Linked Questions
66 questions linked to/from Examples of common false beliefs in mathematics
238
votes
46
answers
90k
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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 ...
107
votes
36
answers
21k
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 ...
113
votes
25
answers
37k
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 2024-04-01 Anything new in 2024?
212
votes
14
answers
60k
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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 ...
58
votes
43
answers
11k
views
What are some mathematical sculptures?
Either intentionally or unintentionally.
Include location and sculptor, if known.
71
votes
28
answers
8k
views
Results from abstract algebra which look wrong (but are true)
There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...
121
votes
4
answers
39k
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 ...
- 10.1k
49
votes
11
answers
7k
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 ...
127
votes
4
answers
32k
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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 ...
53
votes
11
answers
6k
views
What is an important mathematical question?
$\DeclareMathOperator\GL{GL}$Many times I have heard people say sentences like X is an important question/ X is a natural question. I find this very surprising because to me it's all a matter of taste....
48
votes
6
answers
12k
views
Intuition for Integral Transforms
It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace ...
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59
votes
7
answers
4k
views
How closed-form conjectures are made?
Recently I posted a conjecture at Math.SE:
$$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$
where $J_\mu(x)$ and $Y_\mu(x)$ ...
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75
votes
3
answers
11k
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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75
votes
4
answers
6k
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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63
votes
7
answers
5k
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 ...
53
votes
6
answers
13k
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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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38
votes
6
answers
11k
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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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38
votes
7
answers
12k
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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47
votes
3
answers
7k
views
Clearing misconceptions: Defining "is a model of ZFC" in ZFC
There is often a lot of confusion surrounding the differences between relativizing individual formulas to models and the expression of "is a model of" through coding the satisfaction relation with ...
- 2,762
53
votes
4
answers
24k
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 ...
- 12.3k
64
votes
4
answers
8k
views
What is the current status of the Kaplansky zero-divisor conjecture for group rings?
Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good ...
- 1,065
19
votes
5
answers
10k
views
Projection of Borel set from $R^2$ to $R^1$
This should be easy to prove but I have no idea how to do it:
If $X \subseteq \mathbb{R}^2$ is borel then $f(X)$ is borel where $f(x,y) = x$
Thanks
Tobias
- 365
42
votes
4
answers
5k
views
Cocomplete but not complete abelian category
This is a duplicate of the following question to which I did not receive any answer: https://math.stackexchange.com/questions/238247/complete-but-not-cocomplete-category
Let $\mathfrak C$ be an ...
- 2,452
33
votes
3
answers
3k
views
Arithmetic geometry examples
(This is inspired by Algebraic geometry examples.)
I want to collect here (counter)examples in arithmetic geometry.
Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. ...
22
votes
2
answers
8k
views
Geometric vs Arithmetic Frobenius
If an algebraic variety $X$ over a field characteristic p is given by equations $f_i(x_1,...,x_k) = 0$, we can consider the variety $X^{(p)}$ obtained by applying p-th powers to all the coefficients ...
- 1,783
19
votes
2
answers
2k
views
Laurent series in several complex variables
Is there a good generalisation of Laurent series for several complex variables?
I am interested in generalised power series that have some terms with negative powers, but not too many. In single ...
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14
votes
2
answers
3k
views
When is sin(r \pi) expressible in radicals for r rational?
Perhaps this question will not be considered appropriate for MO - so be it. But hear me out before you dismiss it as completely elementary.
As the question suggests, I would like to know when $\sin(...
- 29.2k
25
votes
3
answers
994
views
Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...
- 513
7
votes
3
answers
2k
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Decidable but nonrecursive sets
Until recently, I believed that recursive=decidable,
subscribing to this Wikipedia quote:
"In computability theory, a set is decidable, computable, or recursive if there
is an algorithm that ...
- 151k
16
votes
2
answers
3k
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Clarification of Gödel's second incompleteness theorem
I am sorry for the following question, because the actual answer to this question is in the beautiful works of Feferman and Jeroslow, but, unfortunately, I havn't any time to go into that specific ...
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