There's plenty of disagreement among people who actually think about what it means to prove or disprove a mathematical claim (which is a tiny, tiny minority of mathematicians). See finitism, intuitionism, etc.

There is unlikely to be an agreed-upon mechanism for determining progress in philosophy for the obvious reason that what constitutes philosophical progress is itself a difficult and rather important philosophical question.

The groups you've mentioned set forth their axioms and accepted logic and construct their math accordingly. There's no disagreement here, as there's no "disagreement" between Euclid and Lobachevsky.

There is disagreement about what constitutes an acceptable mathematical proof. E.g., is proof by contradiction a proof?

Yes, it is certainly a proof (in a most commonly-used logic systems)

There may be some logic systems where proof by contradiction is not a proof. The existence of such logic systems does not mean there is any disagreement between mathematicians.

No, it doesn't, but as a matter of fact there has been disagreement among mathematicians on those issues. See e.g. https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert_contro....

How about the philosophy of math? Do numbers exist outside the human mind/culture? Is the universe mathematical, or is it just a map? Does math say anything fundamental about existence?

> Does math say anything fundamental about existence?

I'd say yes. Using the tools of math, we can construct and study (within our mind) infinitely large and infinitely detailed structures that transcend the finite bounds and finite fidelity of our observable reality. This is what makes math beautiful and meaningful to me.

What exactly does that say about existence?