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John Baez (Oct 04 2020 at 21:52):Talking to Colin McLarty just now I learned he's writing two books:
Nikolaj Kuntner (Oct 05 2020 at 13:54):Very nice. Any hint when?
John Baez (Oct 05 2020 at 15:58):A few years before they show up, I guess.
Nikolaj Kuntner (Oct 05 2020 at 16:37):Just yesterday I was asked what philosophies of mathematics are championed currently. I could point to some struggles tying to foundations and practice (and e.g. pointing towards some more structuralist conceptions of the notions of equality), but I feel those only indirectly tie to ontological/epistemological concepts. So yes, there's attitudes and all that comes with it (e.g. type-theory-grown-up folks will naturally live in another math world than, say, Hairer with his stochastic analysis research). But I'm not sure if there's many streams of thought - people that would write about their position explicitly in an argumentative way - among those that do the math. Like Weyl did, say. And the "old" terms (constructivists, Platonists etc.) still seem to be those words of classifications that apply, people still take on the roles as fleshed out 100 years ago?
John Baez (Oct 05 2020 at 16:43):McLarty told me it was Felix Klein who introduced the formalist/logicist/intuitionist distinction in his speech at the World's Fair in Chicago in 1893. (I always imagine him lecturing about mathematics in the midst of ferris wheels, but it seems he gave his speech at the University of Chicago.)
But he considered them types of mathematicians, not philosophies.
John Baez (Oct 05 2020 at 16:46):He starts:
Among mathematicians in general, three main categories may be distinguished; and perhaps the names logicians, formalists, and intuitionists may serve to characterize them.
John Baez (Oct 05 2020 at 16:52):If someone wants to see current issues in the philosophy of mathematics, you might point them to Elaine Landry's edited volume, Categories for the Working Philosopher.
John Baez (Oct 05 2020 at 16:53):I also like David Corfield's book, Towards a Philosophy of Real Mathematics.
John Baez (Oct 05 2020 at 16:54):I think it's really important for philosophers of mathematics to talk about new questions, not just the same old ones.
Reid Barton (Oct 05 2020 at 16:57):John Baez said:
He starts:
Among mathematicians in general, three main categories may be distinguished; and perhaps the names logicians, formalists, and intuitionists may serve to characterize them.
I take it that one or more of these words was used with a different meaning than we might give it today.
John Baez (Oct 05 2020 at 17:03):Probably 'logicians' are a bit like what we call 'logicists', but we could easily read what Klein has to say about it. I haven't read his speech. I should, because I like Klein a lot.
John Baez (Oct 05 2020 at 17:09):Well, it's easy to find:
Among mathematicians in general, three main categories may be distinguished; and perhaps the names logicians, formalists, and intuitionists may serve to characterize them. (1) The word logician is here used, of course, without reference to the mathematical logic of Boole, Peirce, etc.; it is only intended to indicate that the main strength of
the men belonging to this class lies in their logical and critical power, in their ability to give strict definitions, and to derive rigid deduction therefrom. The great and wholesome influence exerted in Germany by Weierstrass in this direction is well known. (2) The formalists among the mathematicians excel mainly in the skilful formal treatment of a
given question, in devising for it an "algorithm." Gordan, or let us say Cayley and Sylvester, must be ranged in this group. (3) To the intuitionists, finally, belong those who lay particular stress on geometrical intuition (Anschauung), not in pure geometry only, but in all branches of mathematics. What Benjamin Peirce has called "geometrizing a mathematical question" seems to express the same idea. Lord Kelvin and von Staudt may be mentioned as types of this category.
John Baez (Oct 05 2020 at 17:10):So, all three of the classifications are actually quite different from the logicists, formalists and intuitionists of later mathematical philosophy!
John Baez (Oct 05 2020 at 17:11):In The Man Who Knew Infinity, the biographer calls Ramanujan a "formalist" - meaning he was very good at inventing formulas. This is a bit like what Klein means by "formalist", though Klein actually says algorithms.
James Wood (Oct 05 2020 at 17:45):The formalist/logician distinction sounds rather like the computer science theory A/B distinction we make these days. I wonder to what extent that's a fair analogy.
Nikolaj Kuntner (Oct 05 2020 at 18:45):Well I think you don't need to know much logic to be a formalist about mathematical objects.
But John moreover just threw logicists into the mix, which I understand to relate to positivists.
John Baez (Oct 05 2020 at 20:20):Logicists like Russell and Whitehead aimed to found mathematics on just "logic". It led to an argument over where "logic" leaves off and where "set theory" starts. For example, to what extent are sets just a way of dealing with predicates? Quine had some strong opinions on what counts as "logic" and why second-order logic is set theory masquerading as logic.
Nikolaj Kuntner (Oct 05 2020 at 23:07):I tend to view set theories as as individually fine tuned higher-order logic as specified (by the axioms of ones desire) from a first-order vantage point.
David Michael Roberts (Oct 06 2020 at 06:51):I'm hoping McLarty is still going to release the paper based on his 2018 ASL talk "Class field theory in exponential function arithmetic (EFA)" ....
David Michael Roberts (Oct 06 2020 at 06:57):FWIW, here's the abstract:
Class field theory was the center of algebraic number theory from Dedekind and Hilbert right up until it led to, and merged with, current cohomological methods. While it is widely considered ‘abstract’ in the sense of being difficult to beginners to relate to arithmetic intuition, and many of its tools look to a naive glance to be third order arithmetic, it is logically quite concrete. This talk presents work on formalizing classical results and methods of class field theory in Exponential Function Arithmetic. This is so weak it lies below the thresh-old of current Reverse Mathematics. So this talk contributes to Harvey Friedman’s Grand Conjecture, saying all mainstream concrete mathematics lies inside EFA, and also to Angus Macintyre’s saying that (full) “Peano Arithmetic is far too strong for mathematics.”
John Baez (Oct 06 2020 at 16:32):I'll try to remember to ask him about that sometime.