Stream: Miscellaneous

Topic: Do mathematicians really care about truth?


view this post on Zulip Karl Palmskog (Dec 08 2022 at 20:04):

if you want "true statement" or "truth", you'll probably end up having to define what that means (Tarski style?), which I don't think classical mathematicians want to do - are there really any non-logic math textbooks that even do this?

view this post on Zulip Paolo Giarrusso (Dec 08 2022 at 20:06):

These seem strange questions... I'd expect most mathematicians to use "true" as if they were Platonists.

view this post on Zulip Paolo Giarrusso (Dec 08 2022 at 20:08):

but let me emphasize "as if"

view this post on Zulip Paolo Giarrusso (Dec 08 2022 at 20:10):

Outside a logic class, I doubt even a formalist would object to "is it true or false that 2 + 2 = 4"

view this post on Zulip Karl Palmskog (Dec 08 2022 at 20:13):

my objection was to that the claim that when doing mathematics, people think they are (or are) figuring out whether some statement is true. I think we have both foundational and empirical arguments that this is not what happens

view this post on Zulip Paolo Giarrusso (Dec 08 2022 at 20:23):

I've skimmed the quote context (beginning of chapter 3), and it seems the text has just finished presenting some model theory (in Chapter 2) and is introducing proof theory (Chapter 3 "Proofs and Deductions")? Quoting more:

To do mathematics, we need to figure out whether our statements are true. In the last chapter,
we already defined what it means for a statement to be true. This definition, however, is hard to
work with in practice.

view this post on Zulip Paolo Giarrusso (Dec 08 2022 at 20:25):

at _that_ point, it might be unfair to nitpick "true" vs "provable" since the text hasn't introduced formal proofs or incompleteness.

view this post on Zulip Yannick Forster (Dec 08 2022 at 20:25):

I'd like to remark that this discussion is very interesting, but more or less entirely unrelated to the question whether you can learn proving and pen and paper proofs like this

view this post on Zulip Notification Bot (Dec 08 2022 at 20:27):

10 messages were moved here from #Miscellaneous > Not being able to write pen and paper proofs by Paolo Giarrusso.

view this post on Zulip Paolo Giarrusso (Dec 08 2022 at 20:28):

@Yannick Forster I've taken the liberty of splitting the discussion, even if I'm not confident about the new title

view this post on Zulip Paolo Giarrusso (Dec 08 2022 at 20:28):

(last time I asked somebody to split threads, people suggested I do it myself, and the split was very easy now)

view this post on Zulip Paolo Giarrusso (Dec 08 2022 at 20:31):

back from meta- to object-level discussion:
I almost agree with @Johannes Hostert — I think mathematicians mostly act as if mathematics is complete "almost everywhere", despite Gödel incompleteness.

view this post on Zulip Paolo Giarrusso (Dec 08 2022 at 20:34):

in part because most independent statements are "logic nonsense" not "real mathematical statements". By now we have exceptions, and people will need more precision to talk about those...

view this post on Zulip Patrick Nicodemus (Dec 09 2022 at 03:16):

I do not think that "true" or "truth" is a word which belongs to constructivism in particular

view this post on Zulip Patrick Nicodemus (Dec 09 2022 at 03:19):

Hilbert said that truth was freedom from contradiction but that was in 1899. I'm honestly not sure he fully understood the ramifications of that. Unsolvability of the entscheidungsproblem, main results of godel etc had not yet been discovered. Is there anyone prominent \ post 1950 that still thought "truth is freedom from contradiction"

view this post on Zulip Patrick Nicodemus (Dec 09 2022 at 03:24):

I think there are some constructivists who are happy to speak of "witnesses", "evidence", "proof", "constructions" etc, the notion of "true"/"truth" is not necessarily a core conceptual term in their paradigm

view this post on Zulip Karl Palmskog (Dec 17 2022 at 19:38):

too good a quote not to post here: https://twitter.com/HLForum/status/1572142089764503554

The panellists are asked if computer proof assistants like COQ/LEAN will influence mathematics? Carlos Kenig says, it will be a help to prove theorems, but will not generate ideas. 'Although it makes me nervous that they can win at Go!' #HLF22

- Heidelberg Laureate Forum (@HLForum)

Last updated: Jan 29 2023 at 18:03 UTC