Which topics in AI require knowing the Peano axioms?
But I think your question is more along the lines of "why do I need Peano arithmetic to build an image classifier?" (or any machine learning application). To be clear, you don't. But without knowledge of some fundamental concepts in mathematics and the theory of computation, you won't understand the power and limitations of any system you may be capable of buidling. As a lay person, that's by the by. As a machine learning researcher, that'd be disastrous.
Unfortunately, as the OP points out, such lack of deep background in AI research is common today, and probably responsible for the absolute quagmire in which machine learning research finds itself.
>ah these parameters/network layout did this, here's a guess as to why
Or just mindlessly gluing stuff to solve a particular problem
>we took x model and trained it to recognise cats
then yes, you probably do not need much mathematical knowledge. However if you are researching fundamentals, neural tangent kernel, neural odes etc, then you will probably be pretty decent at mathematics. You will also probably have been exposed to Peano at some stage in your life, even if you don't remember it. Its just good background mathematical character development.
I would say even if you are more of an applied person, being exposed to the fundamentals probably helps shape your thinking. Yes you can still be a good productive programmer, but maybe you will have something missing. The person with a broader background may be quicker at solving a new problem, they have a bit more to draw on.
"These are mathematics necessary for AI, as in "Artificial intelligence", not as in "machine learning"."
At the risk of being overly presumptuous, the opinion you give sounds like that of someone who does research in such an area. That's what it sounds like to me, at least, because I once argued similarly while I was working from a more mathematical side about the value of understanding things rigorously. And I would issue statements like "well, sure if you just want to put things into PyTorch, _sure_, but that's not _real_ math."
Unfortunately for us (or at least me, because why did I spend so much time learning this stuff?), many top ML or AI researchers have only a vague understanding of fundamentals, and I vehemently disagree that you need rigorous exposure to mathematics to contribute to ML or AI.
My experience is quite the opposite. Lots of people in mathematics / academia are asleep at the wheel. Well, they were at the wheel fifty years ago, and some departments have caught up, but overall mathematical rigor is not what made the field go forward. (D. Donoho's "50 years of data science" seems relevant).
If you are adamant that Peano axioms in particular are necessary, why doesn't your argument that they are necessary extend to things like the axiom of choice? You know what they say: "the axiom of choice is clearly true, the well-ordering principle is clearly false, and who knows about Zorn's lemma?" Or, if the introduction to Folland's Real Analysis is to be believed, if your research project depends on the validity of the axiom of choice -- choose another research project.
All this to say your post makes me recognize a sentiment I once had: mathematical rigor is important. That depends. I realize that more often than not I made that argument to convince myself such knowledge is relevant -- and that creates a dangerous blind spot (at least for an academic who wants to be useful to industry).
But that depends on what you mean by "contribute". Machine learning research has turned into a circus with clowns and trained donkeys and the majority of the "contributions" suffer heavily from what John McCarthy called the "look ma, no hands disease" of AI:
Much work in AI has the ``look ma, no hands'' disease. Someone programs a computer to do something no computer has done before and writes a paper pointing out that the computer did it. The paper is not directed to the identification and study of intellectual mechanisms and often contains no coherent account of how the program works at all[1].
Yes, anyone can contribute to that kind of thing without any understanding of what they're doing. Which is exactly what's happening. You say that "many top ML or AI researchers have only a vague understanding of fundamentals" matter-of-factly but while it certainly is fact, it's a fact that should ring every single alarm bell.
The progress we saw in deep learning at the start of the decade certainly didn't come from researchers with "only a vague understanding of fundamentals"! People like Hinton, LeCun, Schmidhuber and Bengio have a deep background not only in computer science and AI but in other disciplines also (Hinton was trained in psychology, if memory serves). Why should we expect any progress to come from people with a "vague understanding" of the fundamentals of their very field of knowledge? In what historical circumstance was knowlege enriched by ignorance?
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[1] http://www-formal.stanford.edu/jmc/reviews/lighthill/lighthi...