All Questions
12 questions
8
votes
1
answer
388
views
Formalisation of intuitive concepts in the language leading to mathematical progress
In his work, Albert Lautman thinks the genesis of some mathematical works as a dialectic that takes place between opposite notions, like between global and local. He argues that while those notions, ...
- 531
47
votes
7
answers
8k
views
Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians
I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics....
52
votes
6
answers
5k
views
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false
What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false?
One interesting example, and the impetus for this question, is work in number ...
53
votes
15
answers
5k
views
Request for examples: verifying vs understanding proofs
My colleague and I are researchers in philosophy of mathematical practice and are working on developing an account of mathematical understanding. We have often seen it remarked that there is an ...
29
votes
20
answers
7k
views
Modeling in pure math
We all know that models play a major role in scientific practice. (By "model" here I mean any of various kinds of entities that function as representations or descriptions of real-world phenomena. ...
4
votes
0
answers
362
views
Theorems conditional on false conjectures
What is an example of a theorem that was conditional on a conjecture that later turned out to be false?
31
votes
14
answers
4k
views
An example of a proof that is explanatory but not beautiful? (or vice versa)
This question has a philosophical bent, but hopefully it will evoke straightforward, mathematical answers that would be appropriate for this list (like my earlier question about beautiful proofs ...
5
votes
2
answers
1k
views
Is beauty at the high school level even possible? [closed]
This question is a follow up to 74841, and follows from a suggestion by Gian-Carlo Rota that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean ...
104
votes
19
answers
14k
views
Can a mathematical definition be wrong?
This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...
26
votes
9
answers
8k
views
Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]
As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is ...
32
votes
21
answers
16k
views
What are some applications of other fields to mathematics?
It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely:
What are some applications ...
15
votes
10
answers
2k
views
Can you prove equivalence without being able to calculate it?
In mathematics we often seek to classify objects up to an equivalence relation, where two objects A and B are said to be equivalent if there exists a map $f:A\rightarrow B$ satisfying certain ...