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Results tagged with big-picture
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Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and focused, it can be helpful to keep in mind that the design of MathOverflow does not make it a good fit for questions that are too broad.
6
votes
Why does mathematics seem to have a polarity bias?
I don't have any deep insights to offer, but I'd like to suggest that there are two separate questions being mixed together here.
The first question is why there often appears to be an asymmetry betwe …
- 82.7k
10
votes
Do empirical studies have a place in contemporary mathematics research?
coudy's answer, pointing you to the Experimental Mathematics, is the right answer, and there are already other MO questions about experimental mathematics that are relevant, but I can't resist giving …
- 82.7k
13
votes
What is an important mathematical question?
I want to point out that you raised two questions, and in my opinion they are very different questions.
So I really want to know how to decide whether a question is worth studying?
How do I deci …
- 82.7k
4
votes
Foundations of topology
There are different ways to respond to Grothendieck's challenge, depending on how exactly you interpret what he is looking for. As mentioned by ACL in a comment, the theory of o-minimality provides on …
- 82.7k
12
votes
The advantage of asymmetric objects
In light of your stated motivation, the following may not be what you had in mind, but objects with no symmetries are often easier to handle when it comes to computation and/or enumeration. For examp …
- 82.7k
22
votes
Breakthroughs in mathematics in 2021
Having just listened to some of Jacob Tsimerman's Minerva lectures, I became aware of the recent arXiv preprint, Canonical Heights on Shimura Varieties and the André–Oort Conjecture, by Jonathan Pila, …
- 82.7k
46
votes
Logic in mathematics and philosophy
I agree with the commentators that the question is rather too broad, but here's an attempt to answer it anyway.
Readers of MO will likely have less familiarity with non-mathematical logic, so it might …
- 82.7k
24
votes
The use of computers leading to major mathematical advances II
For experimental mathematics as that term is usually understood, I would commend to your attention the paper by Roger Behrend, Ilse Fischer and Matjaž Konvalinka, "Diagonally and antidiagonally symmet …
- 82.7k
23
votes
Accepted
What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?
There's a certain confusion underlying your question, which Andreas Blass's answer is trying to point out. Let me see if I can explain it in different words.
You say, “the negation of Con(ZFC) proves …
- 82.7k
10
votes
Every mathematician has only a few tricks
Terence Tao wrote a paper, Exploring the toolkit of Jean Bourgain. The abstract reads:
Gian-Carlo Rota once asserted that "every mathematician only has a few tricks". The sheer breadth and ingenuity …
- 82.7k
13
votes
Every mathematician has only a few tricks
Scott Aaronson has taken a stab at articulating his own methodology for upper-bounding the probability of something bad. He was inspired by a blog post by Scott Alexander bemoaning how rarely experts …
- 82.7k
4
votes
What is the high-concept explanation on why real numbers are useful in number theory?
Some historical context may be useful here.
In the late 19th and early 20th centuries, many mathematicians informally categorized mathematics into three tiers: arithmetic, analysis, and set theory. …
- 82.7k
18
votes
Theorems that impeded progress
The proof that a particular computational problem is NP-complete can cause people to stop trying to make theoretical progress on it, instead focusing all their attention on heuristics that have only e …
- 82.7k
13
votes
Theorems that impeded progress
Like RBega2 I hesitate to say that this is definitely an example, but the paper "Natural Proofs" by Razborov and Rudich, which showed that certain kinds of proof techniques would be insufficient to pr …
- 82.7k
25
votes
Motivation behind Analytic Number Theory
I will go out on a limb and say that in my opinion, it is the norm, rather than the exception, for a branch of mathematics to be a collection of results that we can prove using the techniques we know, …
- 82.7k