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).

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401 votes
84 answers
185k views

Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...
355 votes
30 answers
75k views

Geometric interpretation of trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandary; Is there a geometric interpretation of the trace of a matrix? This question ...
325 votes
34 answers
92k views

Why is a topology made up of 'open' sets? [closed]

I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
293 votes
8 answers
141k views

Philosophy behind Mochizuki's work on the ABC conjecture

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
249 votes
29 answers
163k views

Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows- An engineer, a physicist, and a mathematician are discussing how to visualise four ...
212 votes
24 answers
45k views

What is torsion in differential geometry intuitively?

Hi, given a connection on the tangent space of a manifold, one can define its torsion: $$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$ What is the geometric picture behind this definition&...
Jan Weidner's user avatar
  • 12.8k
191 votes
34 answers
79k views

What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
182 votes
19 answers
35k views

How do I make the conceptual transition from multivariable calculus to differential forms?

One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...
181 votes
60 answers
42k views

Examples of eventual counterexamples

Define an "eventual counterexample" to be $P(a) = T $ for $a < n$ $P(n) = F$ $n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to ...
160 votes
37 answers
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 ...
Tim Campion's user avatar
  • 60.6k
155 votes
11 answers
19k views

Why are flat morphisms "flat?"

Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason! Is ...
Harrison Brown's user avatar
150 votes
12 answers
41k views

"Philosophical" meaning of the Yoneda Lemma

The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward. Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...
126 votes
13 answers
26k views

Should the formula for the inverse of a 2x2 matrix be obvious?

As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{...
Frank Thorne's user avatar
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116 votes
8 answers
32k views

Zagier's one-sentence proof of a theorem of Fermat

Zagier has a very short proof (MR1041893, JSTOR) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares. The proof defines an involution of the set $S= \lbrace (x,y,z) \...
Keivan Karai's user avatar
  • 6,064
114 votes
32 answers
20k views

What notions are used but not clearly defined in modern mathematics?

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions." Felix Klein What notions are used but not ...
104 votes
11 answers
40k views

What is the exterior derivative intuitively?

Actually I have several related questions, not worth opening different threads: What is the exterior derivative intuitively? What is its geometric meaning? A possible answer I know is, that it is ...
Jan Weidner's user avatar
  • 12.8k
102 votes
10 answers
35k views

What is (co)homology, and how does a beginner gain intuition about it?

This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question ...
Qiaochu Yuan's user avatar
96 votes
17 answers
16k views

What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?

Before you close for "homework problem", please note the tags. Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about ...
95 votes
16 answers
33k views

Most 'unintuitive' application of the Axiom of Choice?

It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis. Even ...
86 votes
9 answers
13k views

Why should I believe the Mordell Conjecture?

It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points. I am interested to know why Mordell and ...
Barinder Banwait's user avatar
86 votes
5 answers
18k views

What is sheaf cohomology intuitively?

What is sheaf cohomology intuitively? For local systems it is ordinary cohomology with twisted coefficients. But what if the sheaf in question is far from being constant? Can one still understand ...
Jan Weidner's user avatar
  • 12.8k
85 votes
12 answers
87k views

Why is the gradient normal? [closed]

This is a somewhat long discussion so please bear with me. There is a theorem that I have always been curious about from an intuitive standpoint and that has been glossed over in most textbooks I ...
Kim Greene's user avatar
  • 3,583
85 votes
9 answers
16k views

Demystifying the Caratheodory approach to measurability

Nowadays, the usual way to extend a measure on an algebra of sets to a measure on a $\sigma$-algebra, the Caratheodory approach, is by using the outer measure $m^* $ and then taking the family of all ...
Michael Greinecker's user avatar
83 votes
24 answers
8k views

Proof synopsis collection

I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself! Definition (Fraleigh): A proof synopsis ...
78 votes
11 answers
20k views

How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?

My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share of math Ph.D.'s, and ...
Bill Thies's user avatar
75 votes
10 answers
17k views

What is a Lagrangian submanifold intuitively?

What are good ways to think about Lagrangian submanifolds? Why should one care about them? More generally: same questions about (co)isotropic ones. Answers from a classical mechanics point of view ...
Jan Weidner's user avatar
  • 12.8k
75 votes
3 answers
3k views

Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?

Here is a couple of examples of the similarity from Wikipedia, in which the expressions differ only in signs. I encountered other analogies as well. $${\begin{aligned}\gamma &=\int _{0}^{1}\int _{...
Anixx's user avatar
  • 9,306
73 votes
10 answers
17k views

Intuition for Group Cohomology

I'm beginning to learn cohomology for cyclic groups in preparation for use in the proofs of global class field theory (using ideal-theoretic arguments). I've seen the proof of the long exact sequence ...
David Corwin's user avatar
  • 15.1k
72 votes
5 answers
8k views

Is there an intuitive reason for Zariski's main theorem?

Zariski's main theorem has many guises, and so I will give you the freedom to pick the one that you find to be most intuitive. For the sake of completeness, I will put here one version: Zariski's main ...
James D. Taylor's user avatar
71 votes
11 answers
8k views

How to introduce notions of flat, projective and free modules?

In the coming spring semester I will be teaching for the first time an introductory (graduate) course in Commutative Algebra. As many people know, I have been plugging away for a while at this ...
Pete L. Clark's user avatar
67 votes
11 answers
10k views

How should one think about non-Hausdorff topologies?

In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" (sequences/...
63 votes
52 answers
23k views

Colloquial catchy statements encoding serious mathematics

As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as ...
63 votes
5 answers
9k views

Intuition about the cotangent complex?

Does anyone have an answer to the question "What does the cotangent complex measure?" Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...
Peter Arndt's user avatar
63 votes
4 answers
11k views

Is there a good way to think of vanishing cycles and nearby cycles?

Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...
S. Carnahan's user avatar
  • 45k
61 votes
3 answers
6k views

Why is there no Cayley's Theorem for rings?

Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a ...
Tom Boardman's user avatar
  • 3,190
59 votes
9 answers
5k views

Examples of back of envelope calculations leading to good intuition?

Some time ago, I read about an "approximate approach" to the Stirling's formula in M.Sanjoy's Street Fighting Mathematics. In summary, the book used a integral estimation heuristic from ...
59 votes
4 answers
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 ...
Timothy Chow's user avatar
  • 78.1k
56 votes
16 answers
14k views

Examples of using physical intuition to solve math problems

For the purposes of this question let a "physical intuition" be an intuition that is derived from your everyday experience of physical reality. Your intuitions about how the spin of a ball affects ...
53 votes
8 answers
19k views

Physicist's request for intuition on covariant derivatives and Lie derivatives

A friend of mine is studying physics, and asks the following question which, I am sure, others could respond to better: What is the difference between the covariant derivative of $X$ along the curve ...
Igor Rivin's user avatar
  • 95.6k
51 votes
6 answers
12k views

What does Mellin inversion "really mean"?

Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
Frank Thorne's user avatar
  • 7,199
51 votes
11 answers
20k views

Intuition and/or visualisation of Itô integral/Itô's lemma

Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve. See e.g. Wikipedia:Riemann sum. The Itô integral has due to the unbounded total variation ...
vonjd's user avatar
  • 5,855
51 votes
6 answers
10k views

How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...
David Corwin's user avatar
  • 15.1k
51 votes
4 answers
5k views

Does anyone know an intuitive proof of the Birkhoff ergodic theorem?

For many standard, well-understood theorems the proofs have been streamlined to the point where you just need to understand the proof once and you remember the general idea forever. At this point I ...
Paul Siegel's user avatar
  • 28.8k
50 votes
6 answers
6k views

Intuition for the last step in Serre's proof of the three-squares theorem

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m (...
Qiaochu Yuan's user avatar
50 votes
4 answers
16k views

What is a symplectic form intuitively?

Hi, to completely describe a classical mechanical system, you need to do three things: -Specify a manifold $X$, the phase space. Intuitively this is the space of all possible states of your system. -...
Jan Weidner's user avatar
  • 12.8k
49 votes
7 answers
8k views

What is a coalgebra intuitively?

How to think about coalgebras? Are there geometric interpretations of coalgebras? If I think of algebras and modules as spaces and vectorbundles, what are coalgebras and comodules? What basic ...
49 votes
2 answers
8k views

Intuition for coends

Let $D$ be a co-complete category and $C$ be a small category. For a functor $F:C^{op}\times C \to D$ one defines the co-end $$ \int^{c\in C} F(c,c) $$ as the co-equalizer of $$ \coprod_{c\to c'}F(c,c'...
K Shao's user avatar
  • 603
49 votes
5 answers
4k views

are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?

I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...
symplectomorphic's user avatar
48 votes
11 answers
8k views

Cures for mathematician's block (as in writer's block) [closed]

What kind of things do you find that help you get the "creative juices flowing," to use a tired cliche, when you're stuck or burnt out on a problem? I've read about some studies that suggest listening ...
48 votes
6 answers
20k views

What is the difference between homology and cohomology?

In intuitive terms, what is the main difference? We know that homology is essentially the number of $n$-cycles that are not $n$-boundaries in some simplicial complex $X$. This is, more or less, the ...

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