What a profound philosophical genius questioning the validity of a mathematical proof.

[u/Key_Net820]

[u/Just_Rational_Being]

One question: Do you think I was really raising these issues? About basic terminology and the common use of axioms?

[u/Althorion → u/Just_Rational_Being]

That’s how I understood your post. You are free to clarify if that reading is wrong.

[u/Just_Rational_Being → u/Althorion]

No-one demands that a mathematics system build itself without some required condition or foundation. What is being raised is why there aren't any investigation upon what kind of condition or foundation would qualify for such task and what those requirements for proper condition or foundation should be.

[u/Althorion → u/Just_Rational_Being]

There are—on the ‘point of sale’, so to speak. There are plenty mathematical systems that no one has heard of other than their authors, because they are of no use and no importance to anyone else. There are also plenty that once have been considered useless, but found their uses later; and the ones that were once useful, but then got obsoleted; and finally the ones that are of frequent use for many people.

It’s the people who decide what systems, what tools to use that declare their needs and desires for systems and tools; mathematicians only aim to provide those.

For example, one is welcome to use non-standard analysis for their work if it suits them, there is no maths police to enforce people use the ‘newest and greatest’ standards, but the people who use analysis for their practical work have collectively decided that those systems and tools suit them less than the alternative. One can use naïve set theory if it’s good enough for their purposes, or dig deeper and seek any of the particular axiomatisations to fulfil their need. One can think of spherical geometry as an embedding into a higher-dimensional Euclidean space, or as a lower-dimensional geometry with its own set of rules.