It isn't possible to prove anything.

For physical theories, this isn't controversial. No matter how many times a theory is corroborated by observation, there is a possibility that the next observation is counter to the theory.

In principle, mathematics is different. A mathematical theory doesn't need to be tested against external criteria. Provided a mathematical theory is internally consistent; then it is a valid mathematical theory.

Unfortunately, proving that a thing is consistent is not possible. The closest we can get is to start by assuming the thing is consistent... until we have evidence that it isn't.

Even more unfortunately, it isn't possible to prove something is inconsistent. Proof and dis-proof are different names for the same thing. It is exactly as impossible to prove something inconsistent as it is to prove something consistent.

{That nothing has been proven to be inconsistent might strike a few people as being somewhere between odd and absurd. There appear to be many, many instances of known inconsistent systems. However, the root problem is that nothing has ever been defined. A proof regarding integers is meaningless if 'integers' has never been unambiguously defined.}

**What we do have**

"Beyond reasonable doubt" and "by a preponderance of the evidence" are the standards that we can actually achieve.

**Why this is relevant**

It isn't possible to prove anything about modern mathematics. It can't be proven right - and it can't be proven wrong.

I would love to be able to prove that axiomatic mathematics is wrong beyond any conceivable doubt. It would make accepting a relationship centric view that much more compelling.

Likewise, being able to prove that a relationship centric view is *the* right approach would shortcut the process of persuading you that it is.

Descarte's "Cogito Ergo Sum" covers this ground.

Briefly: It is impossible to prove that the data we receive through our senses hasn't been tampered with. Further, it is impossible to prove that our personal thought processes aren't being altered from moment to moment.

Absolute, complete, definite knowledge, of any kind, is permanently beyond our reach. It is a fiction. A fantasy.

Yes - it would be nice to be able to eliminate all doubt. But it can't be done.

In practice, everybody accepts "beyond reasonable doubt" and gets on with a life without absolute certainty.