Mathematics Stack Exchange
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Questions tagged [intuition]

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.

To have intuition about a mathematical truth is to have some insight into why it is true, and to understand the motivation for talking about that truth in the first place. This is usually stated in contrast with merely having a superficial knowledge of a mathematical truth as a fact, or only having skills at applying a mathematical truth to solve a problem without having the conceptual understanding of solution.

For a nice explanation of mathematical intuition with examples, and links to other articles on developing mathematical understanding, see Developing Your Intuition For Math on BetterExplained.com.

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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?
Sandbox
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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) MacLane and Birkhoff's Algebra on my own. The…
Michael Benfield
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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? Thanks in advance.
finnlim
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The Intuition behind l'Hopitals Rule

I understand perfectly well how to apply l'Hopital's rule, and how to prove it, but I've never grokked the theorem. Why is it that we should expect that $$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x \to a}\frac{f'(x)}{g'(x)},$$ but only when specific…
user71641
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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 studies -- sequence two of Calculus -- but what I'm…
user24383
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Motivation for spectral graph theory.

Why do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so they must be important. I always assumed that…
Alexander Gruber
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What is the intuition behind uniform continuity?

There’s another post asking for the motivation behind uniform continuity. I’m not a huge fan of it since the top-rated comment spoke about local and global interactions of information, and frankly I just did not get it. Playing with the definition,…
Spencer Kraisler
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Thurston's 37th way of thinking about the derivative

In Thurston's superb essay On proof and progress in mathematics, he makes this observation: Of course there is always another subtlety to be gleaned, but I would like to at least think that I have absorbed the main intuition behind each element…
Zev Chonoles
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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 question has been asked many times on forums such…
Alex Youcis
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Intuition behind conjugation in group theory

I am learning group theory, and while learning automorphisms, I came across conjugation as an example in many textbooks. Though the definition itself, (and when considering the case of abelian groups), it seems pretty innocent, I have to admit that…
Nikhil
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Intuitive explanation of a positive semidefinite matrix

What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors. The definition is: A $n$…
user915
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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 I've seen, algebraic geometers in general) place…
Potato
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Is there an intuitive reason for a certain operation to be associative?

When I read Pinter's A Book of Abstract Algebra, Exercise 7 on page 25 asks whether the operation $$x*y=\frac{xy}{x+y+1}$$ (defined on the positive real numbers) is associative. At first I considered this to be false, because the expression is so…
Zirui Wang
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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 itself. I know that a categorical approach is…
Rachmaninoff
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Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have gained quite some intuition for tensor products and…
Martin Brandenburg
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