There are two notions of proof being considered here, the formal model of a proof as a logically connected sequence of statements, independent of a reader, and the social construct used by one mathematician to convince others. This distinction is what my post is all about.

]]>A question about proofs in general. I’ve been of the opinion that your statement “.. that convince suitably qualified others …” is th mark of a good proof. But then, Prof Devlin’s opening ” a statement S is a finite sequence of assertions S(1), S(2), … S(n)”. seems to suggest that the sequence itself (provided that S(n) follows from S(n-1) and S(n-k)) might be independent of another examiner.

On the other hand, Euclid seems to be trying to convince the reader (or someone): “I say that …. therefore …”

As for “proofs”, this always gives me a chuckle:

“The above proposition is occasionally useful.”

I haven’t read Russell & Whitehead’s Principia, but Russells Introduction to Mathematical Philosophy inspired me to broaden my mathematical thinking.

Best!

]]>Strictly speaking, yes. But that kind of knowledge is way more certain than pretty well anything else. I’m not proposing solipsism; on the contrary, I am trying to be totally practical. Hilbert’s observations notwithstanding, it would be unreasonable to say the theorems in Euclid’s *Elements* are not proved. The argument are so simple and short. But that just is not the case in much of present day mathematics. I have confidence in the Wiles-Taylor proof of Fermat’s Last Theorem and Perelman’s proof of the Poincare Conjecture. (Based on what knowledgeable experts have said.) But I would wager that in both cases the published proofs have errors, and are thus not, strictly, valid proofs. I don’t see how anyone can conclude with total certitude that the claimed theorems are indeed correct. But on a practical level, both are now accepted into the mathematical canon, and reasonably so. It’s all a matter of degree of certainty.

Keith: Would you say then that the Pythagorean Theorem is not proved, but only that we have extreme confidence that it’s true?

(Proof by Appeal to Authority: A U.S. President came up with a proof. Therefore it must be true.)

]]>No, my point has nothing to do with Gödel’s theorems. Those are about the limits of formal systems. I am talking about proofs as used in present-day mathematics.

You can read a proof and conclude, with certainty, “This proof is wrong, and here is why.” But there is no way you can read a proof and say “This is correct.” All a proof can do on the positive side is increase your confidence in the truth of its conclusion. Proofs are evidence that you weigh. Nothing more.

Using computers cannot eliminate this inherent uncertainty. For instance, in 1994 computers with the Intel Pentium 5 chip gave this answer

4,195,835/3,145,727 = 1.333739068902037589

This identity was thus “proved” by a computer. Indeed, it was “proved” by means of an extremely simple argument, namely an arithmetic computation. But a human spotted that it was wrong at the fourth decimal place.

Thanks for the reply, Prof. Devlin.

“[I]t 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.”

So when you say this, are you making a passing reference to Gödel’s incompleteness theorems–that the consistency of a formal system can’t be checked without going outside the system?

Is this also why you say, “you’ve never seen a proof that truly fits the standard definition”?

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