Computer-assisted proof

A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.

Most computer-aided proofs to date have been implementions of proof-by-exhaustion of a mathematical theorem. The four color theorem was the first major theorem to be proved using a computer.

The idea is to use the computer to perform lengthy computations, but to control the round-off and propagation errors through the interval arithmetic technique. More precisely, one reduces the computation to a sequence of elementary operations, say (+,-,*,/); the result of an elementary operation is rounded off by the computer precision. However, one can construct an interval provided by upper and lower bounds on the result of an elementary operation. Then one proceeds by replacing numbers with intervals and performing  elementary operations between such intervals of representable numbers.

Attempts have also been made in the area of artificial intelligence research to create smaller, explicit, new proofs of mathematical theorems from the bottom up using machine reasoning techniques such as heuristic search. Such automated theorem provers have proved a number of new results and found new proofs for known theorems.

Philosophical objections
Computer-assisted proofs are the subject of much controversy in the mathematical world. Some mathematicians believe that lengthy computer-assisted proofs are not, in some sense, real mathematical proofs because they involve so many logical steps that they are not practically verifiable by human beings, and that mathematicians are effectively being asked to put their trust in computer programming.

The reverse question can also be raised: if computers cannot be trusted to carry out lengthy calculations, since human beings are by no means infallible, why do some mathematicians place greater trust in lengthy human reasoning compared to similarly lengthy machine computation?

Other mathematicians believe that lengthy computer-assisted proofs are no more or less valid than any other type of proof, and that the problem of human verifiability can be addressed by proving the proof program itself valid. They reply to their opponents' arguments that computer-assisted proofs are subject to errors in their source programs, compilers, and hardware, can be resolved by multiple replications of the result using different programming languages, different compilers, and different computer hardware.

Another possible way of verifying computer-aided proofs is to generate their reasoning steps in a machine-readable form, and then using an automated theorem prover to demonstrate their correctness. Understandably, the approach of using computer programs to prove other computer programs correct does not appeal to computer proof skeptics, who simply see it as adding another layer of complication without addressing the fundamental issue of the need for human understanding.

Another argument against computer-aided proofs is that they lack mathematical elegance: however, this is an argument that is not restricted to computer proofs, but can also be advanced against any lengthy proof by exhaustion. A similar argument against computer-aided proofs and proofs-by-exhaustion is that they provide no insights or new and useful concepts.

An additional philosophical issue raised by computer-aided proofs is whether they make mathematics into an experimental science, where the scientific method becomes more important than the application of pure reason in the area of abstract mathematical concepts. This directly relates to the argument within mathematics as to whether mathematics is based on ideas, or "merely" an exercise in formal symbol manipulation. It also raises the question whether, if according to the Platonist view, all possible mathematical objects in some sense "already exist", whether computer-aided mathematics is an observational science like astronomy, rather than an experimental one like physics or chemistry. Interestingly, this controversy within mathematics is occurring at the same time as questions are being asked in the physics community about whether twenty-first century theoretical physics is becoming too mathematical, and leaving behind its experimental roots.

As of 2005, there is an emerging field of experimental mathematics that is confronting this debate head-on by focusing on numerical experiments as its main tool for mathematical exploration.

Massively parallel proof-by-exhaustion projects
Seventeen or Bust is a computer-aided attack on the Sierpinski problem currently taking place using distributed computation on the Internet.