Linked Questions

222 votes
46 answers
81k views

Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...
103 votes
34 answers
18k views

Interesting examples of vacuous / void entities

I included this footnote in a paper in which I mentioned that the number of partitions of the empty set is 1 (every member of any partition is a non-empty set, and of course every member of the empty ...
111 votes
25 answers
34k views

Examples of math hoaxes/interesting jokes published on April Fool's day?

What are examples of math hoaxes/interesting jokes published on April Fool's day? For a start P=NP. Added 2022-04-01 Anything new in 2022?
192 votes
13 answers
54k views

Why doesn't mathematics collapse even though humans quite often make mistakes in their proofs?

To begin with, I am aware of these questions, which seems to be related: How do I fix someone's published error?, Examples of common false beliefs in mathematics, When have we lost a body of ...
54 votes
41 answers
9k views

What are some mathematical sculptures?

Either intentionally or unintentionally. Include location and sculptor, if known.
109 votes
4 answers
30k views

Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago, when I studied in university, I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows: All numbers are divided into two classes: those ...
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  • 8,434
119 votes
4 answers
27k views

Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional

A very important theorem in linear algebra that is rarely taught is: A vector space has the same dimension as its dual if and only if it is finite dimensional. I have seen a total of one proof of ...
43 votes
9 answers
5k views

In "splendid isolation"

While browsing the Net for some articles related to the history of the Whittaker-Shannon sampling theorem, so important to our digital world today, I came across this passage by H. D. Luke in The ...
70 votes
3 answers
9k views

Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...
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73 votes
4 answers
5k views

When is a singular point of a variety ($\mathcal{C}^\infty$-) smooth?

If $X$ is a nonsingular algebraic (or analytic) variety over $\mathbb C$ or $\mathbb R$ then it is certainly $C^\infty$ over the reals. The converse is false for a silly reason : in the real or ...
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57 votes
7 answers
3k views

What well known results with countability assumptions can be naturally extended to uncountable settings?

In many of the common categories of spaces (or algebras) in mathematics, one often restricts attention to those spaces or algebras which are "countable" or "countably generated" in ...
45 votes
6 answers
10k views

What does Mellin inversion "really mean"?

Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
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  • 6,931
31 votes
5 answers
9k views

Is there a version of inclusion/exclusion for vector spaces?

I am asking for a way to compute the rank of the 'join' of a bunch of subspaces whose pairwise intersections might be non-zero. So in the case n=2 this is just $\dim(A_1+A_2) = \dim(A_1) + \dim(A_2) - ...
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  • 459
44 votes
4 answers
18k views

When is $L^2(X)$ separable?

I have never studied any measure theory, so apologise in advance, if my question is easy: Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable? In reality, I am interested in ...
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  • 11.4k
35 votes
6 answers
9k views

Limits in category theory and analysis

Is it possible to regard limits in analysis (say, of real sequences or more generally nets in topological spaces) as limits in category theory? Is there some formal connection? Edit ('13): Perhaps it ...
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