Timeline for Circular, or missing, definition in set theory?
Current License: CC BY-SA 4.0
22 events
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May 25, 2018 at 7:26 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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May 24, 2018 at 13:38 | comment | added | Mikhail Katz | OK. Note that my edit did not change the substance of my answer but merely clarified it (the earlier version did not make it clear what "this" was exactly). @ToddTrimble | |
May 24, 2018 at 13:37 | comment | added | Todd Trimble | Okay, in view of your edit, let's leave it at that. | |
May 24, 2018 at 13:21 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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May 24, 2018 at 13:20 | comment | added | Mikhail Katz | ...as his suspicion of the standard dichotomies, namely the menace stemming from the idea that we are merely working with theories which unavoidably have distinct models. Such an idea is a challenge to a monolithic philosophy of mathematics which was Halmos'. @ToddTrimble | |
May 24, 2018 at 13:18 | comment | added | Mikhail Katz | I never claimed that Halmos did not understand these standard dichotomies. What I did claim is that Halmos never overcame his distrust of these dichotomies, and consistently felt that these dichotomies were vague. I would have to disagree with your claim that the business about NSA is a separate matter. As I mentioned in my answer, he is still pouncing on non-standard models in one of his last published articles, and if you look at the context you will probably agree with me that the context does not justify the pouncing. His suspicious attitude toward NSA has the same source... | |
May 24, 2018 at 12:25 | comment | added | Todd Trimble | Meanwhile, the business about NSA is a matter separate from whether he had great difficulty with the dichotomies you speak of -- given that the thread is about understanding those dichotomies, the suggestion seems to be that Halmos didn't understand them, and that I would find too strong to be supported by the evidence. I repeat that it might be right that he was impatient with standard explanations, and I would direct you to the analogy with group theory which I think is quite apt, especially in the advent of Lawvere's thesis which was very clarifying. | |
May 24, 2018 at 12:20 | comment | added | Todd Trimble | and word reductions and substitutions based on relators and whatnot. So I think he's finding standard presentations of logic (propositional, predicate, etc.) as similarly needlessly fussy and complicated, and felt a desire to determine the real algebraic essence of logic (hence, algebraic logic). And I happen to think he's got a good point. (He may have well also felt a strong pedagogical impulse to share what he found with others.) Again I think that it's too strong to claim that he never did understand theory/model distinctions -- I think he likely understood them well after his struggles. | |
May 24, 2018 at 12:14 | comment | added | Todd Trimble | I never made the claim you say I did, so I don't feel compelled to offer you a source for precisely that. My own reading (readers can draw their own conclusions from some relevant passages here: pdfs.semanticscholar.org/f269/…) is that he felt impatience with standard presentations, and I think he explains it well in terms of an analogy: imagine that groups were typically introduced, not as sets with an operation obeying simple axioms, but in terms of presentations (generators and relations), prefaced by a syntactic discussion of words (cont.) | |
May 24, 2018 at 9:13 | comment | added | Mikhail Katz | ...toward Robinson's framework. This included false claims made in his "automathography" about the dating of his first encounter with Robinson's paper on invariant subspaces, which we also document in our article. @ToddTrimble | |
May 24, 2018 at 8:59 | comment | added | Mikhail Katz | Halmos clearly and repeatedly states that he was suspicious of the type of dichotomies I mentioned. Do you have a source for claiming that he developed things like cylindrical algebras to help explain logic to mathematicians? From what I have seen, his motivation stemmed from his discomfort with the dichotomies standard for a logician, which motivated his attempt at algebraization, more than pedagogical concerns. In one of his last articles, he still includes a dig against "non-standard models" which in my mind is a sign of philistinism and intolerance, which was in particular his attitude.. | |
May 24, 2018 at 1:42 | comment | added | Todd Trimble | that's a sound mathematical impulse. Cf. this interview with Lawvere, mat.uc.pt/~picado/lawvere/interview.pdf, page 20, where he says "What is the primary tool for such summing up of the essence of ongoing mathematics? Algebra!" But to return to the answer: it's not at all clear to me that Halmos did not understand the theory/model distinctions; on the contrary, I think he tried hard to understand those types of things better, by mathematizing them in terms of algebraic structures -- this was a motif in one era of his professional life. | |
May 24, 2018 at 1:35 | comment | added | Todd Trimble | could be treated on the same playing field. It seems to me that Halmos was trying to extend his insight about propositional calculus as about free Boolean algebras by algebraizing first-order logic in terms of cylindric or polyadic algebras. Now it may be true that cylindric or polyadic algebras isn't the most flexible or convenient formalism for this program -- I happen to believe Lawvere's hyperdoctrines are much better, by allowing a multi-typed, not a single typed algebraic signature. But in any case the same algebraizing impulse guided much of Halmos's logical work, and (cont.) | |
May 24, 2018 at 1:28 | comment | added | Todd Trimble | My own feeling is that while your co-authored article is interesting and makes some good points, on this particular conclusion about Halmos I think there is overreaching. Halmos in his "automathography" (which you liberally draw on) reports how the scales fell from his eyes where he answered, "What is the propositional calculus?" with "the theory of free Boolean algebras" (p. 206) -- the same kind of insight was developed most penetratingly later by Lawvere who clearly realized how syntax was concentrated in free structures, and thus "theory" (e.g. a Lawvere theory) and "model" really (cont.) | |
May 23, 2018 at 11:56 | comment | added | Mikhail Katz | See this for arxiv version as well as mathscinet link. | |
May 23, 2018 at 11:50 | comment | added | Todd Trimble | Thanks; unfortunately it's behind a paywall. The abstract is certainly provocative and intriguing! | |
May 23, 2018 at 8:54 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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May 23, 2018 at 8:44 | comment | added | Mikhail Katz | See the article linked in the answer. @ToddTrimble | |
May 23, 2018 at 8:43 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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May 23, 2018 at 2:06 | comment | added | Todd Trimble | Can you give grounds for your belief about Halmos? | |
May 22, 2018 at 11:57 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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May 22, 2018 at 11:52 | history | answered | Mikhail Katz | CC BY-SA 4.0 |