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

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Intuitive interpretation of the Laplacian Operator

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 Operator (a.k.a. divergence of gradient)?
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What should be the intuition when working with compactness?

I have a question that may be regarded by many as duplicate since there's a similar one at MathOverflow. The point is that I think I'm not really getting the idea on compactness. I mean, in $\mathbb{R}^n$ the compact sets are those that are closed…
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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 intro Calc course, where the figure has finite…
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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 define it to be $0$. But why characteristic zero? Why…
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Intuition behind Conditional Expectation

I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, please let me know. Let me get more specific. Let…
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What did Alan Turing mean when he said he didn't fully understand dy/dx?

Alan Turing's notebook has recently been sold at an auction house in London. In it he says this: Written out: The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find extremely difficult to understand in spite of it having been the one I …
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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 words as possible? From what I've gathered from the…
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How can I understand and prove the "sum and difference formulas" in trigonometry?

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 \sin \beta \\ \cos(\alpha \pm \beta) &= \cos \alpha \cos…
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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: $$[x,y^{-1},z]^y[y,z^{-1},x]^z[z,x^{-1},y]^x=1,$$ which holds for any…
Alexander Gruber
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Why, intuitively, is the order reversed when taking the transpose of the product?

It is well known that for invertible matrices $A,B$ of the same size we have $$(AB)^{-1}=B^{-1}A^{-1} $$ and a nice way for me to remember this is the following sentence: The opposite of putting on socks and shoes is taking the shoes off, followed…
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What are Some 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 $$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow \infty}(\int f_n \mathrm{d} \mu)$$ or alternatively (for…
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What are good "math habits" that have improved your mathematical practice?

I currently feel like I am not doing maths the best way I could; that is, I'm not making the most out of my time when I'm working on maths problems. The main thing I feel is that I'm not organizing my mind and my derivations as clear as I could,…
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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, logarithmic, exponential, and $n$th roots, and solving…
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Why is the area under a curve the integral?

I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself. However when it comes to the area under a curve for some reason when you break it up…
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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 direction, then for every $z$ in the disc we can…
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