Friday, April 20, 2007

The unreasonable effectiveness …

I posted a version of the following comment on Alexandre Borovik's Mathematics Under a Microscope.
It strikes me that a bigger problem [than why mathematics is effective for physics] is why there are regularities in mathematics itself. I'm not a mathematician, and I don't know why Lagrange’s theorem (every integer can can be expressed as the sum of 4 squares) is true. But it bothers me that it is.
As an aside, I have asked students to write a Haskell program to find those integers. The "greedy algorithm," which builds the set by adding the largest integer whose square is no larger than the remaining sum, doesn't always work. The smallest number for which it doesn't work is 23. The greedy algorithm would generate {4, 2, 1, 1, 1} as a set of integers the sum of whose squares is 23 = 16 + 4 + 1 + 1 + 1. But that set has 5 elements. A set with 4 elements is {3, 3, 2, 1}. 23 = 9 + 9 + 4 + 1. But 4 is not a member of {3, 3, 2, 1} even though 4 is the largest integer whose square is no larger than 23.
The question I'm asking is why such a regularity should exist in the first place. The natural numbers are such a simple thing. Why should something as strange as that be the case?

It seems that no matter how simple a structure, there is always some hidden additional structure within it. Why is that?


Alexandre Borovik said...

I believe into a general principle: never resort to philosophy if there is a (meta)mathematical explanation. Regarding greedy algortihm: it is a linear phenomen (which could be seen, for example, from its close connection to matroids). Why should one expect that it works in a quadratic problem?

Blue said...

That's a fine general principle, but I think you missed my point. My point was to wonder why Lagrange's theorem holds at all not to wonder why a greedy algorithm doesn't find the Lagrange sets.

Cedric Gommes said...

I read recently the essay by R.W. Hamming “The Unreasonable Effectiveness of Mathematics” (The American Mathematical Society Monthly, Vol 87 (2), Feb 1980), which I found even more stimulating than Wigner’s paper with almost the same title.

Hamming’s idea, as I understand it, is that mathematics work because they are to a large extent inspired by our perception of the world we live in.

About natural numbers, what on earth does make them look so natural? I think it is the simple fact that physical objects can be distinguished from one another. If they could not be distinguished, the very concept of counting would not be relevant at all. I don’t mean if the objects were identical, I mean if there was really no way to tell “this is one object and this is another one”. In these conditions, the idea of natural numbers would probably not have come to anybody’s mind. To some extent, the obviousness of natural numbers –and presumably their properties- has something to do about physics rather than mathematics.

In quantum mechanics, identical particles are indistinguishable, even in principle. I don’t know, however, if this means they cannot be counted. For instance, we might be able to say “we know for sure that there are 20 photons captured between these two mirrors, but there is no way we can label them”. We definitely should try to get an able physicist involved in this discussion.

It has often been said that Euclid’s geometry would not have been discovered, had we lived in the curved world of General Relativity. In the same way, I am now wondering whether natural numbers would have been discovered by anybody but the most insane of us, had we lived in the weird world of Quantum Mechanics.