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Results tagged with gm.general-mathematics
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user 766
Questions about mathematics which don't fall into the other arXiv categories. If you have a general question about mathematics but it is not research level, it's off-topic but it might be welcomed on Mathematics Stack Exchange.
94
votes
Accepted
Are there proofs that you feel you did not "understand" for a long time?
As an undergraduate, I learned the Sylow theorems in my algebra classes but could never retain either the statement or proof of these theorems in memory except for short periods of time (and in partic …
86
votes
Mathematical habits of thought and action which would be of use to non-mathematicians
Here are some that came to mind:
Equivalence. Basically, the idea that two things can be functionally equivalent (or close to equivalent) even if they look very different (and conversely, that two t …
73
votes
Still Difficult After All These Years
Difficulty is not additive, and measuring the difficulty of proving a single result is not a good measure of the difficulty of understanding the body of work in a given field as a whole.
Suppose for …
63
votes
Accepted
Are there any fields of academic mathematics whose epistemic status as math is controversial...
There are some speculative mathematical concepts that come to mind, such as the field of one element or motives, though perhaps these are more classifiable as "potential future mathematics" rather tha …
62
votes
Accepted
Why is the Gaussian so pervasive in mathematics?
Quadratic (or bilinear) forms appear naturally throughout mathematics, for instance via inner product structures, or via dualisation of a linear transformation, or via Taylor expansion around the line …
- 114k
58
votes
Accepted
Why do people use "formal calculation" to describe informal calculations?
Formal, adj. Relating to or involving outward form or structure, often in contrast to content or meaning.
In mathematics, a formal argument is one that manipulates the form of an expression with …
57
votes
What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?
This is a bit speculative, and perhaps too challenging for an undergraduate project, but I wonder if an AlphaGeometry type approach might be possible for the task of automatically upper bounding sums …
- 114k
26
votes
Oddities of evenness
The hairy ball theorem is only valid for even-dimensional spheres (or odd-dimensional ambient Euclidean spaces).
Similarly, the strong Huygens principle is only valid in odd-dimensional physical space …
21
votes
Demonstrating that rigour is important
One can rigorously prove that pyramid schemes cannot run forever, and that no betting system with finite monetary reserves can guarantee a profit from a martingale or submartingale.
But there are cou …
19
votes
Where can square roots come from when they are not distances?
The Dirichlet-to-Neumann map on a half-space is the square root of the (positive definite) Laplacian on the boundary of the half-space (see the previous MathOverflow question Characterisation of the s …
13
votes
What are some examples of "chimeras" in mathematics?
The finite simple groups, at least at our current level of understanding, are quite chimeric, in that we have four different "heads" to this beast:
The finite cyclic groups of prime order;
The alter …
10
votes
Heat map of current mathematics
Not exactly the same (this image represents community detection more than intensity of activity), but Roja Bandari took all the programs run by the Institute for Pure and Applied Mathematics here at U …
- 114k