Proofs and ReasonsDate: 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/ |
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