Someone commented just a few days ago about how much new math research is being published.
As a bunch they are really productive.

I think they could do better in philosophy of mathematics.....foundational things like knowing how mathematics exists physically and training the profession accordingly. Especially as we interface with A. I.

As we're setting up curricula to fulfill your own point of view, we should keep in mind that AI itself came from mathematics and the study of computabilty that has not been limited to your point of view. The very advent of the digital computer came from mathematicians who whose creativity with abstractions abounded and redounded to the sciences without having to subscribe to whatever it is you would instruct them about physicality. And also with developments in computer science and the sciences since.

Of course, in the philosophy of mathematics we already have great debates and application of intellect to questions of concreteness, applicability, etc. And we have the field of applied mathematics. And we have people researching the relationship of mathematics with the sciences.

But there are always avenues of improvement. So I'm interested what specific research, what texts and resources you advocate should be course requirements for mathematics students, and what philosophies you think should be inculcated and what philosophies you think should not. Would your curricula for mathematicians include contrasting philosophies across a range in the philosophy of mathematics or would the only required material be study of philosophy that conforms to your own beliefs about mathematics and the physical?

it seems new ideas in
maths are rare — Christoff Montnielsensons

What is your basis for saying that?

Discovering
imaginary, real, natural or end-
less numbers for example,
seems to be nobody's business. — Christoff Montnielsensons

Why do you believe that?

I'm very far removed from any formal mathematics community. Maybe good and bad so I have my own way of doing things.

You have a good overview. Really, I think the people in math have their own view on how things should evolve.

..........off topic a little,
I used to visit our state university math library and there was this little room on the top floor with all their best books. Maybe 20 by 50 feet of shelves. Still lots of books. I would check random books just to see what was out there. Just amazing.

A library is a candy store, temple, sanctuary, and sparkling pool under a waterfall, all in one.

Funny you say that. The book room I was talking about was like a holy of holies. You would walk into the main mathematics department building, down a hall to the mathematics library, past a front desk and main room, into a wing and up a back stairway to the third floor. I think math departments might have their own special collections. It kind of seemed like that. The books seemed like the real math that most people wouldn't know exists. Maybe about year 2000 so before everything was on the internet.
Yes. Candy store.

You can hold in your hands and read every volume of the 'Journal of Symbolic Logic' in its original printing, as at the start of the rows of them is Vol. 1 No. 1 from March 1936 with Church's 'A Note on the Entscheidungsproblem' itself. It's a pretty great feeling.

We see a lot of interest in
physics and the other sciences, but it seems new ideas in
maths are rare . . .What am I really saying? Is research mathematics just
a pointless pastime... — Christoff Montnielsensons

I should comment here. arXiv.org is a repository of mathematical research papers. Even then an author must be recommended by another author who has been approved by the same process.

The last time I looked about 5,200 articles were submitted in the 48 days of the new year. Over a hundred a day. So new ideas in math are anything but rare, since virtually all papers do indeed have new results and not hashing over previous ideas.

Looking at Logic (includes set theory and foundations) one sees perhaps five papers a day. History and Overview category is two or three a day. But when you read the titles and see the abstracts almost all fall under other categories, like the papers titled "Real analysis without uncountable sets" and "The double gamma function and Vladimir Alekseevsky". There is no category of "Philosophy of Mathematics" beyond these two. For philosophy of mathematics one must leave arXiv.org and go to PhilArchive, or a similar repository.

"Pointless pastime"? For the hundred thousand or so practitioners around the world it is a challenging exploration of ideas. For most others it is pointless if it doesn't provide support for physical projects.

it is a challenging exploration of ideas — jgill

To add to that, the practical benefits of abstract research are not always seen at first, but such investigations generate ideas that can lead to practical benefits, and lead to methods and ways of thinking that can be used in wider contexts. Mathematics is a beautiful and creative expression of human intellect and curiosity. I am always amazed when people want to put the kibosh on it.