all of it done:

Understand Grothendieck's Axiom, and why it is (or isn't) important.
Write a decent Binomial Heap in Haskell.
Read Type Theory and Functional Programming or Proofs and Types.
Read these lecture notes on Categorical Logic.
Read all the minor prophetic books (and Daniel).
Read all of the Epistles.
Get comfortable with functors, monads and arrows in Haskell.
Study for GRE. Especially, read Apostal's calculus.
Make headway into the graph problem Dr. Pelsmajer gave me.
Work with Beckman on something interesting...
Pray more!
Read The Tragical History of Doctor Faustus and Das Faustbuch.

There's more, I'm sure... There's always more.
Edit: Oh, dear! I got none of those done... sigh.

From qntm.org

2009-05-02T14:03:00.001-05:00

You will not be able to stay home, blogger.

You will not be able to dial up, log in and cop out.

You will not be able to watch the revolution unfold on your RSS feed because the revolution will not be tweeted.

The revolution will not be tweeted; the revolution will not cost ninety-nine cents from iTunes; the revolution will not appear on Fark, Digg, Reddit or Metafilter, nor be brought to you by Randall Munroe, Ben Croshaw, Jack Thompson, Ron Paul or Stephen Colbert. The revolution will not be tagged "nsfw" or locked for editing by newly-registered users due to persistent vandalism.

The revolution will not have rounded corners because the revolution will not be tweeted.

*****
Read the whole thing here. Also read all of Fine Structure.

laughter and cynicism

2009-04-24T22:21:00.004-05:00

Apparently I laugh a lot. I've known this for a while, but a conversation I had today reminded me. My whole family laughs a lot... probably much more than we have any real reason or right to. This is a bit odd since I think we all consider ourselves to be rather cynical on the whole. I realized a while ago that this is perfectly natural. This is because there are two types of cynic, one of which is always laughing.

First: what is a cynic? Cynicism can be characterized by the overwhelming desire to mock everything, fueled largely by the realization that the world is absurd and quite frankly isn't worth our time. Another key feature is the inability to take the world seriously.

There are two approaches here (actually there are a lot more, but we can make a lovely false dichotomy): scorn and ridicule.

The scornful cynic sees this ridiculous world and yearns for something that is not absurd-- for something meaningful, something consequential, something more worthy. Such a thing is nowhere to be found, so the scornful cynic mocks and derides everything-- he is, of course, above these trivialities, so why should he do anything besides mock them? The world is seen as a tiring mess of painfully evident errors that the cynic can't help but notice. Such a world cannot be taken seriously, but it demands to be, so the cynic lashes back in frustration.

The ridiculous cynic on the other hand takes all this absurdity as an infinite jest. The world is a joke being told by some grand comedian for the ridiculous cynic's enjoyment. There is a discord between the demands for seriousness and the absurdity of the world making these demands, which only serves to make the world more ridiculous-- like a chimp in a suit. As such, when the world demands to be taken seriously, the ridiculous cynic simply laughs in its face-- does that really expect to be taken seriously?

The scornful cynic, it seems, takes himself too seriously (why else let something so useless get to you?), while the ridiculous cynic does not take himself seriously enough. (Why else let yourself become as absurd as the rest of the world?)

The two types of cynics actually bring out the extreme in each other. The ridiculous cynic sees the epitome of absurdity in the scornful cynic: He demands (by taking himself so seriously) to be taken seriously, while refusing to repay the world in kind. On the other hand, the ridiculous cynic has let himself become truly absurd, and is certainly doing nothing worth anyone's time, which of course is what brings on a cynic's scorn.

On kitsch

2009-04-23T07:20:00.003-05:00

Milan Kundera brings up kitsch a lot in The Unbearable Lightness of Being. He defines kitsch (incorrectly):
kitsch is the absolute denial of shit, in both the literal and figurative senses of the word; kitsch excludes everything from its purview which is essentially unacceptable in human existence.

This is not quite correct. Kundera is confusing two types of rejection: denial and exclusion. Denial seeks to forget, while exclusion makes the conscious effort to remember-- with disdain. Rather than deny shit, kitsch excludes shit from the beautiful with such vehemence that if something is not shit, it is regarded as beautiful.

"But!" you may be thinking, "kitsch rejects the truly artistic. This surely is not shit!" And you are correct; but I think Kundera offers a response: since shit is at the opposite end of the spectrum of beauty from art, they are, to use Kundera's words, "vertiginously close." And so, in order to exclude shit, kitsch rejects everything that reminds it of shit, including the artistic. The result is than kitsch accepts only the mediocre.

A pretty bathroom is an excellent example of kitsch. In order to exclude shit fully, all nonshit must be beautiful. The only way a pretty bathroom can be made ugly is for it to be covered in shit, so when there is no shit, beauty is guaranteed.

Half a percent... Just throwing it out there...

2009-03-20T19:24:00.004-05:00

Note: The following post is intentionally incendiary. I don't actually believe all of it-- well, I do, but there's a lot more going on that makes my points rather moot-- I just get frustrated when people get hung up over a single issue. I feel the need to ridicule people who have lost the big picture over a minor detail. Carry on.

A BBC quote:
[A]nalysts say the government could reduce the payment - which is $30bn - by $165m, in order to force AIG to account for the bonuses in another way. (source)

Doing a little math, the $165M in bonuses is 0.55% of the next bailout payout to AIG. For those who didn't quite catch that: that is just over one half of one percent or 55 cents every $100.

Not .55% of the total amount AIG is going to be given, .55% of the next payment to AIG. Since they've received over $180B in "support" from the government, that $165M is less than .1%-- For every $10 AIG has gotten (so far) from the government, all these investors are sharing from a 1 penny pool. Proportionally, that's about the same as 6 drops of soda left in 3 liter bottle-- you leave almost that much sitting in the bottom when you throw it away.

You're damn right it's outrageous! AIG is ripping these guys off... No wonder they're leaving.

I hope this puts things in perspective.

Divine Hiddenness?

2009-03-14T22:47:00.004-05:00

J.L. Shellenberg came up with an interesting argument for the nonexistence of (a Loving and Omnipotent) God. The Argument from Divine Hiddenness (as he calls it) is compelling as arguments concerning the existence of God go. It can be summarized as: If God loves us, he would do everything He could to increase the "good" in our lives. Clearly, if a god like the Christian God exists, this good would be a relationship with Him. However, there are people who, through no fault of their own have not experienced such a relationship (some of whom even undergo emotional duress at the lack of this relationship), and so a loving God must not exist.

This post is a response that is almost as short and incomplete as the summary I gave above.

My response boils down to answering the following question in a different way than Shellenberg: Is nonresistant disbelief something inherently beyond the control of the (non)believer? If it is, than his argument holds sway... a lot of it.
On the other hand, a loving God would allow the beloved the choice to reject Him (I believe Shellenberg admits this.) If nonresistant disbelief is not beyond the believer's control, then the disbelief is still a willful decision to reject God on the part of the believer, and God's actions at that point* cannot be held against Him. If I am in a relationship with you, and you choose to leave, I have no say. Even were I omnipotent, if I loved you, I would value your freedom, and patiently hope for you to return to me. I would even try to convince you to come back, but force has no place in love.**

So, is nonresistant disbelief beyond the believer's control?
More than once I have been on the verge of apostasy. In each case, it was hardly because I was resistant to God, rather, I was feeling something akin to what Shellenberg describes at the beginning of the linked document. I have been one step a way from that discouraged rejection often.
Shellenberg will likely suggest that this is the situation he describes later, concerning God's temporary withdrawal for some sort of spiritual growth, or other higher temporary purpose. This, it would be claimed, is why I was always "one step" away, but I never took that step.
The problem with that response is looking back on each of these situations, I always wonder what I was thinking as I wandered away from God-- he was consistently providing a way for me to come to Him, for me to accept Him again, and grow closer to Him. So why didn't I take these opportunities? I chose to ignore them. Finally, when there was one step left before I did reject Him, I stopped ignoring Him and accepted His hand.

The point is, it was my choice to begin to wander from Him, and my choice to continue to wander. And finally, it was my choice to "walk away", yet I did not make the choice. Certainly, there were influencing factors beyond my control, hence I would consider my apostasy to be "non-resistant", but I was still resisting God, in that I was choosing to separate myself from Him.

And so it is, my experience suggests, with all "nonresistant" disbelief. This does not make nonresistant disbelief any less sad, nor does it mean that people who have so chosen are weaker, or anything else. I have been fortunate enough to always have enough support from community to keep me alive during these times of duress, not all have such a community... a fact which requires constant prayer and action.

****
*I say "at that point" to avoid arguments concerning God's culpability for previous actions... that is a different debate.

**take this sentence with a grain of salt... Again, a different debate.

Father grant me courage...

2009-03-03T07:47:00.001-06:00

"Ah, Sovereign LORD," I said, "I do not know how to speak; I am only a child."

But the LORD said to me, "Do not say, 'I am only a child.' You must go to everyone I send you to and say whatever I command you. Do not be afraid of them, for I am with you and will rescue you," declares the LORD.

On Syntax and Semantics

2009-02-05T19:23:00.007-06:00

This is actually a post about combinatorics, but before we get there, I need to talk about languages. Every expression in any language has two important aspects: syntax-- the structure of the expression, and semantics-- the meaning of the expression. Let's see an example. I'll take Chomsky's; "Colorless green ideas sleep furiously." Syntactically this sentence is "adjective, adjective, noun (subject, plural) being modified by the adjectives, intransitive verb (present tense, active, third person plural) being modified by adverb, adverb." This sentence is what logicians call a "well-formed formula." A well formed formula is any formula which does not violate the grammar of the language. So, we could replace every word in the sentence with another which has the same part of speech, tense, mood, (and every other grammatical term they satisfy that I don't know) and still have a grammatically correct sentence. E.g. "soft deep swords read wildly."

On the other hand, I think we can all agree that without reading too far into either of the sentences, they are both meaningless-- semantically, they are both nonsense, despite their syntactic correctness. Then again, going a step farther we can milk meaning out of them, and I'm sure there's a Zen Koan hidden somewhere in one of those sentences if you know where to look. This search for an expression's meaning, oddly enough, captures the essence of combinatorics.

I'm going to take another detour. Let's look at algebra. Whatever level of algebra you have experience with, this should be true, although it may fall apart a little bit at the higher levels. When you have some statement such as "x+2=y" it means at any point you see y, you can replace it with x+2 and any time you see x+2, you can replace it with y. Equality in an algebraic sense is a rule of transformation. So when you get some long expression, such as "(x+2)*(x+2) + x-2", you can transform it to "y*y+y-2-2" From these transformation rules, you can show the equality of new expressions. So we can say that x*x+5*x+2 = x*x+4*x+4+x-2 = (x+2)*(x+2) + x-2 = y*y+y-2-2 = y*y-4. Notice that these transformations are syntactic changes. You are replacing one expression (which may be a variable, a literal [e.g. 1], or literals and variables combined by operators) with another expression. The semantics of your expression do not change: x+2 has the same value (semantics) as y. This is the idea behind algebraic manipulation: you never change the values, and so you show that the value of some expression whose value you know (e.g. x*x+5*x+2) is the same as the value of some expression whose value you want to know (e.g. y*y-4).

As I'm sure you've guessed, I'm going to assert that in combinatorics, we make semantic transformations. This may seem to be really dangerous at first: how does reinterpreting an expression give us something valuable? You can't just say that "x" means something different because you feel like it! So what's happening here?
To be precise, you don't actually make semantic transformations-- I lied. Instead, you're equating semantic interpretations of an expression. This may still seem problematic-- "colorless green ideas" can mean just about anything you want it to mean. The difference here is that math is significantly more precise. If you say x means y, you don't mean that x gives the emotionally impression that y does, you mean that under some reasonable interpretation of the system, x is interpreted as y. What constitutes a reasonable interpretation is a foundational issue that I'm not going to get into. So, "fine," you say. "I can accept that meaning is stronger in math than English; but what the hell are you talking about?"

Combinatorics works under the the assumption that mathematical expressions are representations of some sort of structural relationship-- some abstraction of a pattern or structure that is commonly found somewhere. And these expressions sometimes codify the same abstract structure. When we can find overlaps like these, we've found two things which are the same.

Ok. So, let's complete this thought with a classic theorem from the first week of any combinatorics course. Let C(n,k) be the number of ways of choosing k objects out of a set of n, without repetition, and where order doesn't matter-- so we want to know how many hands of k cards we can form out of a deck of n. Then C(n,0)+C(n,1)+...+C(n,n) = 2^n, whatever n happens to be.

The proof is as follows:

The right hand side: 2^n is obviously the number of bitstrings of length n: every bit is either 0 or 1 (2 choices) and we have n of them 2*2*...*2 = 2^n.

The left hand side: establish any ordering of the n objects. When we choose k elements, we mark the k we've chosen with a 1, and the rest with a 0. This gives us all bitstrings with k 1's. Now we sum this over all k, this gives us all bitstrings with any number of 1's of length n; in other words, all bitstrings. Since both the left hand side and the right hand side count the number of length n bitstrings, they are the same.

So, what are we doing? we're saying "what does this expression mean?" and finding something... and then finding another way of saying the same thing. It's a very weird way of doing math. A friend of mine once said combinatorial proofs almost seem more "subjective". There's a bit of truth in this.

One thing that combinatorics does is elucidate connections between expressions. Since you're looking at what an expression means, linking the two ideas comes naturally. An algebraic proof says "look you can use these interchangeably", but a combinatorial proof goes a step further, it says "these two concepts are actually the same." It provides a link in your mind between two things that aren't necessarily linked in an obvious way.

And this post happened because I was trying to find a combinatorial proof for C(n+1,2) = 1+2+..+n (which is trivial to prove algebraically).

Edit: The last sentence reminds me of trying to find meaning in "colorless green ideas sleep furiously." Perhaps there is no good reason they are equal. maybe it's a sentence which works, but there is no "deeper meaning". Is this possible? do mathematicians accept this possibility? I'm not sure if they do.

A Note to Most Christians (probably none of whom read this blog...)

2009-01-17T11:41:00.002-06:00

Non-Christians-- especially non-theists and anti-theists are missing the point entirely. And it's our fault.

Well, they're getting the message we're sending, but we're not sending Jesus' message of Love and Truth; we're sending some butchered Pharisaic religion that is closer to the legalism Jesus fought against than what he fought for. We need to change that.

Here. Now.

2009-01-13T15:55:00.003-06:00

Again I'm taking issue with a note in the Scofield Study Bible. On the whole, I like it, but it is very focused on the second advent. The note on Malachi 3:1-5 reads "[...] the next words [...] are nowhere quoted in the N.T. The reason is obviously that [...] the picture in vv. 2-5 of the Lord who suddenly comes to His temple (Hab. 2:20) is one of judgment, not of grace. Malachi [...] saw both advents of Messiah blended in one horizon, but did not see the separating interval described in Mt. 13 which followed the rejection of the King[.] The Church Age was even less in his vision [.] "My messenger" (v. 1) is John the Baptist; the "messenger of the covenant" is Christ in both of His advents, but with special reference to the events that are to follow his second coming."

I don't read the passage the same; and I think their reading, while not necessarily wrong, is liable to enforce the kind of attitude which Christians must fight, tooth and nail, if we are to be taken seriously, and if we want to glorify God; the attitude is the complacent attitude that God's tremendous work is to be saved for the second advent, so we can wait patiently and complacently, go to Church once a week, and seclude ourselves in little "Christian communities" and forget the outside world, and when Jesus comes back, "Hallelujah! We're saved!"

That's not good enough. I would go so far as to say that isn't Christianity; it isn't faith; consequently, it is not grounds for salvation. We are called by God to fight injustice. We are called to be like Him; to be of one will with Him. So if our Lord was "anointed to preach good news to the poor", we also are anointed to preach good news to the poor. We also have been sent to bind up the brokenhearted, to proclaim freedom for captives and a release from darkness for the prisoners. We are called to live as citizens of God's Kingdom, and to live God's justice, God's hope, and God's love. We cannot afford to be complacent when our faith, and the souls of those around us are at stake.

I'm not saying I don't often find myself being complacent: it's human nature. But we have to understand that we must at the very least strive to be above that. It isn't okay, although, thankfully, with God's Grace, it is acceptable; so long as there is genuine, true repentance.

Getting back to Malachi 3:1-5, the NIV reads "'See, I will send my messenger, who will prepare the way before me. Then suddenly the Lord you are seeking will come to his temple; the messenger of the covenant, whom you desire, will come' says the Lord Almighty.
"But who can endure the day of his coming? Who can stand when he appears? For he will be like a refiner's fire or a launderer's soap. He will sit as a refiner and purifier of silver; he will purify the Levites and refine them like gold and silver.
Then the Lord will have men who will bring offerings in righteousness, and the offerings of Judah and Jerusalem will be acceptable to the Lord, as in days gone by, as in former years.
"'So I will come near to you for judgment. I will be quick to testify against sorcerers, adulterers, and perjurers, against those who defraud laborers of their wages, who oppress the widows and the fatherless, and deprive aliens of justice, bud do not fear me,' says the Lord Almighty."
The middle paragraph, I assume, is where the Scofield editors get the impression that only the second coming is being discussed. But who among those who have been saved can say that the process of becoming saved does not burn? Isn't it nearly unbearable? Such warmth, such light, and after such numbing, cold darkness. And surely Christ's coming was shocking and "sudden" to the religious leaders, and surely it burned more than they could imagine; why else would they have him killed, despite the governor's doubts?
And the rest "I will send my messenger" ... "the Lord you are seeking will come to his temple" ... "Then the Lord will have men who will bring offerings in righteousness, and the offerings of Judah and Jerusalem will be acceptable to the Lord" ... "I will be quick to testify against sorcerers, adulterers, and perjurers, against those who defraud laborers of their wages, who oppress the widows and the fatherless, and deprive aliens of justice, bud do not fear me." Does that not sound like the call for justice that Christ placed on our hearts, that Christ made our mission?

We should be moving, acting-- here; now. And not waiting complacently. Hope does not sit by idly, hope acts and strives for the satisfaction of promises.

This is important... NCSoft is ripping off worlds.com

2009-01-04T01:42:00.005-06:00

It appears video game company NCSoft has been making unauthorized use of a patent by worlds.com-- they have been maintaining a highly scalable architecture for a three-dimensional graphical, multi-user, interactive virtual world system. I am frankly disgusted that they would resort to stealing technology from anyone, no less, a reputable pioneer in the development of 3d internet technology. I hope they get what's coming to them...

*****
Seriously, though, what the hell? What is worlds.com? (Some random company that no one would know about if it weren't for this lawsuit) Why has no one heard of them? (Because they haven't done anything). It's infuriating that they would actually seek an injunction against a company for creating "a highly scalable architecture for a three-dimensional graphical, multi-user, interactive virtual world system." I think all game makers should be paying royalties to Atari, for creating "an imaginative system of interactive electronic entertainment" which includes a "software application to be installed on 'client' machines." You wrote a revenge novel? You owe money to the Dumas estate; you're capitalizing on his idea.

The SCO vs [whole free software community] cases were more specific than this, and SCO was standing on something that resembles ground, and they still lost. This suit is absurd. Worlds is scared about the economy, and sees the money pouring in to the MMO market and wants to sustain their livelihood, so they're trying to tap into that fund. It's shameless and awful. Is a lawsuit really what the industry (not to mention the economy) needs right now? 3d mmos have been out for 10 years, are you that dense that you are just now realizing they exist? Or are you wicked and conniving, trying to get some cash out of frivolous lawsuits? Or are you just desperate?

More importantly, who authorizes such bullshit patents? What jackass in the patent office is so busy daydreaming about being the next Einstein that he can't read through the crap and realize they've patented a genre. I need to talk to a patent lawyer here, but I just got a great idea to patent a scalable system of chemical combinations designed to alleviate symptoms of certain diseases. Of course, in a "preferred embodiment", the chemicals would interact with the patient's vital systems and each chemical would correct an abnormality in said systems. I can totally use the same patent, and just reword it a bit. Or would that be copyright infringement?

Why I don't think I'll read "The God Delusion"

2008-12-16T10:54:00.004-06:00

There's a book which has been out for 2 years by Richard Dawkins called The God Delusion. It is, as its name implies, an attack on God. It is, in short, a polemic. I feel that I should read it for the sake of intellectual honesty, if nothing else. But the more I read about it, and the more short excerpts I read, and the more I hear/read from Dawkins, the less I care. And here's why.

Dawkins quite clearly rejects out of hand the possibility that any other world view than his is right. When I say "any other worldview" I don't mean the possibility of God-- that's what he's arguing against-- but any philosophical starting point besides logical positivism. He is a strict materialist, and a logical positivist, and everything else is flatly not worth mentioning.

He makes, I won't doubt (at least until I read his book... if I read it) a convincing argument that a l.p. framework implies that God is a god of the gaps, and that those gaps will quickly disappear, and God with them. But by doing this and only this, he sidesteps the issue completely. He misses the epistemology, which is the most important part in deciding the possibility of God. You cannot tell someone they have come to a false (moreover delusional) conclusion when you refuse to discuss the framework by which they establish truth. The utter arrogance and ignorance of trying to tell me I'm delusional without even so much as mentioning my epistemological groundings makes it impossible for me to take anything Dawkins says seriously.

He (sort of) rebuts such arguments in the preface with "Do you have to read up on leprechology before disbelieving in leprechauns?" This is a fantastic point, but few people are asking him to go through a detailed study of the theology of Aquinas, Luther, Calvin, Lewis and Chesterton to argue that God does not exist. What we are asking is for him to discuss their epistemological basis, rather than a priori tossing them aside. To quote Terry Eagleton "What, one wonders, are Dawkins’s views on the epistemological differences between Aquinas and Duns Scotus? Has he read Eriugena on subjectivity, Rahner on grace or Moltmann on hope? Has he even heard of them?" The question is not about theology, but about philosophy.

Let me expand upon that point; I don't care what Dawkins has to say about scriptural justifiability of The City of God, Fear and Trembling or Mere Christianity. I care about what he as to say about the way Augustine, Kierkegaard and Lewis establish Truth. What do they consider to be evidence of the truth of a statement? What grounds do they use to justify the existence and character of God? Why is this wrong?

Moving past his naive dismissal of several thousand years of epistemology, his caricature of religion shows how little he has even attempted to understand the religious mind. There is far too much to say on this topic, and I could hardly do it justice, so I will limit myself to a one sentence summary of his feelings on the morality of religion: He finds it to be dangerous and evil. If religion is defined to be "irrational, self-righteous, hate-filled zeal", I agree wholeheartedly. But in his religious (in that sense of the word) intolerance he refuses to accept that the source of such religion can be anything besides religion as it is normally defined and he also refuses to accept that religion (as it is normally defined) can create anything apart from such closed-mindedness.

By so vehemently rejecting everything that isn't is worldview, he is making the most absurd straw-man of himself. He claims that atheism will lead to a more peaceful society overall, but he proves himself, through his self-righteous intolerance, to be a poor example of this "truth". So, I'm not going to read The God Delusion unless someone convinces me that his argument does discuss the epistemological "failings" of a religious view, and that he doesn't caricature religion as I claim...

****
My first point reminds me of a quick exchange that was made a while ago... I tend to be bad at "arguing" points, so this is my esprit d'escalier. A friend of mine (who is also very logical positivist, and can't understand other worldviews-- although he tries, at least) made a comment about science solving some sort of subtle and difficult to determine system. Another friend responded with "that's assuming the scientific method is infallible, which it's not." to which the first friend responded "If it were possible to get machines, which cannot make the same mistakes we do, to do it, it would be." Before a rebuttal could be made (or at least before one was made) we got distracted with something else.

Besides the "you're assuming a logical positive, blah, blah..." I realized that the truth of this statement requires that the universe be objectively observable from within the universe-- otherwise there will always be irreconcilable error in any measurement, hence, science is limited. Given our current understanding of the universe, objective observability is absolutely impossible: "Einstein says so." Any observation can only be made with respect to a given object, and at a high enough resolution, everything about that observation is different for any other object.
Further, if we accept a logical positivist view on the universe, and hence, a Fregean view on math, it makes sense that the universe is governed by some mathematical system. I'm getting to a point I've made before, but.. If the universe is governed by a mathematical system, then we must accept Tarski's undefinability theorem and Gödel's incompleteness theorems. In other words, there are limits to how much we can mechanically discover about the universe. Hence, science has limits.

Developing in "impractical" languages

2008-11-24T12:31:00.004-06:00

So... functional languages, in the case of ones I know, Scheme and Haskell, are often called "impractical" for real applications. Due to the small libraries, this is mostly true of scheme, but it is only because scheme libraries are small... It is not true of CommonLisp, or Haskell, or OCaml, or ...

The most common argument is that they run slower. This falls flat for OCaml, which outpaces basically everything (that means you, imperative language) except C; also, CL is one of the faster languages around. But everywhere I look in relation to Haskell has some sort of argument about it being "slow and impractical".

Let's look at the cost ($$) of developing and running a piece of software written in C vs one in Haskell. Also let's assume that it is computationally expensive (uses a lot of CPU power).

Now, everywhere I've looked, suggests developing in functional languages is at least 2 times (actually closer to 3 or 4 times...) as fast as developing in their imperative counterparts. But! We'll give C the benefit of the doubt, and say that developing in Haskell takes 2/3t (e.g. if a C program takes 3 months to develop, it will take 2 months to develop it in Haskell).

Now, we'll continue with a conservative estimate of $40000 a year for a programmer. For a job which requires a BS in CS and "several years" of experience... this is a modest (and, quite frankly, bad) pay... but programmers love their jobs, right?

Now, a decent (but not huge) project is going to take, say, 9 months for 5 programmers to write in C. So 6 months in Haskell. This means the development cost of the C program is 5*40000*(3/4) = $150000. The Haskell program? 5*40000*(1/2) = $100000. Obviously, since we're looking at 2/3 the cost.
Now, we've just saved ourselves $50000. That's more than the salary for a programmer for a year!

But! We've got to take into account the extra amount of computation required by the Haskell program. Since the differences are constant (not asymptotic), and Haskell is ~3 times more "expensive" than C, we're looking at just getting a better computer here... We'll say one that's "twice as good" (whatever that means), which isn't quite enough, but we've been giving C the benefit of the doubt this whole time...

So, getting a computer that's "twice" as good is going to cost the average person less than $1000 extra-- $150 for RAM, $150 for a CPU and $200-$300 for video card: less than $600. Let's say $750.

So, in order for the imperatave program to be more efficient, all around, economically, there need to be roughly 50000/750 = 75 copies...
Hmm... that's not too much, now, is it?

Giving programmers more money (hurray!) increases this number some, but not all that much... However, there are a few things to note:

(1)In reality, every study/anecdote suggests about 2x speed with functional development. This means we have ourselves $100000, so 150 copies... still small...

(2)Giving programmers $50000 a year (on average... this includes project managers, etc) means an extra $25000 (given (1), above)

(3)The cost of hiring good programmers is not decreasing very fast... the cost of upgrading computers is. So in 5 years, this whole thing is going to be stupid.

(4)Most substantial programs take closer to 10-20 people 1 to 2 years (in C). Given (1) and (2), that's at least 400000 (taking 15 for 1.5 years, that's 1.125Mil). In Haskell? For 15 people it will take .75 years, so that's 562500. We've saved just under one million. So we can now sell almost 1000 copies... still not much.

(5)This is Haskell vs C. In reality, development is done in a combination of C and C++. OCaml performs only moderately slower than C (and faster than C++), and CL is only slightly slower than C++. So, working in OCaml will save about the same amount of time, and won't cost much (any) more to run; developing in Lisp will only cost a tiny bit more to run, and won't be much (if any) more than OCaml... especially with Macros, and CLOS.

(6)There are very, very few applications where performance matters that much... Rendering is just about the only common task. So the "It's too slow" argument is dumb anyway.

My point is: People see that Functional Languages are experimental (theoretical) hotbeds, and say "Oh, they must only be good for language theory"... This is not the case. They are, mostly, only a tiny bit slower to run, and are significantly faster to develop in...
Also, pugs, and darcs

Edit: So, I got my orders mixed up... (Actually, I was looking at an incomplete ranking when I said OCaml and Lisp are fast). The best way to look at things is: The Computer Language Shootout! Look at rankings... play around with what does what... etc.

Euler and Haskell

2008-11-02T19:34:00.003-06:00

I just started learning Haskell, which is a fun, and slightly ridiculous language-- it seems to be what happens when you let mathematicians design a programming language without supervision: It's too clever and uses way too much graduate level theory. Any language which makes frequent use of monads, functors and has a wikibook describing its relation to category theory is the result of an evil genius (or several, to be precise).

Anyway, I'm using Project Euler to learn it, at Anne's polite not-actually-a- suggestion. Which is to say, she brought up Project Euler, and I said "Oh! I can learn Haskell!" It's working rather well, and I recommend it.

Balaam

2008-10-23T18:33:00.004-05:00

Balaam is a figure (apparently a prophet, or oracle) in Numbers (22-25) who was summoned by the Moabites to curse the Israelites as they went from Egypt, through Moab to the promised land. He did nothing of the sort, but still did not do as God commanded him to, and in the end, went against Israel. He is most famous for having a donkey which laid down, and then spoke when beaten. "The way" (2 Pet 2:15), "The error" (Jude v11), and "The teachings" (Rev 2:14) of Balaam are all condemned by New Testament authors.

The teachings of Balaam, which come last in Numbers (and aren't explicitly shown to be associated with him until Chapter 31), quite clearly represent an attempt to subvert Israel by intermingling them with Moabites and Moab culture. There isn't much mystery in them, so I will skip them.

The way of Balaam, often confused or associated with the error of Balaam, is typically taken to be greed. The Oxford NIV Scofield Study Bible has, I think, a mostly correct analysis of the error of Balaam: "The error of Balaam was that he could see only the natural morality. A holy God, he reasoned, must curse such a people as Israel. Like all false teachers he was ignorant of the higher morality of vicarious atonement, by which God could be just and yet the justifier of believing sinners." There is, I think more to this story, but it is details, and not incredibly important, so I'll move on.

The way of Balaam, as I said before, is typically taken to be Greed. This comes as no surprise, since both 2 Peter and Jude mention money (Peter says "wages"; Jude, "profit") when characterizing Balaam's problem. I think this is wrong. Firstly, profit and wages are used throughout the New Testament as metaphor for just about everything-- I guess people thought about money as much in the first century Mediterranean world as they do now.
Beyond that, Balaam quite clearly does not go for money. Instead, I think, he is afraid of conflict... afraid to tell people things they don't want to hear. He is too "nice" to go against them. The following are all quotations from Numbers (NIV); I'll try to edit out only what isn't important, but all quotes are "out of context"...

"The elders of Moab and Midian left, taking with them the fee for divination. When they came to Balaam, they told him what Balak [King of Moab] had said.
"'Spend the night here' Balaam said to them, 'and I will bring you back the answer the Lord gives me.'
[...]
"The next morning, Balaam got up and said [...] 'The Lord has refused to let me go with you.'"

Notice that he does not immediately go with them, thought they represent the king of Moab. He instead says he cannot go.

"Then Balak sent other princes, more numerous and more distinguished than the first. They came to Balaam and said:
'This is what Balak son of Zippor says: Do not let anything keep you from coming to me, because I will reward you handsomely and do whatever you say. Come and put a curse on these people for me.'
"But Balaam answered them, 'Even if Balak gave me his palace filled with silver and gold, I could not do anything great or small to go beyond the command of the Lord my God.'"
Again, he does not go for wealth or power.
Later, three times he is asked to curse Israel, by the king Balak himself, and 3 times, he listens to what God says, and tells Balak. Eventually, he is sent on his merry way without gold or jewels.

However, at every point in the story, Balaam is as polite as possible, going as far as he can to satisfy the people who are asking him to do something that God Himself told him he could not do... He is afraid to tell them what they do not want to hear...

insert(Balaam.concluding_paragraph()); //I don't have anything else to say here...

God of Judgment?

2008-10-16T13:53:00.002-05:00

A friend of mine recently posted something about his uncle's response to him being gay. It included "I guess that’s why you decided you really didn’t believe in God. Made it convenient to act in ways He condemns.”
this reflects a sentiment which is rather common in modern "Christianity": that the Christian God is a God of condemnation. A cursory reading of the Bible might lead one to suspect this is so, but a little more depth shows something quite different.

I'll start with the "curses" God pronounces. And he pronounces several. But all of them amount to this "The consequence [not punishment] of this action is as follows. Choose wisely." God pronounces several curses before the choice is made (e.g. Genesis 2:16-17, 1 Samuel 8:11-18), and a few after the fact (as in Genesis 3:6-19).

The first curse, in Genesis deserves particular attention. First he says "If you eat of this fruit, you will surely die." After they eat the fruit (and don't die), he says "All these bad things will happen to you."

But what about death? There are two things: They did die... later. Would they have died otherwise? The passage (Genesis 3:20-22) suggest otherwise. Also there is spiritual death, which we as humans, most certainly do experience.

So, why would God condemn us to this, simply for wanting a fruit which made us "wiser"? There is a lot happening in this curse, but it boils down to 2 things:

Firstly, it is not "I sentence you to this" but "this is the result of your action." God wants us to choose to love Him. Choice requires that we can choose not to love Him. Since choosing "not God" is the same as choosing sin (by definition). Choice means sin can exist. The existence of sin implies the existence of evil. So God enabled the existence of evil. His statement in Genesis 3 is that we have actualized that opportunity.

The second thing is that he wanted Adam and Eve to be happy (because he loved them). To choose to eat of that fruit meant to choose not to trust God; to not trust God is to reject his love. To be unloved is to be unhappy, and so choosing wisdom over love is to choose unhappiness.

Beyond that, God pronounces other curses, which contain similar sentiments: "Love me (or come to me, or repent or ...) or this bad thing will happen." They represent the same idea: This is a result that follows from your actions.

There is more to say on this topic, but it really doesn't matter. It comes down to this: God did not create us to condemn us; He did not send His Son to condemn us. He created us, and He sent His Son to love us. "Love is patient, love is kind. It does not envy, it does not boast, it is not proud. It is not rude, it is not self-seeking, it is not easily angered, it keeps no record of wrongs. Love does not delight in evil but rejoices with the truth. It always protects, always trusts, always hopes, always perseveres." (1 Cor 4-7)

Christianity is missing a lot of things these days, the most important one is Love.

Mystery...

2008-10-03T09:48:00.002-05:00

Do we, as a society, understand the notion of mystery? We explain everything away, so that even that which is "mysterious" to us isn't really a mystery, isn't something mystifying and awe inspiring: its something to coldly study...

The lack of mystery is among the many things that we are missing. Mystery drives us, it fills us with passion; it moves us.

Almost no one feels that mystery. Maybe in the past not many people did... maybe it's only those with above average intellect who actually feel mystery and are inspired by it, and in the past they were the only ones who wrote, so the apparent lack of mystery in our lives is just the result of biased data.

But I think even most of those who "explore" the universe don't feel mystery in their daily lives. I think we have a habit of just accepting the world and moving on...

I think that's a problem.

haha... Good times

2008-08-10T11:32:00.003-05:00

Today's sinfest (10 Aug, 08) contains a brief retelling of the Gospel... In comedic form... with a terminator parody.

On evolutionary models...

2008-08-10T10:57:00.004-05:00

I was thinking about evolution on the way home this morning. It bugs me, because Darwin really didn't say anything at all. There's nothing profound, nothing insightful about it. It is a general model for change. "The object most suitable to it's current environment will be the one that survives best, and small changes accumulate (somehow), such that new objects will have traits which suit the new environment better, thus prospering." The object can be an animal, a theory, or anything else; the environment can be any type of environment; the changes can form and be accumulated based on any set of rules.
In essence, it says things change, and certain things change "better" than others.

This is why there is social darwinism, memetics, Popper's Knowledge evolution, etc. I realize that rather than falsifying evolution it provides some circumstantial evidence, but that's not the point. The point is: it doesn't say anything. What we now call social darwinism actually predates the theory of evolution-- in fact, Darwin made an argument that can roughly be summarized as "We all know this sort of change happens in the economic realm, why not here? It's really the same sort of evolution."

You can apply the idea to literally anything that changes. Philosophy, science, marketing, pop culture, art, etc, etc-- Figure out the (intellectual/cultural/economic/political) environment, and you can see why the "victors" are the things that grew.

Is it easier to believe that the universe precisely follows simple, elegant and human discoverable laws, or that we impose simple and elegant structures on an unstructured (or weakly structured... or, dare I say, inelegantly structured) universe? What happens if the universe is simply, but inelegantly structured? If the different pieces sort of "klunk" together, rather than flowing smoothly? Can we really, actually, imagine our universe working that way? Or working according to rules that humans can't discover?

We do it because we need to impose structure on our universe in order to understand it. We cannot cope with-- cannot advance in-- a world without order. So we impose structures on our universe. Obviously, the elegant ones are the easiest to deal with. And this is where memetics/knowledge evolution come from: we learn to use more advanced structures to model the phenomena we are seeing, thus allowing us to account for more of the intricacies.

It's the smart ones who can impose structures on anything that go crazy and search for codes in the newspaper. The idea that there are a few simple laws in the background of all of the workings of the universe isn't that different from the idea that there are a few people in the background of all the workings of man, is it? What's the difference between Templar conspiracies and the search for the TOE? Humans are fickle, while universal laws can't decide-- they can only act?

***
On another note, I'm finally creating a "science" tag. I'll retroactively place posts in it at some point.

My Servant Whom I love...

2008-08-05T02:24:00.003-05:00

Isaiah 42:1-4:
Here is my servant, whom I uphold,
my chose one in whom I delight;
I will put my spirit on him and he will bring justice to the nations.
He will not shout or cry out, or raise his voice in the streets.
A bruised reed he will not break and a smoldering wick he will not snuff out.
In faithfulness he will bring forth justice;
he will not falter or be discouraged
till he establishes justice on earth.
In his law the islands will put their hope.

***
The note in my Bible says:
There is a twofold account of the coming Servant:
He is represented (1) as weak, despised, rejected, slain; and also (2) as a mighty conqueror, taking vengeance on the nations and restoring Israel. The former class of passages relate to the first advent and are fulfilled; the latter, to the second advent and are unfulfilled.

The chapter goes on with words similar to Isaiah 61: "to open eyes that are blind, to free captives from prison and to release from the dungeon those who sit in darkness."

I'm not sure if I agree with the note in Bible, about the second description being unfulfilled. I think that the idea that the "Coming Kingdom" is outside of our hands allows a lot of laziness within Christians. While that interpretation is not wrong, and is certainly not unBiblical, it does miss something. As with anything, especially the Scripture, there are multiple levels to anything-- multiple interpretations-- all of which must be taken into account. I am a firm believer that the nature of God has not and will not change; So Our Lord must have been a conqueror from the start. We must, as his followers be diligent, and conquer. But not, as those loyal to earthly kings, in a physical way. The battles we fight are not for land or lives or power. Nor are they intellectual battles, whose purpose is to spread knowledge of truth. They are battles of the Spirit, whose purpose is to spread the experience of Love.

Perhaps the note is correct, and that those things are unffulfilled, but if that is the case, it is only because we, as His Kingdom, have not fulfilled the prophesy.

On Lambda Calculus and Ordinal Numbers (Part 2)

2008-07-29T09:48:00.008-05:00

So we're going to take a little detour and forget about functions for a moment. Instead, we're going to talk about ordinals. First, we need to talk about ordered sets really quick.

An ordered set is a set with order. I know...

So, a set is any collection of objects within the universe of discourse (i.e. if we're talking about numbers, we can't have a set with plates; but if we're talking about dishes we can't have a set with numbers). Typically, order doesn't matter, so the set {1,2,3} is the same as the set {2,3,1}. Also, the number of times an element shows up doesn't matter, so {1,1,1,2,3} is the same as {1,2,3} (is the same as {2,3,1}).

In an ordered set, the order does matter. Some relation applies to each set of two differentelements. I don't use the word pair, because in a mathematical pair, (a,b) is not the same as (b,a). Either a < b or b< a.
To be more for formal. Define a relation < on a set S. For any distinct members a,b of S, either (a,b) is in < , or (b,a) is in < .

An ordered set is well-ordered if the set, and every subset (ordered by the same relation) of it has a first element. For example the integers (including negative integers) ordered according to the normal definition of "< " are not well-ordered, because it does not have a first element. Neither are the natural numbers ordered backwards (i.e. {...,3,2,1,0}).
But the natural numbers ordered normally are, since any subset will have a first element.

An ordinal number is a "number" which designates a well-ordered set. (Well, any well-ordered set that satisfies the same properties).

Anyway, avoiding all the definitions and proofs, when you add two ordinals, attach the second set to the end of the first set. and "rename" the elements to avoid repetition. Notice that this is not commutative-- a+b is not necessarily the same as b+a.

Also, before you ask, numbers are defined as the set of all numbers that are before it in the usual ordering of the naturals. 0={} (empty set), 1={0} = {{}}, 2={0,1}, etc.

For example 1+ω (ω meaning infinity) is {0,0,1,2,...}. Renaming all of the elements after the first one gives us {0,1,2,...} = ω
However, +1 gives is {0,1,2,3,...,1}. The last element can't be relabeled to the "next element of ω" because ω has no last element. So we make 1 into 1' (or something) so we have {0,1,...,1'}, which is still infinite but has a first and a last element, so is not ω.

[note: from now on the relabeling of identical elements will be assumed, so ω+1={0,1,2,...,1} will be acceptable)

Now, multiplication works similarly. In a*b, a is appended to itself b times. E.g.
2*4 = {0,1,0,1,0,1,0,1} (with proper relabeling).

Again, 2*ω = {0,1,0,1,.....} Again, we get 2*ω = ω
But, ω*2 = {0,1,2,...,0,1,2,}. This is ω+ω which is not the same as ω, because we have two disjoint maximal subsets with no last element.

That's really all we need to know about ordinals: addition and multiplication are non-commutative, and for any finite a, a*ω = a+ω = ω, and ω*a = ω+ω+...+ω, (a times).

Now that that is out of the way, we can look back at lambda calculus.

The first thing to notice is that everything has a successor and a predecessor (There's a function for that, but it's messy and we don't need to see it.) And if a comes before b, and b comes before c, then a comes before c. Also, the predecessor of 0 is o-- which is to say, 0 is the first element.

What I'm trying to say is that each of the natural numbers in LC represents an ordinal number.
I said earlier that Y might act like an ordinal number. Namely, ω:
ω=Y=λf.(λx.f(xx)) λx.f(xx)
Since I haven't actually described how this works.Yf takes a function and returns f(Yf). That is, it returns itself with an extra f out front; where f is the function it was given. This also means that ff(Yf) -> f...ff(Yf). Hurray! An infinite number of f's.

We need to see if this is actually a valid encoding of ω. So, we need to make sure all of the following hold:
1+ω=ω, 2*ω=ω, ω+1 is distinct from ω, and ω*2=ω+ω.

Let's look back at our definition.
Addition:
plus m n := λm.λn.λf.λa. m f (n f a)
Multiplication:
times m n := λm.λn.λf. m (n f)

1+Y:
-> λf.λa. 1 f (Y f a) -> λf.λa. f (Y f a). There are a few rules of LC that allow us to throw out the a. I'm not sure how to describe them easily. It has to do with free and bound variables... I'll let you figure that out.
Anyway,
-> λf.f(Yf) which is what Yf "reduces" to. So 1+Y holds as expected

Y+1:
λf.λa. Y f (1 f a) -> λf. λa. Y f fa -> λf.λa.f (Yf) fa
The fa at the end don't go in the (Yf) parenthesis, because Y doesn't use them.
So what we have is equivalent to
λf. (Yf) f.
The a can be thrown out, but not the f, since it's the same as the argument to Y. This means, we have an extra f at the end. I.e. Y is not Y+1.

2*Y:
λf. Y (2 f) -> λf. (Y (ff))
This is our own major problem. I'm not entirely sure what to do with this, and it isn't going to reduce to (Yf) unless I'm very much mistaken.

Y*2:
λf. 2 (Yf). -> λf.(Yf)(Yf)

Y+Y = λf.λa. Y f (Y f a). Again, this becomes:
λf. Yf (Yf), rearranging our notation:
λf. (Yf)(Yf)

So we have a tricky 2*Y. Does that mean Y isn't ω? Probably. Does that mean Y isn't infinite? No. Especially since 2*Y is still going to end up infinite.
I doesn't matter anyway, since Delaney was talking about addition, and this clearly holds for Y=ω.

Future things:
Figure out how to do 2*Y. Figure out how to encode sets and cardinals in LC.

On Lambda Calculus and Ordinal Numbers (part 1)

2008-07-27T20:31:00.010-05:00

I'm still annoyed at the guy I posted about on Friday. Mostly, I'm annoyed about how he accuses mathematicians of being contrary to critical thought, and of being authoritarian. Those two things don't work in math-- you can preach something as dogma, but if you don't have proof (based on certain clearly stated axioms, and valid lines of reasoning), your words ring hollow.

The other thing I'm annoyed about is his callous rejection of transfinite theory. He seems to think Hilbert's Hotel was supposed to be an intuitive "proof" that infinities work the way they do, rather than a clarification of how they work. Also, he rejects infinities... absolutely and completely. As a formalist, I disagree with the rejection of anything interesting, but something else is crying out in pain at that idea-- perhaps I'm more of a Platonist than I think.

Anyway, I'd like to clarify my comment on the fixed point combinator (Also known as the Y-combinator), because... mostly because I'm a nerd. But also because I think it acts like an infinite ordinal.

I'm going to write this in two parts. The first will be on Lambda Calculus, the second on Ordinals and how to apply LC to ordinals. Hopefully, I can also discuss cardinals, but that may be a bit of a stretch.

First, a bit on (untyped) Lambda Calculus and Church Encoding: Lambda Calculus is a formalism for dealing with functions. For those who have heard of set theory, think of it as an analog, only everything is a function (or an argument of a function), rather than a set (or element of a set). The basics are as follows: You have some function f, and some argument a you apply the function to the argument and get a result ((f)a) -> b. So b is the result of evaluating f. It is defined based on how f is defined, so if we define ((f)a) -> a, then for any a we use we get the same value back (this function is called the identity, often just I.) When it isn't ambiguous, we can drop the parenthesis:
Ia is the same as ((I)a) which will evaluate a.

You can have any number of arguments:
(f)ab) -> c, or (..(f)a... z) -> A.
But, you can treat f as a function which returns a new function. Looking at the first example above [(f)ab -> c], we can say (f)a -> fa, where fa is a new function, which will take some new argument. In this way, a function taking any number of arguments can be turned into a series of "chained" functions, all returning new functions.

What's useful about this, is we can forget that non-functions exist (until it's convenient to do otherwise), and just work with functions that return other functions. Also, we can abuse our notation and use ((f)abcd...) to be ((((((f)a)b)c)d)...); the first is obviously cleaner.

Now we can also define what Church calls "abstractions". Abstraction is just a way of defining a function. It looks like:
f := λa.b
This is the same as what I showed earlier with ((f)a) -> b, but it's more precise notation (technically, my earlier notation means something different). What it says is f is a function which takes 1 argument and returns b. So the function I (the identity) is I := λa.a
We can do the same thing with multiple arguments:
f := λabc.d OR f := λa.λb.λc.d.
Notice how the second version can be interpreted to mean f takes one argument a, and returns a function which takes another argument, which returns a function....

I won't get into the gritty details, but there are only a few things we're allowed to do in lambda calculus:

*Define a function, using abstraction (and maybe a special notation... we'll cross that bridge when we come to it.),

*Rename variables (this is called alpha-conversion,). There are certain rules for this (which we don't need to pay attention to), but the only real purpose is to avoid confusion, and to make
proofs easier to follow.

*Apply abstract functions. (Called beta-reduction) This only comes into play with abstractions, but since we can only use functions which have been defined via abstraction (or in terms of already defined functions), this can be applied to any function. If we have some abstraction λx.fx (where f is some function we already know), applying the abstraction looks like this:
((λx.fx) a) -> fa.
All that really happens is we return what's after the "." with everything before the dot replaced with everything outside of the parenthesis.

*Replace equivalent expressions with each other (called eta-conversion). So, for example, I and λx.x can replace each other in any expression.
This comes into play when we have, for example ((I)a). To evaluate this:
((I)a) -> ((λx.x)a) , which is an application, that beta-reduces to a. This is, of course, mind-numbingly tedious, but in actual calculation, most of these steps are done implicitly

***
Now, it's a little more complicated than that, but not much. However, we can construct all of our numbers with it. I'll only focus on the natural numbers (0,1,2, etc), but it can be extended to integers, rationals and the reals using messy definitions in much the same way as in set theory.

Now, a "Church Numeral" is an encoding of a number using lambda calculus. Basically, the goal is to use LC to make definitions for all of our numbers (well, counting numbers) as well as multiplications, addition and exponentiation (exponentiation, just because it's easy). We'll ignore subtraction and division because they are actually rather messy. What we need are functions that interact the same way our numbers and operations interact.

Let's start with numbers. An easy way to define a number is to say that a number n takes a function, and applies it n times. This makes sense, but it isn't quite lambda calculus. So, we start with 0:
0 := λf.λa.a The astute reader will notice that what it returns is the identity:
0 := λf.I

1 is defined in the same way:
1 := λf.λa.fa Ignoring the a, we notice that this is the identity. The one difference is that we restrict the valid arguments to function values. (In general, f will always be a function. a can be, but whether it is necessarily a function is dependent on context, and will be clarified)

Now, we can define a successor. That is, the number after the number we're given. It takes (awkwardly) 3 arguments (see above about arguments, and see below about bracket notation):
SUCC := λn.λf.λa.f (nfa) (i.e. ((f) (((n)f)a))
What it says is given any number n, the successor is defined by applying the function argument of n (that is, f) to the other argument of n (that is, a) one extra time.
This is how we get to each number from the previous one. As an example:
[Don't worry if you can't follow my examples, they shouldn't actually be necessary.]
SUCC 2 -> (λn.λf.λa.f (nfa)) λf.λa.ffa -> λf.λa. f ((λf.λa.ffa) f a)
applying the "f a" at the end we get:
λf.λa. fffa; when we count the f's this means 3.

Now, using the same logic as the successor function, we can defined addition:
[M+N] := plus m n := λm.λn.λf.λa. m f (n f a) {remember that m and n are numbers}
The thing to notice about this is (n a) is the "a" in m. I.e. ((λf.λa.f...f a) f) (n a) -> (λa.f...fa) (n a) -> f....f...fa , where the first "f..." is m f's, and the "f...f" is n f's. So we have m+n f's, as we want.
Another example:
[2+3] -> (λm.λn.λf.λa. m f (n a)) 2 3 -> λf.λa. 3 f (2 f a) -> {dropping the leading λf.λa for cleanliness}
3 f ((λf.λa.ffa) f a) -> 3 f (ffa) -> ((λf.λa. fffa) f) ffa -> (λa.fffa) ffa ->{adding the λs back in} λf.λa.fffffa -> 5

Next,
[M*N] := times m n := λm.λn.λf. n (m f) {i.e. ((n) (m f))}
Notice that the function being applied n times is (m f). So, we have (remembering back to 2nd grade) n sets of m f's, or n*m f's. Again, just as we want.

Finally, exponentiation. I don't know if I'll work this into the second part... but we'll see.:
[M^N] := exp m n := λm.λn.λf n m f
This is very much like multiplication, only n is applied straight through (without the little "m f" detour), so What ends up happening is we get n copies of M all multiplied together.

Anyway, tomorrow I'll have a post about ordinals, and how to apply ordinal arithmetic to LC

On One (and Infinity)

2008-07-26T01:41:00.003-05:00

So, I once again find myself puzzled by people who cannot accept that .9... = 1. I'm not sure exactly what it is that draws mathematicians to this puzzle; it very well could be the average person's confusion-- similar to how many theologians felt the need to waste their time refuting The da Vinci Code.

Anyway, I stumbled across this, a mostly well written essay discussing the topic, with the author ultimately coming to the conclusion that he isn't entirely convinced of their equality.

Avoiding the painful first paragraph (which had me expecting an entertaining piece of non-sense), his argument comes down to a rejection of attainable infinities. viz, the philosophical ideal that infinity is outside of the realm of mathematics.

This is, I guess, a valid axiom, assuming of course that you're a platonist (or, as it turns out in this case, an intuitionist). A formalist can choose to reject the infinity axiom (or any axiom dealing with infinity-- thus, the idea of infinity), but he cannot claim it is wrong. Other philosophies of math run into similar ideological problems. Platonists can claim that infinity "doesn't exist", and I guess, looking at the average complaint with the .9... = 1 proofs comes down to "infinity doesn't exist!"-- an argument which only makes sense in a mathematical platonic context. Of course, Delaney's rejection is largely from the intuitionist persepcetive... I'll deal with that specifically in a bit.

Overall, I find it an annoyingly awkward position to take, and cannot believe that someone with the math training that Delaney has would take it. Allow me to explain:

First and foremost, we lose the concept of the decimal expansion (or maybe not...). This is a small loss for Delaney, as he rejects them anyway "With great temerity, I still hold that any decimal expansion is never exactly equal to pi. The decimal expansion is simply an approximation of pi."

Besides the implicit rejection of any non-finite decimal expansion, this seems to be a a confusion between number and numeral. A numeral is a symbolic encoding of a number-- that is a symbol (or group of symbols) which is interpreted to mean a number. It is, for lack of a better description, the "name" of a number. Just as the word "one" is symbolic of the mathematical object 1, any written decimal expansion is a symbolic description of the number under question. So, yes, any written numeral describing pi (either "pi" or "3.14159..." or anything else) will not actually equal the number itself. However, pi is a number with a value in the real (well-defined) number line. And a decimal expansion (as a mathematical object, rather than a symbolic representation) will have the exact value of pi. Whether or not a human can "read" this value on a page is immaterial-- the mathematical object which is a decimal expansion is identical to the mathematical object which is left as a greek letter. Again, they represent the exact same mathematical object. When doing algbera (or calculus, or anything else), you are working with mathematical objects, not any representation of the mathematical object. It is only when the representation is ambiguous (e.g. when a number is rounded) that problems arise, and that is not because of the objects, but because the representation is imprecise. [note: In rare instances, you do work with the representation, but in such pursuits (aptly named "metamathematics"), the representation is treated as a mathematical object, and is subject to it's own rules (like how working with the reals is different than working with vectors).]

Unless I'm mistaken, we are then forced to allow "infinite decimal expansions" as long as we allow the mathematical object they are identical to. If, on the other hand, we reject these objects, we are rejecting the irrationals (a hefty price to pay), as well as many of the rationals: any number which can be represented as p/q, where p is relatively prime to q, and q is relatively prime to both 2 and 5 will be disallowed (which is to say, 4 out of every 10 non-integer rationals)
We are left with a set of numbers which is not closed under any operation-- that is, we do not even have a group, unless we restrict ourselves to the integers... a boring mathematical universe indeed. Unless of course, we "diagonalize", which Delaney also rejects.

The point of that whole paragraph (which may have been lost somewhere) is that we either accept decimal expansions as valid mathematical objects which are [i]equal[/i] to their fractional counterpart, or we reject almost the whole number line.

In the same way as he questions the nature of decimal representations, he questions "epsilon" the elusive little infinitesimal that he (correctly) calls a "logical entity". Now, he suggests that the separation of epsilon between two numbers (e.g. .99... and 1) should be taken into account. Contrary to his likely expectation, I'm not inclined to disagree. The problem, of course, is that saying two numbers are unequal in a continuum (e.g. the real number line) means there is a number between them. There is no number between them. The "difference"-- epsilon-- is a representation of exactly that idea: they are at the same spot on the continuum, thus there is no mathematical difference between them.

If epsilon had any actual properties in the real number system, we as the mathematical community would be glad to take these into consideration, but adding it has the same effect as 0-- that is to say, no effect at all.

His argument (in the Repeating Nines essay) really hinges on these two concepts: Epsilon, and unattainable infinities. Beyond that, he seems to be bitter because he's on "the losing side" of the debate between "logicists and intuitionists". Of course there is no "losing side" in the first place, as there are plenty of intuitionists doing math right now. Delaney apparently just has trouble working in systems with rules different from those he thinks are "right"-- even most Platonists I know can work in a system they think is untrue. Why was Delaney unable to finish?

Beyond that, he tries to claim that "logicists" (I assume this is a blanket term for formalists and platonists) focus on paradoxes as a way to construct a system... How is this possible? As far as I know, set theory is built to avoid paradoxes. Russell's paradox is a bit of blow to Fregean set theory, but Quine's set theory deals with it quite well; it reduces the statement to nothing. Not a null statement-- but something entirely outside of the universe of discourse. Logicians don't focus on paradoxes, however they are, as you are certainly aware, something that must be dealt with when they arise.

I offer, as "evidence for infinity" the fixed point combinator (because a set-theoretical notation will click with him about the same as the rest of the set theoretical notions have). Y Defined (in an untyped lambda calculus) as Y->λf.(λx.f(xx))(λx.f(xx)). I won't go into the details of the operation, but if we are given Yf, we get f(Yf) which becomes f(f(Yf) -> f(f(...(f(Yf))..)) [I know, my parenthesis are backwards from Church's.] So, Yf= f(Yf) = f....(Yf)... No matter how many f's are started with, an infinite number are attained; that is calling y infinity: infinity +1 = infinity + 1000 = infinity. So, would a functional representation, rather than a set theoretic notation soothe his aching soul, and allow him to accept transfinite theory? Probably not; I'm sure he'll describe some other absurdity which "proves" infinite is unattainable.

He can work in a system whose maximal cardinality is Aleph-0 all he wants, and I won't restrict that, but it doesn't mean infinity is any less real than one-- less applicable, yes, but just as real (which is to say, entirely a mental construct, with certain agreed-upon properties.)

I know, it's rambly...

Faithfulness...

2008-07-15T09:32:00.002-05:00

God just pointed out to me a (sort of) new interpretation of Romans 8:35-39:

Who shall separate us from the love of Christ? Shall trouble or hardship or persecution or famine or nakedness or the sword? As it is written:

"for your sake we face death all day long;
we are considered as sheep to be slaughtered."

No, in all these things we are more than conquerors through him who loved us. For I am convinced that neither death nor life, neither angels nor demons, neither the present nor the future, nor any powers, neither height nor depth, nor anything else in all creation, will be able to separate us from the love of God that is in Christ Jesus our Lord.

**
I guess it's not so much a new interpretation as it is a new revelation. Once we have tasted of that love. Once we feel that deep bittersweet longing so acutely-- so much more acutely than the bitter sehnsucht we as humans are cursed with-- it is impossible to forget it. Try as our heart might, and despite any laziness, or any apathy, or hardship or distractions, something will always bring that love back into our heart, and like the eager bride before her wedding, we shake with excitement.

God's faithfulness is amazing. It's so incredible how he seeks us despite everything. If only I could consistently nurture that relationship, instead of getting apathetic, I would see more of the amazing things I've seen...

Hurray, Postings!

2008-06-17T18:41:00.003-05:00

Sorry, it's been a while since I've actually posted anything here. And this post is insubstantial...

link: Landscapes. It's a project I'll be slowly working on with time... So far, only one piece; another will be up tonight. After that, who knows.

Like Clockwork...

2008-05-14T22:31:00.004-05:00

This is something I recently came up with; I wrote it down while I was at breakfast with Ellie (actually on the train back from breakfast). I'll be reading it Friday.

I've been inside that machine-- the one they say revolves the heavens about the earth-- I've watched its gears turn, traced the outlines of their teeth with my fingers; I've even stood on some. I watched a gear crack and fall, a spring loosed to follow its own path.

I looked up through the labyrinth of clockwork-- the endless cacophony of ticking and grinding-- and I thought to myself "This is not the machine which revolves the universe."

Imagination

2008-05-06T08:47:00.005-05:00

I was thinking in the shower about a comment my physics professor made regarding funding for scientific research. "If the government doesn't fund it, who will?" I'll leave an argument for how stupid this question is for someone else to make, because I just don't care, but it got me thinking. We humans are so (read; sooooo) incapable of believing that things can work any way besides the way they do. We have really no imagination.

This set me thinking (since it was a physics professor) about quantum theory, and 20th century physics in general. I have no trouble accepting wave-particle duality, quantization, super-positioning, entanglement, the Uncertainty Principle, or all of the other fun names we have for physics concepts. I can accept them and understand them without the slightest twisting of my brain, because I can say with absolute conviction that a donut makes the same shape as a coffee cup; I can say with absolute conviction that 2*3=1; that 4*4 = 6; And I can say with a straight face that 4 or 5 dimensional space is "easy", and is just a special case of d-dimensional space. Imaginary numbers are as real as negative numbers are as real counting numbers (real in the intuitive sense, not the well-defined). I won't bet that a coin which has landed heads 1000 will land heads or tails on the next flip, unless I'm betting less money than you are.

I can turn a hollow sphere inside out. I can split a solid sphere into two spheres the same size and density of the first. And I have no super powers.

All of these different facts and systems that I've mentioned have different rules, follow different patterns, have different truths. Also, I'm expected to understand all of this before I leave college. So why would it not be the same with physics? Why could it not be the same with funding? Or art? Or the future? Why could it not be true that God follows rules which don't make sense to us?

We humans have such poor, poor imaginations. "Capacity for abstract thought," no! Where is the abstract thought?

There are times when I still think I should have been a physicist. It's so mind-numbingly intuitive; the math is easy, even at the quantum level. Oo! Group theory! I need to know that to get into graduate school; as well as ring theory and field theory, analysis, topology, and anything else you physicists have tried to play with. One of these days physicist will start using category theory to start kludging all these half-baked ideas of theirs together into a "coherent" whole, and that's when the rest of the world will know they're just making it up as they go along.

At lest mathematicians know their making it up; and are expected to.

Maybe I should still be a physicist; Maybe physics need someone who thinks they're all a bunch of idiots.

Combinatorics

2008-04-25T11:37:00.004-05:00

Counting is fun. It's just really enjoyable to sit there and figure out the most "elegant" way to solve a problem. It's really, really amazing that there are so many ways to approach the same problem, and as long as they count the same thing, they're the same.

It also is finally starting to give me a solid grasp of isomorphism. Not that the idea of isomorphism is difficult, just that seeing isomorphisms sitting there in front of you is difficult.

I think graph theory will fully solidify that idea for me.

Functional programming in Perl!

2008-04-19T17:47:00.003-05:00

Perl has Lisp-like lexical closures and it supports currying.

Both of these happen through perlref, a new feature in perl 5. Essentially, how it works is you can make anything a scalar ($) variable... anything. Sort of.

You can make a scalar variable that points to anything, which is, for the programmer, almost the same thing.

This way you can make a variable that is the curry of a function... I'm not going to go into the details, because my understanding of Perl is, well, yeah... It was fun to find out as I was searching for something else, however.

***
Perl, it seems, is a really fun language, for mostly opposite reasons from scheme. Perl's beauty comes from the amount of pre-imposed structure, but a very large language to work with. Pretty Perl is sort of like really well-written blank verse: it is going to have some obtuse words and awkward grammar, but it still flows well.

Scheme, on the other hand, comes from a small rule set, but a language which can be molded and formed as you please. Scheme would be comparable to some of Cummings better works: It follows (almost) no rules of language, but it drags you around the program nonetheless.

...
Or something like that. The poetry = programming analogy really falls flat under any rigorous scrutiny, but I hope I got the point across:
Perl- a lot of rules, but a lot of ways to exploit them
Scheme- few rules, meaning absolute freedom to exploit anything and everything.

Functions are the Numbers of the 21st Century

2008-04-14T19:41:00.003-05:00

I went to two of the talks at Mengerfest today, (at the urging of a professor). They were unenchanting. Ellis's talk was a research talk, and so came with the dryness to be expected, while the material was interesting. Trefethen's talk would have been exceptionally interesting, were I a numerical analyst. Alas, I am no such creature.

After the talk someone commented about Trefethen's talk, and quoted Menger (apparently from a conversation): "Functions are the numbers of the 21st century."

That statement says a lot about the process of abstraction, and mathematical thought in general. Mathematics, (and it seems, with it, all of human thought) is tending toward increased abstraction. Originally integers were the epitome of mathematical abstraction; then rational and real numbers; then the Cartesian plane, complex numbers, and then sets.... With the 20th century developments of general abstract nonsense, topos theory and type theory, mathematics has climbed higher up this mountain than ever before.

There's a certain similarity in rules underlying all of these developments and abstractions-- they all work more or less the same. This implies (at least) one of three things:
1)The way we do math requires structures to exhibit similarities.
2)There is an inherent structure behind math... That is, the universe is design in such a way that there is a tremendously simple, yet elegantly complex structure.
3)The structures we impose are mental constructs. The fact that we are doing math requires these structures-- our mind cannot process abstract data in any other way.

Being me, I tend towards 1 or 3. 2 is too deterministic. 1, I think, presents more interesting applications: Is there another way to approach math, so that other structures develop?

And when I step back and examine the broader implications of 1,2, and 3, I realize, once again that it just doesn't matter. Math will still be conducted the same way; humans will continue to act roughly the same and life will continue as normal-- no interrupt to the daily scheduled programming.

If the different systems (that is, math, or the universe, or anything else) end up acting the same way, we have an isomorphism, right? So the systems can be considered the same, no matter how different the rules.

And I realized that more and more, this is why it's hard for me to give a shit when someone spouts self-righteous or otherwise ignorant nonsense: they'll continue acting the same. The system in which I said nothing at all is more or less the same as the system in which I correct their error, in the name of "Truth."

And at this point I've lost my train of thought... something on everything being the same, and so on and such... and I need to count some orbits.

Cheers.

Philosophy of Math

2008-04-12T00:41:00.003-05:00

I was thinking today about the various philosophies of math. Ideally, I'm a formalist, but more and more, I don't think that is quite right. Various other semi-formal positions seem to be a little closer to what I believe than formalism proper. Anyway, as I was considering this, I came to the conclusion that math would still be done by the same people, in the same way no matter which philosophy turned out to be "correct" (if there is one). With this realization I came full-circle, because it is such a formalist thing to decide: "it doesn't matter what it means outside of the system, because the system works the same way no matter what."

I'm also still bemused with Gödel's second theorem: For any formal recursively enumerable (i.e. effectively generated) theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent. That is, any mathematical system that can express addition can not be proven consistent within the system. If it contains a statement proving its consistency, it means it is not consistent. Gödel's first theorem is likewise entertaining, but the basic idea of the second is just hilarious. It says that a mathematical system saying it is consistent is like someone saying "I have never lied." Everyone has lied at some point. Thus, we know that statement to be a lie... Only it's stronger in math.

Perl... and the pearl of great worth

2008-04-11T13:56:00.003-05:00

Schlomo just sent me the scripts he's using to pick apart Shakespeare plays. They're really not that bad, but since (presumably) he's the only one who has had to deal with them, his documentation consists of #{file does this}. It makes it severely difficult to follow, when there are 10 scripts which assume things about each other. They are fair assumptions, but not helpful when you don't have the context to understand...

****
On another note, God has been active lately. We're all growing closer together, and it seems more people are slowly joining our community. It's really exciting

Romans 8 (NIV)

2008-04-05T13:53:00.003-05:00

Therefore, there is now no condemnation for those who are in Christ Jesus, because through Christ Jesus the law of the Spirit of life set me free from the law of sin and death. For what the law was powerless to do in that it was weakened by the sinful nature, God did by sending his own Son in the likeness of sinful man to be a sin offering. And so he condemned sin in sinful man, in order that the righteous requirements of the law might be fully met in us, who do not live according to the sinful nature but according to the Spirit.

Those who life according to the sinful nature have their minds set on what that nature desires; but those who live in accordance with the Spirit have their minds set on what the Spirit desires. The mind of sinful man is death, but the mind controlled by the Spirit is life and peace;
the sinful mind is hostile to God. It does not submit to God's law, or can it do so. Those controlled by the sinful nature cannot please God.

You, however, are controlled not by the sinful nature, but by the Spirit, if the Spirit of God lives in you. And if anyone does not have the Spirit of Christ, he does not belong to Christ. But if Christ is in you, your body is dead because of sin, yet your spirit is alive because of righteousness. And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies through his Spirit, who lives in you.

Therefore, brothers, we have an obligation-- but it is not to the sinful nature, to live according to it. For if you live according to the sinful nature, you will die; but if by the Spirit you put to death the misdeeds of the body, you will live, because those who are led by the Spirit of God are sons of God. For you did not receive a spirit that makes you a slave again to fear, but you received the Spirit of sonship. And by him we cry, "Abba, Father." The Spirit himself testifies with our spirit that we are God's children. Now if we are children, then we are heirs-- heirs of God and coheirs with Christ, if indeed we share in his sufferings in order that we may also share in his glory

I consider that our present sufferings are not worth comparing with the glory that will be revealed in us. The creation waits in eager expectation for the sons of God to be revealed. For the creation was subjected to frustration, not by its own choice, but by the will of the one who subjected it, in hope that the creation itself will be liberated from its bondage to decay and brought into the glorious freedom of the children of God.

We know that the whole creation has been groaning as in the pains of childbirth right up to the present time. Not only so, but we ourselves, who have the firstfruits of the Spirit, groan inwardly as we wait eagerly for adoption as sons, the redemption of our bodies. For in this hope we are saved. But hope that is seen is no hope at all. Who hopes for what he already has? But if we hope for what we do not yet have, we wait for it patiently.

In the same way, the Spirit helps us in our weakness. We do not know what we ought to pray for, but the Spirit himself intercedes for us with groans that words cannot express. And he who searches our hearts knows the mind of the Spirit, because the Spirit intercedes for the saints in accordance with God's will.

And we know that in all things, God works for the good of those who love him, who have been called according to his purpose. For those God foreknew he also predestined to be conformed to the likeness of his Son, that he might be the firstborn among many brothers. And those he predestined, he also called; those he called, he also justified; those he justified, he also glorified.

What, then, shall we say in response to this? If God is for us, who can be against us? He who did not spare his own Son, but gave him up for us all-- how will he not also, along with him, graciously give us all things? Who will bring any charge against those whom God has chosen? It is God who justifies. Who is he that condemns? Christ Jesus, who died-- more than that, who was raised to life-- is at the right hand of God and is also interceding for us.

Who shall separate us from the love of Christ? Shall trouble or hardship or persecution or famine or nakedness or danger or sword? As it is written:

For your sake we face death all day long;
we are considered as sheep to be slaughtered.

Sylvia

2008-04-05T13:21:00.003-05:00

I
The event of a lifetime is almost underway
The children laugh and sing, let them play.
The eyes of innocence look on and make another wish:
For wishes can come true when they are pure.
A candle burns, wax trickles down to rest on weathered wood
Between to chairs set facing through the glow.
The winter winter wind comes wandering in
To freeze the silent pose.
No matter how it tries, the candle glows.
No matter how it tries...

In one chair sits Sylvia
with sullen eyes and a weak chin,
next to Father Time with his shoulders black and grim.
They've been sitting there with steady glares
unblinking and unchanged,
contemplating something dark and lonely.
They've been sitting there for hours,
days, and, weeks, and months, and years,
waiting for their moment to arrive
The anthem plays a melody that's in and out of time.

The light that flickers only has one point of view
surrounding darkness makes itself a home.
The girl was of fine color and was shaped of gold;
the man was pallid white and set in stone.
The rose with frozen petals never withers in the sun.
The mind that never learns never forgets!
The winter wind comes wondering in
the freeze the heart and skin.
No matter how it tries the candle glows.

Twinkling lights and merry men are dancing in the street
as shining figures bustle to and fro.
The snow is softly falling like a blanket in the cold,
with fakes that warm the hearts of those who let them.
There is one place along the way that's dimly lit and grey,
but no one seems to notice in their joy.
The wintering wind comes wandering in to hear it's own self blow.
No matter how it tries, the candle glows.

In one chair sits Sylvia
with sullen eyes and a weak chin,
next to Father Time with his shoulders black and grim.
They've been sitting there with steady glares
unblinking and unchanged,
contemplating something dark and lonely.
They've been sitting there for hours,
days, and, weeks, and months, and years,
waiting for their moment to arrive.
Finally a slow grin played across her daring face,
and the old bag grew a look of sudden horror.
With that the old man died and smashed his wrinkled, balding head,
and the girl got up and left the room and went ot into the winter wind and...

II
She walked out through the snow.
She left the body there.
The flakes fell to her face,
and her face fell to the ground.
She listened as the chaos swelled around.
While the church ahead began to ring the bells
That thundering sound
lifted her to her feet and
she walked on.

She continued down the road.
The monotony had made restless beyond words.
She found a shovel and dug a big pothole.
Then a carriage crashed and made a big and deafening sound
Police gathered round.
The flakes fell to her face,
while her eyes
turned them to water.

The owner yelled and called her terrible names,
"Whore! Wench! What have you done?"
They cuffed her up and took her shovel away
but then the sun came out
and dried the stuff from her face.
She smiled sweet, apologized, and walked her self away.

Now you're hear in my heart
and you know who you are.
No one's going to clean up this mess:
the wheels have fallen off,
the current's gonna take you where you want,
but you keep on running,
The current's gonna make you who you are.
You know your smile's growing stronger,
while your stride is getting longer:
You're gonna make it out,
on your own.

~By the Butterfly Assassins

Why I'm Not a Scientist

2008-03-30T10:47:00.005-05:00

Science, I've come to realize is not necessarily about "discovering Truth" as most people see it. Rather, it is a way to further the human study of the universe. By this I mean that a good scientific idea asks more questions than it answers-- allowing human questioning to continue forever. Correctness from the scientific perspective does not mean "It is absolutely true." Instead, it means "It fits the data (roughly), and it leads to a new area of study."
As an example (and I realize the absurdity in it), suppose it were proven, absolutely, that God exists, and everything happens only because "God says so"-- there is no discernable logic, no actual pattern to predict in God's plan. It allows for only one question "Why this way?" which is to question to ineffable will of God.
Science, cannot and would not accept this conclusion; our pursuit of knowledge about the universe would be pointless. Doubtless, people would still continue to practice science, creating mathematical models of the phenomenon around them.

What's interesting here is that they would likely continue to find ever more "accurate" models of the working universe, even though they would know that these models do not, in truth, contribute to any understanding of the universe-- it is already understood. This implies a couple of things:
1)We find patterns in everything, whether they exist or not.
2)Any sufficiently complex language can model anything; even if there is no pattern to model.
3)The link between science and math is artificial.

Ramsey theory suggests that in any large random system (which the universe described about would be, for all practical purposes) there will be some sort of pattern, even if it is not inherent in the design or mechanism of the system. The real problem arises when we ask "Which pattern is it?" This question cannot be answered without enumerating the entire system, an obviously impossible task when confronted with the universe.

Science then, would be conducted in absolutely the same way if it were both useless and contrary to truth, yet scientists trumpet Logical Positivism as something that will ultimately unlock the key to Life, the Universe, and Everything.

Correspondences. I

2008-03-26T21:47:00.005-05:00

I have opened a dialog with one of the few people who asks truly probing questions of people, and makes truly profound statements about the same. She has agreed to let our emails become public. I will post them in the form: {email from me; email from Jamie}. This way, Jamie is the first to see my responses. I find it only fair as they're directed at her. Also, any and all editing from the original is simply to present the same words in a more readable format. All typos, miswordings, etc., remain intact. Also, i've gotten rid of any quotation i've deemed superfluous. Jamie, if i've made an error, please correct me. This first one contains the first 4 emails.

Anyway:

Jamie!
This is Cory, How are you?

You're someone I want to keep in contact with, because it's hard to find
someone who so actively digs into the human brain/soul/heart/whatever...
Anyway, 2 things:

Would you be able to compile a list of my quotes that you wrote down
while you were here? It's always interesting to see how you have
impressed others...

I have a blog of sorts; ramblings and rants on various topics. It's at
onag.blogspot.com
It may present to you probing questions to ask.

Cheers,
Cory

*****
"How do you know whether you are hitting your own personal boundaries, or you're hitting a limit where you can't go farther"
"Everyone has some sort of void in their soul...."
"...that the things that are hilarious are taken far too seriously by far too many people..."
"We have no objective way of determining if our reality is the same as someone else's"
I haven't thought about the first quote yet because I'm terrified that there might actually be a limit where ''you'' can't go farther. That would fuck with my current philosophy of the world. Yet the very act of being terrified implies there is such a limit....I guess I'm just reluctant to reconstruct my notions of the world this month. I think at this point you were referring to some kind of inner personal limit however and not such concepts as the universe and different dimensions and whatnot. That makes is slightly easier to answer, yet is it even necessary to ask questions about the expanse of space seeing as how whatever knowledge we get doesn't affect its very nature...(or does it?) or ours?
-side tangent- why do we even look for answers in general? Is there ever some kind of permanent answer? If we do not ever actually know everything-which we cannot- or even real answers, ones that are really the Truth and not some kind of answers we have decided on for our own life construction and stability, it is very improbably that we could ever actually know them...so what is the point of questions (ah ha) of searching? What makes someone decide on a certain set of values on which to base their lives?
Back on semi-track. Is this limit you were referring to some kind of inner limit? Some 'soul' limit?
Your other quotes suffered a tragic (they were introduced to my car before they made it to my journal) death.
*****
Yeah, my language was kind of vague... it wouldn't do in a proof. Let's try again:
"How does one know whether they are hitting the boundaries of their own comfort, or an insurmountable-- universal-- mental limit." (i use the singular, ambiguous "they", because it is less awkward than the alternatives.)

It's a scary thing to think about; we have some desire-- some need-- to know everything; what happens if we can't?
I don't know whether this limit is some sort of "soul limit" some sort of paradox of self-reference-- i.e. a boundary on our capacity for metacognition.

So. On questions. This is one that has bugged me for quite some time. I've managed to sidestep the question (more, push it out of the way so i don't have to figure it out) by saying i like to learn because there's beauty in it. This is, in part, a cop out. There's truth to it, but that's only a part of the drive.

The real reason is that it seems (at least temporarily) to fill the void in my soul. Hereafter, i'll refer to this void as nostalgia or despair; i'll explain this idea later. Anyway, whenever i think of the quest for knowledge, i think of Unamuno's The Tragic Sense of Life, which i recommend you read if you have questions about our quest for knowledge. Essentially, it's about the very existential idea of despair, the source of that despair and how we use knowledge to combat that despair. I either completely agree or completely disagree with everything he has to say, but it's an interesting read either way.

Anyway, despair:
A lot of philosophers have discussed the concept itself directly, and i find it difficult to think that any philosopher has hoped for anything greater than to understand it. We have, as i've stated, a void in our soul. Despair (the state of being) itself is the recognition of that void, and the conflict created by trying to understand the nature of that void, as well as the struggle to fill or remove that void.
The Christian answer to "what causes despair?" is that we have a longing for an intimacy with God-- an intimacy, which, being fallen creatures, eludes us in except in brief flashes and moments of clarity. This is, i think, an adequate assessment. The general idea can be generalized to be compatible with more humanist philosophies by saying "we have a longing for inclusion in something greater than ourselves." All of human action can be reduced to a (perceived) fulfillment of this desire. The Christian has the benefit of having found a somewhat satisfactory explanation of the cause, but this still does not help us with the solution to the question "How do we fill the void?" The answer, of course, would be "intimacy with God," but what does that /really/ mean?
One my list of problems with science is the attempt to explain away this desire for depth as some sort of result of our biology and mechanical construction. I find this to be a dangerous and morbid simplification of the human condition. It contributes nothing to the discussion of despair, nor does it provide some sort of solution to the "problem". It only gives an excuse for ignoring it, to push it aside. Ignoring something as fundamental to the human condition as despair is a good way to find yourself lost and empty. I think this is where science as a discipline and way of knowing is right now. (There is, of course, some confirmation bias in this characterization of science, but it's something to ponder.)

The more general approach (inclusion in something greater) is more difficult to find any answers to. Why do we need this inclusion? What do we mean, "something greater than ourselves"? How do we gain this inclusion? Alternatively, how do we satisfy this desire? Can this desire be, permanently, or at least satisfactorily fulfilled?
I guess these questions are the next problem in our discussion.

On a related, but perhaps more uplifting note:
It has been suggested (by i forget whom) that hope is impossible without despair. There is a lot of truth in that idea. Hope is the desire (perhaps even need) for fulfillment of something which is outside of your control-- the fulfillment of something greater than yourself. This desire brings you, in some way, into this fulfillment. In other words, hope is the desire for the work of, and thus inclusion in, something greater than yourself. It is a (partial?) fulfillment of despair. Thus, without despair, there can be no hope.
The question, then, is: can we rise above this despair->hope->despair... cycle, or is it the case that "the cure for pain is in the pain"? And, if we can rise above, is it really something we should desire?
(the quote is from "The Cure for Pain" by mewithoutYou; i'd recommend looking up the lyrics to many of their songs...)

I may have had something else to say, but I forgot it.

Cheers,
Cory

*****
Yeah, my language was kind of vague... it wouldn't do in a proof. Let's try again

Seriously. I don't know how you could have made such an error.

It's a scary thing to think about; we have some desire-- some need-- to know everything; what happens if we can't?

EXCATLY!!!!! Yet, could it not be argued-well is obviously can-that life as the potentially beautiful thing that it can be is made possible, maybe even defined by the fact that there never seems to be any concrete answers, except the ones that are semi-hypcritical such as "change is the only constant"? And the search for answers that don't really exist, or maybe do for a limited period of time, defines us......I'm going to stop on the incoherent train of though.

I don't know whether this limit is some sort of "soul limit" some sort
of paradox of self-reference-- i.e. a boundary on our capacity for
metacognition.

hmmm.....well, I think in order to even start going about an answer to that question we need to define what a 'soul' is.

So. On questions. This is one that has bugged me for quite some time.
I've managed to sidestep the question (more, push it out of the way so i
don't have to figure it out) by saying i like to learn because there's
beauty in it. This is, in part, a cop out. There's truth to it, but
that's only a part of the drive.

Agreed.

Anyway, despair:
A lot of philosophers have discussed the concept itself directly, and i
find it difficult to think that any philosopher has hoped for anything
greater than to understand it.

Thats very eloquent of you.

We have, as i've stated, a void in our
soul. Despair (the state of being) itself is the recognition of that
void, and the conflict created by trying to understand the nature of
that void, as well as the struggle to fill or remove that void.
The Christian answer to "what causes despair?" is that we have a longing
for an intimacy with God-- an intimacy, which, being fallen creatures,
eludes us in except in brief flashes and moments of clarity. This is, i
think, an adequate assessment. The general idea can be generalized to be
compatible with more humanist philosophies by saying "we have a longing
for inclusion in something greater than ourselves."

I felt like making a counter-point at this point in time, not with the idea of what was said, but merely because I felt the need to disagree....which brings me to a more interesting question. If two people agree on an answer, but from two completly different points of view, is it really the same answer?

can be reduced to a (perceived) fulfillment of this desire. The
Christian has the benefit of having found a somewhat satisfactory
explanation of the cause, but this still does not help us with the
solution to the question "How do we fill the void?" The answer, of
course, would be "intimacy with God," but what does that really mean?
One my list of problems with science is the attempt to explain away this
desire for depth as some sort of result of our biology and mechanical
construction. I find this to be a dangerous and morbid simplification of
the human condition. Itcontributes nothing to the discussion of
despair, nor does it provide some sort of solution to the "problem". It
only gives an excuse for ignoring it, to push it aside. Ignoring
something as fundamental to the human condition as despair is a good way
to find yourself lost and empty.

Nice quote. Yet, by filling this void, or trying to make it 'go away' are we not still trying to ignore it or push it aside?

The more general approach (inclusion in something greater) is more
difficult to find any answers to. Why do we need this inclusion? What do
we mean, "something greater than ourselves"? How do we gain this
inclusion? Alternatively, how do we satisfy this desire? Can this desire
be, permanently, or at least satisfactorily fulfilled?
I guess these questions are the next problem in our discussion.

Or does it need to be filled? What if, op, just kidding, way to address that in the next paragraph. Or, somewhat. What if it is this void that not only gives us hope, an hard quality to try to define,? My current theory is that it is this void, this despair that makes life so incredibly, for lack of a better word and approaching class, amazing and difficult and different. I'm just going to ramble now. For awhile, basically my whole life, I've been searching for a way to fill this void, comfort the despair. But more recently I decided to try to embrace it, live whole-heartedly in it. Do you think such a thing is possible? Its rather strange. I don't know what to think about it. In which case there is no need to rise above it but rather incorporate it into one's being.

On a related, but perhaps more uplifting note:
It has been suggested (by i forget whom) that hope is impossible without
despair. There is a lot of truth in that idea. Hope is the desire
(perhaps even need) for fulfillment of something which is outside of
your control-- the fulfillment of something greater than yourself.

[...]

Also, assuming this conversation continues in the direction it is going,
could I post it online?
Feel free. ..... (novel concept :)

I can tell you have absolutly nothing to say about the world. Its interesting. I've missed alot of your ideas this time around because you include so many in each sentence and I'm late to class, but I wonder if you need a reply to the questions you ask, or merely the oportunity to fully do so. Because there is such a ...life? in the moment of asking questions.

Ellie has probably already asked you a question similar to this one-

but if you could have an answer, figure out any one thing, what would it be?

Lambda Calculus...

2008-03-25T09:29:00.003-05:00

Is weird. Really weird.

Why is it that we can think of sets of sets intuitively, but operating on operators is nearly impossible to conceptualize?

On a related note: It's ridiculous how exactly LISP notation follows Church's function notation. Even things that aren't related (necessarily) to lambda calculus are borrowed from him.

Grace, Childishness and Service

2008-03-22T20:02:00.002-05:00

Sully wrote this a few months ago; it's one of many truly inspired writings that have flown through him. I hope he doesn't mind me posting it here...

On thought, before I post it. At one point, he mentions knowing something versus knowing about something. I bring up this point often, but only I believe it is a crucial concept to understand. German has different words for each idea. "I know something" would be translated "Ich weiss etwas". The infinitive of the word (to know) is wissen. It implies knowledge of a concept or idea; one would also say "Ich weiss Tom" to mean that he knows about Tom-- has knowledge regarding who Tom is, etc.
"I know Tom" as we say in English would be translated "Ich kenne Tom." The infinitive is kennen. It implies a familiarity, an understand of and closeness with. "Ich kenne etwas" (I am familiar with something) has a similar-- although weaker-- meaning to "I grok something."

Sully brings up this point (briefly) in the essay. God calls us to know Him-- Gott Kennen; not to know about Him-- Gott Wissen.
Anyway, Cheers (I know it's long...):

** The definition and presence of Grace **

The foundation of a Christian life is Christ. This means, more than anything, that we are founded on Grace, for that is Christ’s purpose. He came for us who were unworthy of the favor of God’s presence. We were unworthy not by judgment, but by action because we chose to step from that Presence into the darkness that is absent Him. Thereby we drew ourselves away from God, not He from us.
Having stepped into darkness, we were lost and unable to ever find Him there, for though we may or may not be able to persevere the breadth and width of darkness by religion and other machination, God does not exist in that darkness and thereby can not be found by us who do exist in darkness. Christ is God reaching through the darkness to find us and offer to us His hand that we might again walk with Him. For that was certainly his purpose, that God and all God’s love and blessings should come to us who were otherwise eternally incapable of finding that Presence – this is Grace.
By Grace we are able to enjoy a Fellowship and a Blessing that we could never attain by our own strength. By Grace we who are only capable of at best the most righteous unrighteousness, should be considered (and by being considered, made) truly righteous and able to be near Him who can not coincide with unrighteousness. By Grace, through mysteries indefinite and unresolvable, the presence of God comes without the need of our own understanding or thought.
All Grace requires is choice and by choice action: The choice of accepting the hand outstretched through darkness to us. The choice to recognize the presence of that hand and recognize the necessity of that hand for ever finding God. The choice to give up our comfortable familiarity with uncomfortable darkness and step into an uncomfortable unfamiliarity with all that is truly secure and comfortable. The choice to continue laying every step beside God’s after being brought beside him. The choice to not again step back into darkness. And the choice, when again stepped back into darkness, to accept the loving hand instantly reaching down through darkness to pick us back up from our stumble. Grace is attained only by this choice. Grace is attained only by this action. If we do not choose to accept the hand and choose to walk beside God as he leads and choose to let him pick us back up and choose to simply recognize his hands and our darkness, we have no part with Grace. We then have no part with God and no part with ever being brought out of the darkness. We then choose and by action have sentenced ourselves to living eternally absent God and living eternally absent the Heaven where he does exist.
So it is that only by a persistent walk with God and acceptance of Grace and fellowship with Christ we can receive life. As Christians, we say these things and accept these ideas with lips that swear our hearts know that truth. However, on this point we are liars, for the majority of us are not graceful people. It is this absence of Grace that makes us besmear the label of “Christians”, depend yet on our own strength, judge with envy or malice those around us, and ultimately lose salvation by desecrating our faith down to the same worthlessness of all other religions.
And yet we say, “Yes, I am imperfect, I know that. But that’s just the point. I am allowed to be an imperfect sinner and yet have God and Salvation.” And yet, in saying this statement, otherwise full of truth, we are completely and utterly wrong. For this statement only applies to those who attempt to overcome their imperfection. It only applies to those who begrudge and hate their sin and wish it washed away. Essentially, it only applies to those who hate the darkness and long for the light. As we often say it, and as it was said above, what we really say and really believe is that we are free to yet live in darkness and still have the gift of God’s presence. And this is this same as saying that darkness can exist within full and absolute light - a contradiction that a child can easily see and make fun of, and yet we make ourselves as dumb as adults to believe it.
The true nature of Grace and the true and only power of “Being saved by Grace” are caught up entirely in the calling of Levi. Jesus walks through the city, sees Levi sitting at his tax booth, and walking up to him, says, “Follow me.” Levi could never have brought Jesus there. He could never have found the offering of fellowship with him. And that is why it is an offering of Grace that Jesus walks up and asks him to follow. However, the bestowing and acceptance of this Grace only occurs, that is the fellowship of a life spent in the presence of God only occurs, when Levi responds and “leaving everything, he rose and followed him.” If he had not followed he could not have possibly walked with and talked with and known who Jesus was - he would have refused Grace. Certainly he would have ultimately heard an awful lot about Jesus and possibly even developed an awful lot to say in favor of the things he did, but he would not have know him. This is how we have become as Christians today. We have heard a lot of about God and have developed a lot to say about God, but we know him not. And why? Simply because we choose not to accept Grace and walk with him. We choose to keep our place in darkness rather than be lifted up into Light.
All of us who are offended by these words testify by our own offense that Grace is not in us. For Grace can not be offended, for it is the absence of judgment and prejudice and pride. Grace is the administration of God’s love and as such bears all things that love has and carries nothing that love does not. As Christ says through Paul, “Love is patient and kind; love does not envy or boast; it is not arrogant or rude. It does not insist on its own way; it is not irritable or resentful; it does not rejoice at wrongdoing, but rejoices with the truth. Love bears all things, believes all things, hopes all things, endures all things.” And yet, how many of us love like this? If we do not appear to love like this, then it must certainly also appear that Grace is not in us.

** The presence of self-righteousness in the absence of Grace **

Self-righteousness is the opposite of Grace, just as the darkness of sin is the opposite of God’s light. Self-righteousness is the belief that ones’ own efforts can bring them into the presence of God. It is the belief that Light can be found in darkness. It is the perpetual selfish searching for things that do not exist at the cost of what little does exist. And yet, is it so surprising that so many are self-righteous? Should they be judged and dismissed for being self-righteous? No! Rather, those who are neither in Grace nor self-righteous are all the more foolish. For the self-righteous recognizes the need to find the Divine above all else and merely disregards the Only Way. The other is foolish enough to also disregard and lie about their own hunger. If one does not accept Grace then what other strength can you possibly believe will ever bring you life but your own? And, as for those in Grace, they will never look in judgment upon the self-righteous but rather look with eyes of patient and empathetic love, just as they will look upon all who are outside Grace.
The pattern of self-righteousness is this, all who are deemed beneath one’s righteousness are generally ‘loved’ and all those who are deemed unfairly above one’s righteousness are always enviously hated. All who are beneath the righteousness of the self-righteous are loved in the most perverse sense of benign indifference. They serve the self-righteous person in that by being beneath him or her, they are believed to lift him or her up closer to the Divine. As such, the self-righteous are willing to be benevolent to these under-souls and also match their benevolence with a thousand times the evil by stepping on the backs of these under-souls and press-ganging them to abide by whatever boundaries of self-righteousness they suggest will lead to the Divine. Those who will not be press-ganged into the self-righteous person’s particular religion, or otherwise suggest the true unrighteousness of the self-righteous, become competition and are hated.
Those who follow the self-righteous person’s views but appear more righteous can either be hated, or accepted in the same way as the under-souls. By being above they can seem closer to the Divine and thereby offer both an affirmation of the self-righteous person’s methods as well as present a human ladder for the self-righteous person to pull themselves higher. In that case those above are ‘loved.’ Alternatively, those above may be seen as obstacles of separation to the divine and by their presence confirm how unrighteous the self-righteous person truly is. In such a case, it is but the logical choice to hate and envy.
Those people who are both outside the self-righteous person’s beliefs of righteousness, appear to be about equivalently as righteous, and who suggest that the self-righteous person is unrighteous, are hated with passion. This is particularly true if such a person otherwise abides by nearly the exact same patterns and beliefs of righteousness. For, in this case, not only is the self-righteous person in struggle for closeness to the Divine, but they are being told that their struggling is incorrect and thereby become even less righteous than they had thought they were.
Of course, all of these words can be confusing, and on this particular previous point it is perhaps of especial benefit to provide an example: Sam believes that it is ok to drink alcohol with friends and be a Christian because Sam otherwise follows all the proper rules of conduct and goes to church and participates in a small group. Hunter, from Sam’s small group, tells Sam that drinking alcohol is ungodly and sinful and that Sam is a poor Christian unless Sam repents of drinking alcohol. They have an argument on this point and begrudge each other about it. Later, Sam sees their pastor give recognition to Hunter for being an exemplary Christian. Sam now enviously hates Hunter. In this case, both Sam and Hunter are self-righteous and thereby both are absent of Grace and thereby also absent of Christ. They have forsaken Christ’s hand for belief that following the straightest lines drawn in darkness will give them a path to Him. They merely argue about whose line is straighter. And yet, both are in fact in darkness. And yet, the hand of God and Christ’s Grace is right there reaching down to them, begging them to take it again or for the first time, depending on if they ever took it before.
Another example that perhaps will touch the reader’s soul better is this: Sam and Hunter have never officially met but are aware of each other’s presence as members of the same church. Hunter has been working hard helping her church. She has been making bulletins and organizing meetings and orchestrating prayer gatherings and leading a small group. Hunter has never seen Sam do anything for the Church. Hunter hears their pastor say that Sam is an incredible Christian and that Sam means a lot to the body of Christ. Hunter now envies Sam and while she will not admit it, actually hates her simply by the fact that she refuses to love her. In this case it appears that Hunter is the only one that is self-righteous and the only one whose monstrous efforts do more to help her reject Grace than bring her close to Christ.
For a final look to expound upon this point of self-righteousness and the absence of Grace, I will return to the example of Levi: Jesus looks across the market and sees Levi, a tax collector, and walking past Rabbi’s and the most ‘righteous’ and worthy of men, goes up and invites the rather ‘unrighteous’ Levi, not the Rabbi’s and Scribes, to walk with him and absorb his love. This simple fact that God is ignorant of all personal effort is what makes grace into Grace. And does this seem wrong? Well, from the framework of self-righteousness it is wrong. If you believe that your personal effort is necessary then you will begrudge those who have God’s love despite lack of effort. That exchange of love is proof that you and all your pride are wrong and that your ideas and efforts are worthless.
The trouble with all this is that it leaves us at the point of the contemporary church, which has accepted that it is not by our efforts but by God’s effort that we know Him. And again, this is true. What is wrong is that we have extrapolated this to mean that Faith only requires God’s effort and not our own. Which really means that we have rejected the idea that we need to take God’s hand to be lifted up by Grace. This leaves us at the point of being able to stay in sin and conduct our lives of darkness, living in richness and sloth saying, “If God intends differently, then He will make it so.” And the truth is that God will make it so. He will overwhelm Earth by the presence of Heaven and the old will go. The new will come – and we will not be a part of it because we did not step from the old when he asked us to walk with Him. We will be in Hell for having rejected Grace.
Now, if we have rejected Grace, then it would seem, as mentioned before, that most of us would fall back into self-righteousness. And that has been outlined somewhat above and many will be able to prove or lie well enough that they are not part of that self-righteousness. However, for us Christians that have rejected Grace by the false belief that we need make no action and that it is all about God, we are left with an entirely different sort of self-righteousness. We come to the point of saying that we have Grace simply by believing in it. That is, we think that it is what we think that grants us God’s love - so we have descended to a self-righteousness not of physical actions but of thoughts.
We argue theology day by day. We pick and choose our spiritual heroes. We form lines in the sand over how we understand particular points. We become upset and horrified at people thinking differently than us - And we are self-righteous and absent Grace and know not Christ.
Now these points are particularly subtle, extraordinarily so. This is why this is the latest and greatest of Satan’s defeats to our Faith. For is it necessary that we believe in Christ and the necessity of Grace and in the love of God? Yes, most definitely – And yet we can believe such words and not have any of those! In fact, we can believe many many more words than all that and have even less!
The trouble comes down to the fact that all these thoughts still have not accepted Grace. You can believe all you want that Christ’s hand is reaching down to you and that it is necessary for you to take it, but if you do not take it you simply do not have it. The act of taking that hand does requires the thought to take that hand and we have failed by equating simply thinking about taking that hand with actually taking it. So in this way we are left in a world of thoughts and absent Grace.
Our self-righteousness becomes one of imagining that if we think hard enough and strong enough and with enough conviction about the means and method of taking God’s hand and the shape of that hand and about where that hand is going to lift us, we will have attained what that hand offers. In simpler terms, if someone invites me and my friend to a walk in the park and we say no and instead imagine walking in the park, have we actually walked in the park? I might make a better guess than my friend at what the weather is like in the park and my friend might make a better guess at how long the walk will be and we both might well enough predict where the walk would traverse, but have either of us actually walked in the park? And yet, we delude ourselves into believing that if we imagine well enough and accurately enough about the walk in the park it is the same as walking in the park! We as Christians believe that if we imagine and believe and know enough things about God and Christ and Grace we will have those things!!!
And so it is that we argue and envy and despise anyone who contradicts us or our thoughts on Christ. We hate and judge in proportion to how overtly contradictory their assertion is to whatever ideas we have accepted. And in all of this, we know the least part of Christ and Grace. To be fair, we may even know accurately about these things, but we do not know these things! Do you meet someone’s strange assertions about God with curiosity and love, or at the very least sympathy? If not, then you are intellectually self-righteous and know not grace. For if you knew Grace you would not need your self-righteousness and would not need to violently argue or judge your fellow human brother or sister. You would not feel threatened. You would not feel frustrated. You would not be absent of love. You would not be absent of Christ. – And yet, even here, in this error, Christ waits with loving arms, not just hands, waiting to pick us up and let us truly know him and love!
To see how incredible this Grace is, let us return again to Levi. When Jesus walked up to Levi, was it necessary for Levi to know this was Jesus? If Levi did not know that this was Jesus. If in contradiction to all other evidence he thought it was some random bum off the street, and yet followed, would he have received any less of the blessings of Christ? Of course not! You see that the Grace is in the following, not in the knowing or thinking. Now, if Levi were to have made believe that Jesus did not exist and that no one was calling him to follow, would he have missed Grace? Yes. However, as long as we recognize Christ’s presence and follow, whether we believe him teacher, beggar, liar, or even Jesus, nothing else matters – there is the same benefit for all perspectives! Levi has shown us that all that matters is following, and that as long as we follow, false beliefs and thoughts can do nothing to us! Such is the wonder of Grace!

** A note on the usefulness of correction **

All of the discussion presented so far has relied greatly on what Grace does not look like to provide proof of its absence. Henceforth, it is of interest to talk, in more hopeful and brilliant life, about what Grace is and how it benefits. In both the former and proceeding cases of negative and positive, the contradictions presented are for the purpose of showing the need to seek out and accept Christ and Grace. On this point, it is the nature of Grace and Christ to point out errors and fallacies, which we Christians certainly do well enough. However, the entire urgency of Graceful pointing is absolutely for the purpose of directing hearts to Christ. By pointing out lack of love to the Pharisees, Christ was pointing out the need for love and thereby the need for himself. It was out of love and the desire to lead others into His Grace that Christ corrected others. This is the only useful means of contradiction. To contradict as we do today, simply for the sake of ‘asserting truth’, is to waste breath and at its best, is merely a means of sustaining self-righteousness. To contradict for the sake of opening another’s eyes that they might see Christ and then long after Him better, not just for ‘opening another’s eyes’, is the only benefit of contradiction. Contradiction as a means purely of judging another and finding them incorrect is a motivation of pride and sin. In one is self-service, in the other love and selflessness.
Further, the nature of Christ does not actually care about the correction of things pointed out. That is, errors are irrelevant in contrast to the need of Christ they point out. In this way Grace does not seek to instruct others in a manner of correcting appearances, but in the manner of seeking Christ. If anything is amiss, Grace instructs ignorance of that error and focus on Christ and from the focus on Christ correction of that error (and correction of many other errors unmentioned and unperceivable) will come. In this way one does not say, “You are not caring for you neighbor” with the intent of the offender actually then going and caring for their neighbor. Rather, they say it with intent that they go and accept the Grace of Christ. And then following this acceptance, the offender will be filled with such love that can not but help caring for their neighbor, (and their brother, and their parents, and those living in distant cities, and all of creation.) All of this is the manner of Christ’s correction, and it cares least of all about focus on correcting any error pointed out. It abides that error quietly in love for the sake of love that the one in wrong may find God’s love. This is Grace, and if it were not this way, none of us could ever stand under the burden of all our faithlessness.

** The necessity and birth of Childishness by Grace **

Grace is God giving us his love and blessings without any other requirement upon us than that we acknowledge and follow Christ. The blessings of God are the assurance of his ultimate rescue through all things, the assurance of his ultimate eternal presence, and the assurance of his ultimate unending sustenance in the realm of Heaven. In a word, Security. The blessings of God are Security. They are a security unlike any security the world offers. This Security is what people long after when they seek the Divine through the plans of all their self-righteousness. And, quite simply, it is for this reason that Grace frees us from self-righteousness and any effort of our own to attain Security. We have Security as a perpetual guarantee written upon our hearts and are thereby free to give up all the struggles of the world.
Now, if we have Security without the need of self-righteousness, what things does this means we can give up and cast off? Mostly, it means that we can give up all the rules imposed by the self-righteous and we can give up the need for strength. For truly, the only purpose of gathering strength in this world is for self-righteousness. In the world of Christ’s Light, He desires to be our strength and wash our feet and give us ease. This does not mean that we will not gain strength, for actually we gain far more strength by Grace than we ever would have by our most violent efforts – again, such is Grace. What this simply means is that we need not be focused on gathering and accounting for strength.
The greatest antonym known today for strength and pedantry of rules, is childishness. And what is childishness? Childishness is the naïve belief that one is protected, secure, can play whatever, can run free without a single care and all this for merely trusting completely and again naively in the absolute power and love of a parent. Effectually, Childishness is then Christianity, for God is our Perfect Parent.
Now, some will say that there are still rules even here, and that will be gotten to, but for the mean time consider that all of this is exactly what Christ intended when he called us to meet him like children. We were called to fear not the world’s judgment, to play each day of our life, and to depend completely and ultimately on him with the same clinginess of any child. For that is exactly what it means to be a child. And if we do truly believe in Christ and accept Grace and have Security, then why should we not be this way? As everything we are to have comes through Grace by God’s power and not the world’s, then we really do not need fear anything the world can do to us. If we are freed by Grace from the struggles and efforts of attaining Security, then what is left to do but play each and every moment of our lives? And if all of this does come by Grace from Christ, then how can we not ravenously and fearfully check every moment to see that Christ is always there beside us within grasp? This is all the pattern of a true Christian life. These are the trappings of Grace.
Childishness can not exist within the world of strength and self-righteousness. Pure childishness requires the complete and perfect service of another. If self-righteousness is all about what we can do for ourselves, and a child is meant to trust to have to do nothing for themselves, then self-righteousness seeks to kill the child. And how does it do this, rather, how do we in all our self-righteousness do this? Day by day we demand and require children to learn how to support themselves. We tell them day by day to play less, to do more for themselves, and to behave in a manner of fearing what the world will do to them. We do not teach them that they can be children and have Security, but rather that Security is only attained by killing the child in them. We teach them this because we were taught this because the world taught us this. And that is exactly the point; the world taught us this, not Christ. If we truly had Grace, we would not teach this to our children and to each other. We would simply have no need to teach and propagate this living death of self-righteousness. We would see how much better it is to become children. More than that, we would be instantly transformed by Grace into children as that is all that would be left to us after being lifted out of darkness.
In the world of the self-righteous, the weak serve the strong. For the strong force the weak to serve, doling out strength and the promise of Security as a reward for service. Strength is a necessity to force one’s way to the top and hopefully somehow attain Security. And no weak or strong person can do anything to change this system because there is always someone stronger enforcing the system.
In the world of Grace, the strong serve the weak. For the strong have no need of serving themselves since they (and the weak) have Security. And as the strong are strong, it makes little sense that it should be any other way than that they should serve the weak. Quite simply, the strong are much better at serving the weak than the weak are at serving the strong. By the presence of Love through Grace, it is the desire of everyone to serve everyone else. There is no frustration in this service and no conflict over serving - as that is not the loving desire to serve another but the hateful and selfish desire to serve one’s own plans and pride. Graceful service bears total patience, total empathy, and total joy. No one bemoans their weakness. No one bemoans the modesty of their service - for cares about the gloriousness of one’s abilities and the grandness of service are cares only of the self-righteous who still struggle for Security. Those in Grace do not only think they are freed, but are actually freed from this care. And, in the world of Grace, since each person has Security, each person rarely has any need, and play predominates and play becomes service. In truth, play is nothing more than the desire to serve expressing itself through each player pretending to have a need to be served. Each pretends to be less than they are that the other might be more. This is true playfulness and true Childishness. All of this is true Grace.
Now, as far as rules are concerned within the freedom of Childishness in Christ, rules are the commands of God. However, they are no different than the rules of earthly parents: “Look both ways before you cross the street”, “Brush your teeth before you go to bed”, “Don’t hit your sister”. Each of these rules is intended for our protection and uplifting and each of these rules requires no more strength or effort than we already have. And this is the difference between the world’s rules and the rules of the true God. They are not burdensome, are not filled with malice, and do not demand our ‘growth’ and the death of our childish nature. As such, these rules are nothing more than God in all his strength serving us who are weaker, and require in no way self-righteousness or strength in trade for love. If they did, they would be rules of the world and not gifts of God.
Grace creates the Childishness commanded by Christ. Without this Childishness, there cannot be Grace. And if there is not Grace, there is not Christ. And yet, with Grace, the constantly offered Grace, look what life there is! Look, to be a child, to be free, to have Security, and to have Christ, all for ever! And what does Christ ask!? Merely that you follow him and accept Grace!

CommonLisp; Also, The Silver Chair... and Abstract thought

2008-03-21T11:45:00.004-05:00

So, OOP has been too ingrained in my skull for me to properly use Scheme for an AI project I have to do, so I've decided to use CommonLisp- It's still Lisp, right? I now know full well why people say "CommonLisp is more powerful than Scheme." There are some really useful things you can do easily with CommonLisp. However, I also know full well why people say "CommonLisp is ugly and convoluted compared to Scheme." Line-noise like this: (format t "~{~a:~10t~a~%~}~%" cd) is why I don't want to learn perl... And I now have to deal with it when programming in lisp. Granted, it does a lot, but is it really worth memorizing all the syntax? Probably, but also definitely not.

***********
On a (probably) more interesting note, I really, really love the scene in the Silver Chair during which the Queen tries to convince the children, the prince, and Puddleglum that Narnia is a dream. Her arguments, which fall absolutely flat when you're outside the scope of the novel are used so often. Of course we can't explain God's presence, it isn't something we have a concrete understanding of.
Questions:
How does someone with Synesthesia describe the shift and change of colors as they hear music? How does one describe the sun to a blind man? Music to the deaf? Snow to someone from the tropics? A computer to a tribesman?
Answer: In metaphor and simile.
But of course metaphors and similes will present the object in question as a twisted version of the concrete object it is being compared to, rather than a distinct object with a similar sensation.

tangentially,
I can (sort of) understand how a synesthesiac thinks, because I have a similarly abstract and incommunicable thought-process. The best way, I think, to describe it is as imagism. I perceive, with all 5 senses and abstract thought an "emotional and intellectual complex" whenever reading, writing, drawing, seeing, doing math, coding, ...
My thought process is a string of Pound-ian images, which shift, move, fall, rise, appear and disappear in conjunction with everything that enters my brain through any medium. There are uncountably many stories, scenes, colors, sounds, tastes, ideas and so much more that pass through my mind every second, and over any period of time, and also distinct from time.

So, how does this relate to synesthesia? If I equate these images to colors (or whichever other sensation the synesthesiac perceives), I see the same thing.

Moving back to my string of questions, is there any way to truly describe my thought process? I've given a quick description; everything I said is true, but there is so much more to the mental "image" than the scenes and Pound-ian images I've described.

The Weight of Glory

2008-03-13T11:14:00.002-05:00

I've been thinking quite a bit lately about The Weight of Glory by CS Lewis. It is a truly excellent analysis of the Christian experience. But from the second or third day after we read it in Small group, something has bugged me about it. I think I finally have a coherent enough "argument" at this point to actually put it online.

Let me preface this by saying that I really do admire and respect CS Lewis. Not only was he a great apologist, but he was clearly a man abounding in faith-- a man who seemed to sit and enjoy God's presence constantly.
That being said, we Christians have an obsession with the afterlife. Not that the afterlife is bad, or anything we shouldn't look forward to, but God's presence and spirit here, now, is so beautiful and so amazing, and so ignored. Throughout the essay, Lewis talks about the "proper reward" of various things. The "proper reward" of love is marriage; the "proper reward" of fighting is victory; he likewise discusses food in metaphor, and (he may actually say this) were he to name the "proper reward" of food, I can hardly imagine it would be anything but health.

Does something seem missing in all of these cases? I submit that there is. You do not fall in love in order to marry; there is no justice in deciding to fight solely because you will achieve victory; food is not merely a way to avoid illness, weakness and death.
Marriage is the proper reward of correct and sustained love, but love comes with more immediate, more beautiful and tangible rewards. Victory is the end reward of proper combat, but fighting for victor as an end is perverse. Health is the end reward of eating, but if you eat solely for sustenance, you will never enjoy food, and you are quite truly eating wrong.

There are, to every action, an immediate reward-- which I will call the excitement-- and an eventual reward-- which I will call the fulfillment. Any action done with no consideration of the fulfillment is sinful-- aesthetic as Kierkegaard puts it, hedonistic would be the more common word. Likewise, any action done with no consideration for the excitement is self-denial-- the actions of the knight of infinite sorrow, according to Kierkegaard, ascetic is a more understood term.
This latter action, self-denial, is the "unselfishness" that Lewis speaks out against in the fist paragraph of his essay. You are denying your immediate excitement because you know that the fulfillment of your action is "good".

But! There is so much more to what God has for us than heaven. There is so much on earth, so many beautiful things that God has given, and so many incredible ways He spends time with us, and enlightens us. We are missing God completely if we hope only for the fulfillment-- even the "selfish" fulfillment of heaven. God is here, right now! And He wants you to be excited in Him. Do not simply drudge along waiting for heaven, because you will get bored and distracted and all will be lost. How long can you eat food only for sustenance before you despise the sight of food? But if you eat food not only for sustenance, but also for enjoyment, how much more will you want food! Then, you will acquire a taste for so much finer, more complex and exciting foods that you would have missed completely if you had neglected your taste-buds.

So it is with God. Seek God's beauty out, and leave heaven at the back of your mind, except when God brings it to the forefront. And you will develop a taste for beautiful things God has done and will continue to do that you would have misunderstood, or disliked, or missed completely.

God is beautiful! God is exciting! And that excitement is tangible and real and all around us!

On Abstraction and Education

2008-03-05T00:13:00.005-06:00

I had started writing a rather in-depth post a while ago, and sort of lost the drive... Here's how far I got. It is a complete thought (mostly), but it isn't everything I wanted to talk about... Ce'st la vie.

Abstraction...
Abstraction is at the heart of knowledge, at the heart of learning, and thus, education. It works as follows:
Start with something we have a concrete understanding of. Now, decide there are certain properties of this object that we are interested in, and everything else is unimportant. Now create (in your mind) a conceptual version of that object with only those properties you are interested in. This is the abstraction step. The beautiful thing is that you now can analyze those properties that are interesting. The other beautiful (and perhaps more useful) thing is that you can now find another concrete object which shares the interesting properties, and you will now know things about it. It doesn't matter whether any other property is different, there will be properties of this new object which you have already analyzed.

This is how education works. Abstraction. Take something the student knows and abstract the object so that you only pay attention to the properties which will tie it to the next thing they need to learn. Through this abstraction, the student has a lens through which to view the new, unknown object. A good educator will not only make an abstraction clear (not necessarily explicit), but will show the abstract relation between the new object and multiple known objects.

There are, of course, other very important things in education. I realized a few of these tonight as I was helping Danielle learn to program. Perhaps more important than clear abstraction is do not let the student take a passive role. This is reiterated everywhere, and I'm sure my readers (all 2 of you) will have heard it somewhere. But it tends to be meaningless babble if it isn't explained. So I will explain, using my experience this evening as an example.

First, you must force the student to think. This is, perhaps, the hardest part. It is especially difficult when the student is frustrated with the subject material (as is often the case when tutoring someone). A frustrated or disinterested student will not want to put in the mental effort required to draw connections to previous understanding (read: to abstract). If you can force the student to think about the material, she will move from the known ideas to the abstract without you, and you can focus on shaping the lens, and pointing out key differences between the previous and new material. So, how do you get someone to think?

An answer (among others) is the Socratic Method. Pose questions which lead the the student toward the knowledge you are trying to pass on. If she is stuck, either explain a little more information that is related to the answer, and re-ask the original question, or ask a new, more specific question. In this way, you provide them a context and framework with which to understand the new knowledge. Looking at our example: At the beginning of the night, Danielle was not only lost, but not even trying. As the night continued on, I first began explaining what I was doing to change what she had done (basically copied), and then began to ask her questions, leading her to write her own code, in her own style.

By the end of the night, she was finishing her code by herself with almost no prompting by me, and I was able to focus on correcting small details and providing notes that would make her code cleaner or more efficient. By forcing her to actively think, my position as instructor changed from "I'll show her what I'm doing" to "I'll provide additional knowledge to help her do this better." In essence, I changed from a position of carrying her along, to a position of pointing out paths for her. Granted, she still has a lot to learn, but I was helping her for less than 2 hours, and she went from knowing nearly nothing, to anticipating what I was going to tell her. The amazing thing is that by putting in that extra care and effort, it actually made it easier for me and in the end, easier for her. It was harder for both of us at the beginning but once she started to pick it up, it went very smoothly and efficiently. She accomplished more in the last 10 minutes than in the first 45.

If We Taught English the Way We Teach Mathematics...

2008-03-02T14:04:00.003-06:00

This was originally posted by "Coryoth" (interesting, same name) on K5. It's an interesting approach to the subject, but I think it's pretty good. There's probably more to say on the subject, but he gets the point across: [Edit! It seems it was first posted by Leland McInnes at The Narrow Road, which I highly, highly recommend you read, as it is even better than the following]

Imagine that your only contact with "English" as a subject was through classes in school. Suppose that those classes, from elementary school right through to high school, amounted to nothing more than reading dictionaries, getting drilled in spelling and formal grammatical construction, and memorizing vast vocabulary lists -- you never read a novel, nor a poem; never had contact with anything beyond the pedantic complexity of English spelling and formal grammar, and precise definitions for an endless array of words. You would probably hate the subject.

You might come to wonder what the point of learning English was. In response perhaps the teachers and education system might decide that, to help make English relevant to students, they need to introduce more "Applied English". This means teaching English students with examples from "real life" (for varying degrees of "real") where English skills are important, like how to read a contract and locate the superfluous comma. Maybe (in an effort by the teachers to be "trendy") you'll get lessons on formal diary composition so you can better update your MySpace page. All of that, of course, will be taught using a formulaic cookbook approach based on templates, with no effort to consider underlying principles or the larger picture. Locating the superfluous comma will be a matter of systematically identifying subjects, objects, and verbs and grouping them into clauses until the extra comma has been caught. Your diary will be constructed from a formal template that leaves a few blanks for you to fill in. Perhaps you might also get a few tasks that are just the same old drills, just with a few mentions of "real world" things to make them "Applied": "Here is an advertisement for carpets. How many adjectives does it contain?".

In such a world it wouldn't be hard to imagine lots of people developing "English anxiety", and most people having a general underlying dislike for the subject. Many people would simply avoid reading books because of the bad associations with English class in school. With so few people taking a real interest in the subject, teachers who were truly passionate about English would become few and far between. The result, naturally, would be teachers who had little real interest in the subject simply following the drilling procedures outlined in the textbooks they were provided; the cycle would repeat again, with students even worse off this time.

And yet this is very much how mathematics tends to be taught in our schools today. There is a great focus on the minutiae of the subject, and almost no effort to help students grasp the bigger picture of why the subject might be interesting, and what it can say about us, and about the world. Mathematics has become hopelessly detail oriented. There is more to mathematics than mindlessly learning formulas and recipes for solving problems. And just like our imaginary example, the response to students lack of interest in mathematics has only served to make the problem worse. The "applications" and examples of using the mathematics in the "real world" are hopelessly contrived at best, and completely artificial at worst, and still keep a laser like focus on formulas and memorizing methods without ever understanding why they work.

Of course the opposite situation, with no focus on details, can be just as bad. Indeed, that is where English instruction finds itself today, with students never learning the spelling, formal grammar, and vocabulary needed to decently express the grand big picture ideas they are encouraged to explore. What is needed is a middle ground. Certainly being fluent in the basic skills of mathematics is necessary, just as having a solid grounding in spelling and grammar is necessary. What is lacking in mathematics instruction is any discussion of what mathematics is, and why mathematics works as well as it does.

The discovery and development of mathematics is one of the great achievements of mankind -- it provides the foundation upon which almost of all modern science and technology rests. This is because mathematics, as the art of abstraction, provides us the with ability to make simple statements that have incredibly broad application. For example, the reason that numbers and arithmetic are so unreasonably effective is that they describe a single simple property that every possible collection possesses, and a set of rules that are unchanged regardless of the specific nature of the collections involved. No matter what collection you consider, abstract or concrete, it has a number that describes its size; no matter what type of objects your collections are made up of, the results of arithmetic operations will always describe the resulting collection accurately. Thus the simple statement that 2 + 3 = 5 is a statement that describes the behaviour of every possible collection of 2 objects, and every possible collection of 3 objects. Algebra can be viewed the same way, except that instead of abstracting over collections we are abstracting over numbers: elementary algebra is the combination of objects that represent any possible number (as numbers represent any possible collection with the given quantity), and the set of arithmetic rules for which all numbers behave identically. Numbers let us speak about all possible collections, and algebra lets us speak about all possible numbers. Each layer of abstraction allows us to use an ever broader brush with which to paint our vision of the world.

If you climb up those layers of abstraction you can use that broad brush to paint beautiful pictures -- the vast scope of the language that mathematics gives you allows simple statements to draw together and connect the unruly diversity of the world. A good mathematical theorem can be like a succinct poem; but only if the reader has the context to see the rich connections that the theorem lays bare. Without the opportunity to step back and see the forest for the trees, to see the broad landscape that the abstract nature of mathematics allows us to address, it is rare for people to see the elegance of mathematical statements. By failing to address how mathematics works, how it speaks broadly about the world, and what it means, we hobble children's ability to appreciate mathematics -- how can they appreciate something when they never learn what it is? The formulas and manipulations children learn, while a necessary part of mathematics, are ultimately just the mechanics of the subject; equally important is why those mechanics are valuable, not just in terms of what they can do, but in terms of why they can do so much.

So why is it that this broader view is so rarely taught? There are, of course, many reasons, and it is not worth trying to discuss them all here. Instead I will point to one reason, for which clear remedies to exist, and immediate action could be taken. That reason is, simply, that far too many people who teach mathematics are unaware of the this broader view themselves. It is unfortunately the case that it is only at the upper levels of education, such as university, that any broader conception about mathematics becomes apparent. Since it is rare for people going into elementary school teaching to take any university level mathematics, the vast majority of elementary teachers -- the math teachers for all our children in their early years -- have little real appreciation of mathematics. They teach the specific trees outlined in textbooks, with no real idea of forest. A simple but effective measure that could be taken is to provide stronger incentives and encouragement for prospective elementary school teachers to take extra math; whether it takes the form of courses, or math clubs, doesn't matter, the aim is to get teachers more involved and better exposed to mathematics in general so that they can become familiar with the richer world beyond the specific formulas and algorithms. This exact approach was tried in Finland as part of their LUMA project starting in 1992. As a result the number of teachers graduating with higher level had increased dramatically by 1999. And the results are also clear: Finland finished first, showing continued improvement in mathematics and science, in the 2003 PISA survey of the reading, math, and science skills of 15-year-olds in OECD countries (Finland finished second, just behind Hong Kong, in the mathematics section). Finland has continued to do extremely well in other more recent (though less major) studies.

Whether you view mathematics as an important subject or not, it is hard to deny that, currently, it is being taught poorly in many countries around the world. With such scope for improvement, and clear examples such as Finland showing the way, isn't it time that we took at least some of the obvious steps toward improving the quality of mathematics education?

Tell them it's movement, movement, movement!

2008-02-25T16:49:00.002-06:00

So, God sparked a flame over the weekend, among the people who went to winterfest. It's going to be exciting to see where this goes.

Through Sully and Katie today, God softened my heart. I'm still stubborn and prideful. I'm still rather hard-hearted, wicked, and foolish, but God is working, and it is exciting. The spiritual pit I've been in for a while seems to be gone, and it's exciting!

Anyway,
God bless.

Cum hoc ergo propter hoc

2008-02-18T00:29:00.003-06:00

There's a common error that's bugged me for a while. It's related to "post hoc ergo proter hoc" (after this, therefore because of this), and apparently is called a spurious relationship statistics.

Anyway, the argument that bugs me goes something like this:

A happens,
Later B happens;
they seem to be related, thus A leads to B.

Now, I see this all over the place.

I recently read an article that suggests that if people (particularly women) take AP calculus in high school, they're more likely to be engineers. This is true, but she was using this fact to encourage women to take AP Calc, in order to increase their chances of become engineers.

The problem with this argument, is that she ignores the (10 or) 11 years of schooling prior to taking AP calc. The reason women who take AP calc go onto be engineers more often is that women who take calc in high school are better at math than at other subjects-- this was the conclusion of her study. People who are better at math than other subjects go on to engineering related careers more often than those who don't.

Event A: Woman takes calc in high school.
Event B: Woman becomes an engineer.
Cause C: Woman is better at math than english.

Somehow, this cause which she argues is forgotten. If she encourages more women who are "well-rounded" students to take calculus, the relationship between C and A weakens; hence, the relationship between A and B weakens.

The general idea has always baffled me, especially since spurious relationships almost always come up as a way to rectify problems with the cause. How can you argue a causal relationship and then forget about the implications of the cause? That's what a causal relationship is!

A Short Bit on Faith

2008-01-20T22:05:00.001-06:00

I started writing this a few months ago, and forgot to actually finish it until today. I understand that it is all over the place, but I'm not terribly worried about that...

For all of history, people who pushed for something greater have been mocked and ridiculed; even today, when much of America claims to be Christian, nearly anyone who dare profess the absurdly idealistic love that Christ represents would be ignored, or, at best, called a hopeless romantic who needs to awaken to the real world. When I hear this, I think of Puddleglum's speech:

"One word, Ma'am. One word. All you've been saying is quite right, I shouldn't wonder. I'm a chap who always liked to know the worst and then put the best face I can on it. So I won't deny any of what you said. But there's one thing more to be said, even so. Suppose we have only dreamed, or made up, all those things-- trees and grass and sun and moon and stars and Aslan himself. Suppose we have. Then all I can say is that, in that case, the made-up things seem a good deal more important than the real ones. Suppose this black pit of a kingdom of yours is the only world. Well, it strikes me as a pretty poor one. And that's a funny thing, when you come to think of it. We're just babies making up a game, if you're right. But four babies playing a game can make a play-world which licks your real world hollow. That's why I'm going to stand by the play-world. I'm on Aslan's side even if there isn't any Aslan to lead it. I'm going to live as like a Narnian as I can even if there isn't any Narnia. So, thanking you kindly for our supper, [...] we're leaving your court at once and setting out in the dark to spend our lives looking for Overland. Not that our lives will be very long, I should think; but that's a small loss if the world's as dull a place as you say."

The key word in this whole speech is "important". He does not say that the made-up things are more "fun" or "interesting" he says they are more important. This choice of words changes the whole speech from a confession of willful delusion to a statement of purpose. Puddleglum is not deluding himself into believing that the Overworld exists, he is saying that, since that world is more fulfilling, more alive, it is more real, whether it is tangible or not. We shouldn't stop fighting for true love, for brotherhood, for God, just because people call them romantic concepts designed only to give us hope; Oh, no! They are of greater consequence, of more value than any end on this earth, so even if they do not exist, they are still worth more than this hollow, dark, empty pit of a cave through which we wander hopelessly.

What is it, exactly, that makes them more important? Just before Puddleglum makes this speech, the queen, Jadis, is telling them that all these things are silly exaggerations of real things: Aslan, the large lion, is just a house cat; the sun is merely a large lamp, the sky just a large roof, and so on. The things she mentions are the things we need in order to live: Shelter, light, someone/something to comfort us and keep us company. The things Puddleglum mentions fulfill all the same functions, except they are more fulfilling. The sun is warmer and brighter than any houselamp, the light more comforting; Aslan is like a house cat, only larger, warmer and gentler; Narnia is just like the large underworld, only more just. Likewise, God's love, the kingdom of God and pure romance are more fulfilling than the "real-life" equivalent. Note, however, that this is not a Utopian, happily-ever-after sort of fulfillment; it is satisfaction, which is the reward of action. Effort is required for satisfaction: Anyone who tells you Christianity does not require tremendous force of will has never felt God; anyone who says that a relationship is not difficult has never committed to another person; anyone who says there is no stress in loving your fellow man has never helped out a friend in need. But this love, this faith, has more power, more value than any end on this earth. Is there any loss in living for these things? You will miss whatever the "normal" purpose of life is, but to quote Puddleglum again, "It's a small loss if the world's as dull a place as you say."

What's even more important than anything Puddleglum says is that the Overworld does exist. Puddleglum is in fact not a dreamer who has committed himself to a non-existent cause, but rather a lone seer, who has stepped out of Plato's cave into the warm sunlight. How, then, can we be assured that we are right, that the world is blind and stubborn, while we walk in Truth? How can we convince others that we walk in the Truth? Unfortunately, there is no certain way; there is no proof that God exists, no proof that there is a higher purpose, and the cave is as empty and meaningless as the Underworld. However, stronger and more convincing than any clever display of logic is the experience of these things. Those of us who proclaim the power of God have seen His light; those who hold love as a greater end than anything on this earth have loved and been loved in such a way that they cannot deny its power.

Once again: how can we convince others of this? Only by showing the same that we've seen. We have to spread love, God, and brotherhood, not by word of mouth, but by deed. We cannot just talk, convincing people of the Truth because Truth is not a reasoned conclusion. It is not a statement of the form "p therefore q." God is not something you know, it's something you feel, something you understand. In Stranger in a Strange Land, Heinlein invented the "Martian" word grok to describe this concept. Literally, grok means "drink", and it should be no surprise that Jesus used the same term to describe living in faith; when you drink something, it becomes a part of you, a part of everything you do, and every decision you make. We must drink in God's love until it flows out of us like a fountain. Only by spreading this, by sharing the same beautiful experience of God's presence can we hope to expand God's kingdom.

There is no knowledge, no action, no speech which can by itself convince anyone of God's existence and love. Even if someone understands philosophically that God exists, that does nothing for them. "You believe that there is one God. Good! Even the demons believe that- and shudder." (James 2:19) Faith is not an intellectual adherence to doctrine; it is a statement of purpose a commitment to live "as like a Narnian as possible, even if there isn't any Narnia." Once you experience God's love-- once you have that first sip of grace-- you cannot be content with anything less than the whole cup. "If only you knew what it would have meant to drink the thimbleful of fire. But, alas! you swear you already know its taste! and yet your belly remains intact."

If there is no way to convince people of the Truth, we must then live our lives to the fullest-- live in God's joy and freedom all hours of the day-- and pray and wait for them to reach up and grab the thimble from Jesus' hand. When that happens, and only then, will they understand the fire that consumes the believer. We cannot wait for our lives to end, for Heaven, "the afterlife" to experience God's Grace. There is so much more on earth than we see around us, so much more than we accept and feel content with. Why then are we sitting here waiting to die? We must reach out, and search for Narnia! We must go out and find Aslan! If God is there, reaching for our hands, why would we sit here and wait for him to? Shouldn't we strive with all our heart, soul, mind and strength to reach up and grab at his gentle strength?

And you, will you look and think and pray? Or sit, and simply wait?

Spirituality, laws, and love

2008-01-14T23:29:00.001-06:00

I've been in an interesting spiritual state for the last month and a half. On the one hand, I feel much closer to God than I ever have (barring a few short bursts of inspiration); on the other, I feel like I'm exactly the same-- with the exact same flaws-- as I've always been. Perhaps it'll always be that way. Paul himself struggled with the personal disgust that's associated with sin.

Anyway, I've read in the last 2 months, from Genesis through Kings. There are times when I'm reading and I'm trembling with awe at God, but it typically ebbs after I close the book (or just as often, leave it lying open on the floor by my bed). I've perpetually been just below that state. It's extremely frustrating because I think "What can I do to come closer to God," but I know (and it's so hard to actually live it) that it is not through our own ability, but through God's grace that we come closer to God. I want Elijah's spirit. I want to be able to talk to God the way Moses did. So I pray for the presence of God; I pray to be struck dumb with his beauty and glory; I pray for this more frequently than I've probably ever prayed in my life.

Today, Sully and I were talking and he talked about how annoyed he is when someone thinks they have the way; that if only people would listen to them, do things like they do, they'd be righteous-- as if our knowledge and ability save us. I listened, and agreed, but was struck with how much God was pointing a finger at me through him.

I'm so quick to judge legalists, so quick to pass off everything they say with a "Your mind is in the right place, but your heart is empty" that I've slipped off the other side of the mountain as I backed away from them. I am a whore for what Kierkegaard calls "the aesthetic". I really need to take one large step back toward those people I've so long tried to distance myself from.

On a related note, perhaps I am closer to God than I think. Katie broke up with me today, and I'm surprisingly fine with it. It's not that I actually don't mind. It's just that I know it's something God has called her to do. I've been hoping otherwise, but it's becoming increasingly obvious (especially over the last week) that we are being called in different directions. Suppose it had not happened now, but in another 6 months... It would be so difficult for either of us to give it up for what we've been called to do. Anyway, I think she's having more trouble with this than I am, and the "gentleman" I am, I just want to be there to help her through it... which may or may not have the opposite effect.
I can deal with talking to someone I want a romantic relationship with, but the one thing I'm dreading is telling people it's over: I'm afraid that other people won't understand, or will think it's just a cop out.

On video games, and adult content

2008-01-14T19:00:00.001-06:00

This was supposed to be a post here

So... I've been playing video games for longer than I can remember. I'm sure I've played every game that any activist has ever said should be banned. As someone who is studying math and computer science, and plans on either doing research into fields that most of the world hasn't even heard of, or teaching at a Christian school in some random country, I really, really just don't see where the outrage comes from.

Lot's daughters rape him; Noah passes out drunk and naked; David (A man after God's own heart!) has his way with a woman (it never says how compliant she is, but since they marry, we can assume she didn't object) and kills her husband; Prophets rain fire from heaven; there are prostitutes; there are "male temple prostitutes"; there is a vile amount of sex and violence in the Bible. And yet, most of us here (myself included!) would encourage a child to read it. Why?
1) It represents a strikingly accurate picture of how vile and corrupt we humans are.
2) It is a song of praise to the glory of God.

Crime and Punishment contains a graphic depiction of a murder. 100 Years of Solitude contains multiple rapes, sex scenes, murders, massacres, ... Frankenstein is about a vile corruption of nature. Therese Raquine is horrifyingly gruesome
Many, many, classical books contain the same. Graphic depictions of sex and/or violence are an inherent part of literature.
And yet, most people here (myself included!) would encourage a child to read these things. Why?
There is an intellectual stimulation in reading literature; vocabulary improves, writing ability improves, speaking ability improves, etc., etc., etc.

Now, what we must remember is that sex is not evil; The portrayal of sex and the portrayal of violence are not necessarily evil, if in their proper place.

Now, mature people (I refuse the word adult) can handle portrayals of sex and violence, and either can see through them to the message (as in the Bible), or understand the entertainment or intellectual value of a story, which-- for immersion-- must necessarily contain sex and/or violence (as in Literature).

If you don't believe that there is intellectual stimulation in video games, find the name of a game and google search "[name of game] tactics" or "... strategy" or "... walkthrough". Look at the depth in the results you find.

Remember that there are no restrictions on ordinary literature, only on "dirty magazines". Remember that there is no *legal* restriction on the sale of R movies/tickets, or Parental Advisory music. What makes games any different?

There was a war with comic books. There was a war with movies. I'm sure the many centuries ago, there was a war with papyrus that sounded something like this:
Those rebels are using papyrus to write messages that blaspheme our gods and slander our king. We must control the use of this artificial memory.

I would have put something about "sex and violence", but that wouldn't be historical: the two things have been an integral part of nearly every civilization since the dawn of man. And certain people have the arrogance to claim that our "traditional morals" are being attacked? If those values are Elizabethan, Roman, Egyptian, Greek, Babylonian, Indian, Turkish, Germanic, Frankish, Nordic, Chinese, Arab...
If those morals were the morals of the crusaders, the zealots, the pharisees, the Conquistadors, or slave-owning Americans...
Well, if any of those are the case, I really don't think I mind seeing any of those corrupt, broken moral systems corroded.

Instead, I will continue to study group theory, Galois theory, and combinatorics; I will continue reading about those two harlots- Oholah and Oholibah- and when I need a break, I will read about gruesome murders and outrageous rapes in books that have been praised for centuries, or I will play video games- probably violent ones, or ones with "sex scenes" that are on prime time TV- and I will feel confident that when my kids experience these things, they will have been raised well enough to understand them.

Edit: it has been brought to my attention (see comments) that I have forgotten to point out how wrong the original article is. Not only is there no rape (according to everyone I've heard from who has played the game), but the "sex scene" only has (quoting my roommate) "a few seconds of ass."

Since someone posted as a response to the original article a youtube capture of the video in question (I did not watch it), I cannot accept that the game is as raunchy as Mr. McCullough makes it seem. I would like to call him a liar, but I'm much too fair, and a little wise for such name calling. The question is not a question of misrepresenting data to fool others, but a question of misinterpreting the data your senses feed to your brain-- He's irrational, not malicious.

You see, when someone has not been taught to critically analyze information they receive, they automatically jump to a conclusion based in part by what they do see, in part by what they expect to see, and in part by what they have been taught to see. Mr. McCullough (and others who have advocated censorship of interactive content) expect games to be such filth, so when they see the slightest evidence of it, they cache that-- confirmation bias. It then exaggerates itself, so that these people actually believe the nonsense that spews out from their mouth and pen. I could go on for a while about the state of an education system which does not teach one to think, but I want to make progress in 1st Chronicles; I want to make progress in an abstract algebra book, and I want to start a story (ironically and coincidentally, it's about preconceptions)

Mr. McCullough is wrong. He is wrong about what happens in Mass Effect; Were he right about the nature of Mass Effect, he would still be wrong.

Haunting

2008-01-12T15:09:00.002-06:00

I often have to explain to people that I do math, not for it's scientific utility, but it's aesthetic value. Many people either get confused by this idea, or think of pretty math pictures or mystifying symbols. This is not at all what a mathematician is thinking of when he calls math "pretty".

A much closer representation would be Pope's question "Ah, Why, ye Gods, should two and two make four?" Why, indeed! There is a simplicity, a finality that is striking and counter-intuitive in math. It is obvious that 2+2=4, and saying otherwise will confuse people (even when you're correct.)

But mathematically obvious and intuitively obvious are two very different things. Mathematically obvious means "a proof is trivial". Intuitively obvious means "the assumption that it is true is so ingrained within us that another system is difficult to comprehend." 2+2=4 is not mathematically obvious... until you define addition, and define equality. 2+2=4 is so often used because of this fact. It relies, almost solely on the foundations of mathematics. If math were a building-- one built over millennia, with thousands of architects-- 2+2=4 would be the buttress opposite 1≠0.

There is something mystifying in this idea, something almost haunting. More haunting yet, is Euler's Identity. eⁱ^π+1=0. This may be meaningless to my audience...
e (2.718281...) is the "natural base" It has a number of fascinating properties, perhaps the most important of which is that the the slope of the line e^xat any point is also e^x. This number is primarily important in calculus (where it is the most important number): real analysis and the study of differential equations. π (3.14159) is the ratio of a circle's circumference to it's diameter. It is the most important number in trigonometry, and therefore all of geometry. Consequently, π, like e, shows up everywhere in math. i is the imaginary number √-1. Complex analysis is the study of this number.

We have, then, undoubtedly the 5 most important numbers in math: 0, 1, e, i, π. They come together in one simple equation-- 7th grade math-- eⁱ^π+1=0. When you see this equation, you are simultaneously looking at all of analysis, as well as analytic geometry. If this doesn't leave you in some way uncomfortable, you have missed something: take a math class.

As you learn the required material-- certainly more work than 7th grade math-- the equation changes from meaningless symbols to utterly counter-intuitive. Mathematically, it's obvious; there are proofs which require no more than high school math; but intuitively, it cannot be. How can everything be related so simply?

If you still do not see it, if you still cannot feel it. Think of the way a good song can almost bring you to tears, or the way certain paintings refuse to be forgotten, or the power in a great piece of literature. That's what a mathematician feels when he sees this equation.

"Why must e to the i pi negate 1? And how, O God, does it steal my sleep?"

On Medieval Paintings

2008-01-07T20:32:00.000-06:00

Heather Bamford writes a blog that I sporadically check. In her most recent post, she asks a series of questions that essentially amount to "How can surreal sources present reality to us, and how can we use these to understand the real nature of the world?" Unfortunately, I can understand only the English in the post, but I will try to respond anyway.

The answer, I think, lies in the emotional and subjective nature of human experience. We are not rational creatures, and we do not have an objective tie to the world we live in. While most people will see a very similar world, each person's experience of this world is affected by past experiences. Thus, to answer her actual question: the images in the LA help us to understand the medieval psyche and medieval perspective. Through these pictures, we can begin to understand how the world impressed medieval persons.

Flos Duellatorum (or Fior di Battaglia) proves to be a similarly surreal instruction manual. In answer to a somewhat rhetorical question posed by Heather in an earlier post, the text has corresponding illustrations (not the other way around). In Fiore's manual at least, the illustrations seem to be for the purpose of refreshing a a knowledgeable student's memory, providing key points for comparison; The student is expected to fill in the missing parts based on the text and prior knowledge. There are, of course, differences between the LA and FD, but the art, and the relation of the art to the text seem to be similar. Perhaps my untrained eye sees non-perspective illustrations and expects them to be related; I do not count this possibility out.

The next research question, then, would be "why do medieval artists see the world as they do?" What about the pictured positions, people, locations, etc. makes them more important than others?

I try to stay out of it...

2008-01-06T01:07:00.000-06:00

Among the very long list of discussions I try not to waste my time with is the Evolution/Creation debate. I haven't seen any new developments in the debate since 9th grade... and I first started to care in... 9th grade. That was ~6 years ago now. So, for more than 5 out of less than 6 years, I've seen the same ideas expressed in the same words, and I just don't care any more. Now... for at least 4 of these nearly 6 years, I've realized that the question is not one of "logical conclusion", but of analysis from core values. Allow me to explain.

Science is an attempt to objectively observe and hence explain our universe. Since the only objectively observable part of our universe is the natural, science is conducted in the naturalist realm. Further, since a mathematical proof requires agreed upon axioms, and well-defined operations, and these cannot be established in a concrete, non-theoretical system (read: the universe), the strongest support for a theory is its repeated observation in nature.

Observation of a theory requires more than observing a specific phenomenon; it requires the prediction of related phenomenon, along with valid reasoning as to why the theory can account for both. Also, it is required to not contradict currently accepted theories.

An example. The constant cycle of the sun, moon, stars and planets around the earth would lead one to initially believe that the celestial objects rotate around the earth. There is, here, one unexplainable phenomenon: the planets do not circle the earth in any sort of logical pattern. If we then say the earth circles the sun, which creates days (one phenomenon), years/seasons (a second), and the planets revolve around the sun (a third), we see that this theory is "stronger". That is, it accounts for more with fewer observations. A stronger theory is not necessarily a more correct one; however, when Newton's Laws and Kepler's Laws are taken into account, the theory is nearly uncontestable.

Another example, Both Einsteinian Relativity and Quantum Mechanics were difficult to accept due to their clash with (then) currently accepted theory. Their strength and consistent observability are the factors which allowed for their acceptance.

Science, as a way of knowing, suffers from the fact that it is a strict rationalist system with no possibility for proof. For a naturalist, then, the only theory which can begin to account for the evidence in the fossil record and otherwise, is evolution. The core values, the system of analysis they use, forces the acceptance of evolution.

Faith, on the other hand, is also an attempt understand the universe. It is, however, a vastly different way of knowing. Rather than empirical evidence, it is based on a priori revelation, and subjective experience. It suffers from other flaws than science. The flaws of faith are much more complicated, although hardly more numerous than those of science, and relate primarily to the subjective nature of faith.

Faith, unlike science, does not attempt to objectively collect data, so is not bound to accept a naturalistic, or for that matter "observable" theory. Thus the discrepancy. Further, when a person of faith attempts to view the evidence with a scientific eye, they start, naturally with their faith-based assumptions, rather than scientific assumptions. Evolution, then, does not present itself as a stronger theory, but creation does. It is not an unscientific way of thinking, but an unscientific starting point that "leads creationists astray".

It would be wonderful if people on both sides of the debate would take the time to actually research where their opponents' views come from, rather than just assuming everyone thinks the same way, and calling each other idiots for not seeing the "obvious" conclusion.

On another note, most of what Darwin said (before The Descent of Man) is really not that hard to accept, even from a creationist perspective. Mendel's gene theory is also absolutely acceptable. Modern Synthesis, then, is also acceptable. It is when modern DNA theory, and genetic mutations begin to enter the picture that things get shaky. Even speciation through mutation (assuming the data fits) can fit within a Christian perspective, the problem is once you accept speciation through mutation, there is nothing, save an argument by authority, preventing the conclusion that everything has evolved.

Anyway, You'll not see me write anything on this topic again for years, because, well.... I've been hearing both creationists and evolutions refused to try to understand what the other camp is trying to say for almost 6 years now, and it's a waste of breath...

Dijkstra

2008-01-03T12:28:00.000-06:00

Yesterday, I read both A Case Against the Goto Statement and On the Cruelty of Really Teaching Computing Science, by Edsgar Dijkstra. They were both rather eye-opening. One amazing thing about Dijkstra, is that he did not even own a computer until very late in his life, and rarely used them throughout his life... but he is one of the most important computer scientists in history. Why? Because "Computer Science is no more about computers than astronomy is about telescopes." Just like telescopes for astronomers, computers are the tool used by computer scientists to engage their science with the real world. Fortunately for us, Computer Science is abstract theorization, rather than concrete study of physical phenomenon. Because it is abstract theorization, a computer scientist is allowed to spend no time actually using a computer.

The point is that computer science is not a discipline, but a branch of mathematics. A branch that deals specifically with the generating, solving, and analyzing algorithms. Some people may try to disagree with me on this. "How is a professional [insert type of program here] an algorithm we can analyze?"
Firstly, this argument is ridiculous, because all a computer can do is manipulate symbols by a concrete, predefined set of rules. That is, all a computer can do is process an algorithm. If you don't believe me, read anything on the theory of computation (typically by mathematicians), specifically pay attention to Turing's work. Anything a computer can do must be describable by an algorithm... Would you be surprised, then, if I told you that all programming languages are just systematized algorithm languages? All compilers do is translate these algorithms to a form the computer can use.
Secondly, I would hardly call most of today's programming an exercise in computer science. To continue Dikstra's analogy, putting a satellite in orbit is not a job for astronomers. A computer scientist lays the groundworks: creates algorithms, creates languages, etc, and then the programmer uses these as tools for a finished product. Just as a physicist doesn't need to know why group theory works, just that it does, or an aerospace engineer doesn't need to know or why wings work, just how they work.
The why and how are for theoretical disciplines, primarily math and computer science (I know, I'm biased away from sciences). The engineer, or applied scientist doesn't need to know why,they need to know what and sometimes how.

Cheers

Game design

2008-01-02T13:48:00.000-06:00

I've been thinking about video game design again lately; I think just like movies, game designers are trying too hard to make worlds understandable, and not hard enough to make them unique and alive. I want to see a fantasy world with surreal landscapes; you should see Dalí or Kush everywhere. I want characters like Marquez's. I want creatures like Tolkien's-- not in appearance, but in imagination and originality.

Castes need to be broken; roles need to be changed. Turn our ideas on their heads. We gamers talk about games as art, but where are our artists? Gears of War? Assassin's Creed? Beautiful games, to be sure, but we can achieve a whole new level of beauty-- a whole new type of beauty-- with video games. Who is doing that? I see cloud ships and sky harbors; I see cultures that represent these things; I see a player making a drastic change in these cultures; where are they? When will we see our beloved games wrapped up in literature and art in a new way? Where are the cubist and surrealist game designers?

Game makers need to spend less time making games look amazing, and more time making them feel amazing. Is there tremendous detail in Van Gogh's work? Or Picasso's? How about Dalí? Then fuck detail: we need worlds that come alive and strike us with awe. Until someone steps out and does that, we will be stuck playing tired old FPSs that play exactly the same, RPGs with the same classes, and fighting games with a Bruce Lee look-alike. Some companies have been making small steps in this direction, and have been welcomed, but we still need something drastic. We need someone to step out and fail, keep pushing and succeed in everything that failed, and perfect everything that worked.

I know I've been ranting for a while now, but it just struck me as I imagined a harbor city on a cloud. Can you imagine piloting a ship from the clouds, through a pirate attack, and down to an ocean harbor? Then why hasn't this game been made?

A sudden realization.

2008-01-02T13:46:00.000-06:00

I think I'm going to make a conscious effort to let my stories become more surreal. I've always found a certain level of reality in surreal works-- an awe-inspiring mythical atmosphere-- something that's missing in realism. For the most part, my writings fit under the classification of surreal, but even Breviary, which has a extremely surreal plot, is framed as a series of dreams. It's as if I'm trying to rationalize my works, bring them into reality. I need to stop doing that. I feel like I'm trying to make my writing something it isn't; trying to add something in order to make it better, or more literary or something. I'm killing it when I do that. I need to just let it flow.

On another note, I'm definitely going to start drawing and painting again soon. It's a lot of fun, and I've spent far too long away from it.

Ideals, Scheme, and useful kludges: 3 unrelated topics

2007-12-22T18:29:00.000-06:00

So... Today I met with a friend I've seen once (maybe) since highschool. We were never really that close, but 2 years age difference will do that... I've often complained about women, drama, etc, but I do find myself often preferring the company of women. I've long struggled to find an answer, and I think I've begun to understand.

It hinges around my idealistic hopes for the world; I have an almost kitschy ideal of human relationships, and there are certain types of interaction I find stimulating. I'm still having trouble pinning these down, but, I digress.

The longing after my ideal place in the world is something I've comet o call nostalgia, however (in)correct that term might be. It is certainly related to the concept of sensucht, angst, despair, nostalgia, discussed by many philosophers of the 19th and 20th centuries. Anyway, when I feel some sort of fulfillment of this ideal, I experience the emotion nostalgia, in all of its painful beauty.

Now, how does this relate to my preference for the company of women? Nostalgia, faith, romance, platonic love, and empathy all fall under a class of emotions; all of these ideals trigger the emotion nostalgia (as well as the emotion they are associated with.) I find myself able to sit down and converse with female friends much more often than male friends. I'm not sure why this is, but a personal connection is easier for me to establish with a woman. Something tells me that I find this connection easier to make because of a desire for romance with these women, but I find this very difficult to accept. Tiffany, Rachel, Katie, Kate, Haley, Tessa, Melissa, Sasha, Adriana, Adie. Of these, I've only felt a "crush" on Tiffany, Katie, Melissa and Adie. Of those, only Katie has developed into something I'd classify as romance. Tiffany was an emotional security blanket, Melissa was the last person I had an engaging conversation with before leaving the springs, and Adie was the first person I truly felt close to in Chicago, so it follows that I'd have had a short-lived crush on each of them.

What really makes sense, is that all of these girls represent an approximation of my ideals. They evoke nostalgia, which allows my to feel a solidarity with them. I haven't met nearly as many guys who do this, perhaps because men make friends in a fundamentally different way than women; the push for alpha status has to be set aside before this sort of friendship can develop, or it has to be clear that we think in a similar way for me to be able to relax around another guy; I guess I'm afraid of the guys "kicking me out".

Katie, then, seems to represent the closes approximation of this ideal. Perhaps it's her passion and faith; perhaps it's the warmth she radiates, but I can't seem to interact with her without getting excited...

*****
On another note, I'm having problems with my cell update functions for the Game of Life, so I've decided to sit down and force my way through the tedium of a whole Scheme tutorial, while of course supplementing it with "oo! I think I can do this!"
I'm still struck my how easy it is to do things in functional style. It's truly breathtaking.

*****
And finally, it ahs been explained to me why someone would want to use the kludges taught by koots. Granted, it doesn't excuse their use as a teaching element: good habits should be taught first, useful, yet inelegant kludges should be taught only after the concept they are useful for. So perhaps he's not a terrible coder, just an inelegant one: an engineer, not an artist.

Bad Coding, Bad teaching. OR: Why I want to be a professor

2007-12-12T21:46:00.001-06:00

Edit: It seems this site hates my whitespace.... Sorry

I've been helping a friend with CS homework the last few weeks. He's taking "Object Oriented Programming I", which is the intro to programming class for CS and related majors, taught in Java.

I've been helping him because his professor is a hack, an absolute hack. Were he a brilliant man, but a bad teacher (e.g. Calinscu), or an idiot who knew how to teach, it would be acceptable, but he cannot code well and he cannot think well.

Firstly, my friend said "I do not know what object programming is. I don't know how object oriented programming languages work." Investigation shows that the problem is not on my friend's end, because he's understood everything I've taught him. It's that the subject has not come up in his lectures. He has mentioned "template classes", which is a bit redundant....

But I digress. Code samples and critique! These are the professor's, not my friend's... (Some variable names have been changed, but the logic is precisely the same.) The numbers in parenthesis mean I'm going to comment on the line after the segment.

First, a BubbleSort (Copied from a handout):

public int [] bubbleSort (int [] array){
(1) boolean done=false;
(2) int [] arraySorted=array;
(1) int p=0;
(1) int temp=0;
(3) while(!done&&p<=arraysorted.length-1){ (4) done="true"; (5) for(int j="0";j<=arraysorted.length-p-2;j++){ if (arraysorted[j]>arraysorted[j+1]){
temp=arraysorted[j+1];
arraysorted[j+1]=arraysorted[j];
arraysorted[j]=temp;
done=false;
}
}
p++;
}
return arraysorted;
}

1: booleans initialize to "false" and ints initialize to '0'. explicitly setting these values is unnecessary. Perhaps a good habit, but unnecessary. p is also in no way descriptive. It isn't following the lisp '-p' convention; why can't he use 'i' or 'count'? 2: Also unnecessary. the parameter "array" does not point to the same location as the array that was sent to it, so the line "int [] newArray = bubbleSort(oldArray);" will create a copy of oldArray, and pass that to bubbleSort().
3:This 'done' flag is absolutely unnecessary... It does nothing. To make matters worse, he handles the flag how drunken frat boys handle glassware: throws it around till it hurts something:
4:Every iteration of the outer loop sets "done" to "false". This means the operation is made n times in the outer loop, where n is the number of loops. It is also set to "false" in every inner loop (as long as flipping happens), this adds ~nlog(n) operations to the procedure. The operations that are done to this boolean flag (which I remind the reader, does not need to be there) have a complexity of O(nlogn). He has effectively turned a flag, which, if properly used (assuming it did need to be there) will take exactly n+1 [O(n)] operations into a flag which will take between 2n and (2n)^2 operations.
5:This one is fun. On the first page of the "how to write readable code" handbook (which doesn't-- but should-- exist) is the following statement:
"Do not use magic numbers!" In CS there are 3 numbers: 0,1,n. Any number besides 0,1 number should be specifically labeled because it's being used for something specific. Better, he has "<=/*some number*/-2" rather than "