Question Can the "real world" provide evidence that mathematical knowledge is legitimate? I think its many peoples' intuition that the successful application of math to science and engineering (e.g., that we can use math to build bridges) shows that math is true.

Accepted:
October 12, 2007

Comments

The question is whether what we find in the physical world could tell us whether math is true. Let's consider two sorts of cases. One is what we might call mathematical laws -- 1+1=2 is a particularly simple example. An algebraic law like x2 - y2 = (x+y)(x-y) is another. The second sort of case includes things like Newton's law of gravitation -- F12 = G(m1m2)/r2 -- or some mathematical description of the characteristics of steel beams used for bridges. This may be closer to what you have in mind.

Start with the first sort of case. Suppose we have two 1-liter beakers of water. We pour them together, measure the volume and find that it's two liters. Have we confirmed the mathematical claim that 1+1=2? If so, what do we make of the fact that if we put pure alcohol rather than water in one of those beakers, when we put the two together we get about 1.94 liters? Does that count against 1+1=2?

It's pretty clear that neither experiment tells us anything about whether 1+1 equals 2; we already know that it does. But we do discover that mixing volumes of liquid isn't always additive. Have a look at question 1759 for related discussion.

What about the second kind of case -- physical laws and such stated in mathematical form? Here we're using the language of mathematics and the rules of algebra, calculus and whatnot, to say things about and to reason about about the world. But if we find out that the statements we make aren't true, that doesn't reflect on math itself. Here's a statement in mathematical form:

b = p2

The "b" represents the number of brothers of Allen Stairs and p is the number of parents of Allen Stairs. It may come as no surprise that p = 2. And so the statement apparently tells us that I have 4 brothers.

As it turns out, the "equation" is wrong. In reality, b = 1. And so, substitution gives us

1 = 22

That's plainly false. But this silly example doesn't show that arithmetic breaks down when applied to the Stairs clan. All we have here is a humdrum falsehood decked out in mathematical dress.

Scientific cases are generally a lot more interesting, but the point remains: we can use the language of mathematics to say things about the world that aren't guaranteed to be true. When they do turn out to be true, what we've discovered is that the non-mathematical world fits a certain mathematical description. When they turn out to be false, what we've discovered is not that mathematics needs revising, but that if we want to describe things correctly, we need to revise our mathematical description -- we need to pick a different equation or revise our guess about the value of a constant or something of that sort. When we do this, we take for granted that the laws of algebra and the like are correct. If we didn't, we wouldn't know how to say the things we use the language of mathematics to say.

Question Can the "real world" provide evidence that mathematical knowledge is legitimate? I think its many peoples' intuition that the successful application of math to science and engineering (e.g., that we can use math to build bridges) shows that math is true. Accepted:October 12, 2007

Question Can the "real world" provide evidence that mathematical knowledge is legitimate? I think its many peoples' intuition that the successful application of math to science and engineering (e.g., that we can use math to build bridges) shows that math is true.

Accepted:
October 12, 2007