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While I agree that some amount of mathematical maturity is useful, I think it is a far reach to claim one has to know the Peano axioms in particular, at least i

While I agree that some amount of mathematical maturity is useful, I think it is a far reach to claim one has to know the Peano axioms in particular, at least in their set-theoretic form. I assume you don't mean knowing and using them implicitly, since you seem to be arguing for explicit mathematical training. 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).

YeGoblynQueenne 5y ago
>> 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. 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? ______________ [1] http://www-formal.stanford.edu/jmc/reviews/lighthill/lighthill.html http://www-formal.stanford.edu/jmc/reviews/lighthill/lighthi...

YeGoblynQueenne 5y ago

>> 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. 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? ______________ [1] http://www-formal.stanford.edu/jmc/reviews/lighthill/lighthill.html http://www-formal.stanford.edu/jmc/reviews/lighthill/lighthi...