This is a reflective investigation concerning reasons we believe things. It might be called an engineering approach to epistemology. I mention a few reasons in no particular order. Authority, logical proof, comfort, implicit, utility.

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.

Naïve Sense Impressions

Most philosophical notes on belief spend most of their space on this obvious yet difficult subject. I thereby spend less space. The role of optical and other illusions and how we learn to compensate for them is an important aspect of this.


Blaise Pascal’s argument for the existence of God, often referred to as “Pascal’s wager”, amounted to arguing that it was useful to believe in God considering that there was some probability that he really existed in which case the positive consequences of belief were very great. Others have argued that society is better off if people believe in God even if there is none. This assumes that one can will himself to believe something in the first case and can persuade others to believe in the 2nd case.

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.


The church has been an authority for many people for many centuries. I believe that Pluto is in orbit about the Sun not because I have seen it, even thru a telescope, but because there are people whose methods I understand and admire, and they tell me that Pluto is there. I have worked with some of these people and their skepticism is scarcely less than my own. (Actually I suspect that I know no one with direct knowledge of Pluto but I am confident that I know people who know people with such direct knowledge.) There is precious little science that I take on other than authority. My limited experience in the laboratory corroborates my belief in F=ma and vector addition of forces.

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.


Wishful thinking is common I will not dwell on it here.


The idea of proof is remarkable. Someone that you do not trust tells you a proof and causes you to believe the thing proven. The advent of lying expanded the role of proof; yet proof protects not only against lying, but error as well.

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.

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 book is about the philosophy and politics of computer proof.

I remember very clearly one morning in about the third grade a talk by a visitor to our class. She was telling us about nutrition. I had not yet learned the meaning of “propaganda”, at least in its narrow sense, but the concept was already very familiar to me. She was giving us no reasons for believing what she was saying. I think that my skepticism arose from several factors. What she was saying was inconvenient to believe. She provided no conceptual glue (logic) to relate the assertions. It wasn’t very coherent. She did not even claim that there were reasons for believing what she said. To this day have I have difficulty believing assertions about nutrition. They come without proof, or even rumor of proof.

See this.

Evidence with Logic

Part way between proof and sense impressions is an important sort of information acquisition. This paper chronicles the tale of a web site that noticed a manipulated news photograph from a major commercial news source. The discovery and reasons for believing that the image was manipulated were readily available. The web provides a new and uniquely effective medium for such memes.


I strongly believe that there is an array of corner reflectors on the Moon placed there by one of the Apollo missions. It is because I think that NASA would have been foolish to fabricate the idea especially since it can be verified by someone with a telescope and certain laser equipment. In fact I believe that there have been no fabrications of space missions by NASA because “that is not their style”. I frequently disbelieve what the CIA or NSA says. I suspect that I would sometimes agree with their reasons for lying. I can imagine situations where I would want them to lie to us.


To believe something tentatively is a sort of belief and not a reason for believing. It is often a stage in coming to believe something. Often we believe that something is possible as a step towards believing it. Only recently it occurred to me that normal logic is suitable for both mathematics, where Theorems always hold, and also in describing observed patterns in the world that only appear somewhat more than ‘you might otherwise expect’. This dual efficacy of logic gives logic an evolutionary boost. It relates somehow to The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Computer Simulation

I believed that a test particle orbits a mass point in an inverse square force field by reading several authorities and even following the difficult mathematical derivation of this fact. When I wrote a fairly simple computer program to track successive points in an orbit and it showed an ellipse, I felt a rush and came to believe the proposition more thoroughly. The computer can attend to details that we humans must gloss over.

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 general Skeptic

A fascinating site that deals in specific issues.