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.

A Difference Between Friends and Acquaintances

I have had reason within my own personal life to ponder the difference between being friends with someone and being merely acquainted with them. One major difference can be summarized as follows:

• Friend: Someone you want to spend time with and are willing to actively work to make time for as it is possible to do so.
• Acquaintance: Someone who you spend time with incidentally, not intentionally or with whom you are not willing to make time for beyond the incidental. People you only see in class or at work would could as such.
  

The sticky bits are with respect to

1. a person you simply do not have time for regardless of whether or not you would work to make time if you had any. I would say this is more a friend than an acquaintance.
2. People you have been acquainted with over an extended period. Then this could be said to have grown into friendship.
3. People you were friends with but now no longer spend time with.

So as even I must admit, the above distinction leaves something to be desired. But I never said it was the whole definition.

Playing with Knives

I'll admit I ha five stitches put in today after doing just that while working on my sailboat. Rather embarrassing really, it's a knife I carry on a daily basis. I'm starting to hypothesize that any tool you haven't hurt yourself with recently you don't sufficiently respect.

Thanksgiving Meals

While I live in Minnesota, I'm originally from parts farther south and am amused by the seemingly monolithic nature of traditions surrounding the Thanksgiving meal. It seems to go something on the order of:

1. Turkey (or ham)
2. Sweet Potatoes
3. Stuffing
4. Mashed potatoes
5. gravy
6. bread
7. pumpkin pie
8. various fruit pies
9. salad
10. Cranberries

While such a meal is certainly good, I find it amusing that the list is so rigid and so seemingly widespread. Everyone seems to do some subset of it (and in some cases superset).

Here I am wishing for ribs, cornbread, gumbo, barbecue, seafood . . .

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.
    

Why I'm a Beer Snob Part 1

What does it mean to be a beer snob? Arguably it's in the same class as people who are wine snobs. The critical thing that sets apart a real (and arguably good) snob from your average
snobbery-as-a-social-indicator-type snob is a real, deep, and abiding passion for beer.

I started drinking beer regularly with a friend who is definitely passionate about his beer--so much so he home brews. My first beer was a Guinness on tap in Scotland, my second a Summit Great Northern Porter. Hooked then and there. Since then I've had Heineken (not a fan),
Summit's IPA, EPA, Grand; Sam Adam's Boston Lager, Strongbow, Sierra Nevada's Pale Ale, and a couple of different home brew varieties.

Also since that day I've done three batches of home brew. Two are in the fermenter right now.

Passion about beer brings something new to the table: the pursuit of the perfect beer. Anyone can sit down and have a decent--even a good beer--enjoy it, and move on. Someone really passionate about beer takes it to the next level. Every beer is an experience, a test, a meeting of
minds, a criticism . . . it's something that can't be duplicated. It's a little bit of life right there in a bottle.

Infinity

Infinity fascinates me. The German word for it translates literally as "without end". To someone who studies mathematics, infinity is always a special case but also a powerful tool and construct. We take limits to infinity, sums of infinite terms . . . and it can yield some bizarre results. And yet our finite minds cannot actually handle infinity itself.

For example, take one common example of infinity (to a math person anyway) the natural numbers 1, 2, 3, . . . There is no biggest natural number. They go on without end, or to infinity. But infinity is larger than any natural number, so many think of it naively as the largest natural number. It's the only way to get a handle on it.

Another is to stare into a part of the night sky without a star in it. Of course, in all probability there IS a star there somewhere, but we can't see it and so the darkness goes on forever, to infinity, without end. Just thinking about flying beyond every star to a place where there are no more before you forever in unimaginable. Our little minds just can't handle the concept.

This isn't really that surprising. We can't handle lots of finite things either. Try visualizing 4 trillion of something. Or a fourth dimension of space. Look at you computer for a moment and try to think of everything that had to happen to get it manufactured and to you. First you had to buy it and get it home. That means a vehicle and a credit card (probably). How does a car work? It had to be bought, but before that assembled from parts which were bought ready-made. But by whom? How did they do it, how did they know to? Swiftly we reach a point where the sheer complexity of the tangled mesh of requirements to put the computer under your gaze overwhelms the mind.

You Know You're A REALLY Nerdy Math Major If . . .

It's a Friday evening and I'm posting to my blog. Need I really say more?

Neat Energy Efficiency Article

http://hardware.slashdot.org/hardware/07/11/15/1522208.shtml

I love /., largely for reasons like the above article. It demonstrates some amazingly easy ways to reduce your energy usage, electricity bill, and emissions footprint without a major hassle. I suspect I'll be doing some of these things myself, but it will be a little harder on my Linux box.

Beauty of Mathematics Part One--No Right Way to Think About Mathematics

I call this Part One because it will probably be a recurrant theme in my posts for a VERY long time. The rest of the title should be self-explanitory.

Saying there is a right way to think about mathematics is a little like saying there is a right way to think about sculpture or any other form of art. It just doesn't work. Film (to pick a relatively ubiquitous medium for the modern human) speaks uniquely to all who view it. The Godfather Trilogy, for instance, is to me a very sublime commentary on life and a moving tragedy in three parts. To another it may just be another crime film. Math is the same way. For many of my aspiring engineer friends, math is a set of tools for getting a job done. To me it is, amoungst other things, art.

Needless to say, it takes a particular breed to be passionate about what to many is a collection of formulas or a set of rules for balancing your checkbook.

But mathematics transcends this simple formulation as sculpture transcends the aftermath of attacking a rock with hammer and chisel. Think of something classical, like the Pythagorian Theorem. In its simplest form, it speaks of the length of the hypoteneus of a right triangle. In a broader sense, it is the shortest distance between two points in Euclidean space with the usual Euclidean distance. One geometric proof derives it from properties not of triangles, but circles. It has intimate connections to sine and cosine--the length of the opposite side of over the square of the length of the hypoteneuse is by definition sine. In calculus, it is critical in solving certain classes of integrals. It's the simplest non-trivial form of the dot product from vector analysis of a vector with itself (which is just the square of its usual Euclidean length). Each context worries about a version of what could be called a Pythagorean Theorem, which is simply a special case of more general ideas about metric or distance and each is equally valid.

Why I Use Public Key Encryption

There are three reasons why I use Public Key Encyption (PKE).

First, it's just plain cool! I mean, as a math nerd, it's awesome!

Reason number two is privacy. Here in the states anyway there is some argument that sending an email in plaintext is like sending a postcard--no expectation of privacy. Even if the legal eagles thought it should be private, the plain truth is anyone sniffing packets can read an unencrypted email (or thousands of them). Digital signatures are like sending your letters in a tamperproof envalope and encryption is the digital equivilent of sending it by Kourier (yes, a Snow Crash reference).

Reason number three is integrity. Used right, when I send someone an email that I have signed and encrypted, they will know for sure I'm the one who sent it and vice versa. Truth be told, this is pretty cool when all electons look more alike than not.

Bayesian v. Frequentist

This debate is a century old, and is between people who believe what you believe going in should be accounted for (Bayesians) and those who think the only thing that matters is the data you observe (Frequentists).

Here's my position: Anything that fits Kolmogorov's axioms of probability can be treated probabilistically. That means all the usual things like the odds of a strait flush and somewhat amorphous things like what you think the odds are of a person bluffing on a given hand.

The biggest thing is the nature of probability and statistics. Yes, it is usually used to describe things we see and experience around us, but is usually formulated in a much more abstract setting. This means while the people who use it are thinking poker hands, the theory works equally well for predictions, etc. So long as what we want to talk about meets a few very broad requirements, the math will still work.

The irony is that Kolmogorov was a frequentists, but his theory (built on measure theory, set theory, and analysis) and his theory grounds both schools of thought.

Telling People I Brew Beer

"I brew beer. "

Those three little words have probably started more non-geeky conversations than any other statement I know. This is really cool, because as you have probably already noticed, I can be pretty geeky.

Interestingly, I usually get one or both of the following reactions: (a) Oh cool! How does that work? or (b) Oh cool! I or some person I know brews/brewed beer.

When someone has brewed before, the conversation quickly turns to best practices, recipies, and fond memories of good or bad batches. On the other hand, when someone has never brewed before, it's more of a toss-up.

One thing is for sure: everyone who drinks beer wants to know how it tastes.

Webcomics

I realized the other day just how much like real life a web comic can be when I began categorizing people I know by which character from Questionable Content they are like.

I'm just hoping I can't ever make effective match-ups with Dilbert characters.

Community Standards

Recently at a local school (the University of St Thomas) there was a racism incident. Reading the news and looking at the reaction as a strictly outside observer, I have to ask an uncomfortable question: How does a rally and a few letters to the community or the editor change anything?

I'm going to point to one example I know pretty well--the Ubuntu forums. They are what they are in large part because of the willingness of the community to set and enforce community standards. If that commitment wasn't there, I shudder to think what it would be like.

By the same token, a university or town or pick-a-community-unit which sets and enforces good standards is going to have a climate which is good or bad in proportion to how good the standards are and how effectively they are enforced.

I know there are genuine racists out there still and by no means am trying to downplay the significance of the incident. But if someone walked by and saw someone writing the comments, the right answer would be to smack them upside the head, berate their ignorance and racism, and perhaps hold their attention while the appropriate authorities are summoned. Until enough members of the community have the will and courage to do things like that, these kinds of issues will be a much more regular fact of life than otherwise.

New Grading Scale Proposal

What are two seeming constants in the debates about education? Assessment and grade/gpa inflation. As a student I find the whole debate very deeply interesting for personal reasons. Intersting enough to put together the following rough sketch of what I would impliment if I got to pick how grades where assigned.

As a student, I want my grades to measure how capable I am with the material. Sadly, that's pretty hard to condense into one of five letters.

Thinking like an admissions officer, I would want grades to transparently reflect performance and provide a common measure across applicants from different backgrounds.

So what's wrong with the tried and true A-B-C-D-F system? For one, it condenses everything about performance into a single value thus hiding HUGE amounts of potentially useful information like class size, class difficulty, etc. Secondly, it's easy to inflate even in the face of all but the most draconian standards.

It is my belief we can alleviate the second by addressing the first.

Rather than a simple letter grade (or any other single number) the final report for a class should look like this:

• Raw Percentage
• Five number summary (min, Q4, Median, Q2, max) for the class
  
• Five number summary across all classes
• Class Size
• Some form of objective letter grade
• Subjective letter grade/Overall letter grade
• Prof's comments (?)
  

In short, include a basic set of summary statistics with the overall letter grade. The first four items allow anyone looking at the transcript to gage things like how well the student did relative to their peers in the class and against everyone who took the class (which should average out differences across sections). Not perfect, but being trapped in a harder than normal section will show up (as will easier than normal sections and grade inflation). The objective letter grade might be as simple as which quintile the student's raw score falls into, and the overall letter grade and perhaps some simple comment codes would help take into account things like effort, etc.

F1R$T P0$T

The first glory of your own blog is trolling it.