Proofs and Reasons

Date: 06/18/2002 at 01:41:00
From: Kaitan Sharma
Subject: proofs and ethics of mathematics

I know that a proof is a line of reasoning, step-by-step, using 
formal logic to arrive at a conclusion, or a theorem. When I see 
proofs, some questions, however, occur to me.

Is a proof a reason? For example, the proof that odd plus odd is 
even is rather simple: 2r + 1 + 2k + 1 = 2r + 2k + 2, which is 
divisible by 2, so it must be even. Is this the reason that odd and 
odd is even, because when added, they're a multiple of two? This is 
what it seems to me -- that this is the actual reason for the 
theorem, and the proof merely shows that theorem.

But in cases of induction, it doesn't seem we're showing 
the "reason" behind the proof but just that the proof works.

Date: 06/18/2002 at 09:05:33
From: Doctor Peterson
Subject: Re: proofs and ethics of mathematics

Hello, Kaitan.

A proof is a statement of the reason something is true, in the sense 
that it should convince any reasonable person of its truth. It can be 
stated in various forms (and there can be entirely different proofs 
of the same fact). Often there are many details left unstated, 
particularly in a proof as simple as your example. We might expand 
your proof like this (and even here I will be leaving out details):

    Given that two integers A and B are odd.

    By the definition of "odd" (or more likely a simple theorem,
      starting from a definition that an odd number is not
      divisible by 2, and a definition of what that means),
      there exist integers p and q such that A = 2p+1 and B = 2q+1.

    Then the sum A+B = (2p+1)+(2q+1).

    By the associative property of addition, this is equal to
      2p+1+2q+1.

    By the commutative property of addition, this is equal to
      2p+2q+1+1.

    By the distributive property, this is equal to
      2(p+1)+2, and again to 2(p+q+1).

    By the definition of evenness, A+B is even.

This proof, however stated, shows that the theorem is true. I'm not 
sure I understand your distinction between a "reason" and a "proof"; 
the proof shows that there is a reason to believe the theorem. Do you 
just mean that the "reason" can be the basic idea behind a proof, 
whereas the proof is a formal statement of all the details? I could 
show the "reason" for our theorem in a picture:

    o o o o +   * * * * = o o o o * * * *
    o o o     * * * * *   o o o * * * * *

    7(odd)  +   9(odd)  =    16(even)

The conceptual "reason" is that the "odd", or extra, units "fit 
together", leaving no extras in the sum. This is insufficient as a 
proof, but might in fact be the idea in the mind of a mathematician 
as he works out the details of the proof. 

Often we have such a feeling about how something works, so that we
believe in the theorem (that is, expect it to be true) before we have
proved it; but the proof is necessary because "feelings" or vague
"understandings" can be wrong. The sense of understanding the reason
may then guide us to a proof, formalizing what we sense; or it may
only guide us in stating the theorem, which we can then prove by
indirect methods. 

But the understanding is also important, because by developing a sense
of how and why things work, we are better able to develop further 
theorems, as well as to apply those we have to new problems. If we 
only learned the formal proofs, and never thought about "why", we 
would have an impoverished understanding.

Now, inductive proofs often don't show an underlying mechanism, or 
conceptual reason "why" something is true, but merely prove that it 
is true. We may need to use an inductive proof because it is too 
difficult to express (or even to find) a deeper "reason" why it 
should be true; or we may do it just because it is easier than a 
direct proof. The proof still gives us a "reason" to believe the 
theorem is true; but it is not as satisfying because it doesn't give 
us a feel for "why" it is true. And in that sense, a proof may not 
show the ultimate "reason" for a fact. I think that is what you are 
trying to suggest, and you are right. Not all proofs are equally 
clear in showing what is really going on.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/