Question Most of our modern conceptions of math defined in terms of a universe in which there are only three dimensions. In some advanced math classes, I have learned to generalize my math skills to any number of variables- which means more dimensions. Still, let's assume that some alternate theory of the universe, such as string theory is true. Does any of our math still hold true? How would our math need to be altered to match the true physics of the universe?

Accepted:
June 25, 2011

Comments

Let's start with a quick comment about string theory. My knowledge is only journalistic, but it's clear that string theory is a mathematical theory and states its hypotheses about extra dimensions using mathematics. And as your comment about additional variables already suggests, there's nothing mathematically esoteric about higher dimensions. When variables have the right sort of independence, they represent distinct mathematical dimensions in a mathematical space, though not necessarily a physical space. (Quantum theory uses abstract spaces called Hilbert spaces that can have infinitely many dimensions. But these mathematical spaces don't represent space as we usually think of it.)

Of course, it might be that getting the right physics will call for the development of new branches of math. Remember, for example, that Newtonian physics called for the invention of Calculus, and though earlier thinkers had insights that helped pave the way, Calculus was something new. Just what sort of new mathematical ideas science might lead to is something we'll have to wait to see. But you've raised another question: if sound physics calls for new math, would the math we have now "still hold true" as you put it?

An example might help. General relativity tells us that the geometry of space-time isn't Euclid's geometry. It's something more complicated called pseudo-Reimannian geometry. Does that mean that Euclidean geometry isn't true?

A good answer calls for making a distinction. As a mathematical construction, there's nothing wrong with Euclidean geometry and there are lots of true statements that go with it. From the axioms of Euclidean geometry, it follows that the square of the hypotenuse of a right triangle is the sum of the squares of the other two sides. Briefly, it's true that Euclidean triangles satisfy Pythagoras's rule. However, this is a statement within math itself, so to speak. Whether physical space fits Euclid's axioms isn't a mathematical question but an empirical one, and the answer turns out to be "No" (or at least "not always.")

Here's a way to look at it: math gives us ways of describing possible structures. (Euclid's axioms describe a very general sort of possible structure.) We can construct abstract proofs about those structures whether or not they fit anything in physics. A theory in science might say that one kind of structure rather than another (this geometry rather than that, this probability distribution rather than that, this kind of differential equation rather than that...) gives us the best model of some part of physical reality. But changing our mind about which mathematical structures are good models for the world doesn't amount to changing our minds about math itself.