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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Law of Excluded Middle Controversy

I was reading an introductory book on logic and it mentioned in passing that the Law of Excluded Middle is somewhat controversial. I looked into this and what I got was the intuistionists did not ...
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Why is $x^2 \equiv 1 \pmod{x+1}$ for $x > 0$?

One day my mind wandered off and came upon the following. $x^2 \equiv 1 \pmod{x+1}~\forall x>0, x \in \mathbb{Z}$. My markdown might be a little bit broken :) I tested this out in Python for the ...
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How does scaling $\Pr(B|A)$ with $\Pr(A)$ mean multiplying them together?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I modified the following's source for concision. $1.$ Now look at ...
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4answers
2k 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 ...
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Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
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19 views

If the $\Pr($hypothesis) is due only to chance, then what is the problem?

Source: p 224, Think: A Compelling Introduction to Philosophy (1 ed, 1999) by Simon Blackburn. I capitalised miniscules, which the author uses for variables. I pursue only intuition; please do not ...
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19 views

Why is the accuracy of $\Pr($hypothesis) in Bayes's Theorem less important than apparent?

Source: p 224, Think: A Compelling Introduction to Philosophy (1 ed, 1999) by Simon Blackburn. I capitalised miniscules, which the author uses for variables. I pursue only intuition; please do not ...
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738 views

Intuitively, why does Bayes' theorem work?

Why does Bayes' theorem work? I'm not looking for a cryptic math demonstration. Rather, I'm interested in the intuition behind the theorem that reveals the a posteriori probability given the prior ...
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1answer
78 views

What is the intuition behind the Cauchy-Schwarz inequality in the real numbers?

The Cauchy-Schwarz inequality states that $$\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).$$ The proof, with the discriminant argument, is ...
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24 views

Dimension of a space VS dimension of a function in this space [on hold]

A colleague and I ran into a problem when we realised that we had a complete different understanding of dimensions. If we consider this function: $z(x,y)=x^2 + y^2$ Person A believes this function ...
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22 views

How can you picture Conditional Probability in 3D?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I modified the following's source for concision. $1.$ Now look at ...
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1answer
20 views

How can you picture Conditional Probability in a 2D Venn Diagram?

I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I pursue only intuition; do not answer with formal proofs. Which ...
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1answer
48 views

An intuitive explanation of how the mathematical definition of ergodicity implies the layman's interpretation 'all microstates are equally likely'.

I'm self-studying Statistical Mechanics; in it I got Fundamental Postulate of Statistical Mechanics and that took me to ergodic hypothesis. In the most layman's language, it says: In an isolated ...
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802 views

Intuition behind the Definition of Conditional Probability (for 2 Events)

What is some intuitive insight regarding the conditional probability definition: $P(A\mid B) = \large \frac{P(A \cap B)}{P(B)}$ ? I am looking for an intuitive motivation. My textbook merely gives a ...
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1answer
1k views

Intuition for gradient descent with Nesterov momentum

A clear article on Nesterov’s Accelerated Gradient Descent (S. Bubeck, April 2013) says The intuition behind the algorithm is quite difficult to grasp, and unfortunately the analysis will not be ...
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1answer
30 views

Intuition behind exponential sum convergence

My textbook states without proof that the summation: $$\sum_{x=0}^{\infty} \frac{1}{x!} e^{ax}$$ converges for all real $a$. I am trying to understand this. I assume the reasoning is that the ...
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412 views

Is there a more intuitive way to understand matrices and linear algebra?

When learning linear algebar I often blindly follow algorithms to perform operations with matrices without fully understanding concepts behind them. It helped a lot when I started dealing with ...
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70 views

The significance of failure of uniqueness in differential equations

The nonlinear ODE: $y'(t)=y(t)^{1/2}$ with initial condition $y(0)=1$ has two solutions. Non-uniqueness is not surprising because of the failure of Lipschitz continuity in the $y$ term. While this ...
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33 views

What is the intuition for adding vs multiplying probabilities?

Caution: I modified this original answer to simplify the examples. You add probabilities when the events you are thinking about are alternatives (eg: A soccer team scores 0 goals or 1 goal or 2 ...
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1answer
26 views

Why does the boundary of a region $D$ have enough information to dictate the value of an integral over $D$?

There are many theorems which say something along the lines of the title: The FTC: $\int_a^bf'(x)dx=f(b)-f(a)$. Green's Theorem: Let $F=(P,Q)$, then $\oint_{\partial D}Fds=\iint_D(\frac {\partial ...
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How to intuitively understand that an open subset of the reals can contain the rationals and have finite measure?

A question that one could ask is the following: if $U \subset \mathbb{R}$ is an open subset such that $\mathbb{Q} \subset U$, then is the measure of $U$ infinite? The answer is no, as the ...
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The definition of metric space,topological space

I have read some books in analysis. All of them define metric space, topological space or vector space directly, without any reason. Therefore, I want to know the background of the definition - the ...
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10answers
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Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the ...
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1answer
41 views

Limit point Compactness does not imply compactness counter-example

I think that I understand why compactness implies limit point compactness: Suppose $A \subseteq X$ has no limit points. Then $A^{\prime} \subseteq A$. Thus, $A$ is closed. Then for all $a \in A$, ...
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Intuition of LHS in Green's thorem for a velocity field

I am trying to get an intuition for the LHS part of Greens theorem. For a potential field like gravity the LHS part is work, but if the vector field is a velocity field then what does the LHS ...
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17 views

compactness requirement for the tube lemma of a product space.

The tube lemma: Let $X,Y$ be topological spaces s.t $Y$ is compact. Let $X_0 \in X$ and let $N$ be an open set in $X \times Y$ so that $x_0 \times Y$ is contained in $N$. Then there exits a ...
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144 views

How to picture a first countable space?

I find myself forgetting what it means for a space to be first countable on a frequent basis. This is unlike say other terminologies such as "Hausdorff space", where you can picture balls separating ...
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65 views

integrals with no analytic answer - intuition and proof

the following integral has no analytic solution it appears: $$\int_0^\pi e^{\sin(x)} \, dx$$ intuitively, what is the reason for this integral having no analytic answer? (is there a way to prove it ...
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110 views

How was Zeno's paradox solved using the limits of infinite series?

This is a not necessarily the exact paradox Zeno is thought to have come up with, but it's similar enough: A man (In this photo, a dot 1) is to walk a distance of one unit from where he's standing to ...
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Explain for students: Why does 0 mod n equal 0 (zero)?

I told my students that the mod operator basically gives the remainder of division, so upon seeing: 0 mod 10 Some students (apparently) reasoned that, "10 goes ...
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Geometric meaning of vanishing of higher cohomology of quasi-coherent modules over affine schemes

One of the basic vanishing results about quasicoherent (sheaves of) modules over affine schemes is that their non-zero cohomology vanishes. My only geometric intuition for sheaf cohomology is via ...
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42 views

Degrees of Freedom in Covariance: Intuition?

If we say $Var(x)$ has $n-1$ degrees of freedom which are lost after we estimate $Var(x)$, this matches how $n-1$ observations are now constrained to be sufficiently close to the remaining observation ...
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183 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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Elements of bounded distributive lattice belonging to same prime ideals are equal?

I have read in a paper that by an easy application of Zorn's lemma one may show that two elements of a bounded distributive lattice are equal iff they are contained in exactly the same prime ideals of ...
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Geometric intuition for conjunctive spaces

A topological space $S$ will be called conjunctive if for each open set $A$ containing a point $p$, there's a point $q\in S$ satisfying $\overline{\left\{q \right\}}\subset A\cap \overline{\left\{p ...
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Limits Outside of Mathematics.

The concept of the limit is one upon which the whole math of calculus is built. can anyone, however, think of an example in which the concept is applied outside of mathematics? Just anything that ...
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What is meant by finite (infinite) bernoullis?

I came across the following quote while searching for intuition behind Poisson distribution - think about a Poisson process. It really is, in a sense, looking at very, very small intervals of ...
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Why do natural transformations express the fact that a vector space is canonically embedded in its double-dual but not in its dual?

I've been struggling for quite a while to understand why a vector space is considered to be "canonically embedded" into its double dual, but not its dual. As has been remarked in many other places, ...
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1answer
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What is the intuition behind the proof of Abel-Ruffini theorem in abstract algebra?

Is there a way to explain this proof in Wikipedia without knowing the abstract algebra too much or deep function experience? In addition, I don't how the abstract algebra work even after I look at an ...
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Intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory?

iWhat is a intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory? I was thinking that textbooks make the proofs over complicate. If ...
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1answer
50 views

Question about geometric interpretation of modules

I would like to understand the accepted answer to this MO question about the geometric interpretation of modules. In particular, I would like clarification on the following excerpt. Let $R$ be the ...
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290 views

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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449 views

University-level books focusing on intuition?

I help some students with difficulties in Mathematics and Physics (especially math, physics, and engineering majors). While in high school they usually don't study, or are not interested, etc., in ...
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1answer
118 views

Initial and Final Objects in a Category

I understand the definition of initial and final objects in a category: that an object $\frak{I}$ is initial in a category $\frak{C}$ if for every other object in $\frak{C}$ (we'll just call it ...
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816 views

How to get intuition in topology concerning the definitions?

Most topology texts go on directly to give definition of topology, then they give some examples and that's it, like they directly tell you right Let $X$ be a set and let $τ$ be a family of subsets ...
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Intuition for the construction of the product topology and its equivalence to the euclidian metric

While I have been provided a proof for the previous statement, I still cannot fully grasp why the euclidian metric [ $d(x,y)=((x_1-y_1)^2+...(x_{n}-y_{n})^2)^{1/2}$] generates the same topology as the ...
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Intuition behind the Frattini subgroup

I am trying to get a better feel for what the Frattini subgroup really is, intuitively. Let $G$ be a group and denote its Frattini subgroup by $\Phi(G)$. I know that $\Phi(G)$ is the intersection of ...
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39 views

More intuition on the curl formula

I have a question regarding this quesiton. It says that $3$ simple fields that describe rotations around $x,y,z$ axis are: $$H_1(x,y,z)=(0,−z,y)\\ H_2(x,y,z)=(z,0,−x)\\ H_3(x,y,z)=(−y,x,0)$$ but why? ...
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134 views

What is the geometric meaning of representability?

Representable functors play a large role in algebraic geometry when developed through the 'functor of points' approach. One finds schemes represent Zariski sheaves and this gives access to the great ...
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31 views

What is the Lagrange remainder in a Taylor series expansion

I know what a Taylor series expansion is and I know how to find the Lagrange remainder but what does it mean intuitively? I need an explanation of what the Lagrange remainder represents in terms of ...