One of the things that philosophy is actively working on is determining what "right" and "good" actually mean. There are epistemic/analytic senses ("the answer on the test was right", "all bachelors are unmarried"), practical senses ("it's good to run"), normative senses ("you ought to run"), and so on. A great deal of contemporary philosophy is invested in further delineating these senses and explaining how they relate to our ontology, our metaethical views, etc.

I've heard more than one person say, too, that philosophy progresses by creating new subjects. When you consider that pretty much all of logic, political thought, and biology come from the tradition started by Aristotle, that claim sounds reasonable.

Encouraging. And how is that going so far? If I meet a fellow Philosopher now, can I prove him wrong in a way he'll accept?

It depends on what you mean by that. If someone presents a view that is inconsistent with the accepted facts, then you you can "prove them wrong" in that sense.

But that's not really the point of philosophical inquiry -- the point is to flesh out each view entirely, and look for the best possible explanation of the subject. Those kind of explanations don't always (or even usually) come from outright rebuttals -- they come from synthesis, refinement of the situations considered, improvements in clarity, etc.

In math, don't proofs only show whether something is consistent, not that something is 'better?' I'd wager the credit you are assigning math is based more on its utility when applied rather than to anything inherent to proofs. So, maybe the question to ask is 'Have there been changes in how philosophy is usefully applied?' Seems like there could be. For example, I'm wagering that there's plenty of applied philosophical work to be done in the ethics of automated machinery like, say, automobile braking systems.

The point is there's no disagreement among mathematicians wrt the mechanism of (dis)proving claims. It has nothing to do with usefulness, consistency, or the set of axioms assumed.

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?

It's not poorly designed, it's just too open ended. You have tools like logic to to assess arguments, but there isn't a limit on what questions can be asked or what avenues investigated for areas that fall outside other well defined disciples. Questions such as what makes life worth living, is this wrong or right, what is the nature of reality, how do we know, etc.

As they say, science answers questions, but philosophy answers which questions should be asked in the first place. Science is weaponized philosophy: once a field or subfield of philosophy matures enough, it turns into a science. If you accept this premise, then that is one way to measure progress in philosophy. It seems that this is exactly what the author has in mind.