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Callan McGill (Sep 28 2020 at 22:55):If I am remembering correctly there is an old email from Lawvere on the FOM mailing list about the meaning of a foundations. Does anyone happen to know the email and have a link to it?
David Michael Roberts (Sep 29 2020 at 00:11):@Callan McGill how old? Searching here (1994-1999): https://www.mta.ca/~cat-dist/catlist/ I find only one topic that mentions 'foundation'. From 2000 onward one can maybe try some google-foo on these pages: https://www.mta.ca/~cat-dist/archive/ .
Or else one can delve through https://github.com/vcvpaiva/categories-mailing-
Callan McGill (Sep 29 2020 at 01:42):@David Michael Roberts Thank you, I might be misremembering. I recall Lawvere talking about the different meanings of foundations, one being something we can in principle encode another theory within and a different notion being one which can be the foundations for how we think about something.
David Michael Roberts (Sep 29 2020 at 01:48):With sufficient skill, one could extract all the emails by Lawvere, and then scour those, but sadly I do not have that skill. I'm afraid. Searching my own inbox, I think the message would have to be pre-October 2011.
Jonas Frey (Sep 29 2020 at 01:50):@David Michael Roberts I think the original question referred to the FOM mailing list, not the categories mailing list
Callan McGill (Sep 29 2020 at 01:51):I was just hoping I could job the memory of someone who read the same thing.
David Michael Roberts (Sep 29 2020 at 01:53):@Jonas Frey how did I miss that! :-/
David Michael Roberts (Sep 29 2020 at 01:55):I don't have the patience to wade back though the mess that was the flame war in which Lawvere's name arose many times.
David Michael Roberts (Sep 29 2020 at 01:56):I just read one message of Friedman that did not do my blood pressure any favours, I'm not looking at more.
Jonas Frey (Sep 29 2020 at 01:59):I never read the FOM list, but I heard they also had a great discussion about Voevodsky's thoughts on consistency :-D
Jonas Frey (Sep 29 2020 at 02:01):But if Lawvere ever posted there, I'd be also interested in reading that
Nikolaj Kuntner (Sep 29 2020 at 10:48):I never read FOM either, which gives me FOMO.
Fawzi Hreiki (Sep 29 2020 at 10:55):I don’t know about the FOM list, but the appendix to Sets for Mathematics has some interesting ideas on what ‘foundations’ and ‘logic’ mean. Likewise, the intro to the Perugia Notes (which you can find online) has some comments on what he means by ‘axiomatic method’
Fawzi Hreiki (Sep 29 2020 at 10:56):Also, despite the target audience being high-school and first year university students, his book Conceptual Mathematics also has some pretty radical ideas about the philosophy of math
Robert Seely (Sep 29 2020 at 21:07):Callan McGill said:
If I am remembering correctly there is an old email from Lawvere on the FOM mailing list about the meaning of a foundations. Does anyone happen to know the email and have a link to it?
Not an entirely satisfactory reply, but ... there is an archive of FOM postings, arranged by month, and within each month, by author (and by thread, subject , and date): https://cs.nyu.edu/pipermail/fom/
I also recall such a post, although to be honest, I recall perhaps more clearly, the "noise" created by those who disagreed that category theory had anything to contribute to FOM at all (!). I don't have the energy to go through these posts, month by month, but if you have that, best of luck! (Of course, if his post was before Sept 1997, it's possible you won't find it ... I no longer recall when FOM started.)
Fawzi Hreiki (Sep 29 2020 at 21:31):https://www.google.com/search?q=lawvere+site:cs.nyu.edu this seems to work
David Michael Roberts (Sep 29 2020 at 22:30):Sadly, I can't see an email from Lawvere in those results (I checked all the pages, not too many to do so!)
David Michael Roberts (Sep 29 2020 at 22:32):@Nikolaj Kuntner you aren't missing out. Basically, in the late 90s Harvey Friedman (together with his posse) takes every possible opportunity to insult category theory and category/topos theorists over a massive multi-thread discussion that lasts for more than a year. Colin McLarty and others tried to patiently set out the mathematical facts and answer bad-faith questions for a long, long time, with no results.
Nikolaj Kuntner (Sep 29 2020 at 22:57):Sounds horrible. But I might enjoy the drama, haha.
John Baez (Sep 29 2020 at 23:00):Or you can just watch the US presidential debate tonight.
Nikolaj Kuntner (Sep 29 2020 at 23:04):It's in 2 hours, right? 3am Vienna time. While I got into the habit of going to bed at that time, I'd not want to sit trough it I'm afraid.
John Baez (Sep 29 2020 at 23:06):I forget when it is. I'm not sure if I'll watch it.
I definitely enjoyed reading those old FOM posts. They're great if you like categorical logic and want to feel superior to set theorists who don't understand what categorical logic is for.
Fawzi Hreiki (Sep 29 2020 at 23:31):I think theres just a fundamental incongruence between how people here and on the nLab and so on use the term 'foundations' and how those at FOM use it.
Dan Doel (Sep 29 2020 at 23:43):I think the incongruence extends beyond nlab. :smile:
Fawzi Hreiki (Sep 29 2020 at 23:51):I'd imagine that most mathematicans don't think that their work relies at all on how it is coded in ZFC, or any other formal system for that matter.
Dan Doel (Sep 29 2020 at 23:52):Not just mathematicians, either. ZFC is useless to me.
Dan Doel (Sep 29 2020 at 23:52):The axiom of choice doesn't compute, so I can't use it as part of calculating what web page to display.
Nikolaj Kuntner (Sep 30 2020 at 00:42):Restrict everything else and then choice isn't so bad :^)
Dan Doel (Sep 30 2020 at 01:20):Well, that can be true, but mathematicians will point out that the usual, acceptable way to have it makes it kind of silly. The 'axiom of pre-chosen choice.' When you get to something like HoTT that can state it in an interesting way, it becomes a problem again.
Max New (Feb 23 2022 at 17:36):Resurrecting this old thread, it's not Lawvere, but what you describe sounds a lot like section 1 of Paul Taylor's "Foundations for Computable Topology": http://www.paultaylor.eu/ASD/foufct.pdf
Todd Trimble (Feb 28 2022 at 21:28):I've only just seen this thread. Here is an interview where Lawvere explains what he means by "foundations" (which is opposed to what he calls "speculative foundations" which considers that math is founded on ZFC, etc.). For Lawvere, "foundations" encompasses the giving of basic principles which clarifies how mathematics is actually practiced, or that concentrates its essence in judicious axiomatics. See especially around pages 20-21.