Consistency, physics, and coinduction
Seminar Room 1, Newton Institute
AbstractIn the first part of this talk we discuss the consistency problem of mathematics. Although we have very well verified mathematical proofs of $\Pi_1$-statements such as Fermat's last theorem, we cannot, because of Gödel's 2nd incompleteness theorem, exclude the existence of a counter example with absolute certainty -- a counter example would, unless there was a mistake in the proof, prove the inconsistency of the mathematical framework used. This uncertainty has similarities with physics, where we cannot exclude that nature does not follow the laws of physics as determined up to now. All it would imply is that the laws of physics need to be adjusted. Whereas physicists openly admit this as a possibility, in mathematics this fact is not discussed very openly.
We discuss how physics and mathematics are in a similar situation. In mathematics we have no criterion to check with absolute certainty that mathematics is consistent. In physics we cannot conduct an experiment which determines any law of physics with absolute certainty. All we can do is to carry out experiments to test whether nature follow in one particular instance the laws of physics formulated by physicists. In fact we know that the laws of physics are incomplete, and therefore not fully correct, and could see a change of the laws of physics during the historical evolving of physics. Changes of the laws of physics did not affect most calculations made before, because these were thoroughly checked by experiments. The changes had to be made only in extreme cases (high speed, small distances). In the same way, we know by reverse mathematics that most mathematical theorems can be proved in relatively weak theories, and therefore would not be affected by a potential inconsistency which probably would make use of proof theoretically very strong principles. In both mathematics and physics we can carry out tests for the axiom systems used. In physics these tests are done by experiments and as well theoretical investigations. In mathematics this is done by looking for counter examples to theorems which have been proved, and by applying the full range of meta mathematical investigations, especially by proof theoretic analysis, normalisation proofs and the formation of constructive foundations of mathematics.
As the laws of physics are empirical, so are $\Pi_1$-theorems (phrasing it like this is due to an informal comment by Peter Aczel). As one tries in physics to determine the laws of physics and draw conclusions from them, in logic one tries to determine the laws of the infinite, and derive conclusions from those laws. We can obtain a high degree of certainty, but no absolute certainty, and still can have trust in the theorems derived.
In the second part we investigate informal reasoning about coinductive statements.