Quote from book Yannick linked:
To do mathematics, we need to figure out whether our statements are true.
This sounds like constructivism to me. Isn't the usual classical view something like: "we need to ensure that our statements are not inconsistent"
I‘m not sure whether a classical mathematician would see a difference between „true“ and „not inconsistent“ statements.
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?
These seem strange questions... I'd expect most mathematicians to use "true" as if they were Platonists.
but let me emphasize "as if"
Outside a logic class, I doubt even a formalist would object to "is it true or false that 2 + 2 = 4"
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
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.
at _that_ point, it might be unfair to nitpick "true" vs "provable" since the text hasn't introduced formal proofs or incompleteness.
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
10 messages were moved here from #Miscellaneous > Not being able to write pen and paper proofs by Paolo Giarrusso.
@Yannick Forster I've taken the liberty of splitting the discussion, even if I'm not confident about the new title
(last time I asked somebody to split threads, people suggested I do it myself, and the split was very easy now)
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.
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...
I do not think that "true" or "truth" is a word which belongs to constructivism in particular
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"
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
too good a quote not to post here: https://twitter.com/HLForum/status/1572142089764503554
Last updated: Mar 02 2024 at 14:01 UTC