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.

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Understanding Abel-Ruffini

I'm wondering of anyone can point me towards a proof of why we can't have a quintic formula, using concepts from basic group theory. In particular, I understand that there is some connection with ...
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Motivating $y'=y \implies y=Ce^x$

Is there some intuitive reason why one should think that a function which is its own derivative should be of the form $Ca^x$ for some number $a$? Of course I can prove that the unique solution set to ...
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How to see that $\text{gcd}(a,b) = \text{gcd}(a-b,b)$?

I'm trying to understand why $\text{gcd}(a,b) = \text{gcd}(a-b,b)$. What is clear to me is that the $\text{gcd}$ divides $a,b$ and also $a-b$ (let's assume $a\ge b$). But then it seems to me we ...
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Is there a more intuitive way of coming up with matrices for problems like this? E.g. Find two matrices B and C with AB = AC, and B does not equal C

Let A = $\begin{bmatrix}1 & 0\\1 & 0\end{bmatrix}$ Find two matrices $B$ and $C$ with $AB = AC$, and $B$ does not equal $C$. I always have trouble with problems like this. Here, I ...
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Is there an intuitive way of viewing the Law of Total Expectation $\mathbb{E}\big[\mathbb{E}[X|Y]\big]=\mathbb{E}[X]?$

Law of total expectation If $\mathbb{E}\big[|X|\big]$ finite then for any $Y,\;\mathbb{E}\big[\mathbb{E}[X\mid Y]\big]=\mathbb{E}[X]$ I remember reading this for the first time and ...
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Summation of a constant using sigma notation

Apologies if this is a silly question, but is it possible to prove that $$\sum_{n=1}^{N}c=N\cdot c$$ or does this simply follow from the definition of sigma notation? I am fairly sure it's the ...
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108 views

Learning Galois theory geometrically?

Recently I started poking at algebraic geometry and commutative algebra. My background is basic category theory and basic algebraic topology. I don't know a lot of other mathematics. I noticed Galois ...
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172 views

How does the internal language of a topos come to be?

There are several books and articles on topos theory which mention the internal language, but I can't manage to see the big picture from any of them. I would like a soft explanation of how the ...
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25 views

Bijections and Visual Representations

Throughout math we learn about the cardinality of sets. We learn that the existence of a bijection between two sets imply that the cardinality of these two sets are equal. For each set there is ...
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What is an intuitive definition for “conjugate” in Group Theory?

In Abstract Algebra, I learned about "conjugation" in the context of a group $H$ being a 'normal' subgroup of $G$ if the element $xhx^{-1}\in H$ for any $x\in G$. But this is not the first time I've ...
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53 views

What is the Newton's general theory of diameters?

I was reading a book on Mathematics, which contained this topic. I was not able to grasp the concept. There was not much info on internet also. It was as follows: Let an $n$th order curve be ...
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105 views

More intuitive/easier explanation for Knuth's division algorithm needed

Knuth's division algorithm, here refers to the D algorithm mentioned in TAOCP written by Donald Knuth in Volume 2 (Semi-numerical algorithms) Section 4.3.1 I have been trying to understand the ...
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108 views

Left invariant Vector Field on $S^2$

How intuitively look like all left invariant vector fields on this manifold: the 2 dimensional unit sphere $S^2$ with the smooth structure inherited from $\mathbb R^3$? Why all left invariant vector ...
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44 views

What does Poincaré mean for intuition of pure number?

To what does Poincaré refer in this article http://www-history.mcs.st-andrews.ac.uk/Extras/Poincare_Intuition.html speaking about the intuition of pure number? My answer is that he may refer to a ...
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96 views

Intuitive or visual understanding of the real projective plane

If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is ...
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108 views

Factorial in power series; intuitive/combinatorial interpretation?

It is well known that the terms of the power series of exponential and trigonometric functions often involve the factorial function, essentially as a consequence of iterating the power rule. My ...
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132 views

Intuition behind universal arrow construction of adjoint functors

I try to understand adjoint functors in category theory. I like the idea of thinking of a left adjoint as 'best approximation from above' and a right adjoint as 'best approximation from below.' More ...
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Higher homotopy groups meaning

I am developing intuition for higher homotopy groups but it's very hard for me to visualize what $\pi_2$ represents (and $\pi_n$ for that matter). I know that $\pi_2(S^2) \cong \mathbb{Z}$ and can ...
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Deterministicaly stops winning, yet if conditioned on time there is positive probability to win in the future.

In the answer to question http://mathoverflow.net/questions/12462/limsup-and-liminf-for-a-sequence-of-sets "has2" gives a concrete example for the use of $\limsup$ and $\liminf$ in form of a "card ...
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97 views

How are irrational numbers, fixed points on the number line?

Please, while answering/reading this question, only keep in mind my point of view only. The question is, that how come an irrational number on a number line is a fixed point. To make things more ...
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65 views

Alternative Geometries

In our world, the distance between two points (in 2d) is defined as $\sqrt{(\Delta x)^2 + (\Delta y)^2}$. Suppose that in an alternative geometry, it was defined as $\sqrt[p]{|\Delta x|^p + |\Delta ...
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What does it mean for a set to be countably infinite?

Why distinguish between countable and uncountable? What advantages does this property have? I haven't studied much set theory but I am writing about the set of algebraic vs transcendental numbers and ...
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Why is abelianness such a precious property?

My abstract algebra teacher said the other day that constructions like ideals and cosets and normal subgroups are "trying to capture a little bit of abelianness." He has used phrases like "magic ...
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Is an example of a PDF $\lambda e^{-\lambda t}$ or $\lambda e^{-\lambda t}\mathrm{d}t$?

The reason I ask is because according to this source: the $\fbox{$\color{blue}{\mathrm{PDF}}$}$ for the sum of two Exponential Density Functions is $$\rho(x_1,x_2)\mathrm{d}x_1 ...
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Which general physical transformation to the number space does exponentiation represent?

Addition and multiplication may be defined in two ways, one specific and one general: Addition specific: addition is repeated incrementation. This is specific and sub-optimal as while $2 + 4$ is ...
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Why does the fundamental theorem of calculus work?

I've known for some time that one of the fundamental theorems of calculus states: $$ \int_{a}^{b}\ f'(x){\mathrm{d} x} = f(b)-f(a) $$ Despite using this formula, I've yet to see a proof or even a ...
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What is the difference between a Poisson and an Exponential distribution?

For a Poisson distribution: $$\mathsf{P}(X=x)=\frac{e^{-\mu}\times \mu^x}{x!}$$ where $\mu$ is the mean number of occurrences. For an Exponential distribution: $$f(x;\lambda) = \begin{cases} ...
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24 views

The Banach–Tarski Paradox [duplicate]

Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original? What is the true nature of this paradox ? I don't really understand this ?
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84 views

How to see symbol manipulation from an intuitive perspective in math?

I have recently started to develop my mathematical intuition. In the past I saw math as a mere game of symbol manipulation, whosoever was able to see patterns and cram formulas and apply them upon ...
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52 views

Geometric interpretation of monotone operators on a Hilbert space

Recall that a monotone operator is defined by the relationship as follows: $$\langle y - x, F(y) - F(x)\rangle \geq 0, \quad \forall x,y \in X$$ ($X$ is a Hilbert space) What is a good geometric ...
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168 views

Proof/intuition that any number can be expressed in binary form and every number will have a unique representation?

I was just thinking lately that how do we know that literally every number can be expressed in binary? And that too, with a unique representation? Clarification: With numbers, I mean whole numbers. ...
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56 views

Vector Integration - Intuition

I understand that an integral of a scalar valued function can be visualized as "signed area under the curve". But what about integration of a vector valued function by its parameter? Is there a ...
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52 views

What is the origin of the formula: $\rho_x (x)=\left|\frac{{d}x}{{d}\alpha}\right|^{-1}\rho_\alpha(\alpha)$ that relates random variables?

I'm trying to understand the origin of a certain formula used in the solution to the following question: This question relates to the position probability density for a classical particle ...
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Developing Mathematic Intuition

I'm an engineering student, currently working my way through the fundamental mathematics courses. I've done reasonably well so far—mostly A's and a couple of B's in Algebra, Statistics, ...
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Are all instances of torsion special cases of the same concept?

The concept of 'torsion' pervades mathematics. As far as I know the origin of the word is in algebraic topology where it was used to describe chains $\gamma$ which are not boundaries but such that ...
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Question about: How many partitions of $12$ have parts of size at most $5$?

If the parts are of size at most $5$, why are considering numbers greater than $5$? For example, why can $z_5$ take on $10, 15, \dots$?
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Is there some sort of trick to show naturality?

This is about natural transformations in category theory. Almost always, I somewhat know why some defined maps or homomorphisms behave naturally, but I am almost never entirely sure (if things get ...
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38 views

A question concerning Jacobians of coordinate transformation

Apologies for perhaps a very trivial question, but I'm slightly doubting my understanding of Jacobians after explaining the concept of coordinate transformations to a colleague. Basically, as I ...
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30 views

How do you actually write out the terms in a Cauchy sequence?

For example for $\epsilon>0$ there exist $N$ such $n,m>N$ implies $|s_m-s_n|<\epsilon$. I understand that intuitively, we don't need to know a certain limit and thus this definition for ...
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34 views

What is an intuitive way to think about uniform continuity? [duplicate]

I've been having trouble understanding uniform continuity, as it doesn't seem to describe the same kind of thing as regular continuity. I know what the formal difference between the two is and I know ...
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83 views

Does a retraction really “retracts” something?

I wonder what is the intuition behind the definitions of a section and retraction in Category Theory. From Awodey's book: Definition 2.7. A split mono (epi) is an arrow with a left (right) ...
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153 views

What is the intuitive way to understand Dot and Cross products of vectors?

Suppose, we are crossing a river with heavy current using a speed-boat. If $x$ is the vector for the current and $y$ is the vector for the speed-boat, then what do $x \cdot y$ and $x \times y$ ...
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Geometric / Intuitive construction of the rotation axis of a 3D rotation matrix?

I have been looking without success for an intuitive / geometric construction of the rotation axis of a given 3D rotation matrix. To put the problem in more familiar terms, let's assume you have the ...
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69 views

Intuitive way of thinking of reduced homology

I know the definition of it, but is there a more intuitive way of thinking of it that even a layman could (kind of) understand? The $n$-th homology group, for example, can be thought of as the number ...
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185 views

Why is $\ln 1 = 0 $? [closed]

Yes I know, and believe, and have used it for all the time I have done mathematics as fun as well as a subject. But why is it that $$\ln 1 = 0$$
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What is a dual space?

I've started studying differential geometry by myself and I ran into dual spaces in a section on 1-forms. I'm not very well versed in linear algebra so any help is much appreciated.
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Non-analytic proof that Beta-Binomial $BB(1,1,n)$ (i.e. $Binom(n,p)$ with $p\sim Unif(0,1)$) is discrete uniform on $[0,n]$

This is equivalent to showing that $$P_n(k)=\int_0^1\binom n kx^k(1-x)^{n-k}\,dx=\frac 1 {1+n}\tag 1$$ for $k\in [n]$. A single integration by parts yields $P_n(k)=P_n{(k+1)}$ for $k\in[n-1]$ which ...
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Geometric justification for the prime spectrum and “generic points”

I realize there have been plently of discussions about this, but most of them are over my head and I never understand the geometric intuition behind them. I'm trying to make a big list of ...
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68 views

Intuition behind Vacuous proofs

My book says we can quickly prove the conditional statement $P \implies Q $ when we know $P $ is false. This much I'm fine with as I can show it with a truth table. But then I'm asked to, using ...
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55 views

How to read formal proofs

I'm in discrete math and I see the following notation $\forall x (P(x) \implies Q(x))$ and I read it as "for all x, P of x therefore Q of x". I'm struggling to follow proofs with this kind of ...