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)

0
votes
3answers
166 views

Understanding the use of the Cartesian Product in the proof of $|\mathbb R\times \mathbb R|=|\mathbb R|$

Where the Cartesian Product of two sets $\mathbb A$ and $\mathbb B$ is such that $\mathbb A\times \mathbb B=\{{ (a,b)|a \in \mathbb{A}, b \in \mathbb{B}\}}$ In trying to understand the proof that ...
2
votes
1answer
47 views

matrices with determinant equals to one

we already know what does it mean the determiant of a matrix is null, it's not invertible ! but what about matrices with determinant equals to $1$ ?! I know that the determinant of matrix is the ...
1
vote
1answer
71 views

What is the practical meaning of derivatives? [closed]

I mean practically integration means sum of all components, and the integral can be visualized as the area below a curve. Is there a similar intuition or geometric meaning of the derivative?
7
votes
6answers
120 views

Intuitively understanding $\sum_{i=1}^ni={n+1\choose2}$

It's straightforward to show that $$\sum_{i=1}^ni=\frac{n(n+1)}{2}={n+1\choose2}$$ but intuitively, this is hard to grasp. Should I understand this to be coincidence? Why does the sum of the first ...
2
votes
2answers
99 views

Proofs of theorems, where picture is sufficient

A while ago I have had the pleasure to come across those lectures of Topology & Geometry by Dr Tadashi Tokieda (I do recommend watching at least the first lecture, both parts). My question is ...
1
vote
1answer
40 views

Geometric intuition of the mean value theorem of several variables

Mean value theorem Let $f:U\to \mathbb R$ be defined in the open set $U\subset \mathbb R^n$. Suppose the segment $[a,a+v]$ be contained in $U$ and the restriction $f|_{[a,a+v]}$ be continous ...
20
votes
5answers
2k views

Where does the constant increase by 2 of differences between integer square values come from?

$1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$, etc... Looking at the difference between those square values, we get: 3, 5, 7, 9, etc... The difference from one (integer) square to the ...
3
votes
2answers
53 views

Improve/extend my attempted intuitive explanation for why terms in determinant calculations have alternating signs

The determinant of a shape defined by points $(a,b)$ and $(c,d)$ as labelled in the gif below is $\left|\begin{matrix}a&c\\b&d\end{matrix}\right| = ad-bc$ The following process is the ...
2
votes
1answer
72 views

Why are the Cauchy-Riemann equations in polar form 'obvious'?

In my book on complex analysis I'm asked to prove the Cauchy-Riemann equations in polar form, which I did. However, at the end of the question the author asks why these relations are 'almost obvious'. ...
2
votes
6answers
264 views

Intuition: “If P then Q” = 'Not P or Q' [closed]

I already understand, and so ask NOT about, the Conditional Law: $P \Rightarrow Q \; \equiv \;\lnot P \vee Q$. But what's the intuition? Because I ask only for intuition, please do NOT prove formally ...
1
vote
0answers
20 views

Linear functionals defines projection operators?

The way I always understood linear functionals on a vector space $V$ is to consider then as measuring objects which give projections when they are given vectors. Now I wanted to make this a little bit ...
1
vote
5answers
95 views

Intuition: Why is the biconditional true if both statements are false?

I already know that a false statement implies anything. Because I ask only for intuition, please do NOT prove this or use truth tables (which I already understand). Source: p 333, A Concise ...
3
votes
0answers
72 views

How to think/see point-set topology abstractly?

I've started learning point-set topology this semester. I've learned basic material about: topology on a set topological space open sets closed sets clopen sets closure neighborhoods interior point ...
1
vote
3answers
84 views

When is it sufficient to use logic as proof for an intuitive answer

Say I have the following limit $$\lim_{x\rightarrow\infty}\frac{3^x}{e^{x-1}}$$ In this case it's simple enough to write it as $\lim_{x\rightarrow\infty}{3^x}{e^{1-x}}$ and then show it approaches ...
1
vote
0answers
45 views

Intuitive probability theory!?

Recently I saw something where someone had a paper with several lines on it and a needle. The length of the needle was the same as the distance between the lines. They then proceeded to say that when ...
3
votes
1answer
60 views

Why is a cartesian morphism called cartesian?

I am reading about fibred categories. After reading the definition of "vertical" morphism, I can imagine why they are named like that. What about "cartesian" morphisms? What is cartesian about them? I ...
84
votes
6answers
10k views

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 ...
0
votes
1answer
60 views

Barendregt's Substitution Lemma (lambda calculus)

I am struggling to put words on an idea used in Barendregt's Substitution Lemma's proof. (available here) The lemma states that: If x≠y and x not free in L and M, L are $\lambda$-terms: then ...
2
votes
1answer
37 views

Interpretation of alternative group structures on a given group

Let $(G,e,\circ) $ be a group with $e$ the identity element and $a \in G$ and $\circ$ the group operation. Then we can form a new group $(G_a,a,\circ_a)$ with the same underlying set as $G$ and $x ...
0
votes
0answers
35 views

Antiderivative of a position function.

Let's say you had some sort of position function, i.e. a moving ball, $f(x)=-2x^{2}+3x-7$. What is the intuitive (relates to the situation) definition of the antiderivative of this function? It ...
4
votes
2answers
58 views

Index in complex analysis is an integer : Intuition?

In complex analysis course, we prove that given a closed path $\gamma$ and $a\notin \gamma^*$ the following number: $$ \frac{1}{2i\pi}\int_{\gamma}\frac{dz}{z-a} $$ is an integer. The integral can ...
4
votes
0answers
46 views

How to show $k$-algebras are isomorphic in practice

I am working through some problems which require me to show when some $k$-algebra ($k$ a field) maps are isomorphisms. Unfortunately, I've got myself a bit confused with definitions and the like, and ...
5
votes
4answers
283 views

Why are we defining the norms on certain vector spaces the way they are?

What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers? I'm asking because I've been asked to find a norm on the ...
6
votes
2answers
485 views

Is a circle classified as an ellipse?

I read that an ellipse had $2$ focal points. So, I thought if a circle had $2$ points that were simply infinitesimally close together wouldn't it be classified as an ellipse? Help would be ...
8
votes
1answer
96 views

What the curvature $2$-form really represents?

Let $(E,\pi,B)$ be a principal bundle with structure group $G$. The connection $1$-form can be thought of as a projection on the vertical part. It allows us to characterize the horizontal subspaces as ...
1
vote
0answers
50 views

connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian

In the context of solving linear programs, the big-M method refers to adding additional variables to the problem such that there is, as far as I understand it, a trivial basic feasible solution. In ...
4
votes
2answers
97 views

How to intuit 'only if'?

I already know, and so ask NOT about, the proof of:   $A$ only if $B$   =   $A \Longrightarrow B$. Because I ask only for intuition, please do NOT prove this or use truth tables. My problem: I try ...
0
votes
0answers
55 views

Motivation For Tensor Product of R-Modules

I have recently learned about tensor products of modules,specifically the material in Dummit and Foote chapter 10 section 4. My understanding is that the construction of tensor spaces is important ...
2
votes
1answer
66 views

pointwise convergence of a filter on $\mathbb{R}^\mathbb{R}$

In my topology lecture we have defined pointwise convergence for filters on function spaces, say $\mathbb{R}^\mathbb{R}$. A filter $\varphi$ on $\mathbb{R}^\mathbb{R}$ converges pointwise to ...
4
votes
3answers
71 views

What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
1
vote
1answer
79 views

intuition about cubic splines vs quadratic splines (degree 3 vs degree 2).

my intuition about quadratic(degree 2) splines is that by the help of its three variables (in each sub-interval) you can make a piecewise differentiable function on the whole interval. in the process ...
3
votes
1answer
109 views

What is an example of a proof that explicitly relies on the law of excluded middle?

I was talking with a friend about logic and I realized she might be an intuitionist. I was looking online for a proof that explicitly uses the law of excluded middle to see if she would have an issue ...
5
votes
1answer
118 views

Intuition behind the connection one-form

When we define a connection on one principal bundle $\pi: P\to M$ with structure group $G$ we define it as an association of one subspace $H_pP\subset T_pP$ for each $p\in P$ such that $T_pP = H_pP ...
7
votes
3answers
120 views

Where did the idea of hermite interpolation came from?

I am given the Hermite interpolation formula directly in my text book without ANY explanations about how it was first made (obviously it was somehow constructed for the first time with some sort of ...
0
votes
0answers
49 views

How to prove that $2^\sqrt2$ is irrational [duplicate]

I know the result is intuitive, but is there a way to mathematically prove it. I tried taking logarithm or tried proving by contradiction, but I don't know how to proceed. Thanks.
0
votes
1answer
34 views

What does the power of an ideal *mean*?

I am stumped trying to understand Silverman's definition of $\operatorname{ord}_P(f)$, the (normalized) valuation on $\bar K[C]_P$ (which denotes the localization of a curve $C$'s coordinate ring at ...
0
votes
0answers
16 views

$h$ invariant of forms and its relation to Birch rank

Let $k$ be a field. Given a form $F_i \in k[x_1,..., x_n]$ of degree $d$, the $h$-invariant $h_k(F_i)$ is defined to be the least positive integer $h$ such that $F_i$ can be written as $$ A_1B_1 + ...
3
votes
4answers
128 views

How to visualize permutations?

I'm getting a warning that this is a subjective question, and it very well probably is. But nevertheless, it is still a valid question that helps in the studying of mathematics from my point of view. ...
0
votes
0answers
26 views

Explanation of Gauss sums' preference for the North-East?

Define the Gauss sums algebraically as follows: $$\mathcal{G}_q = \sum_{a=0}^{q-1} e^{2 \pi i a^2 / q}$$ Then the result ends up being $\sqrt{q}, 0, i \sqrt{q}, (1+i)\sqrt{q}$ depending on $q \equiv ...
6
votes
2answers
131 views

Intuition about $v\otimes w$

In Physics and Differential Geometry usually tensors of type $(k,l)$ on a vector space $V$ over $\mathbb{F}$ are defined as multilinear functions $$f : \underbrace{V\times\cdots\times V}_{k \ ...
0
votes
0answers
40 views

Intuition behind spectral radius of a graph

Suppose that I have a graph G, along with its respective adjacency matrix A. The definition of how one can compute the spectral radius of this graph is not hard to grasp, but I was wondering about the ...
5
votes
2answers
155 views

$\int_{0}^{\infty}e^{-st}h(t)dt=0 \Rightarrow h(t)=0.$

Suppose $h(t)$ is continuous function and $\int_{0}^{\infty}e^{-st}h(t)dt=0 ~\forall~ s>s_{0}$, then prove that $h(t)=0$. I know "if a function is continuous, non-negative or non-positive, and its ...
3
votes
2answers
62 views

Intuition behind non-continuous derivatives

Although the intuition behind continuity and derivatives seem pretty obvious to me, I cannot figure out what it means to have a differentiable function with a non-continuous derivative. I'm trying to ...
0
votes
3answers
113 views

'If…then…' and '…if…' and '…only if…' and 'If… only then…' statements?

Suppose you have two statements A and B and "If A then B". I am trying to think of what this implies and alternative ways of writing this. I think "If A then B" = A$\rightarrow$B = "A is ...
3
votes
1answer
54 views

What is an intuitive explanation for Birkhoff's ergodic theorem?

If I'm not familiar with measure theory, what is a good way to understand the idea behind the definitions involved, the interpretation of the theorem, and the proofs thereof? Particularly, it's not ...
24
votes
14answers
973 views

How to explain to a 14-year-old that $\sqrt{(-3)^2}$ isn't $-3$?

I had this problem yesterday. I tried to explain to the kid this: $$\sqrt{(-3)^2} = 3,$$ and he immediately said: "My teacher told us that we can cancel the square with the square root, so it's ...
0
votes
3answers
67 views

How is the following a CW complex

My professor today draw on the board a sphere and attached to half a circle of the sphere half of the boundary of a disk so the shape looked like you glue a curvy half disk to a sphere. He then said ...
2
votes
1answer
99 views

Notions of consistency / heterogeneity in sets of vector values

The problem Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$ I ...
9
votes
0answers
131 views

What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices ?

Given a set of points $x_1,x_2,...,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m$ x $m$ Euclidean Distance Matrix $D$ where $D_{ij}={||x_i-x_j||}^2$. We know a little bit about these ...
7
votes
3answers
238 views

What is the intuition behind the definition of the differential of a function?

What is the intuition behind the definition of a differential of a function in differential geometry? i.e. $$df(p)(v_{p}) =v_{p} (f)(p) $$ where $v_{p} \in T_{p} M$ is a vector in the tangent space to ...