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.

learn more… | top users | synonyms (1)

1
vote
1answer
47 views

How to rigorously determine whether two events are independent?

Consider the following question: Tim has lost his pet in either forest A (with probability 0.4) or in forest B (with probability 0.6). If his pet is in forest A and Tim spends a day ...
2
votes
0answers
29 views

How would look a harmonic and p-harmonic series vector in a multidimensional space?

Learning about the harmonic series divergence and p-harmonics series convergence, I tried to manipulate them globally as follows. $$\lim_{n \to \infty}A \cdot I = \lim_{n \to \infty} \begin{pmatrix} ...
9
votes
1answer
97 views

Intuition behind the “infinite velocity” of a falling ladder

In Calculus there is a "classic" related rates problem involving a falling ladder. Say the ladder is $25$ ft tall and is leaning against a wall. The bottom edge of the ladder is pulled away from the ...
0
votes
1answer
39 views

Abstraction and/or concreteness - What should be emphasized

Alexandar Grothendieck was probably a mathematician focusing on theory developement and abstraction much much more than focusing on concrete examples and/or problems. In his biography, he wrote: ...
1
vote
1answer
43 views

Intuition: groups, quotient groups, cosets, homomorphisms.

If we start with the group of rational numbers $\mathbb{Q}$ and the subgroup of $\mathbb{Q}$; $\mathbb{Z}$ the integers, and then form the quotient group $\mathbb{Q}$/$\mathbb{Z}$ we have that this ...
1
vote
0answers
20 views

Concerning Approximations, Multiplicative Order, and Residue Number Systems…

I'm interested in particular representations of reals in residue number systems. Specifically, if we are given a real $0 \le n \le 1$, we wish to represent that number as a fraction in a residue ...
7
votes
3answers
71 views

Approximating $x=\sqrt{2}+1$

Suppose $y>1$ is some approximation to $x=\sqrt{2}+1$. Give a brief reason (not a proof) why one should expect $(1/y)+2$ to be a closer approximation to $x$ than $y$ is. After testing this ...
2
votes
3answers
93 views

Is there a geometric meaning associated with the condition “dot product equals $1$?”

Consider $x,y \in \mathbb{R}^n$. Then the condition $x \bullet y = 0$ is easy to understand; it just means that $x$ and $y$ are orthogonal. Question. Does the condition $x \bullet y = 1$ have an ...
1
vote
1answer
29 views

What are the examples to understand L-process and R-process?

R-processes and L-processes R-process on $ [0,\infty) $ to signify a process all of whose paths are right-continuous on $[0,\infty)$ with limits from the left on $(0,\infty)$. R-function or R- ...
1
vote
0answers
36 views

Quotient topology from Delta complex,

A $\Delta$-complex structure on a space X is a collection of maps $\sigma_{\alpha} : \Delta^n \rightarrow X$ with n depending on index $\alpha$ such that 1)The restriction ...
1
vote
2answers
39 views

Relation between Poisson Distribution and Process?

I think I'm confused by the fact that they have similar names and both involve i.i.d. exponential RVs. From what I understand, Poisson distribution: the probability of a particular number of events ...
1
vote
0answers
34 views

Group theory: Intuition as to what a group is [duplicate]

In group theory the group is an algebraic structure consisting of a set which has elements associated with definite finitiary operations. Can an intuitive explanation be provided as to what this ...
4
votes
0answers
47 views

Finding how large $p$ needs to be to have $n$ unique factors…

If we take a prime $p$, how large does $p$ have to be so that $p-1$ has at least $n$ factors between $f_1$ and $f_2$? (Note that the factors can be prime or composite) Note that I'm looking more for ...
2
votes
1answer
43 views

concentration of volume of hypersphere

I am reading about features of volume of hyperballs, where I see two theorems, Most of the volume of the d-dimensional ball of radius r is contained in an annulus of width $O(r/d)$ near the ...
1
vote
1answer
72 views

The role of visualization and intuition in graduate and postgraduate math and developing it

In Visual Complex Analysis's preface, the author gives an analogy with pseudo-deaf musicians and follows the same to mathematics. Mathmatics today, he argues, is mostly build on abstract symbolic ...
1
vote
2answers
54 views

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 ...
2
votes
1answer
36 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 ...
2
votes
2answers
26 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 ...
2
votes
2answers
92 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 ...
1
vote
0answers
31 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 ...
1
vote
1answer
37 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 ...
1
vote
2answers
52 views

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 ...
6
votes
2answers
156 views

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 ...
5
votes
2answers
92 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 ...
2
votes
1answer
38 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 ...
0
votes
2answers
49 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 ...
2
votes
1answer
29 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 ...
4
votes
0answers
57 views

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 ...
2
votes
2answers
79 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 ...
0
votes
1answer
52 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 $a \in A$, ...
0
votes
0answers
14 views

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 ...
2
votes
1answer
32 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 ...
5
votes
2answers
152 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 ...
2
votes
2answers
79 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 ...
1
vote
4answers
169 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 ...
1
vote
1answer
72 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 ...
1
vote
0answers
20 views

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 ...
1
vote
0answers
17 views

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 ...
0
votes
0answers
15 views

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 ...
3
votes
2answers
106 views

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, ...
1
vote
1answer
66 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 ...
1
vote
2answers
63 views

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 ...
0
votes
0answers
44 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? ...
1
vote
3answers
70 views

Order of an Equivalence Class and the number of Coset.

Let $ \mathcal{M} $ be the set of all subsets of finite group $G$ which have $p^{\alpha} $elements. Thus $ \mathcal{M} $ has $ {p^{\alpha}m \choose p^{\alpha}} $ elements. Given $M_1 ,M_2 \in ...
1
vote
2answers
76 views

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 ...
3
votes
0answers
31 views

Moving from sheaves over spaces to sheaves over sites

The first example of a sheaf that I have consciously come across is the sheaf of continuous (real) functions on some topological space. The fact it is a sheaf is equivalent to the pasting lemma, which ...
2
votes
2answers
51 views

Improper integral involving trigonometric function

I was wondering what happens when evaluating an improper integral involving a trigonometric function where the denominator is a rational function with a zero at $x=0$. The example I have in mind is ...
10
votes
1answer
179 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 ...
0
votes
1answer
24 views

Multiplication principle and permutation

Definition of Multiplication principle in Principles and Techniques in combinatorics by Chuan-Chong, Khee-Meng is given as: Let $$\prod_{i=1}^rA_i=A_1\times\dots\times A_r=\{(a_1,\dots,a_r) | ...
-4
votes
3answers
200 views

What does $3+2i$ apples mean? Can the simple counting analogy with apples be extended to complex numbers? [closed]

Please read, or at least skim the question. Past attempts at answering the question have ignored vital constraints provided below. Natural Numbers Imagine I have $n$, a positive natural number, of ...