What requirements? And isn’t this just wild overgrowth in the fields of nonsense, even within mathematics? And more mildly: is there, or could there be, any application for this in physical reality?

Whatever requirements people decide on.
You can argue about whether natural numbers are some kind of idea that exists in some world of ideas, but mathematicians aren’t interested in that. You can define a number such that if it is multiplied by itself you get -1, and that’s it. For the definition to be acceptable, it only has to be coherent (here, for example, you’d apparently also need to find a broader definition of multiplication so that it applies to such numbers too). Of course, you can define lots of boring things with no implications and mathematicians won’t bother with them, but if a certain definition opens up something interesting, they will. Imaginary numbers are also useful in various calculations (though that doesn’t interest most mathematicians all that much).

This has nothing to do with nonsense except sharing the name. It is a precise and consistent definition that has many interesting theoretical results and lots (!) of practical applications. But there are also mathematical fields with no application, and even so they really do not fall under the term “nonsense” by my definition. This comparison is just a misunderstanding.

What is the point of defining definitions and deriving things from them? Just to amuse ourselves and build worlds for ourselves? To uncover every true thing simply because it is true (intrinsic interest)? Because it brings us closer to knowing the whole truth, which brings us closer to God / the good / the whole, etc. (a religious motive)? Or in the hope that one day it will be useful to us, as happened with a number of areas of mathematics (a practical motive)?
I’m worried about what I called “intrinsic interest,” because things of that sort can be the invention of illusory values (values that stand on their own and are unrelated to us), arising from confusion between that value and other values that are logically derived from values directly connected to us, and that seem to me to be existentially ingrained in human beings (from the little I’ve seen and heard), such as personal advancement, the advancement of humanity, and the aspiration toward infinity (which are fairly overlapping values).
What is dangerous about values of the illusory type (as I defined them) is that they can be any value one can think of, because once I choose an “intrinsic value” I no longer need to justify it to myself or to society, and I’m not sure whether it can be tested against the “eyes of reason.”
Which raises another question: what are the ideas that can be tested against the “eyes of our intellect,” and are there things we are incapable of touching at all (as I think regarding values of the illusory type, if such things indeed exist).
And how can we ever reach a conclusion about whether values detached from us are illusory or not? After all, if we “make an effort” to believe in them and then succeed in “seeing them,” we could say either that we’ve risen to a higher level and really acquired the ability to sense them, or that we simply deluded ourselves.

There is a collection of claims and definitions here that I find hard to address all together in this jumble. You started with mathematics and ended up in a barrel.

So I’d be glad if the Rabbi could address the mathematics part (up to the paragraph that begins with the words “Which raises another question”), and regarding the matter of ideas I’ll look for posts and questions on that topic.

I’ll address only the first paragraph. There is intellectual value in mathematics, as in philosophy (mathematics is a branch of philosophy), even apart from applications. It is impossible to explain what value there is in intellectual fields. It broadens and deepens the mind. Study for its own sake.
As for whether this is building worlds or uncovering them—that is a matter of worldview (Platonism). I am a Platonist, so in my view it is uncovering.

Broadening and deepening the mind doesn’t sound like study for its own sake; it fits very well with the utilitarian idea of self-advancement and the advancement of humanity.
And that also makes a lot of sense. I don’t see value in intellectual study beyond advancing yourself and advancing humanity, spiritually and practically.
I once heard a claim similar to what you’re saying: that whatever people have occupied themselves with throughout human history must contain something worthwhile, since it’s not for nothing that we are predisposed to deal with those things. There the intention was intrinsic interest, and I think it may be self-deception. Intellectual pursuits have clear purposes, which I mentioned above; there is no need to turn them into transcendent and unclear things, and I also don’t understand why one would do that.