Epistemology and Mathematical Proofs gets blurrier
Do we learn anything fundamentally new, or do we just know the answer to the question?
Elgindi
I’ve written about the problems of epistemology and using modern computing to solve problems before. Mathematical proofs are have long lived in a world of rigorous pure thought and universality. Computer assisted proofs are more recent constructions that have been used for a variety of questions with differing purposes and results – but largely on smaller problems.
Two mathematicians recently used computer assistance in their proof that Euler fluid dynamics equations have singularities that ‘blow up’ in certain badly behaved conditions. They developed some really interesting techniques to solve difficult numerical accuracy problems and deal with the numeric problems around mathematical singularities. They broke tons of new ground – inventing and creating a lot of new techniques in the fluid flow investigation. Despite all this and an interesting proof, the questions started up again.
What really qualifies for a proof? According to the mathematicians in the article, many say a proof has to convince other mathematicians a line of reasoning is correct. Others argue that a proof must also improve the understanding of why a particular statement is true, rather than just simply validate the reasoning is correct. “Did we learn anything fundamentally new, or do we just know the answer to the question.”
Constantin says, “A computer can help. It’s wonderful. It gives me insight. But it doesn’t give me a full understanding. Understanding comes from us.” Eigindi still hopes to work out an alternative proof by hand but says “I’m overall happy this exists, but I take it as more of a motivation to try to do it in a less computer dependent way.”
I think this recognizes two important points. First, did we just answer this question, or a universal question? Secondly, did we unlock some fundamental new understanding about the problem – often revealing some new first principles – or just answer a yes/no question? These questions are ones of epistemology which tells us that some answers are better, or provide different, kinds of knowledge.
Traditional mathematical proofs are universal. Once we have them, we know anything built upon them will stand the test of all time and all conditions anytime, anywhere, and in any forms in the universe. It’s the bedrocks that we build our mathematical, scientific, and engineering systems on. Proofs built from first principles often reveal deep truths about how a system operates. An example is describing planetary orbits in terms of mass and gravity forces. This system reveals fundamentally new understanding to the problem based on the first principles of gravity and mass as the basis for why orbits happen. At higher levels, simply answering a yes/no question can disprove a theory/proof or help us know if there is/isn’t at least one solution. This is good information and helps us at least put some bounds on problems that may or may not be actually provable; but they are not as useful as a rigorous proof.
I think it’s worthwhile for everyone, especially scientists and engineers, to understand the different kinds of knowledge and their inherent limitations and that we have solid discussions on computer assisted proofs. Otherwise, we might end up building knowledge upon sandy foundations that might get washed away in the future. Or worse, cost lives when the engineering, scientific, or economic systems we built on them break down catastrophically.
