Questions tagged [intuition]
Questions asking for the intuition behind some definition, conjecture, proof etc. In other words, questions designed to improve or to acquire understanding on a conceptual or intuitive level, as opposed to on a technical or formal level. When asking such a question it can be helpful to include a rough description of ones understanding of the subject at hand (on a technical level).
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Examples of eventual counterexamples
Define an "eventual counterexample" to be
$P(a) = T $ for $a < n$
$P(n) = F$
$n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to ...
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The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$
The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
There is a convoluted proof that $P(ab>c)=\frac12$. But since the ...
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Why are flat morphisms "flat?"
Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason!
Is ...
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Densities, pseudoforms, absolute differential forms and measures, differential forms, etc
Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here.
Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
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What is so geometric about symplectic geometry?
Symplectic geometry is often motivated by the Hamilton's equation which in turn are a reformulation of Newton's third law. But the subject itself is of independent mathematical interest. What I don't ...
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Should the formula for the inverse of a 2x2 matrix be obvious?
As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{...
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Compositional inversion and generating functions in algebraic geometry
The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV ...
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Intuition behind Alexander duality
I was wondering if anyone could offer some intuition for why Alexander duality holds. Of course, the proof is easy enough to check, and it is also easy to work out many examples by hand. However, I ...
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Demystifying the Caratheodory approach to measurability
Nowadays, the usual way to extend a measure on an algebra of sets to a measure on a $\sigma$-algebra, the Caratheodory approach, is by using the outer measure $m^* $ and then taking the family of all ...
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Meaningful interpretation for fixed point of a probability generating function
Suppose $f$ is the probability generating function for the Galton-Watson branching process.
What intuition makes the fact that $f(s) = s$ is the probability of extinction obvious? Moreover, can one ...
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Daunting papers/books and how to finally read them
Most people throughout their career encounter at least one paper that seems especially daunting to them.
I'm interested in real stories of how you successfully overcame that to extract the knowledge ...
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The Giraud-Benabou construction for splitting fibrations
I'm currently reading "Revisiting the categorical interpretation
of dependent type theory" and they give a very terse description of the Giraud-Benabou construction:
For a fibration $p : \mathbb E \...
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What is torsion in differential geometry intuitively?
Hi,
given a connection on the tangent space of a manifold, one can define its torsion:
$$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$
What is the geometric picture behind this definition&...
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Conceptual reason why the sign of a permutation is well-defined?
Teaching group theory this semester, I found myself laboring through a proof that the sign of a permutation is a well-defined homomorphism $\operatorname{sgn} : \Sigma_n \to \Sigma_2$. An insightful ...
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What notions are used but not clearly defined in modern mathematics?
"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
Felix Klein
What notions are used but not ...
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How to think about Beilinson's gluing data?
Let $X$ be a complex manifold, $D$ a divisor (that is globally the zero locus of a function) and $U$ its complement. Recall Beilinson's "how to glue perverse sheaves":
Given a perverse ...
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Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?
Here is a couple of examples of the similarity from Wikipedia, in which the expressions differ only in signs.
I encountered other analogies as well.
$${\begin{aligned}\gamma &=\int _{0}^{1}\int _{...
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Dévissage for a stratification in Grothendieck's Esquisse d’un programme: What is it?
I have a question about the the intuition of what Grothendieck proposed as tame topology in his "Esquisse d’un programme" as a "better suited" geometric structure in order to have ...
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Meaning of the coadjoint representation and its orbits
Given a Lie group $G$ there is a natural representation of $G$ on the dual of its Lie algebra $\mathfrak{g}^*$ given by the coadjoint representation. This representation is obtained by differentiating ...
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Is there a good way to think of vanishing cycles and nearby cycles?
Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...
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Most 'unintuitive' application of the Axiom of Choice?
It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis. Even ...
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Why is the gradient normal? [closed]
This is a somewhat long discussion so please bear with me. There is a theorem that I have always been curious about from an intuitive standpoint and that has been glossed over in most textbooks I ...
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Proofs without words
Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I ...
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What is the motivation and purpose of the Floretion group?
When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...
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Interpretation of the action in classical mechanics
In classical mechanics the dynamics on a manifold $M$ are characterised by the minimisation of a functional
$$ \min_{q \in C^\infty(\mathbb{R},M)} \int_{\mathbb{R}}L(q(t),\dot{q}(t))dt, $$
where $L:TM\...
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What is the exterior derivative intuitively?
Actually I have several related questions, not worth opening different threads:
What is the exterior derivative intuitively? What is its geometric meaning?
A possible answer I know is, that it is ...
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Justification of modular law in allegories
The modular law in modular lattices can be described as an isomorphism between opposite edges of the square $(a \land b), a, b, (a\lor b)$. A fancier way of saying this is an adjoint equivalence with ...
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Intuition for the last step in Serre's proof of the three-squares theorem
Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m (...
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Algebraic P vs. NP
I recently attended a lecture where the speaker mentioned that what he was talking about was connected to the algebraic version of the $P$ vs. $NP$ problem. Could someone explain what that means in a ...
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What heuristic arguments support Montgomery's conjecture for primes in short intervals?
I have a question regarding a conjecture due to H. L. Montgomery on the number of primes in short intervals. The conjecture apparently arises from probabilistic reasoning upon assuming the Riemann ...
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Meaning of a quantum field given by an operator-valued distribution
I am trying to grasp the basics of rigorous quantum field theory. Let me summise how the setup of non-interacting quantum field theories look like to me.
Let $\mathcal{H}$ be a Hilbert space in which ...
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Intuition behind existence of moduli space of stable curves
I'm not entirely sure that the title is what I'm looking for. What I'm really asking is for intuition as to why $\bar{\mathcal{M}_g}$ is the compactification of $\mathcal{M}_g$. I'm sure this is ...
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Why is addition of observables in quantum mechanics commutative?
I am no expert in the field. I hope the question is suitable for MO.
Background/Motivation
I once followed a quantum mechanics course aimed at mathematicians. Instead of the usual motivations coming ...
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What is Bernoulli umbra philosophically?
Well, Bernoulli umbra is an umbra whose moments are the Bernoulli numbers.
But what is it philosophically?
For instance, we can consider imaginary unit $i$
an umbra with moments $\{1,0,−1,0,1,\ldots\}$...
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What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?
Before you close for "homework problem", please note the tags.
Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about ...
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Intuition about ordinal fixed points $\alpha = \aleph_\alpha$
I wanted to ask for your intuition about ordinal fixed points $\alpha = \aleph_\alpha$, where $\aleph_\alpha$ stands for the $\alpha$-th Aleph number in the Aleph sequence of cardinalities.
For ...
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Is orientability a miracle?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$This question is prompted by a recent highly-upvoted question, Conceptual reason why the sign of a permutation is well-defined? The responses made ...
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Why does non-decreasing entropy imply actual convergence to that max entropy distribution?
Let $X_n$ be i.i.d with finite variance. Let $\bar X_n=\frac 1n \sum_{i=1}^nX_i$. It is a famous result that the continuous/differential entropy of the normalized average is non-decreasing. $$\mathrm ...
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What's the intuition for weighted limits?
I am reading Fosco's Coend Calculus and Emily Riehl's Categorical Homotopy Theory, Riehl's book motivates it in the following way,
Abstraction 1: Classical limits in terms of cones: Cones from an ...
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What does Mellin inversion "really mean"?
Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
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How rigorously can we apply the data supplied by this nonstandard attack on Kuratowski's closure-complement problem?
Suppose a student assigned an advanced version of Kuratowski’s closure-complement problem to solve—one that leaves out the standard hint about the finite upper bound of $14$—decides to look for the ...
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"Philosophical" meaning of the Yoneda Lemma
The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward.
Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...
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Meaning/origin of Seiberg-Witten equations/invariants
Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from.
We take ...
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Intuition behind the notion of distance between curves
Let $(M,g)$ be a Riemannian manifold and let $p$ and $q$ be two points on it and define $d(p,q)$ as the length of the minimizing geodesic between them. Now given two rectifiable paths $\gamma_1$ and $\...
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Is there any intuition of why the both, regularized logarithm of zero is $-\gamma$ and the regularized logarithm of Bernoulli umbra is $-\gamma$?
If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...
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Visualization of Riemann–Stieltjes Integrals
The Riemann–Stieltjes integral $\int_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating Riemann–...
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Intuition and/or visualisation of Itô integral/Itô's lemma
Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve.
See e.g. Wikipedia:Riemann sum.
The Itô integral has due to the unbounded total variation ...
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Flat manifolds are local geometric objects
In an article called synthetic geometry in Riemannian manifolds M. Gromov says that
tori (and in general flat manifolds) must be seen as local geometric objects.
He does so after making an example ...
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Intuitively, what makes Bernoulli umbra so similar to the zero divisors in split-complex numbers?
Notation. Here I will denote Bernoulli umbra (its moments are Bernoulli numbers $B_n$) as $B_-$, $B_-+1$ as $B_+$ (an umbra with moments being Bernoulli numbers except $B_1=1/2$).
I will denote the ...
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Why, and how badly, does the proof of "no percolation at the critical point in half-spaces" fail for full spaces?
The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a ...
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