How The Game Is Played

The Incredible Abstractness of Mathematics

Hello All,

Its been a bit I know since my last blogging effort. What with E3 and project issues at Sun I've rather had my hands full. Over the July 4th shut down I've been rebuilding the back of my house-- basically tearing out a very bad attempt to enclose a deck and doing it as a proper addition with stud walls and a real roof. I also had to replace the deck itself as it had totally rotted out. Its a comparatively small space, about 4' by 10', but its kept me busy over break.

Which, oddly enough, brings me to my next topic. A somewhat philosophical musing this time.

The other day I was explaining to my wife as we looked over my framing that, even with all the right tools and leveling, perfect 90 degree angles just don't exist in the real world. Even in a factory situation with jigs you only get within some fraction of a degree of tolerance to the intended angle. This brought back something I've talked to friends about before....

Kesselman's Contention: Math is the most perfect AND the most abstract science.

These two things go together. Math is a perfect science because it is entirely internally consistent. Even when someone throws away a postulate, such as the inventors of non-euclidian geometry did, all that happens is the body of theorems that is math extends itself. There is no such thing as a failed prediction in math as long as you do your proofs right.

In contrast, most other sciences have to periodically back-track and re-adjust their entire structure when they run into failed predictions. A failed prediction is when the theory predicts an event that does not match with what actually ocurrs in the real world. Its how science knows its made a mistake.

But math is totally abstract and has nothing to do with the real world and so never has to check itself against the real world. Now I hear a whole bunch of mathematicians out there starting to raise their voices in protest to this statement but hear me out.

Take the example above. We commonly talk about perfect 90 degree angles in math. Trigonometry is based on them. But perfect 90 degree angles do not physically exist in the real world. In fact, no exact measure exists as there is always some degree of error, however slight.

More basically, almost all of math is based at its root on a few fundemental concepts. One of them is equality. "1 apple equals another 1 apple" according to the mathematician. But equality itself does not exist in the physical world. I challenge anyone to show me two apples that are exactly the same. The same weight, the same shape -- it just doesn't exist. The world is about variety and difference, not sameness. And when you reduce all the real measurable and differing quantities of those two physical objects to a single concept of "apple-ness",you have just crossed over the line from the real, physical, measurable world into the entirely abstract world of human concepts. That is the world where math and mathematicians really dwell.

Note that the mathematical concept of equality is not the same mathematical concept as identity. Identity is "1 apple equals itself." Identity exists, but equality does not.

And if equality is an abstract concept that has no physical reality, neither does anything built ontop of it, which is the entire body of mathematics.

Mathematics is beautiful. Mathematics is pure. I love that about it. But I also contend that its beauty and purity come from the fact that it is entirely artificial, a logical construct of the human mind, with no basis in the physical world at all.

A perfect abstract science.

Posted at 10:51PM Jul 08, 2005 by gameguy in General  |  Comments[8]