This s lovely stuff, Jon, above my understanding and beyond my reach to learn in my current circumstances. Thank you for both.
I know Fotini distantly, from brief overlap at SFI; I didn’t understand that this was the particular thing she had done, though I knew this was the general area of her work. I have also been able to talk to her about why she left professional math to do design. It is not the saddest disappointment in what our culture should offer people and sometimes fails to, but it is a contender. I am currently watching a debate or learning session, between a dutch philosopher and mathematical logician who specializes in intuitionism, and a younger mathematician (maybe from MIT?, currently working in documentary film!) who knows category theory well, and some philosophy of math, and is trying to learn in the conversation how intuitionism fits into the landscape. I don’t use names because I don’t know whether the existence of the exchange should be left as a private correspondence protected from traffic analysis. But the positions are interesting. The younger cat-theorist, who is reading philosophy of math, presents a picture much like the one you describe, with pluralism of several dimensions and no strong attachments. The dutchman asserts that there are ongoing interests in what we want from notions of truth, and holds that the formalist/intuitionist polarity is one of the more important ones on that question. The idea that there is no “winning strategy”, in a Jaako Hintikka-sense, is what interests me, as something illuminating about our aspiration for a truth-notion, and how perhaps inadequately we have been able to pin one down after millennia of quite sophisticated efforts. That is why I expect the formalist-constructivist dialogue on the psychology topics to be persistent. Both discussants in the math conversation seem to agree that, in some sense, the formalists didn’t declare a full victory, but at most a severely qualified one. The incompleteness theorem ended the Hilbertian hope for a self-contained formalist program, and they both seem to agree (I have no knowledge or background to say myself) that even the formalists came to some degree to admit that there were sectors of their reasoning that did appeal to a kind of demonstrative semantics of the kind that intuitionists pin a lot on for number theory of finite numbers. My witness of this exchange lies behind my earlier remarks. I wish I had the mind to understand the issues for myself. Eric > On May 20, 2020, at 10:39 AM, Jon Zingale <[email protected]> wrote: > > EricS, > > You write: > I bring up this debate in mathematics because it seems significant to me > how long and how intensely it has been going on, with both sides wanting a > notion of “truth”, and neither being able to claim to have achieved it in > terms > satisfied by the other. If the intuitionists had never been able to build a > real system around their position, the formalists could just declare victory > and go home. But the debate seems still live, even within math and not only > in > philosophy, with clear trade-offs that there are proofs that each side will > accept that the other rejects. > > It would surprise me to meet a mathematician who feels intensely one way > or an other about a particular choice of topos. For mathematical-logicians, > what seems more interesting are the geometric morphisms between toposes. > I would argue that the formalists to some extent did just declare victory > many times over and that their are still pockets of scientific/mathematical > culture that believe everything can be reduced to bits. Still, and not just as > with the intuitionists, richer toposes are there to be found and explored. > > My two favorite examples come from algebraic geometry and from > quantum cosmology. In the former case, Grothendieck arrives at the > idea of a non-boolean topos while writing the foundations of algebraic > geometry. In the latter, Fontini Markopoulou-Kalamara > <https://en.wikipedia.org/wiki/Fotini_Markopoulou-Kalamara> develops her > non-boolean topos in the context of quantum gravity†. > > Jon > > †) Tangentially related to other parts of the overall discussion, Fotini > is also a design engineer working on embodied cognition technologies. > -- --- .-. . .-.. --- -.-. -.- ... -..-. .- .-. . -..-. - .... . -..-. . ... > ... . -. - .. .- .-.. -..-. .-- --- .-. -.- . .-. ... > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > archives: http://friam.471366.n2.nabble.com/ > FRIAM-COMIC http://friam-comic.blogspot.com/
-- --- .-. . .-.. --- -.-. -.- ... -..-. .- .-. . -..-. - .... . -..-. . ... ... . -. - .. .- .-.. -..-. .-- --- .-. -.- . .-. ... FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com archives: http://friam.471366.n2.nabble.com/ FRIAM-COMIC http://friam-comic.blogspot.com/
