53
votes
32answers
6k views

What are some 'conceptualizations' that work in mathematics but are not strictly true?

I am having an argument with someone who thinks that it's never justified to teach something that is not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is ...
5
votes
1answer
80 views

What makes “the topos $\mathbf{M}_2$” such a good counterexample?

I'd like to ask this question sooner rather than later; it might be a bit half-baked. So I'm sorry. It's just that there's a chance I'll be side-tracked from Topos Theory for a couple of months (with ...
20
votes
8answers
2k views

Statements with rare counter-examples [duplicate]

This is a soft question. I'm searching for examples of mathmatical statements (preferably in number theory, but other topics are also fine), that seem to be true, but are actually not. Statements ...
11
votes
3answers
212 views

Natural uses for the co-product of sets?

I had come across countless uses of the (Cartesian) product of sets long before I first ever met the concept of a "co-product"1 of sets. In fact, anyone who has learned basic analytic geometry in ...
1
vote
1answer
74 views

Examples for Conditional Expectation (modern probability theory)

I'm in the process of learning about conditional expectation in the framework of modern probability theory. The sudden change brought about by the notion of conditional expectation being a function on ...
8
votes
3answers
242 views

Examples of non-Riemann integrable functions that appear “in nature”?

I am teaching an honours calculus class, and am looking for examples on non-integrable functions that occur somewhere real in mathematics. (The standard example of 1 on $\mathbb{Q}$ and 0 elsewhere ...
10
votes
1answer
324 views

What are some examples of hard theorems in category theory?

I'm currently learning some category theory, but so far I've used it only as a handy way to talk about various related concepts in algebra and topology with some nice, easy-to-prove lemmas like "left ...
1
vote
0answers
17 views

Does the notion of “commutative algebraic theory” generalize to many-sorted signatures?

I think that the notion of commutative algebraic theory only makes sense for unisorted signatures, and cannot sensibly be generalized to the case of more than one sort. Does anyone know of a ...
13
votes
1answer
181 views

What's so cool about local compactness?

As I study more algebraic number theory, I hear more and more often about local compactness: locally compact fields, locally compact topological groups, Stone-Čech compactification of locally compact ...
4
votes
3answers
138 views

abstract algebra example book

It's very exciting when you can use the theory to solve "lower level" problems. For example, I'm looking forward to understanding why the quintic equation is not solvable. In the undergraduate ...
14
votes
4answers
788 views

Can we get un-obvious results by defining sophisticated topologies?

What I originally found so interesting about general topology was how general a type of thing a topology is, and how the terminology open, closed, compact, continuous, convergence et cetera means ...
4
votes
1answer
189 views

List of proofs of non-trivial theorems which were unnoticed to be wrong for at least a few years

For example, the Weber's proof of Kronecker–Weber theorem. I would like to know such proofs. It seems to be important for me to remember that a widely accepted proof might be wrong.
7
votes
1answer
256 views

The perception of mathematics

In my work I wrote the following sentence. "...there is a negative perception of mathematics and mathematicians, both within and outside of academia." Err. Right. So, I believe that this is ...
3
votes
2answers
271 views

Interesting Problems for NonMath Majors

Sometime in the upcoming future, I will be doing a presentation as a college alumni to a bunch of undergrads from an organization I was in college. I did a dual major in mathematics and computer ...
14
votes
3answers
466 views

Are there interesting rings without unity?

There are several introductory textbooks which define a ring without any reference to a unity. However, nearly all of the rings one encounters in various branches of mathematics are endowed with a ...
5
votes
2answers
159 views

Analysis without algebra

I once heard someone say that analysis is $99 \%$ algebra. He was, of course, referring to the amount of algebraic manipulations in the exercises from any calculus course. I know that in topology, ...
8
votes
1answer
234 views

what's the role of fiber bundles play in understanding the base space?

Suppose $\pi: E\to B$ be a fiber bundle, and assume $B$ is connected, I want to know: 1, What do the fiber bundles (e.g., the covering spaces) on $B$ help understand the topology, or the structure on ...
26
votes
1answer
782 views

Counterexample Math Books

I have been able to find several counterexample books in some math areas. For example: $\bullet$ Counterexamples in Analysis, Bernard R. Gelbaum, John M. H. Olmsted $\bullet$ Counterexamples in ...
8
votes
2answers
236 views

What was Klein working on when he “replaces his Riemann surface by a metallic surface”?

I am reading The Value of Science by Poincare, and the following paragraph from Chapter I seems rather interesting: Look at Professor Klein: he is studying one of the most abstract questions of ...
4
votes
4answers
295 views

Examples of partial functions outside recursive function theory?

My math background is very narrow. I've mostly read logic, recursive function theory, and set theory. In recursive function theory one studies partial functions on the set of natural numbers. Are ...
29
votes
17answers
5k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
1
vote
1answer
204 views

Reference: Compendium of interesting graphs

I've been writing a little about some results on graph theory, and I want some nice examples of applying the results to some interesting finite connected graphs to show how the results might be ...
17
votes
3answers
834 views

Examples of math contest problems that can be solved in a 'cheap' way?

What are some examples of math contest problems that can be solved by using a nonrigorous, 'cheap' shortcut? For instance, a problem on the 2011 AMC went: A raft and a motorboat left dock A and ...
4
votes
0answers
224 views

Interesting applications of the cofinite topology?

Background: I'm doing some expository writing on intuitionistic logic and I have been toying with the idea of demonstrating its applicability via models where the denotations are taken from a Heyting ...
8
votes
1answer
272 views

When can we say that a theorem has been proven?

I'm taking a Data Structures and Algorithms course for a CS program. The introductory material was all mathematics, mostly a series of formulas that we are to remember. I can work through the formulas ...
172
votes
30answers
12k views

A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...