1
vote
4answers
150 views

What if we removed all the irrational numbers from the real number line? [closed]

(1) Imagine drawing the real number line and tippexing out all the irrational numbers. What would the resulting shape look like- would there even be a line? (2) And what about if we ...
3
votes
1answer
25 views

Intuition behind prism operators to prove homotopy invariance of homology

I'm trying to understand the proof of homotopy invariance of induced maps on homology. However, I do not really understand the intuition behind this proof and especially what the prism operators (as ...
1
vote
1answer
65 views

“Geometric” proof of Rouche's theorem on the number of zeros?

I understand the analytic proof of Rouche's theorem as presented in Stein and Shakarchi's complex analysis - $|f(z)| > |g(z)|$ on the boundary circle C ensures that the argument principle can be ...
1
vote
2answers
95 views

Abstract Objects in Logic

I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, ...
2
votes
2answers
64 views

On prime(less)ness and composite(less)ness of 1

I was sitting in my room when suddenly my cousin came and asked me, "Why is $1$ neither prime nor composite". Well ofcourse, i was never given an explaination of that in school, it was just a ...
9
votes
1answer
139 views

A graph of all of mathematics

In mathematics, one often makes (proves) statements on the basis of: Previously proven statements Axioms I like to think of these dependencies as a directed graph, with edges from the accepted ...
1
vote
1answer
61 views

Mentally visualizing functions of complex numbers

I've recently been learning about functions of complex numbers (to complex numbers), and I can't quite fit them into my head. When I think about real functions, I tend to mentally visualize them as ...
0
votes
1answer
36 views

Borsuk–Ulam theorem for $n=2$

How one can intuitively prove the following statement: At any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures. What about a rigorous proof?
2
votes
3answers
132 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 ...
0
votes
0answers
35 views

Intuition analysis-deconstruction-reconstruction.

The following question is a refinement of this question, which caused a lot of people to give answers that were missing the point entirely, probably because the question was not clear. Being human, ...
0
votes
1answer
101 views

Books (and supporting material) that are useful in deconstructing one's intuition?

I recently came across the following problem from Paul Zeitz's book The Art and Craft of Problem Solving. Given the image below, can you find a way to connect corresponding blocks (i.e. A to A, B to ...
7
votes
2answers
119 views

How would one arrive at the formulas for divergence and curl?

It has been some years since I've taken multivariable calculus now, but there's something I really never understood: how people would discover the expressions for divergence and curl. I mean, the ...
8
votes
1answer
209 views

“Easy” (maybe not) question about dual spaces (Lineal Algebra).

Hi everyone is my first time reading about dual spaces and in one part of the notes that I read, says: The dual of the quotient space $V/U$ is naturally a subspace of $V$, namely the annihilators of ...
7
votes
4answers
245 views

How do I convince someone that $\mathbb{R}^2$ and its copy inside $\mathbb{R}^3$ are different?

One of my friends is taking a first course in linear algebra now, and one of the problems on his latest homework was to explain why $\mathbb{R}^2$ and $\{(a_1,a_2,a_3) \in \mathbb{R}^3 \mid a_3 = 0\}$ ...
12
votes
2answers
178 views

How can you describe topology to a non-mathematician without using continuous deformations?

One of the most frequently used ways to describe topology to non-mathematicians is that it studies the properties of objects that are preserved under deformations where ripping or tearing is not ...
5
votes
2answers
91 views

Intuition for differentiating beneath the integral

I apologize in advance for a vague question. There is a theorem: If both $f(x,s)$ and $\partial _sf(x,s)$ are continuous in $x$ and $s$, then $$\partial_s\int_a^bf(x,s)\,dx=\int_a^b ...
4
votes
4answers
188 views

What does “area” really mean?

My professor had an interesting statement at the beginning of first year integral calculus. What does area really mean? How do we know that the area of a circle is $\pi r^2$? Archimedes used ...
2
votes
3answers
64 views

“Asymmetric” results in maths analogous to “Parity violation” of the weak force?

Disclaimer: I'm not a physicist and I don't claim to be one so if I have any mistakes I’ll be glad to be corrected. One feature of the standard model of particle physics is that the weak force is not ...
3
votes
0answers
60 views

Proofs involving Disjunctions [Velleman, Chapter 3.5]

$\Large{{1.}}$ Are proofs using strategies $P136, P143$ always easier than those using $P140$? In the former two, only one statement (either $P$ or $Q$) must be proven. In the latter, both $P$ and ...
4
votes
1answer
121 views

Solvable and nilpotent groups, normal series and intuition

I'm reading Hungerford's algebra and I'm on Nilpotent and solvable groups chapter. Hungerford starts with: Consider the following conditions on a finite group G: i) G is the direct product ...
11
votes
1answer
233 views

“Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?

Obviously this question is a little open-ended. A lot of complex analysis seems to work primarily because we can view $\mathbb{C}$ as a finite-dimensional $\mathbb{R}$-algebra, and apply analytic and ...
6
votes
3answers
280 views

How can people understand complex numbers and similar mathematical concepts?

In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural ...
0
votes
2answers
97 views

How was Integral Calculus discovered/derived? [closed]

Can you please explain in layman terms. I don't know if this is a duplicate, but if I find one I will delete this question. The thing I don't get the most is differentials.
3
votes
0answers
40 views

Henri Poincaré writings

I have heard that Poincaré writings were very intuitive in its approach and not very formal in the arguments. I'm searching for something like this to complement my study of dynamical systems. I ...
6
votes
3answers
164 views

How much topology for graph theory?

I am writing a thesis in the context of descriptive complexity in theoretical computer science and therefore need to study a little bit of graph theory. My background is not mathematics but computer ...
0
votes
1answer
82 views

Are there any natural phenomenon that cannot be explained by mathematics? [closed]

I'm writing an essay about mathematics and it would help to know whether there are any natural phenomenon that math cannot describe. I know that math can describe what goes on inside black holes and ...
26
votes
6answers
898 views

How to develop intuition in topology?

Is there any efficient trick (besides doing exercises) to develop intuition in topology? The question is general but i would like to add my view of things. I started to teach myself topology through ...
6
votes
0answers
109 views

How strong is the analogy between spectra and abelian groups?

I am led to understand that spectra are some kind of $\infty$-analogue of (discrete) abelian groups, or perhaps more accurately, some kind of generalisation of chain complexes of abelian groups. How ...
4
votes
1answer
221 views

Explaining math solution to a 9-year-old

I have a problem called "The Prison Problem" that I need to explain to my 9-year-old cousin. I would think that he has just started learning about divisors and perfect squares, and as such, I have a ...
3
votes
2answers
129 views

Good at abstractions bad with numbers

Ever since I had an interest in math I was aware that what I'm good at and what really pulled me was the abstract thinking. My intuition for even the simplest number related concepts (modulo ...
16
votes
1answer
409 views

Learning Math Efficienctly and Succeeding in Grad School

I'm currently a second year Ph.D. student studying pure math. I've recently come to the conclusion that I must be studying wrong. Actually, more to the point, I must be thinking about mathematics ...
5
votes
1answer
126 views

Physical Meaning of Symplectic Vector Fields

The mathematics of symplectic (as well as Hamiltonian) vector fields is something that has been quite clear to me for some time, but recently I have been thinking much more about what certain ...
2
votes
1answer
91 views

Why does $e$ seem to be an intuitive number? [closed]

I often find two numbers roughly "in the same ballpark" if they are within a factor of about $e$ of each other. For example, if I know computers generally cost upward of $\$1000$, then $\$2700$ would ...
1
vote
4answers
245 views

What really is an indeterminate form?

We can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ or ...
3
votes
0answers
61 views

understanding maths by translating it to real world? Any learn to formulate Initiatives? [closed]

I am CSE graduate, I was very much interested in physics. I had several theories during my higher secondary school like i thought of vacuum as a special medium than nothingness and like gravity is ...
24
votes
2answers
586 views

Intuitive reasoning why are quintics unsolvable

I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, ...
0
votes
1answer
110 views

When does infinite intersection preserve a closed property?

There are two statements well known in Math and Computer Science: Intersection of infinite number of regular languages is not regular. Intersection of infinite number of convex sets is convex. ...
0
votes
0answers
91 views

Understanding degrees of freedom in a standard linear model

So I'm taking a class in applied regression analysis right now, and it is being taught with no theory whatsoever. I am TOTALLY lost on the concept of degrees of freedom. All I've been told / can find ...
9
votes
1answer
166 views

Could we reasonably classify finite groups?

So I have been reading some work on $p$-groups, and I noticed a particularly disturbing sentence: "There is no hope of finding a finite set of invariants that will define every p-group up to ...
6
votes
5answers
151 views

How would one define a “manifold” object in prose writing?

I have a question that I fear may raise some objection to the fact that it has been posted here, but I cannot think of a more appropriate place to pose it. I am not a mathematician; I'm a historian, ...
1
vote
1answer
99 views

Is there a way to mathematically describe “surprise”?

Let's say that there are ten people entered into a random drawing, the winner gets some large prize. If I were one of those ten people, and I were to win, then I would be pleasantly surprised. If ...
64
votes
12answers
5k views

Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
8
votes
4answers
234 views

Confused about differentiation

I'm new to calculus and have been taught that $\displaystyle \frac{dy}{dx}$ is the rate of change of y with respect to x. Does $\displaystyle \frac{dy}{dx}$ show how much the variable y changes as x ...
8
votes
2answers
186 views

What is a fiber bundle? (for non-mathematicians)

How can I explain the concept of a fiber bundle to someone with no mathematical background?
18
votes
3answers
625 views

How to Garner Mathematical Intuition

Motivated by Why Is Intuition so Important to Mathematicians but Missing from Mathematics Education? $^{1}$ by Leona Burton, I would like to learn about specific ideas or strategies to attain ...
11
votes
4answers
921 views

mathematical maturity

So, I finished my undergrad with a degree in applied mathematics, but when reading some graduate level texts and/or papers, I often find myself struggling. I eventually get there, but I often feel ...
37
votes
8answers
2k views

Intuitive meaning of Exact Sequence

I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all. Can anyone explain them for me? ...
6
votes
2answers
179 views

What happens after the cardinality $\mathfrak{c}$?

While having measure theory this year the following came in my mind: When we go from finite objects to infinite we "lose" a lot of properties. For example the summation isn't well defined ...
2
votes
0answers
59 views

Intuition behind criterion for an irreducible Markov chain to be transient

I have been looking over my notes for Markov chains, and I have come across the following: Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ ...
6
votes
4answers
284 views

Intuitive Explanation of Morphism Theorem

Is there an intuitive explanation for the morphism theorem from introductory abstract algebra? First Morphism Theorem: Let $K$ be the kernel of the group morphism $f: G \to H$. Then $G/K$ is ...