Why do we need the foundations of mathematics and what is their relation to philosophy? What is the most complete and widespread approach to the basics of mathematics?

One Answer

1. Since everyone is silent again, I'll have to, but I'm not a real welder, so I can get something wrong �

   In order to be able to do math at all, you need to decide on a number of basic questions. These questions precede mathematics, are therefore mathematically unsolvable, and require a philosophical solution.

   What are these questions?

   First, axiomatics. A set of axioms that do not require proof, something from which we will deduce theorems according to the rules of logical inference. And here we are waiting for the first problem. Some of the axioms seem obvious, but not all of them. The axiom of choice, for example. On the one hand, quite a few proven theorems are proved using the axiom of choice. On the other hand, its adoption allows us to prove completely counterintuitive things that, as many people think, cannot be true. Rejection of the axiom of choice entails the need to re-prove many theorems without relying on it, which is quite a serious work, not the fact that it is feasible. But you can potentially take any of the unsolved mathematical problems, that is, a statement that looks empirically correct, but has no evidence, and declare it an axiom – and here, too, the question arises-why can't you? Or can I? And what will be the consequences of such a decision?

   Secondly, why do we draw conclusions from axioms at all? Based on what? We are guided by the rules of logical inference; we consider something proven when we demonstrate exactly how a given statement can be necessarily derived from our axioms. Where do the output rules come from? Why are they like this? Can they be different? Do they necessarily exist? If so, what is their nature? Are they part of the properties of the universe? Or are they the result of the evolution of thinking and selection for performance? And where is the guarantee that evolution selects exactly what is true, that true and useful are synonymous? Or is their nature supernatural? Let's take intuitionistic mathematics as an example. It does not have the law of the excluded third, but not as in Hegel's dialectic, when A and -A may be true simultaneously, but on the contrary, they may not be true at the same time. It's not raining, but it's not raining. As a result, a proof to the contrary is not applicable in this math. It is not enough to prove that one of the statements is wrong, you need to prove that the other is true, and this does not follow from the first one. Not everything can be proved in intuitionistic mathematics, but what is proved is proved much more strictly than in ordinary mathematics.

   Third, and perhaps most important, what is the epistemological status of mathematical knowledge? Is it true or just conventional? Does mathematics study objects that actually exist (or is it more correct to say whether these objects have an independent, albeit non-physical existence), or are they just abstractions generated by our culture? Can we trust evidence that we don't have enough time to read in our entire lives, but that the computer has checked and found to be correct? Is mathematics just another science, that is, it can only claim to have plausible models, or is it something more?

   All these questions are at the intersection of epistemology, ontology, philosophy of logic, metaphysics and methodology of science. They are fundamentally important for mathematics, and they are, of course, philosophical.

   At the moment, the most widely used Zermelo-Frenkel system, which appeared as a result of the discovery of the inconsistency of naive set theory described by Russell's barber paradox. The remaining systems are rather experimental in nature, but some are of great interest and have made a significant contribution to the development of mathematics.

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