Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.
136
votes
9answers
9k views
What's an intuitive way to think about the determinant?
In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
120
votes
27answers
7k views
Too old to start math
I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could.
This is a question I've been asking myself now for some ...
73
votes
5answers
4k views
Intuition for the definition of the Gamma function?
In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity
$$n! = \int_{0}^{\infty} t^n e^{-t} dt$$
coming from ...
72
votes
5answers
3k views
In (relatively) simple words: What is an inverse limit?
I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me.
The inverse limit. I tried to ask one of ...
59
votes
16answers
6k views
How do you explain the concept of logarithm to a five year old?
Okay I understand that it cannot be explained to a 5 year old. But, how do you explain the logarithm to primary school students?
49
votes
39answers
5k views
What are some examples of a mathematical result being counterintuitive?
As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition.
My first and favorite experience of this is Gabriel's Horn that you see in ...
48
votes
8answers
2k views
Different kinds of infinities?
Can someone explain to me how there can be different kinds of infinities?
I was reading "the man who loved only numbers" and came across the concept of countable and uncountable infinities, but ...
46
votes
4answers
3k views
What are the Axiom of Choice and Axiom of Determinacy?
Would someone please explain:
What does the Axiom of Choice mean, intuitively?
What does the Axiom of Determinancy mean, intuitively, and how does it contradict the Axiom of Choice?
as simple ...
46
votes
12answers
3k views
Intuitive explanation of Cauchy's Integral Formula in Complex Analysis
There is a theorem that states that if $f$ is analytic in a domain $D$, and the closed disc {$ z:|z-\alpha|\leq r$} contained in $D$, and $C$ denotes the disc's boundary followed in the positive ...
44
votes
8answers
2k views
Intuition in algebra?
My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) ...
40
votes
4answers
2k views
Why “characteristic zero” and not “infinite characteristic”?
The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...
38
votes
1answer
730 views
How to think of the group ring as a Hopf algebra?
Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
36
votes
12answers
4k views
I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2 [closed]
I didn't know how to phrase the question properly so I am going to explain how this came about.
I know Math is a very rigorous subject and there are proofs for everything we know and use. In fact, I ...
36
votes
8answers
7k views
What does matrix multiplication actually mean?
If I multiply two numbers, say 3 and 5, I know it means add 3 to itself 5 times or add 5 to itself 3 times.
But If I multiply two matrices, what does it mean ?
I mean I can't think it in terms of ...
36
votes
13answers
1k views
Surprising Generalizations
I just learned (thanks to Harry Gindi's answer on MO and to Qiaochu Yuan's blog post on AoPS) that the chinese remainder theorem and Lagrange interpolation are really just two instances of the same ...
35
votes
7answers
724 views
Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra
I am taking a course next term in homological algebra (using Weibel's classic text) and am having a hard time seeing some of the big picture of the idea behind homological algebra.
Now, this sort of ...
34
votes
4answers
941 views
Algebra: Best mental images
I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on ...
33
votes
8answers
1k views
What makes elementary functions elementary?
Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
32
votes
1answer
556 views
How does one see Hecke Operators as helping to generalize Quadratic Reciprocity?
My question is really about how to think of Hecke operators as helping to generalize quadratic reciprocity.
Quadratic reciprocity can be stated like this: Let $\rho: Gal(\mathbb{Q})\rightarrow ...
31
votes
0answers
1k views
Does a four-variable analog of the Hall-Witt identity exist?
Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125):
An amazing commutator formula is the Hall-Witt identity: ...
30
votes
3answers
2k views
Do you prove all theorems whilst studying?
When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my ...
30
votes
4answers
2k views
Why do we care about dual spaces?
When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don't really understand why we want to ...
30
votes
3answers
546 views
Why do we look at morphisms?
I am reading some lecture notes and in one paragraph there is the following motivation: "The best way to study spaces with a structure is usually to look at the maps between them preserving structure ...
28
votes
7answers
1k views
Why do we restrict the definition of Lebesgue Integrability?
The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.)
Why is it we ...
27
votes
5answers
1k views
Dominoes and induction, or how does induction work?
I've never really understood why math induction is supposed to work.
You have these 3 steps:
Prove true for base case (n=0 or 1 or whatever)
Assume true for n=k. Call this the induction ...
26
votes
4answers
3k views
Connection between fourier transform and taylor series
Both fourier transform and taylor series are means to represent functions in a different form.
My question:
What is the connection between these two? Is there a way to get from one to the other (and ...
25
votes
4answers
1k views
Can someone explain the Yoneda Lemma to an applied mathematician?
I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood ...
24
votes
13answers
8k views
Formerly good at math, but after 12 years I've lost most of my skills. Now I need them once again. Any advice to grow them back?
I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the ...
23
votes
8answers
625 views
Intuitive meaning of Exact Sequence
I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all.
Can anyone explain them for me? ...
23
votes
5answers
2k views
What is an intuitive explanation for $\operatorname{div} \operatorname{curl} F = 0$?
I am aware of an intuitive explanation for $\operatorname{curl} \operatorname{grad} F = 0$ (a block placed on a mountainous frictionless surface will slide to lower ground without spinning), and was ...
23
votes
4answers
1k views
Understanding the intuition behind math
I'm currently a Calculus III student. I enjoy math a lot, but only when I understand its beauty and meaning. However, so many times I have no idea what it is I am learning about, althought I am still ...
22
votes
3answers
566 views
Why the emphasis on Projective Space in Algebraic Geometry?
I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed.
Why does Miranda (and from what little ...
22
votes
3answers
511 views
Intuition on fundamental theorem of arithmetic
I'm sorry ahead if time if this is overly trivial for this site.
Currently in school, much of what I enjoy is number theory - based. Currently, I lean pretty heavily on the FTA for a good deal of my ...
22
votes
4answers
521 views
The Meaning of the Fundamental Theorem of Calculus
I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
21
votes
3answers
1k views
Why is the Jordan Curve Theorem not “obvious”?
I am horribly confused about Jordan's Curve Theorem (henceforth JCT). Could you give me some reason why should the validity of this theorem be in doubt? I mean for anyone who trusts the eye theorem is ...
21
votes
6answers
735 views
Trying to understand why circle area is not $2 \pi r^2$
I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below:
The area of a square is like a line, the height (one dimension, length) ...
21
votes
3answers
533 views
Intuition behind Snake Lemma
I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even ...
21
votes
4answers
1k views
Simple explanation of a monad
I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are ...
21
votes
3answers
391 views
Intuition for the Importance of Modular Forms
I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things.
I was wondering if the following interpretation of ...
20
votes
6answers
4k views
How can I understand and prove the “sum and difference formulas” in trigonometry? (cos(a ± b) = …, etc.)?
The "sum and difference" formulas often come in handy, but it's not immediately obvious that they would be true.
\begin{align}
\sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha ...
20
votes
5answers
423 views
Motivation for spectral graph theory.
Why do we care about eigenvalues of graphs?
There must be some reason. There is an entire mathematical discipline about them.
I always assumed that spectral graph theory is an extension of graph ...
19
votes
4answers
863 views
You are standing at the origin of an “infinite forest” holding an “infinite bb-gun”
I use stories like these to develop intuition... or perhaps to destroy it. I have my own answers in mind, but I want to see if I have made any mistakes...
You are standing at the origin of an ...
18
votes
4answers
1k views
Tricks to remember Fatou's lemma
For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality
$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow ...
18
votes
5answers
2k views
Intuitive interpretation of the Laplacian
Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian (a.k.a. divergence of gradient)?
...
18
votes
3answers
353 views
What's the connection between derivatives and boundaries?
The (second) fundamental theorem of calculus says that
$$\int_a^b f'(x) dx = f(b) - f(a)$$
which can also be stated, if one knows enough about what's coming next, as:
The integral of the ...
18
votes
2answers
951 views
Geometric intuition behind the Lie bracket of vector fields
I understand the definition of the Lie bracket and I know how to compute it in local coordinates.
But is there a way to "guess" what is the Lie bracket of two vector fields ? What is the geometric ...
18
votes
1answer
431 views
What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?
What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?
Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
17
votes
13answers
1k views
Intuitive Understanding of the constant “e”
Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate.
I know there are many ways of calculating (or should I say ...
17
votes
4answers
5k views
Laplace transformations for dummies
Is there a simple explanation of what the Laplace transformation do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm ...
17
votes
1answer
200 views
Understanding what $\sqrt{p}$ means for an event of probability $p$
Say I have a random event $E$ with probability $p$. There is a natural interpretation in terms of $E$ for the probability $p^2$: it's the probability that $E$ occurs twice if I perform two independent ...
