93
votes
32answers
15k views

Most harmful heuristic?

What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
69
votes
32answers
47k views

Why do we teach calculus students the derivative as a limit?

I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students? Something a teacher ...
70
votes
21answers
17k 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.
55
votes
7answers
4k views

When does Cantor-Bernstein hold?

The Cantor-Bernstein theorem in the category of sets (A injects in B, B injects in A => A, B equivalent) holds in other categories such as vector spaces, compact metric spaces, Noetherian topological ...
29
votes
12answers
9k views

Elementary / Interesting proofs of the Nullstellensatz

Is there an easy proof of the Nullstellensatz that avoids the standard Noether-normalization techniques? One proof I know proves first the 'weak' Nullstellensatz which ensures that maximal ideals ...
54
votes
2answers
16k views

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

Ten years ago when I studied in the 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 ...
41
votes
8answers
8k views

Why are powers of $\exp(\pi\sqrt{163})$ almost integers?

I've been prodded to ask a question expanding this one on Ramanujan's constant $R=\exp(\pi\sqrt{163})$. Recall that $R$ is very close to an integer; specifically $R=262537412640768744 - \epsilon$ ...
45
votes
2answers
4k 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 ...
17
votes
5answers
3k 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) - ...
24
votes
4answers
6k 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 ...
18
votes
7answers
1k views

Extensional theorems mostly used intensionally

Some theorems are stated and proved extensionally, but in practice are almost always used intensionally. Let me give an example to make this clear -- integration by parts: $$ \int_a^b f(x)g'(x)ds = ...
9
votes
5answers
3k views

Projection of Borel set from $R^2$ to $R^1$

Hello 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
35
votes
4answers
3k views

Cocomplete but not complete abelian category

This is a duplicate of the following question to which I did not receive any answer: http://math.stackexchange.com/questions/238247/complete-but-not-cocomplete-category Let $\mathfrak C$ be an ...
39
votes
1answer
3k 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 ...
8
votes
2answers
2k 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 ...

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