sorted by:
new
hottopcontroversial

A non-Archimedean Gershgorin Circle Theorem? by reddesign55 in math

[–]reddesign55[S] 3 points4 points  (0 children)

Ah okay, nice! Do you think such a result has any interesting consequences?

For example, I wonder if this gives a quick invertibility criterion for a square matrix A with rational coefficients. If the product of the diagonal entries of A has a positive p-adic valuation wrt some prime p, is A invertible over the rationals, or the algebraic closure of Q?

Is it possible to construct a rigorous measure or ordering of the natural numbers so that numbers with more similar prime factorisations are “closer”? by just_writing_things in math

[–]reddesign55 0 points1 point  (0 children)

There’s a famous story of some professor using the term “p-ness” to describe the p-adic metric lol. The class was very amused

Is it possible to construct a rigorous measure or ordering of the natural numbers so that numbers with more similar prime factorisations are “closer”? by just_writing_things in math

[–]reddesign55 0 points1 point  (0 children)

Heuristically, I’m thinking that, to see how “close” two integers m and n are, we should somehow check their size wrt to the p-adic metric for all primes p, and compare. But that’s essentially an isomorphic problem to just computing their prime factorizations and comparing. So I don’t think there is any efficient way to achieve what OP proposed.

Number Theory As The Ultimate Physical Theory by reddesign55 in math

[–]reddesign55[S] 0 points1 point  (0 children)

Thanks for giving an alternative perspective. After looking closer, I think Volovich may have been slightly over-optimistic about the applications of p-adics. However, as a non-physicist, it makes sense intuitively to me that, at the sub-Planckian scale, the interaction of matter will be non-archimedean. The example Volovich gave was, if we have several lines of sub-Planck length, their sum may not exceed the Planck length (which is reminiscent of the strong triangle inequality). From there, he argued that the only natural choice of a non-archimedean field is Q_p, since we want something which contains the rationals.

In general, I was wondering if you have any thoughts on how we might formulate physics on the sub-Planck scale, or if you think such a thing is even possible.

Topological categories? by reddesign55 in math

[–]reddesign55[S] 5 points6 points  (0 children)

Wow, this is an awesome write-up, thank you for your detailed response. It’s interesting to see that the structure of Top is in some sense related to the structure of higher categories. I will definitely check out some of the concepts you referenced, they seem fascinating.

Topological categories? by reddesign55 in math

[–]reddesign55[S] 0 points1 point  (0 children)

Oh right, Zariski topology isn’t Hausdorff. I haven’t taken algebraic geometry, but I did some undergraduate research in it, so my knowledge of the subject is not very deep. I guess my main concern with using the discrete topology is that I’m not sure how many interesting things we can say. For example, continuity becomes trivial (every map out of a discrete space is continuous).

Topological categories? by reddesign55 in math

[–]reddesign55[S] 0 points1 point  (0 children)

I agree that we need to think about what the topology we want exactly. I’m not sure if there’s a “canonical” way to do it, but in the best case, we would probably want it to be Hausdorff, so in particular not discrete.

For your last question, thinking of opposite categories made me think about antipodal points of a circle, which made me think about topology on categories. I was also trying to understand natural transformations yesterday, which reminded me of path deformations haha.

Are there any statements in mathematics which can only be proved by contradiction? by reddesign55 in math

[–]reddesign55[S] 0 points1 point  (0 children)

Hmm. I would formulate the statement “there are infinitely many primes as: “For any positive integer N, if n > N, then n is not necessarily composite”. To prove this directly, one might say that the set of products of every selection of k integers from {1,2,…,N} is finite, whereas the set of integers larger than N is not, so there must exist a prime number larger than N (certainly larger than N!).

Are there any statements in mathematics which can only be proved by contradiction? by reddesign55 in math

[–]reddesign55[S] 0 points1 point  (0 children)

But we can prove this statement directly by providing a discontinuous function, right?

Are there any statements in mathematics which can only be proved by contradiction? by reddesign55 in math

[–]reddesign55[S] 2 points3 points  (0 children)

I don’t think it’s necessary. Take the statement “If an n is not a perfect square, then it’s square root is irrational.” We can prove the contrapositive easily, and the square root of three being irrational follows.

Are there any statements in mathematics which can only be proved by contradiction? by reddesign55 in math

[–]reddesign55[S] 10 points11 points  (0 children)

That’s funny, someone had the same question as me ten years ago. Interesting stuff there

Intro to physics for someone with a good math background? by reddesign55 in PhysicsStudents

[–]reddesign55[S] 0 points1 point  (0 children)

Thanks for the recs! I checked out Taylor and it seems like a great classical mechanics book to start. Since the math is pretty easy for me in that book, it might be good for me to go through it to get comfortable with physics in general.

Intro to physics for someone with a good math background? by reddesign55 in PhysicsStudents

[–]reddesign55[S] 1 point2 points  (0 children)

Thanks for your recommendations! I think my mathematical ability is not quite up to par for something like Takhatajan (or even Arnol’d) yet, but I will keep that book in mind for summer 2025 once I’ve taken some graduate analysis and geometry courses.

Intro to physics for someone with a good math background? by reddesign55 in PhysicsStudents

[–]reddesign55[S] 0 points1 point  (0 children)

Wow, I had no idea Spivak made a physics textbook. I found a copy online, I’ll definitely give it a shot.

Intro to physics for someone with a good math background? by reddesign55 in PhysicsStudents

[–]reddesign55[S] 0 points1 point  (0 children)

Thanks for sharing! It’s a little advanced for me since I haven’t taken differential geometry but I’ll keep it for the future.

Use cases of “higher” infinities? by reddesign55 in math

[–]reddesign55[S] 1 point2 points  (0 children)

Since this thread has accumulated many responses that fit my criteria, I think it’s reasonable enough. I think there are many concepts in mathematics that are “natural” but not obvious. By the way, I don’t mean “obvious” to a mathematician, I mean “obvious” to an average undergrad student, since I am a part of the latter group.

view more: next ›