> it's not true for mathematicians

It actually is true for mathematicians.

Most of us work in ZFC. Some people choose to work in constructive logic. Some people choose to only use countable choice. But we always know what logic we're working in and which theorems we can invoke. E.g. if you choose not to accept the axiom of choice you also have a sense of which results depend on Zorn's lemma and have to work around them. All of this is normal background for mathematicians.

So there's no need to allow the foundations of mathematics to vary unless you're working in logic and need to quantify over foundational systems or compare them. You would certainly never start a problem and then switch the logic midway through. That sort of thing wouldn't even be coherent.

You seem to have a very limited understanding of mathematics. You're welcome to stick to it but I'd recommend learning what mathematics is actually about before thinking you can reduce it all to proof search in a single & fixed finite axiomatic system.

> You seem to have a very limited understanding of mathematics. You're welcome to stick to it but I'd recommend learning what mathematics is actually about

You'd think I'd know what mathematics is about since I have a Ph.D. in it...

> fixed finite axiomatic system

ZFC is not a finite axiomatic system. nowhere did we specify that axiomatic systems have to be finite

> single...fixed

As I said above, there are people who study different logics. However the topic of conversation is individual problems. An individual problem does not switch logics mid stream and anyone working on a problem works in the context of a logic.

You don't need a degree to understand mathematics but plenty of people w/ degrees think they understand what it's all about. Seems a bit odd you have a degree in mathematics but lack the basic understanding of what counts for mathematical activity. I don't think you should be proud of that but then again I don't know what they teach in PhD departments these days so maybe that's what they told you you needed to think to get the degree.

I see. It sounds like you're coming at this from an outsider perspective as someone who hasn't formally studied math and doesn't know what research level mathematicians are taught.

There's nothing wrong with taking an interest in a subject as an amateur. I thoroughly support it. But knowing what is taught in PhD departments and what mathematicians do is a precondition for understanding whether what AlphaEvolve is doing is similar to it.

If I wanted to know whether some AI tool was doing what humans do for, say, song writing, the first thing I'd do is figure out how song writing is taught, how professionals think about song writing, etc. And this is true regardless of the fact that I enjoy, support and often prefer outsider musicians.

I think you're making too many assumptions. Something they should have taught you while you were getting your degree but again, I'm not sure what they teach these days b/c it's been a while since last time I was listening to a graduate lecture on mathematics in a classroom.