Hungarian math education

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!)

Structuralist philosophy and methodology in mathematics

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.

Categories of paths; functors and natural transformations...

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?

Categories of paths; functors and natural transformations thereon

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

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

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.

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(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.

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(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.

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(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.

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(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.

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(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.

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(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

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:

1. remove the "co" from the word.
2. Draw the diagram that represents the word you just found.
3. 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 fundamental group functor

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.

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.

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.

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?

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.

Euler and Haskell

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.

On Lambda Calculus and Ordinal Numbers (Part 2)

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)

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)

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...

Imagination

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.