M16,

What the *hell* is that?

Looks to me like M16 *really* does math, not just the arithmetic that most people call "math". I took a graduate Abelian Algebra course during my senior year of college. They talked about the Sieve of Primes of the Kernels of Homomorphisms, or something like that. I had always been pretty good at proofs, but I realized in this course that I had been working before with a blindfold on. Because the undergraduate proofs I had been asked to do were like walking around a small room and bumping into the furniture, I could handle it. This graduate class, however, was more like being asked to find a teacup on a football field. Without a better mathematical intuition, some way to remove the blindfold, I couldn't do it. Being stoned all the time didn't help either. I dropped the class.

The above definition is from

this book at Amazon.

The author was my prof for two mathematical physics classes. One day during a review session he asked for students to write-in suggestions for the class because he was using a proof version of his undergraduate

text and he wanted some feedback. Some of the students wrote in that it would be better to use more numbers in the examples. He seemed to get slightly agitated and went into a discourse on

*why* numbers really didn't matter that much when you are doing math and physics. He wrote the number 57 on the board and asked what information or usefullness that conveyed. Answer: Not much. Then he wrote an equation on the board and asked what information that conveyed. Answer: A lot! And you could stick any set of numbers in the equation and get numerical answers or derive further important results without using numbers. Being able to manipulate the symbols and derive further results are the important things.

Of course, numbers

*are* important for real-life applications. I don't think he was implying that they weren't - but most math and physics and any science derived from these do not need numbers until the final result.

I note that the blog entry focuses on arithmetic, which I find too mechanical. If I can avoid crunching numbers I do - I don't think it requires much analysis. Once you learn the algorithm it just becomes tedious to keep track of what numbers go where.

It's interesting that while some of my profs would grade just as hard for an incorrect numerical answer (for example, I always make numerical mistakes in matrix multiplication), others wouldn't care at all just as long as the derivation was correct. After all,

*anyone* can do arithmetic.

When I used to teach (as a teaching assistant) a 100-level physics lab, the students would always get hung up on using numbers first. For example, take Hooke's Law as it relates to the standard weight-on-a-spring experiment:

F = ma = -kx

Given the mass, acceleration and the distance the spring is stretched, what is the spring constant, k? Well, they'd start getting confused by trying to start plugging in numbers. No! Solve for k first, then stick in the numbers. Don't worry about the numbers until the very end! (This would confuse some of them - it seemed it wasn't arithmetic as it was the simple algebra that got them.) Once I got it through to them how to get 'k by itself' they were able to solve, with a little nudging, other algebraic equations.

--------------------------------------------------------

Here's a bunch of tutorials on very theoretical probability. No numbers here, either. (This stuff is beyond me - maybe someday I'll learn enough to start studying it.)

http://www.probability.net