Delusional systems are logically coherent sets of beliefs. They merely fail correspond to the outside world.

See Nexus on Belief.

An implicit belief is one that one is unaware of. We must be constantly on the alert to discover these implicit beliefs, and we shall never finish. Before one can have beliefs one must have an ontology. Belief in god is more properly an ontological stance. We shall neglect this fascinating arena here. See tacit knowledge too.

It may be effective to believe that the computer against which you are playing chess has a plan, even when you otherwise take the stance that the computer is merely executing a program.

There is a curious and important category of propositions that are true merely because people believe them. For instance “diamonds are valuable”, The zocalo is popular on Saturday night. Similarly words mean what they mean because people believe they mean what they mean.

Another thing that increases my belief in most science is its coherency. I have studied many scientific theories and I am impressed by the logic of most of them. There are few simple reasons to believe in evolution but many complex arguments leave my mind unable to doubt the general outlines of evolution. To me it amounts to overwhelming circumstantial evidence.

I believe that the square root of two is irrational. I can hold the entire proof in my head at once. I believe, a bit less, that all polynomials have roots partly because I passed an exam based primarily in recapitulating the entire algebraic proof. I do not remember the proof but I remember remembering it. I met a mathematics student who felt that she could not really believe a theorem whose proof she could not comprehend simultaneously. She, like all mathematicians, understood big proofs only a little at a time.

I believe the four color theorem. Some mathematicians deprecate the recent “proof” of this theorem because it is too long for a person to comprehend, even piecemeal, having largely been produced by a computer. I think that they mostly believe the theorem but many of them believed it before. To many mathematicians the existence of a proof is more important than the truth of a theorem. Mathematicians also take on the authority of other mathematicians the existence of proofs. Some feel obliged to understand the proofs of the theorems that they presume in their own work. But others assume as given, theorems with published proofs that they have not studied. One of the many mathematicians who first proved a theorem necessary to the recent proof of Fermat’s last theorem, told me that there were several theorems upon which his theorem depended, whose proofs he had not studied.

A proof for the Robbins conjecture has been found by a computer. In contrast to the 4 color proof, the size of this proof is moderate and can be followed by mere humans. The size of the search was exponential — the size of the proof is merely large. The formulae in that proof were unlikely to have been explored by a human mathematician before the computer explored them. The lemmas are truly obscure; the proof is deadly dull reading.

I believe the Riemann hypothesis. Computers have provided large amounts of circumstantial evidence for the hypothesis. Large books have been written based on the hypothesis and such books may be viewed as a failure to disprove the hypothesis. Many results such as the distribution of primes “corroborate” the hypothesis. Yet no proof has been found.

I believe that two computers will compute π to one million decimal places and get the same results even by different methods after their programs have been debugged. This is quite remarkable for computers are of this universe and not the normal Platonic universe of mathematics.

And then, of course, Gödel came along and convinced me of something that has no formal proof; but that does not impugn the proofs that formal logic does express. Some note that Gödel’s proof can be formalized, but I claim that that process involves creating a new proof framework and that this creation process is itself informal.

I would believe that a digital system has some property if I had in my possession a large proof of that property (say one million steps) if I were able to write and run a simple proof checker that verified the proof. The proof need not have come from someone that I trusted—indeed it might have come from someone who stood to gain by deceiving me. The proof would have to be given in some particular form of symbolic logic. This is a statement of my confidence in the fidelity of some common forms of symbolic logic to our natural logical capabilities and also my confidence that computers run for significant periods without error. It also expresses my confidence in my ability to write simple programs and at least weed out bugs resulting in false positives. If I had not studied logic in school for several years I suspect that I would not have this confidence. I can remember only gradually gaining confidence that the formal logic matched my intuitive logic. From the beginning, however, it was clear that the formal proof was more conservative than my intuitive proof notions. Axioms like the axiom of choice, however, are something to worry about.

This seems to be a recent proof of the Jordan Curve Theorem in a formal system for which there are automatic proof checkers. Two papers by Thomas Hales: a very interesting paper on computer proofs and on Jordan’s own disputed proof, including the proof. This fascinating book is about the philosophy and politics of computer proof.

See Logic, Evolution, Math etc. too.

Here we describe epistemology as merely a useful biological hack.

Roger Bacon commended experience to argument and proof in 1268.

Wikipedia gives a larger list of a similar nature.

A fascinating site that deals in specific issues.