Why Do Proofs

Proof is an essential part of mathematics, but it’s sometimes not clear why we prove things in mathematics.  By way of example, let’s consider a simple problem:  Multiplying two even numbers.

If you were a scientist, you would collect data. For example:  $4 \times 6 = 24$$2 \times 8 = 16$$10 \times 10 = 100$, and so on.  After a few hundred pieces of evidence, you would form an hypothesis:  The product of two even numbers is even.  You might then perform some more experiments to test the hypothesis:  $82 \times 14 = 1148$$14 \times 32 = 448$$11 \times 4 = 44$ (then sheepishly realize that $11$ isn’t even, so we’ll ignore that), $24 \times 4 = 95$ (better recheck that…yup, it’s supposed to be $96$), and so on.

But if you’re a mathematician, you’d next try to prove that the product of two even numbers is an even number.  The first thing to realize is this:  No one ever tries to prove anything they don’t already believe to be true.  If you asked me to prove the product of two even numbers is an odd number, I’d say “But that’s not true…why should I waste time trying to prove it?”  What this means is that we don’t prove things because we doubt their truth:  proof isn’t a way of obtaining truth, because in some sense, we already have it.

So why do we prove things?  Let’s consider what we’d have to do.

• First, what do we mean by “the product of two even numbers?”  We need to clarify what we mean by an even number:  “I know it when I see it” isn’t good enough.  We come to the conclusion:  An even number is any number that can be written as the product of 2 and some other whole number $m$.
• So now we multiply two even numbers:  $(2m) \times (2n) = 4mn$.
• But we need to write this as a product of 2 and some other number, so we have $4mn = 2 (2mn)$.

There’s our proof.

But stop and smell the roses.  Halfway through, we found the product of two even numbers was $4mn$.  This means that the product of two even numbers is in fact a multiple of 4.  If we go back to our data, we’ll see that it’s there…but it might not have been obvious at the time.

So what has this effort of proof given us?

• We had to dig down deep and elaborate on what we man by “even number.”  In general, proof requires us to define our terms in a meaningful and useful way.
• We had to use a little bit of algebra.  In general, proof requires us to review mathematics we’ve already learned.  (In fact, since most terms in mathematics have a standard definition, the first step also requires a review of the mathematics we’ve learned…or the mathematics we should have learned)
• We discovered something that we might have missed the first time around.