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Results tagged with mathematical-philosophy
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user 3106
Philosophical aspects of logic and set theory; truth status of mathematical axioms; Philosophy of Mathematics; philosophical aspects of mathematics in general; relation of mathematics to philosophy; etc. Consider also posting at http://philosophy.stackexchange.com/, where philosophy-of-mathematics is one of the most popular tags.
96
votes
What if Current Foundations of Mathematics are Inconsistent?
Contrary to popular opinion, there is no single foundation for mathematics. Probably you're referring to ZF or ZFC, but most mathematics can be developed on the basis of axioms that are logically muc …
58
votes
Accepted
Why is integer factoring hard while determining whether an integer is prime easy?
What I think you're asking for are examples of search problems that seem to be hard, while a corresponding decision problem is solvable in polynomial time (but not totally trivial). It is true that s …
- 82.7k
51
votes
Request for examples: verifying vs understanding proofs
Don Zagier has a well-known paper, A one-sentence proof that every prime $p\equiv 1\pmod 4$ is a sum of two squares. An undergraduate mathematics major should be able to verify that this proof is cor …
49
votes
Is PA consistent? do we know it?
EDIT: I have written a paper that greatly expands on my answer here, and that in particular contains sketches of Gentzen's proof and Friedman's proof, as well as a discussion of formalism.
I have alr …
47
votes
In what respect are univalent foundations "better" than set theory?
This is a question that has been discussed a lot on the Foundations of Mathematics mailing list (unfortunately with more polemics than necessary IMO—though I confess that I may have been guilty of sto …
46
votes
Logic in mathematics and philosophy
I agree with the commentators that the question is rather too broad, but here's an attempt to answer it anyway.
Readers of MO will likely have less familiarity with non-mathematical logic, so it might …
- 82.7k
45
votes
Proofs of theorems that proved more or deeper results than what was first supposed or stated...
The example given by Wojowu in the comments seems worth posting as an answer.
In the NOVA special The Proof, Ken Ribet says the following.
I saw Barry Mazur on the campus, and I said, "Let's go for a …
44
votes
The enigmatic complexity of number theory
What references, especially books, have been devoted to specifically addressing the source of the deep roots of the diversity and complexity of number theory?
To a first approximation, I would sa …
37
votes
Axiom of choice, Banach-Tarski and reality
The other answers don't seem to have said much about why the axiom of choice is widely regarded as plausible. Let me try to address that question.
First let's dispose of some non-reasons. In respon …
- 82.7k
34
votes
Metamathematics of buts
In a paper entitled "Contrastive Logic" (Logic Journal of the IGPL 3 (1995), 725–744), Nissim Francez introduced something he called bilogics, which are logics intepreted over a pair of structures ins …
- 82.7k
33
votes
Has philosophy ever clarified mathematics?
In order to address this question, I think it is important to first take a step back and examine with a critical eye something that we normally take for granted, namely the professionalization and com …
32
votes
Are there any fields of academic mathematics whose epistemic status as math is controversial...
There are several possible dimensions to the question, "Is it math?"
Does it belong in the mathematics department? I think you mostly want to exclude this dimension, because of your comment about pur …
30
votes
Contemporary philosophy of mathematics
It's not entirely clear to me what you mean by a "position." Under the most obvious interpretation, things like Platonism, logicism, formalism, intuitionism, finitism, etc., are "positions." However …
30
votes
Request for examples: verifying vs understanding proofs
Ivan Niven has published A simple proof that $\pi$ is irrational. Verifying that the proof is correct requires only elementary calculus. On the other hand, to "understand" it, a professional mathema …
29
votes
Accepted
Mathematical fictionalism
As I suggested in response to a related MO question, one difficulty with answering this type of question is that most mathematicians outside of logic and set theory lack well-developed "positions" on …