Balaam

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

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

Mathematical insight...

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...

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

Cruel Irony.

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

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...

Is so formulaically modernist...

Rhapsody in Blue

(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

$\displaystyle|\mathcal{F}|^k\leq \prod_{i\in I} |\mathcal{F}_i|^k$
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)

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)

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

(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...

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...

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

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...

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?

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

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

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

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

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

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...

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?

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...

"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

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...)

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.

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

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?