For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still are relevant to this site. Please be specific about what you are after.

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2
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0answers
73 views

Background & Advice for a self-learner of Descriptive Set Theory

A rather straight to the point soft-question: What kind of background should have somebody who wants to study properly descriptive set theory? More specifically, how much analysis should she/he ...
1
vote
4answers
94 views

Mathematical concepts that permeate algebra, geometry, and analysis? [closed]

Arguably, the concept of a linear space penetrates algebra, geometry, and analysis because we can find examples in these subfields. Except for the number field concept and the linear space concept, ...
2
votes
2answers
52 views

Ice cream issue in Lem's 'Extraordinary Hotel'

Could you clarify the ice cream issue mentioned at the end of the story The Extraordinary Hotel (pages 189-190 here)?
2
votes
2answers
128 views

Is mathematics invented or discovered? [closed]

In physics for example, and in science in general, facts are "discovered" in the sense that they arise from observing nature. A particle is discovered if we can measure its existence in nature. A law ...
2
votes
0answers
38 views

Question regarding pattern recognition in a table

Knowing this is a seemingly weird question, I do believe I should provide some additional context . One of our university professor's has a terribly peculiar habit . Namely, each year, the students ...
8
votes
1answer
388 views

How to introduce type theory to newcomer

I want to introduce (dependent) type theory to some friends having background in mathematical logic and set theory. To make this introduction easy I would like to give an informal presentation that ...
6
votes
1answer
87 views

“Visualizing” Mathematical Objects - Tips & Tricks

It has been a while since I am kind of stuck with my skills concerning the visualization of mathematical objects. Here there is the problem. First of all, let me point out that I am completely ...
0
votes
1answer
34 views

How does one learn to be creative? can one learn that?

Lately I've been wondering whether or not one can "learn" to be creative in math, i.e solve unusual questions and have unique ideas on how to approach proofs/study/etc.. This is the whole nature v.s. ...
1
vote
1answer
103 views

Can I Start Analysis? Seeking Your Advice on My Journey to Mathematics!

I am a college sophomore with double majors in mathematics and microbiology, and I have been doing independent research in the mathematical/computational biology, which really led me to love the ...
4
votes
1answer
75 views

I'm taking rings and fields this semester but don't remember group theory. Is it necessary to review everything?

I took group theory a year and a half ago so I don't remember anything. I'm taking rings and fields this semester and I'm worried I won't be prepared (group theory was already a struggle back then). ...
0
votes
0answers
25 views

What other words are good replacements for 'heat' in the phrase “heat equation” (the famous PDE), apart from 'diffusion'?

Historically, the term 'heat' has value as it hearkens back to the context in which Fourier studied the heat equation. Pedagogically, the term 'diffusion' has value as it imparts a more general, ...
4
votes
2answers
90 views

Applications (“in everyday life”) of graph theory

EDIT another idea someone gave me was to consider flows in a network that would not only depend on the node at the beginning and at the end of a vertice but also about the vertice itself, like a ...
18
votes
1answer
857 views

List of generally believed conjectures which cannot all be true

There are some conjectures which most leading experts believe in albeit no one can prove it yet. For example: $\mathcal{P} \neq \mathcal{NP}$, the Riemann hypothesis or the Collatz conjecture. My ...
18
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8answers
245 views

Intuitively, why should the coefficient of the derivative of $x^n$ be $n$?

I am able to differentiate $x^n$ with respect to $x$ from first principles using the definition of differentiation. Also it seems natural that the gradient of a finite polynomial will be one order ...
0
votes
0answers
38 views

Proof using Archimedean property and Bernoulli's inequality

I am trying to prove the theorem below (using both the Archimedean property and Bernoulli's inequality). As usual, I would like to write a highly intelligible proof. Any constructive feedback is ...
-1
votes
0answers
133 views

Does anyone know the bulgarian mathematician Georgi Bonev and his sequence?

This is a topic of seminarian work and I need some information about bulgarian mathematician Georgi Bonev and his sequence. I don't know where else I should ask. Here is some infos: The sequence ...
48
votes
13answers
3k views

How to stop forgetting proofs - for a first course in Real Analysis?

I am taking my first course in analysis. I like the subject. I study it almost on a daily basis. I try to prove theorems on my own without even looking at the hints. If I really get stuck I just read ...
2
votes
1answer
54 views

Motivating the compact-open topology

It has been a while since I studied algebraic topology, and I wanted to revisit homotopy theory. Determined to take a more sustainable approach, I started by questioning and verifying every result in ...
4
votes
1answer
75 views

Better Notation for Partial Derivatives

I'm constantly seeing questions here where people are confused about the notation $\frac {\partial f}{\partial x}$ or $\frac {\partial f}{\partial x} (x,y)$ or $\frac {\partial f(x,y)}{\partial x}$. ...
-2
votes
0answers
25 views

How can I make sure the problem I want to ask has not been presented before? [migrated]

e.g. I have a question on proving an inequality statement. If I google it Google will not recognize it and if I use Math StackExchange'search engine, it will not respond. Thank you for your help.
7
votes
3answers
405 views

Learning Mathematics using Khan Academy

I am in my late 20s learning mathematics using Khan Academy. I have always passed my math with borderline grades. Since last year I have joined Khan Academy, I have learned voracious and re-learn ...
1
vote
3answers
65 views

Which quadrant is the “first quadrant”?

In the coordinate plane split into four quadrants by the $x$- and $y$-axes, I learned (educated in a public school in the U.S.) that the "first quadrant" was the one with both $x$ and $y$ positive, ...
3
votes
1answer
80 views

Euler Vs. Diderot

I'm reading The Music of the Primes by Marcus Du Sautoy and I came across a page with the following excerpt about Leonhard Euler: "The role of the court mathematician is perfectly illustrated by a ...
2
votes
1answer
80 views

Modern algebra and set theory: ZFC vs. NBG

This may be somewhat of a philosophical question and is probably nitpicking, but it is also one that has always bothered me a little: Is it not more natural consider NBG set theory as the foundation ...
-2
votes
1answer
83 views

Disproving cantor's diagonalisation [closed]

Cantor diagonalisation says that set of all sequences is 0 and 1 are uncountable. But, I believe if we can show it's 1 to 1 correspondence with integers, then it is countable. For any finite ...
0
votes
0answers
22 views

Why in unanswered questions there are answered questions? [migrated]

Is it really normal to find answered questions here ? Look at this : Shadock
1
vote
0answers
67 views

How to derive this step in a book called Brownian motion calculus?

How to derive the step which result in the magnitude of slope of the path in section 1.8.2? I know it is not the definition itself.
5
votes
5answers
414 views

Is $1234567891011121314151617181920212223…$ an integer?

This question came from that one and from that talk where it's noted that "integers have a finite count of digits", so that the "number" in the title is not at all a number (not integer nor rational ...
4
votes
2answers
72 views

Multiplication Operation

I am a father of two young boys and I looks forward to exploring mathematics with them for as long as they will let me :-). I would really like for them to have a deeper understanding of mathematics ...
2
votes
0answers
81 views

Who First Considered This Generalization of the Fibonacci Numbers?

I am looking for the author who originally researched a generalization of the Fibonacci numbers, which Koshy, in Chapter 7 his book Fibonacci and Lucas Numbers with Application refers to as the ...
17
votes
1answer
190 views

How are long proofs “planned”?

I just graduated with my bachelors in mathematics last year, so I have little experience in writing huge, very involved proofs. The longest proof I've ever written was about 10 pages, but it wasn't ...
0
votes
0answers
20 views

Are there general conditions under which minimal generating sets can be expected to exist?

There exist algebraic structures $X$ with no minimal generating set. For example, $\mathbb{Q},$ viewed as an Abelian group. There also exist algebraic structures whose every generating set includes ...
2
votes
1answer
45 views

Moduli spaces, stacks and homotopy theory

For my final-year project (not this academic year but next) I'm hoping to write a relatively complete account of the basic theory of schemes used in modern algebraic geometry. My supervisor thinks ...
1
vote
3answers
41 views

judging probability by intuition is not always correct?

Suppose I have a question which goes like,"A pack of cards is counted with face downwards, and it is found that one card is missing. Two cards are drawn and are found to be spades. The odds in favor ...
0
votes
2answers
37 views

The set of the roots of all polynomials in one variable with integer coefficients

Is the following set countable:: The set of the roots of all polynomials in one variable with integer coefficients. Please show the mapping between $N$ and the above set if the set is so I think ...
67
votes
9answers
4k views

Besides proving new theorems, how can a person contribute to mathematics?

There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example: Organizing known results into a coherent ...
2
votes
1answer
101 views

Changing field: from PhD to PostDoc?

Is it possible to change the field you're working in to something other than what you did during your PhD? For example, is it possible to do a PhD studying Riemann surfaces and complex geometry ...
3
votes
1answer
106 views

Is there a system of mathematics where everything is a function?

I was wondering if there is a system of mathematics where everything (except sets) is a function. For example, 3 would be the 3 function $x \mapsto 3$. There would be basic operators, such as $+$, ...
2
votes
1answer
112 views

Real analysis “theory book” similar to Andreescu's Problems in Real Analysis: Advanced Calculus on the Real Axis

I am going through Andreescu et al.,Problems in Real Analysis: Advanced Calculus on the Real Axis and I am very impressed: the style of the book seems really modern and the material covered includes ...
8
votes
2answers
101 views

Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large \frac{1}{z}$ by definition discontinuous at $0$?

Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large{\frac{1}{z}}$ by definition discontinuous at $0$? Personally I would say: "no". In my view a function can only ...
2
votes
0answers
55 views

Generalizations of de l'Hospital rule

Are there any useful generalizations of de l'Hospital rule? Could you point out some references?
1
vote
1answer
67 views

English or French translation of a paper in German

Is there an English or French translation of the following paper (or at least a work in these languages that summarize its main results)? Stolz, O. "Ueber die Grenzwerthe der Quotienten." Math. ...
2
votes
2answers
537 views

Why the fundamental theorems of calculus are fundamental?

I can tell why the fundamental theorem of arithmetic and the fundamental theorem of algebra are fundamental, but, indeed, I cannot convince myself why the fundamental theorems of calculus are ...
13
votes
0answers
139 views

To what extent were mathematicians in previous centuries aware of the lack of rigour in their methods?

By modern standards, much of pre-modern mathematics isn't rigorous. Famous examples include Euler's solution to the Basel problem or literally anything involving sets before Cantor and Russel came ...
1
vote
1answer
29 views

Two different matrix representations of complex numbers

There are two different ways to represent a complex number with $2 \times 2$ real matrices: $$ \rho: \mathbb{C} \rightarrow M_2(\mathbb{R}) \qquad \rho(z)=\rho(a+ib)= \left[ \begin{array}{ccccc} ...
13
votes
2answers
226 views

Is it possible to use physics or other form of non-canonical reasoning to study functions?

It is well-known (see, for example, the books New Horizons in geometry, Maxima and minima without calculus and The Mathematical Mechanic) that it is possible to use some forms of "physical reasoning", ...
8
votes
0answers
135 views

A question on popularization of math: inspiring the beauty of mathematics while making New Year's wishes

Many fellow students of mine today shared by various means the following picture: . I was told that this picture is supposed to communicate the beauty of math in a funny way, but I really can't ...
1
vote
0answers
49 views

Spivak Calculus vs Abbott

I have a copy of both Spivak calculus and Apostol. I use both from time. However, I find some of Spivak's problems just boring. I was thinking of maybe trying Abbott. Especially for continuity ...
1
vote
2answers
68 views

Which discrete math book would you recommend?

I am deciding whether to buy Rosen's Discrete Mathematics and its Applications or Knuth's Concrete Mathematics. Can you guys tell me the pros and cons of each? Thank you.
1
vote
0answers
44 views

Open problems for which all cases except one have been solved

Keller's conjecture states that in any tiling of Euclidean $n$-space by identical hypercubes there are two cubes that meet face to face. The conjecture has been shown to be true for $n<7$ and ...