# Questions tagged [intuition]

Questions asking for the intuition behind some definition, conjecture, proof etc. In other words, questions designed to improve or to acquire understanding on a conceptual or intuitive level, as opposed to on a technical or formal level. When asking such a question it can be helpful to include a rough description of ones understanding of the subject at hand (on a technical level).

372 questions
Filter by
Sorted by
Tagged with
69 views

### Intuition/interpretation: maps into snowflaked quasi-metric spaces

Fix a positive integer $d$ and $\alpha>0$. Consider the following two cases: "Familiar" Case: Suppose that $f:(\mathbb{R}^d,\|\cdot\|_2)\to (\mathbb{R}^d,\|\cdot\|_2^{\alpha})$ is $1$-...
• 4,775
1k views

### The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$. There is a convoluted proof that $P(ab>c)=\frac12$. But since the ...
• 2,947
252 views

### Densities, pseudoforms, absolute differential forms and measures, differential forms, etc

Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here. Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
• 741
17k views

### Daunting papers/books and how to finally read them

Most people throughout their career encounter at least one paper that seems especially daunting to them. I'm interested in real stories of how you successfully overcame that to extract the knowledge ...
167 views

### How to think about Beilinson's gluing data?

Let $X$ be a complex manifold, $D$ a divisor (that is globally the zero locus of a function) and $U$ its complement. Recall Beilinson's "how to glue perverse sheaves": Given a perverse ...
• 5,555
270 views

### Dévissage for a stratification in Grothendieck's Esquisse d’un programme: What is it?

I have a question about the the intuition of what Grothendieck proposed as tame topology in his "Esquisse d’un programme" as a "better suited" geometric structure in order to have ...
• 5,984
70 views

### Justification of modular law in allegories

The modular law in modular lattices can be described as an isomorphism between opposite edges of the square $(a \land b), a, b, (a\lor b)$. A fancier way of saying this is an adjoint equivalence with ...
• 1,091
372 views

### What heuristic arguments support Montgomery's conjecture for primes in short intervals?

I have a question regarding a conjecture due to H. L. Montgomery on the number of primes in short intervals. The conjecture apparently arises from probabilistic reasoning upon assuming the Riemann ...
• 415
2k views

### Meaning of a quantum field given by an operator-valued distribution

I am trying to grasp the basics of rigorous quantum field theory. Let me summise how the setup of non-interacting quantum field theories look like to me. Let $\mathcal{H}$ be a Hilbert space in which ...
• 1,201
389 views

### What is Bernoulli umbra philosophically?

Well, Bernoulli umbra is an umbra whose moments are the Bernoulli numbers. But what is it philosophically? For instance, we can consider imaginary unit $i$ an umbra with moments $\{1,0,−1,0,1,\ldots\}$...
• 9,500
279 views

• 3,680
7k views

### Is orientability a miracle?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$This question is prompted by a recent highly-upvoted question, Conceptual reason why the sign of a permutation is well-defined? The responses made ...
• 79.4k
15k views

### Conceptual reason why the sign of a permutation is well-defined?

Teaching group theory this semester, I found myself laboring through a proof that the sign of a permutation is a well-defined homomorphism $\operatorname{sgn} : \Sigma_n \to \Sigma_2$. An insightful ...
• 62k
4k views

### Archiving mathematical correspondence

What are great examples of comprehensively archived mathematical correspondence (including both handwritten and electronic items)? Context: polished papers usually don't reveal the full process that ...
1 vote
109 views

• 313
643 views

### Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?

I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under ...
• 741
1 vote
162 views

### Intuition of the "work" done by random variables in Monte Carlo methods (incl. MCL)

(I've tried Math SE, but have so far come up empty handed, so I'm trying my luck here.) I would like to get a better intuitive understanding of why Monte Carlo works so well in approximating a ...
• 91