Translation Notice: The Korean version of this resource has been translated with the assistance of Gemini. As a result, some phrasing might feel unnatural. We actively welcome improvements and corrections! Please refer to our CONTRIBUTING.md guide to see how you can contribute to making this resource better.
Assuming this means the Korean is the original, I'm curious how much effort was put into the Korean text and whether it reads naturally and maintains thematic unity. A serious book on this subject matter would fill a valuable need right now
Wittgenstein didn't find logical flaws in the Principia and deeply admired it. He found flaws in Russell's follow up work on Epistemology, "The Theory of Knowledge."
This is commonly believed, but Gödel didn't identify a logical error at the heart of the whole enterprise, he proved astonishing theorems revealing limitations of any sufficiently powerful formal system. One can kind of think of the Principia as a science experiment to find the extent to which known mathematics could be proven from foundational axioms that could be thought of as "laws of logic". To make their system work, Russell and Whitehead themselves had to add extralogical axioms, such as their Axiom of Reducibility [0] and the Axiom of Infinity, giving empirical evidence (but not a proof) that "laws of logic" alone were not enough. They were also aware of limitations in their own system, such as the inability to define the cardinal $\aleph_\omega$ [1].
Like the article says, what they did was ahead-of-its-time, and a monumental influence on all subsequent work on formal systems, including Gödel's work, regardless of whether Russell and Whitehead achieved their initial aims.
The flaw was the belief that it was possible (though they were not sure that they two could pull it off). Gödel showed that it was not an impossibly difficult task but an actually impossible task.
It cannot be overstated how flawed your own statements here are. You shouldn't be talking about logical error regarding a work of logic when you don't even know enough logic to know what a logical error is.
And there was no "flaw" in attempting to resolve foundational paradoxes in set theory, primarily to address Frege's error and deal with Russell's paradox, or in attempting to prove that all mathematics can be derived from pure logic. Seeking to demonstrate a hypothesis that is later shown to be erroneous is not "flawed", else the entire knowledge-seeking enterprise is "flawed".
And switching from "logical error" to the very vague notion of a "flaw" is goalpost moving that looks a lot like bad faith.
A failed hypothesis is not the same thing as a logical error. Would they have been more free of flaws if they believed it was impossible and thus didn't try? We'd all be the poorer for it
Tao does state his hopes in the article: "My hope is that the winning submissions will capture the most productive techniques for solving these problems, and/or provide general problem-solving techniques that would also be applicable to other types of mathematical problems."
I think your suggestions are actually complementary. Distillation of the larger networks capable of solving these problems and study of the layers could be part of the process for generating the cheat sheet.
It seems like the lift in the open-source models is being used as a proxy metric, and the core goal is a human understandable yoga [1] for approaching these kinds of equational proofs in universal algebra.
I spent some time in industry working on ML-based credit risk modeling. In my experience, successful shops that have a genuine interest in applying their models to practical decision making with real stakes care deeply about uncertainty quantification and decision theory. Things can get messy very fast though and the challenges faced are often too hyper-specific to one's situation to make sense as part of an academic research program. I think it's been for the best that academic research has tended to focus on the development of general algorithms intended to be broadly useful. Businesses are already well incentivized to take the best of what academia produces and try to get the decision theory right for their particular problems.
Yes, people try this. Check out dynamic tonality. It doesn't necessarily need a system. Experienced guitar players often find themselves unconsciously making little microtonal adjustments through bends and other techniques when playing leads. I found myself doing this just because it sounded better to me. I didn't even notice there was a consistent pattern until I eventually learned the math. For example I'd always want to bend minor thirds slightly sharp and bend the neck to slightly detune major thirds.
I was planning to make a similar comment. Conjecturing that some theory in the string theory landscape [0] gives a theory of quantum gravity consistent with experiments that are possible but beyond what humans may ever be capable of isn't as strong of a claim as it may first appear. The intuition I used to have was that string theory is making ridiculously specific claims about things that may remain always unobservable to humans. But the idea is not that experiments of unimaginable scale and complexity might reveal that the universe is made up of strings or something, it's just that it may turn out that string theory makes up such a rich and flexible family of theories that it could be tuned to the observed physics of some unimaginably advanced civilization. My impression is that string theory is not so flexible that its uninteresting though. There's some interesting theoretical work along these lines around exploring the swampland [1].
I'd say that I care deeply about the meaning behind theorems, but just find results which swing widely based on foundational quirks to be less interesting from an aesthetic standpoint. I see the most interesting structures as the ones that are preserved across different reasonable foundations. This is speaking as someone who was trained as a pure mathematician, moved on to other things, but tries to keep up with pure math as a hobby.
Yes, but most mathematicians do not seem to make this distinction between sturdy and flimsy truths. Which puzzles me. Are they unaware? If so, would they care if educated? Or do they fully commit to classical logic and the axiom of choice if pushed? I can see it go either way, depending on the psychology of the individual mathematician.
I don't think they usually make the distinction in a formal sense, but I think most are aware. The space of explorable mathematics is vastly larger than what the community of mathematicians is capable of collectively thinking about, so a lot of aesthetic judgment goes into deciding what is and what isn't interesting to work on. Mathematicians differ in their tastes too. A sense of sturdiness vs flimsiness is something that might inform this aesthetic judgment, but isn't really something most mathematicians would make part of the mathematics. Often, ones interest isn't the result itself, but some proof technique that brings some sense of insight and understanding, and exploring that often doesn't make much contact with foundational matters.
No one not working on foundations has any problem with axiom of choice. It has weird implications but so what? Banach Tarski just means physical shapes aren't arbitrarily subdividable.
My 2 cents is they do justify it by the interest of the consequences, as Tychonoff or Nullstellensatz. I wouldn't call that faith: Best practices is to state Tychonoff as "AC implies Tychonoff" and that last is logically valid. Sometimes the "AC implies..." is missing, buried in the proof or used unawaredly or predates ZFC, and is a bad thing. But very ofen one now see asterisks on theorems needing it.
AC makes things much easier as it allows to play God powers. Negating AC is not significantly different from constructing mathematics that avoids AC (no assumption about validity of AC). And that makes things way harder with longer proofs and only in sub-cases of classical theorems.
Simply assuming the negation of AC is boring, as negations often are. But there are stronger statements, implying the negation of AC which might be as useful. I think for instance one could assume all subsets of the plane to be measurable. Seems convenient to me.
Same with law of the excluded middle. Tossing it out we can assume all functions are computable and all total functions in the real are continuous. Seems nice and convenient too!
Vladimir Arnold famously taught a proof of the insolubility of the Quintic to Moscow Highschool students in the 1960s using a concrete, low-prerequisite approach. His lectures were turned into a book Abel’s Theorem
in Problems and Solutions by V.B. Alekseev which is available online here: https://webhomes.maths.ed.ac.uk/~v1ranick/papers/abel.pdf. He doesn't consider Galois theory in full generality, but instead gives a more concrete topological/geometric treatment. For anyone who wants to get a good grip on the insolubility of the quintic, but feels overwhelmed by the abstraction of modern algebra, I think this would be a good place to start.
Looks like a nice book, but what's up with his assertion on page 148 (164 of the .pdf) that the integers don't form a group under addition?
If he defines integers as "natural numbers excluding zero," that seems goofy and nonstandard but also interesting. Is that a Russian-specific convention?
It seems like a typo where "integers" is used when the intention was to write "natural numbers". That is the solution to exercise 194 part a) which asked if the set of natural numbers is a field.
Whether 0 is a natural number is still fairly ambiguous; I remember being taught (1990s UK) to be specific about which definition was being used, or to prefer another name such as 'positive integers' or 'non-negative integers'
Translation Notice: The Korean version of this resource has been translated with the assistance of Gemini. As a result, some phrasing might feel unnatural. We actively welcome improvements and corrections! Please refer to our CONTRIBUTING.md guide to see how you can contribute to making this resource better.
Assuming this means the Korean is the original, I'm curious how much effort was put into the Korean text and whether it reads naturally and maintains thematic unity. A serious book on this subject matter would fill a valuable need right now
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