Exponential Distributions and Tag Clouds

I just noticed this and I have to wonder if it's a common thing or not: When looking at the number of posts with a given tag on my tag list, it has a vaguely exponential distribution. Discrete, of course, but it has that same downward slope and everything when the tags are ordered by number of posts. I wonder if the histogram of number of tags with a given post count is Poisson?

I may just take the current counts of this page, run them through R, and post the results. It would be really cool if lots of other people did this and posted the results below or emailed theirs to me for inclusion.

I wonder if it says anything about human behavior? Do we tend to clump most things into the same few bins and have lots of smaller ones?

LaTeX in a WYSIWYG World

Most of the non-mathematically-inclined people I know don't use LaTeX and probably never would even if paid. I've even had people say with the advent of point-and-click formula editors for MS Word, there's no point in bothering with it anymore anyway.

Yet I can't shake the sneaking suspicion LaTeX is still better. Maybe it's just elitism from knowing a system most would find esoteric at best and downright weird at worst (outside of mathematics and related fields, of course).

But I still snicker at the thought of sending my humanities profs .tex files when asking them to review a draft of a paper, or using beamer to give a presentation.

Why I Can't Be an Engineer or Scientist

Quite simply, I hate units. More specifically, I hate worrying about them.

Whenever someone says, "This formula takes kilograms and meters/second, but we're working with metric tons moving at kilometers/hour, so you'll need to convert" I think "Why didn't we just measure it in kilograms and meters per second up front and save ourselves the effort of converting back and forth?" Whenever someone says "And now we need to convert to use the next formula" I think "Why don't we convert the formula once rather than the inputs every time?"

Is this a little petty of me? Yeah, it probably is. On the other hand, UNITS ARE ARBITRARY ANYWAY, and I like mine that way.

My First Instinct: Sledgehammer

I have an applied mathematics class wherein the professor has promised little of the math we will need is more advanced that a solid grasp of basic matrix algebra, its meaning, and what it is doing.

Yet whenever he asks for a solution to a problem (despite his prior assurance an elementary solution exists), I consistently throw out a more advanced concept. He asks for how to check the invertability of a matrix, I say determinants. He asks for how to compute the inverse, I say the adjoint. He says make an orthogonal matrix, I say pick some vectors and run Graham-Schmitt.

Pretty often I'm getting "That's way too complicated for what we need".

I think there's a theme here.

Which Came First: Theory Or Application?

This is a question I've had for about a year now: Which came first, theory or application.
I know this is equivalent to asking "Which came first, the chicken or the egg?" However, it's still intriguing.

Last semester I took three of theory courses. This semester I'm taking two which are more application than theory, another which is more balanced (numerical methods), and a theory. My current problem is jumping back and forth between application and theory within the same context. It could just be mathematical immaturity, but I have to wonder if some people are just meant to silo themselves either into pure theory or pure application without spending too much time trying to bounce back and forth. Perhaps that's why many research projects take years.

So which came first, the chicken or the egg?

Research as an Undergrad: Why I'm Thankful For It

One of the things I've been very thankful for has been the opportunity to do research in mathematics and economics as an undergrad. It's been an experience I'll always look back on fondly. People talk about taking learning beyond the classroom, but in reality it's been my experience the learning which results from research can completely overshadow any related work done in the classroom. This isn't simply technical material (and there's plenty of that to learn before real research can begin), but an entire philosophy about work in general.

One of the biggest ways in which learning from research overshadows classroom learning is that in the classroom it is usual for all presented problems to have been previously solved or to fit into a given mold. In research neither of these things are true. The point of doing research is (usually) to do something which hasn't been done before in a given way and the first question may well be "does this resemble anything else I already know/is anything I already know applicable here?"

The second major way in which research overshadows classroom learning is it requires a far greater level of self-motivation, maturity, creativity, and determination. There isn't some manual that tells you how to solve the problem and most of the competition is against yourself. Nor is it designed to make a given point in a "reasonable" amount of time after "reasonable" effort. It's on the individual researcher to find it within themselves to keep going and working and thinking until they find the insight to make more progress until the next plateau. It goes in lurches: you lurch forward, then stall only to at some unknown point lurch forward again.

At the end of the day, I can't really express how much I value the experience research has given me. All I can do is try to keep working and learning.

Philosophy No Less A Game

I wrote a bit ago about Hilbert's comment on mathematics as a game. While I'm admittedly critical of the implication that because mathematics is a game it is without merit. However, one natural follow on question asks which other disciplines are equally game like.

Candidate number one is philosophy. I'll pick on ethics for my example as it it closer to home for most of us, but the argument generalizes well.

To recap, any consistent system of thought is built on three things:

1. axioms
2. definitions
3. a system of logic

Admittedly, any system of logic is itself built on axioms and definitions, but for simplicity let's abstract away from that.

One thing that needs to get defined in any ethical system is the nature of good and a mechanism by which to classify things as good or bad. Yet this mechanism must come from either an axiom or a definition. But what makes our axiom or definition appropriate (or, if you like, good)? Suppose we argue for our choice of definition, then for that argument to be logically valid it too must follow from some set of axioms and definitions. Repeat this argument a few times and it quickly becomes apparent asking for a rigorous basis for something like the definition of good or a mechanism for making the decision quickly mires down.

So if we can't ever get to a fundamental set of principles which underly everything (assuming for the moment God is not interjected into the conversation), then this leaves ethics--or any other branch of philosophy--as based on an arbitrary choice of first principles, or at least cannot be rigorously shown as better than any other. When I took an ethics class last semester this usually came out whenever the instructor said the words "You could make an argument for . . . " and then filled in the blank.

The point here is this: whatever ethical system you chose to follow on any basis, so long as it is consistent it is just as objectively valid as any other consistent ethical system as there is no objective mechanism for assigning one as better than any other.

In short, ethics is no less a game than mathematics.

Mathematics as Just a Game

There is a quote attributed to Hilbert that " Mathematics is a game played according to certain rules with meaningless marks on paper." It is perhaps the most cynical characterization of the whole field I have ever encountered.

Pure mathematics, in its modern form, is a race to find prove theorems on the basis of some set of axioms, some set of definitions and a given system of logic. Unlike a science or medicine in which a result must be repeatedly checked against experience and observation (and there is a good career to be had in doing so), in mathematics once a theorem is proven there are only three ways to work with it:

1. To prove another theorem.
2. To get some applied work done.
3. Find another, better way to prove it.

While this last looks on its face like verification, note the term "better"--the new proof must add some new insight, not simply verify as one would accept in physics or medicine. On first glance, Hilbert has a point.

On the other hand, those theorems can describe all manor of objects (and in fact describe anything which meet the hypotheses). One simple example, set theory, works fine for piles of sand or stacks of money. Geometry we see around us all the time. The Navier-Stokes Equations govern, on the applicable scales, the flow of every fluid known to man.

The point here is simple: given a system of logic, a set of axioms, and a set of definitions power sets of theoretical tools for describing not only other theoretical objects but myriad real world systems can be effectively analyzed. While, yes, mathematics for its own sake does resemble a game on paper with meaningless symbols, it is those powerful tools which set it apart.

Thoughts on the AMS/MAA Joint Meeting 2008 Part Two: How to get mistaken for a Professor at 22.

This is really a tongue-in-cheek guide to the dress code at the Joint Meeting. There are two basic axes: age and formality. From there most people can be accurately classified. However, there are some notable exception.

First is age. There are three broad bins: 24 and under, 24-32, and 32+. These roughly line up with undergrads/first year grad students, grad students/job seekers, and profs.

The second is formality (of dress). This is a much broader scale, but on the low end is jeans and a tee-shirt and on the other is a full suit or equivalent. I'll lump things into three categories nonetheless: casual, business casual (slacks, polo, sport coat; shirt and tie, no jacket), and business formal (shirt and tie with jacket, suit).

How to spot a:

• Prof: 32+ in either casual or business casual clothes. If it's business casual, it's probably not new or gently used unless they're presenting.
• Grad student/job seeker: Business casual most of the time as they're interviewing or speaking. Business formal means interviews or a presentation where they expect to be performing for future employers (not always true). Usually 24-32.
• Undergrad: Under 24, business formal will only be seen if presenting, but will present in business casual and up. Otherwise seen in jean and a tee shirt or slacks and a polo (but not much of the latter).

Wildcards:

• Exhibitors: If young and well dressed, probably a book representative/salesperson. Fortunately, they're labeled on their name badges.
• Undergrads who don't like to look like slobs. That would be me. Simply put, I walked around in business casual (polo and sport coat) or business formal (shirt, tie, sport coat) all week and was constantly mistaken for a prof or graduate student.
  

Thoughts on the AMS/MAA Joint Meeting Part One: Hostelling

First, my most sincere compliments to the AMS, MAA, the San Diego Convention Center Staff, and everyone else who made the 2008 AMS/MAA Joint Meeting happen. It was, in short, completely awesome. Most of the talks I saw were very well done, especially the AMS Special Sessions and the more focused AMS topics sessions.

Next, as for accommodations: I stayed in the listed hostel four blocks or so from the convention center. As a hint to those on a budget and considering doing the same, this is probably a big toss-up experience wise, but I had a blast.
Pluses:

1. Much more contact in informal/social settings with other conference goers (I was in a 10 bed room and all of us were attending).
2. Great networking and "insider's view" of graduate school as most of those staying there were grad students looking for jobs.
3. Very collegiate/dorm atmosphere.
4. Very budget conscious. I spent $100 for four nights. Best price at a convention hotel was around $150/night after taxes, etc.
5. Fully stocked kitchen.
6. In the heart of the Gaslamp district.
   

Downsides:

1. Little private/personal space. See 10-person room comment above. If you booked far enough in advance, however, private rooms were available.
2. Minimal storage space, but the space was lockable. Basically, there was plenty of room to lock up valuables like laptops, wallets, etc, but not enough to store things like suits. No real space to hang anything.
3. You are NOT the normal customer.

Overall, I'd say the hostel experience was great for me (as a fourth-year undergraduate), but is probably not something I'd recommend to everyone.

Off to Conference Land

I'll be at the AMS-MAA Joint Meeting in San Diego. I'm flying in tomorrow afternoon and out Wednesday afternoon. I have a 10 minute contributed paper, which is completely awesome even though it takes no work to get one.

Should be fun. While I have to front everything, my university is going to pay me back for basic living expenses. I completely expect to look at them and say "Look, I don't expect you to pay all of this, so pay what you consider reasonable". Dangerous, but I know who will be putting this thing together and they don't want to screw me any more than they want to get screwed. It should be okay.

Not sure if I'll go to the beach. Never have been a fan of that place.

Prepping for a Math Final

It's taken seven semesters, but I think I've finally figured it out. While it's a great thing, I wish I knew a while ago what I know now.

The basic process is like this:

1. Find during the semester another textbook in the area. Classic books in the area are usually a good choice.
2. The prof (or experience) should indicate what the spread of material is for the test.
3. Go through the other book, identify each relevant section, and work as many problems as possible from that book, using its examples and exposition.

If done right, this should result in you having retaught yourself the entire course in a week or two from a different but similar perspective.

The remaining problem is textbook selection. This is tougher. For my compex analysis course, we used Saff and Snieder's book and I prepped out of Churchill. It turns out Churchill is so cannonical that Saff and Snieder is in some sense equivilent to it. This turned out really well since the prof followed the book very closely and specified the problem areas in a fairly granular fashion. Ergo, Churchill became a source of problems more than alternative viewpoint.

On the other hand, for a course where the prof is more prone to do somewhere a bit off the beaten path with tests, it would probably be better to find a book which approaches the area in a different way than the origional so as to be more or less orthogonal in both approach and problems. Then working the other book is very much like relearning the entire course and may expose one to an additional perspective that may come in handy come test time. Having worked out of Royden for my second semester or reals, I would probably use Baby Rudin as my "orthogonal" text.

One of My Favorite Comics

If you havent read my (very short) list of favorite webcomics, xkcd is on the list.

It's actually pretty eclectic, but usually has a pretty nerdy under- (or over-) current.

For instance, this one (which I am conviced is the litmus test for mathematical insanity).

http://xkcd.com/230/

Here's how it works:

Do you find this:

1. Pathetic: who works on math while having sex?
2. Funny, but would never interupt sex to work on a proof.
3. Funny, and would be so distracted it would be hard to enjoy the enconter until you write it up.
4. Funny, and would interupt sex, but not write on your partner's body if short of paper.
5. Funny and you would absolutely do it.
6. Not funny--this is legitimate, normal behavior.

I would say if answer number one, you're probably normal. If you answer six, well, I'm not sure even someone like Nash would have gone that far.

I'm about a 4.

Why I'm Now A Bayesian Who Will Use Frequentist Methods

Why I'm a Bayesian: Absolute Continuity with respect to a probability measure. The Improper Prior does not have it, and you need it for Bayes Theorem to make sense.

Why I'll Still Use Frequentest Methods: I'd say there are two reasons. First, convenience. Frequentist methods by removing the question of prior are certainly easier. Second, for large samples or very weak priors, Frequentist methods are a reasonable approximation, especially where the prior is unknown or would not contribute much.

Climbing the Putnams

The Putnam Competition. Six hours, 10 math problems. Mode score: 0. Median Score: Usually 0 or very close to it. Grand prize: Harvard Scholarship. Do I expect to win it? Not at all.

This begs the question: What kind of person (or even math major) will volunteer to compete in a competition they have almost no hope of winning? My answer is this: Because it's there.

Of course, having now taken them (and completely zeroed out) twice, I can honestly say the only reason I take them is because they're there.

Axiomatization

Axiomatization, or development of a set of ideas from first principles, is certainly a popular technique in mathematics and to some extent in areas like physics, etc. The irony is that across my humanities classes, few want it in any of them, even philosophy.

It is true some fields of endeavor don't lend themselves to it particularly well (the study of natural languages comes to mind--most languages have rules but they also have exceptions, etc). On the other hand, an axiom by definition is something which is true by assertion and so whenever something is simply asserted as true, that is an axiom. At the same time, whenever one says "these are the conclusions desired", it automatically forces them to build an argument in support of them with certain premises which in term have premises and sooner or later one hits axioms. The difference is merely they are enumerated last.

Meaningful discourse can certainly be had without explicit axiomatization, but axioms are the logical foundation of discourse (along with a specified system of logic which is itself axiomatic). Thus to fully understand the other and the foundation of their argument, one must understand--at least implicitly--their axioms.

Coding Irony

I went through the part of the code which was most suspect vis-a-vis the errors I was getting and found the flaw: a typo. The irony of this situation is not to be underestimated. Insofar as I can tell, all the algorithmic stuff is right, and in fact everything else is right, but this little tiny thing in wrong.

I'm just hoping it doesn't unveil some larger flaw lurking in the shadows, obscured by the heinous nature of what I found.

My Little Coding Nightmare Revisited

It's even worse than I thought initially. It turns out that the error is systematic and not sporadic (as I should have immediately guessed from its repeatability). It turns out that there is always a sort of inflation of the values of the sum of squared differences coming from somewhere.

This means that while the script looks like it is working great and doing what it should, it is completely bunk and will need to somehow be fixed to avoid this issue. I am thinking about trying to put a series of commands in to reset the values of different variables to zero in hopes of clearing out whatever error is compounding on me.

The next option if that fails it to post the relevant files to the R mailing list and hope someone is kind enough to tell me where I went wrong . . . which I somehow doubt.

Coding Nightmares

My coding nightmare tonight has been discovering an error in the way a piece of code executes I can't duplicate outside it's native environment, but I can repeat natively.

Here I am, working on an R script to do some (rather elementary) computational geometry and I want it to compute $(a-c)^2 + (b-d)^2$ (to use the TeX notation) where $a,b,c,d \in \[-2,2\]$. A little examination shows that the sum should always be less than or equal to 16. I was getting 16.8. Not good.

The problem is that while I can duplicate this kind of MAJOR error when computing this number as part of the script, I can't get it by starting R up cleanly and manually imputing one example. Then it works fine. I have to wonder what in the world is going on.

The worst part is most likely the solution is simple and would be obvious to a professional programmer but to me, an amateur, is far from it.

Grrrr . . .

Some Signs You May be Living In A World of Math

Not an exhaustive list, but fun anyway. . . .

1. You wonder if something is measurable.
2. You want everything to axiomatic.
3. The ultimate argument stopper is a counterexample.
4. The person you're talking to respects (3) as conclusive proof they are wrong.
5. Inheritance is more than just what people get when a relative dies.
6. You classify china patters according to their symmetry types and group.
7. A donut is just a really tasty coffee cup.
8. Induction is either weak or strong, never ambiguous.
9. Choice of coordinate system is arbitrary.
10. When someone talks about a group your first question is "Under what operation?"
11. Complex analyses are usually easier than a real ones.
12. Everything is a conjecture until proven.
13. Hypotheses are what you assume going in, not what you're trying to prove.
14. Examples are just special cases of things that might not hold in higher dimensions/more generality.
15. You tried pouring coffee in your donut.
16. You pour coffee into a cup but onto a Klein Bottle.
17. You've got vector space, subspace, null space, column space, and row space, but no shelf space (and probably no floor space either).
18. When people talk about a kernel of truth you get confused.
19. No matter how much stuff you have, it's still okay because it's a compact set.
20. Any question you know the answer to is trivial.
21. You understand at least half of the above.