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

]]>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.

“[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”?

]]>“[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. 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.”

Prof. Devlin, please feel free to stop by and comment!

John

]]>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?

]]>