What is Math?

Posted on 2010-01-04 by Curt Sampson :: comments enabled

On his blog Good Math, Bad Math, Mark Chu-Carroll tried to answer the question What is math? He’s rather lyrical, going on about music being mathematical and so on, something with which I used to agree wholeheartedly. But the comments were interesting and eye-opening.

There’s a great deal of back-and-forth about how mathematical music is, and whether the math found in music is interesting.

In comment 3, woupiestek writes:

As mathematical logician I must disagree that analyzing Bach and jazz and cubist painting is mathematics. Math is the science of laws and concepts, and their logical consequences. A mathematician does not study ideas, but works out the ultimate consequences of the concepts and laws he derives from them. This is the only way, that mathematics can ever be what it is named for: that which we can know.

Later, in comment 11, Joe Shelby disagrees with this, and notes that formal systems are in use in music (Schoenberg’s 12-tone system being a particularly obvious example), but I can’t say I find the argument very convincing. Self-imposed rules within musical compositions are so frequently broken that that they really seem more like suggestions to me; mathematicians would never get away with that sort of thing in a proof. And lots of people write very popular bits of music using no formal mathematical training or analysis at all.

The discussion continues in comments 14 (Chris Whitman), 19 (Mark C. Chu-Carroll), 21 (Richard Leon), and it goes on.

In comment 22, cfranc mentions a fascinating article by John Baez, which does apply a lot of fairly advanced math (group theory!) to music. But, while entertaining, I’m not sure I see the difference between this and numerology. It’s inevitable, given our innate biological sensitivity to patterns and relationships in the physical world, that the art we produce will embody these to some degree or other. (It’s no surprise that we may prefer to hear 440 and 660 Hz together over 440 and 665 Hz when we can naturally hear the interference patterns.)

I think woupiestek, in comment 34, finds the difference here:

When someone presents the analysis of a mathematical structure, like real analysis is an analysis of real numbers and real valued functions, and proves a counterintuitive theorem, then I will say that my intuition is wrong, and that the real numbers are what the axioms imply.

When someone presents the analysis of a musical structure, the opposite happens: a counterintuitive conclusion would lead to the rejection of the analysis, not of the music. This is how mathematics is different from music, or anything else. And even though music can be interesting for its structure, it isn’t mathematics.

So the difference (in rather over-dramatized form, perhaps) is this:

  • Finding parallels between the patterns in real-world objects (be they musical compositions or anything else) and mathematical ideas is a fun and entertaining pastime for many.

  • Relying on math to tell you when you need to change your real world behaviour is a serious business that people use for, among other things, saving lives.

In comment 6, Paul W. Homer is a little less dismissive:

Looking at math in terms of its structure is a reasonable approach, but I still tend to see it in the way that Hofstadter presented it: as a formal system.

And a formal system is an axiomatic approach to modeling something in abstraction. Ultimately we create these ‘formalized’ intellectual models and then relate them back to the real world somehow.

This appeals to me, in part because it seems very much what we do in computer programming. (And the abstraction, manipulation and de-abstraction seems to have a parallel in category theory, particularly with sections and retractions–or perhaps I’ve just got that on the brain at the moment because it’s the next chapter in the book I’m reading.)

And perhaps that’s related to beauty, or lack of it, in computer programming. Computer programming often seems closer to the music side than the math side as described above, in that, though we always work within truly formal systems, for the vast majority of programmers outside of academia, we’re doing this as a map of a real-world system. Even if we use excellent models (and Lord knows, most programmers do not), we still have the issue of mapping the real world into our model and the results of the model back into the real world. (The second is often rather easier than the first, but not by any means always easy.)

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