I feel it has more to do with perception. You present a person with a proof given out by an established professor from a good college. The gut feel is to go with the proof if it makes mathematical and logical sense.

]]>Personally, I use the argument that the point is not to get the right answer, but to show others that your answer is correct, to justify showing each step for my freshman and remedial math students. ]]>

In another place, you mentioned software programs. There’s been a fair amount of work on proving programs correct, but since any program that does anything useful is bound to be thousands of lines long, with millions of possible paths, and an almost infinite number of initial conditions, I don’t think that’s going to happen any time soon.

In fact, we can’t even prove whether a program will eventually stop, or run forever (in an infinite loop). I think it was Turing who established that.

That seems to say that the halting problem is a different sort of beast from the mathematical proof.

On another level, how do we deal with the fact that some machine proofs found better solutions to complex problems than had been established by mathematicians?

]]>Understanding, I noted, is what mathematicians primarily look to proofs to achieve.

I did, however, address the issue of proof checkers when I responded to the first round of comments. I guess it’s possible the later commentators did not read through those initial exchanges. Since almost all in that second wave accused me of being unaware of proof checkers, it is also clear they had not read my 2005 article about Goerges Gonthier’s use of Coq to obtain a verified proof of the Four Color Theorem.

Since a Google search on “keith devlin proof checker” brings that article up (for me) as number 2 behind this current blog post, it appears those afficonados of automated proof-checking do not seem to have the same affinity for automated web search.

Even more strange, those second wave proof-checker fans also did not bother to click on the second Web-link *in my original post*, which brings up my 2003 article “When is a proof?” in which I discussed Thomas Hales’ Flyspeck Project for verifying his proof of the Kepler Conjecture.

Other than making specific references to what are arguably the two best known, and spectacular, examples of automated proof checker use, I am, frankly, at a loss as to how to effectively acknowledge the role that can be played by proof checkers. (As it happens, I met and talked with proof-checking pioneer Larry Wos many years ago when I was a wide-eyed graduate student in logic, but that is not public knowledge, and I doubt Wos remembers the occasion.)

But to get back to the issue of knowing whether a proof is correct, it makes no difference whether the argument is checked by a human or a machine, we can never achieve 100% certainty that it is logically correct. The most we can obtain is greater confidence.

What this means is that proofs are * evidence that humans evaluate* in order to reach a conclusion about the truth of a mathematical statement.

If we proceed carefully, and in sufficient detail, we can often achieve so much confidence in our conclusion that we no longer harbor any doubt about the truth of a particular theorem. (Almost all the proofs encountered in a typical undergraduate mathematics course are of that nature.)

By far the most common approach mathematicians take to gain that greater confidence is to subject the proof to repeated human attempts to find an error. In general, the longer it stands up, the more confident we become.

Very occasionally, that approach proves inadequate, however, and then, as the Gonthier and Hales examples illustrate, our increased confidence may result from the use of automated techniques.

But either way, history has shown us repeatedly that we should never make the mistake of believing we, or our machines, have achieved infallibility.

]]>It never occurred to Euclid or indeed anyone back then that this was something that requires proof. In fact, it required the construction of the real number continuum, which came centuries later. So in this case, filling in the missing details was a lot more than a few little niceties.

This does not render Euclid’s arguments logically incorrect. They were not. But just because they were logically correct does mean they are rigorous *proofs*, since they omit steps that the intended reader (say, a professional mathematician) would find non trivial.

If we are going to allow proofs that miss out significant steps, then I can give you a logically correct proof of Fermat’s Last Theorem that will fit in a tweet, let alone a margin. Here it is:

*Let n>2. Suppose there are positive integers x, y, z such that x^n + y^n = z^n. Then 0 = 1. This is a contradiction, proving FLT.*

That sequence of statements is logically correct. It just happens to miss out some steps. Some fairly hefty steps, as it turned out.

Most of us would say my tweet is not a *proof*, and rightly so. But now we are talking about providing enough steps for the reader to be able to follow it, perhaps with some effort, filling in any missing steps as she goes. Andrew Wiles likely requires a lot less filling in of missing steps than the rest of us. I would need fewer steps than the average high school student. And so on.

In Euclid’s time, it was not recognized there were any missing steps in his arguments, so what he wrote was regarded as a proof. Back then, it was. But it surely was not a proof in our present day sense. We would probably say it is a strategy for constructing a proof. It can be built out to yield a proof. The standard mathematical way is to construct the real (Euclidean) plane.

I don’t believe I am disagreeing with what you say. But I think the different interpretation I put on it is significant if we want to understand what mathematicians today (really) mean by *proof*.

So we are still in the realm of stories told by mathematicians to mathematicians, but in some cases only a handful are able to fully understand those stories.

There’s also a leap of trust involved with a computer proof. Let’s not forget the Pentium FDIV math bug.

Certainty is still restricted to death and taxes. Mathematics does not make the cut.

]]>BTW, I have never read either paper. I take it on trust that the theorem is proved.

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