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by cnity
935 days ago
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My understanding of the flaws of PM is in its attempts to avoid self-reference, which was sort of folly from the beginning as proven by Godel. I learned this from I Am A Strange Loop, and I'm not sure how accurate it is historically. But Godel's Incompleteness Theorem is one of the most interesting things I've ever read about. |
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The elementary vernacular foundations of modern mathematics is (more or less) naive set theory; the starting tools of serious foundational work (as arcane as it is even within maths as a whole) are logic and more rigorous set theory, perhaps with some computability mixed in; the main tool of a mathematician who wants to reason in great generality is category theory (with some handwaving in the direction of the previous point about universe hierarchies and whatnot). There are some signs of convergence between these (and of mainstream mathematicians starting to once again take foundations seriously), but at the basic level those are what you’ll be dealing with.
None of them existed in their current form at the time PM was written, even logic (no Kripke semantics! no forcing! and no Gödel of course). Some did not yet exist at all. Some changed quite drastically in direct response to PM. And of course PM is the origin of (embryonic) type theory, which is the inspiration of the unified approach I referenced above. So as a historically important text, sure, if that’s what you want, but as a gateway to understanding more interesting maths it’d be terribly inefficient.
In that respect TAoCP was uniquely lucky. It was also a self-obsoleting book: it ceased to be comprehensive months after it was published, exactly because it told you everything there was to know about algorithms to date. Yet none of the stuff that’s in it is itself obsolete, there’s just immesurably more stuff now. PM, on the other hand, was attempt at “rationalization” in the 19th-century sense, and mostly a failed one except for serving as fertilizer for all of the later ones.