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2
votes
1answer
304 views

True but unprovable?

I would like to ask a question about Gödel's Incompleteness Theorems which I've had in the back of my head for some time. Since I'm a student working in a completely different area of maths (my usual ...
2
votes
2answers
62 views

How to make a sum vanish?

This is a very very basic question but I just cannot think of a way to tackle it for some reason. Say I have three numbers $a,b,c$ with the sum $a+b+c\neq1$. Now if I want to make this sum equal to 1 ...
5
votes
1answer
527 views

Is the book “Naive Set Theory” from P. R. Halmos still up-to-date?

My question is, if Halmos' book "Naive Set Theory" is still up-to-date concerning contemporary mathematics, that is, is it outdated or not? I really love the books so far, and while it's clear the ...
4
votes
2answers
155 views

What happens to a great mathematician's unpublished works when they die?

When a great mathematician dies, they often leave plenty of unpublished and incomplete works in their manuscripts. As we assumed that they were a really good mathematician, most of the ideas in these ...
6
votes
2answers
244 views

Pre-requisites and references for $K3$ surfaces

I would like to know the "roadmap" to study $K3$ surfaces. Perhaps, my background might be helpful: I am an undergraduate student, who knows the basics of Differential Geometry, Topology, Complex ...
0
votes
5answers
279 views

Guessing on the SATs, is it ever better to leave it blank than to guess?

On most SAT questions, there are 5 answers of which exactly one is correct and exactly four are wrong. If one answers correctly you get $1$ point. If you answer incorrectly, you receive $-\frac14$ ...
8
votes
4answers
142 views

Is abstract algebra (mostly?) restricted to $2$-ary operators?

This may be due to my own pure ignorance but it's my experience that all abstract algebra I've been introduced to, both in actual courses and in self-studies only exclusively deals with algebraic ...
4
votes
2answers
45 views

Does $A^i \cap A^j = \emptyset, $ if $ i \neq j$?

I'm doing a bit of set theory and, of course, I'm confused. How true is it that if we have a series of cartesian products of a set, say $A^n, n< \omega$, then it necessarily holds that $A^i \cap ...
17
votes
4answers
346 views

Value in retracing mathematicians' steps (specifically Galois)?

So I'd like to learn Galois Theory, which I am probably not "qualified" for in an ordinary sense (I've never done abstract algebra, and I'm just now learning linear algebra in my vector calculus ...
14
votes
8answers
334 views

Mathematicians' manual of style

I know that there are many styles to write citations and footnotes and that they are all equally good (as long as the reference is complete), but I would like to know if mathematicians follow some ...
1
vote
0answers
110 views

Mathematics only with physics? What about biology and chemistry?

In The Mathematical Mechanic, the author "reveals how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways ...
3
votes
2answers
200 views

Will computers one day start creating and proving math for us? [closed]

I suspect in the future we might be able to build computers that research math for us. And I also suspect they will probably be way more efficient at doing research than we are. I do think this ...
1
vote
1answer
78 views

Why quotient space is needed?

I was wondering why quotient space is so important? Let say for vector space why quotient space is needed? Please explain!
0
votes
0answers
144 views

Normal distribution in nature

I applied for a job as a mathematician. In one of the test questions they asked the following: Why normal distribution is so common in nature? What do you think?
0
votes
0answers
15 views

Resource Request- Good classifcation of shapes by number of vertices?

There are many websites that classify various aspects of mathematics. For example: Integral table, a huge list of integrals oeis.org/ www.encyclopediaofmath.org/index.php/Main_Page ...
1
vote
2answers
94 views

How can we explain the discrepancy between $\rightarrow$ (IF-THEN) and $\setminus$ (A-BUT-NOT-B)?

Let $\mathbb{B} = \{0,1\}$ denote the Boolean domain, ordered in the usual way. Then $\mathbb{B}$ is a lattice. It has a join operation $\vee$ that coincides with "OR," a meet operation $\wedge$ that ...
3
votes
1answer
141 views

Is Linear Algebra the foundation of Applied Mathematics?

I've lately taken an interest in foundations of my field. While there are many important areas that contribute to Applied Mathematics (differential equations, probability & statistics, numerical ...
4
votes
0answers
215 views

Quantifying infinitely large sums such as $\sum_{x\in\mathbb{R}^+} x$

I thought of this as a student in calculus years ago, and it may be a silly kind of question. I wondered if there were notions of different sizes of infinity a series might sum to, which then lead me ...
1
vote
0answers
20 views

Is there a relationship between the integrand in Green's Theorem and the test for finding an integrating factor for a differential form?

Green's Theorem has the formula $$ \int_C Mdx+Ndy=\int\int_D\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy $$ There is also a well known test for finding an integrating ...
2
votes
1answer
81 views

Problem about problem solving

I am having some problems on how to solve a problem.When I read a chapter on say group theory or real analysis,I feel that I have grabbed the concepts quite well,but when I start solving exercises ...
0
votes
1answer
149 views

Platonist research on the cardinality of the reals

Apologies to any formalist! Here's the basic thought: $\mathbb{R}$ is a well-defined concept with unambiguous meaning in reality. Everyone can imagine an infinite series of digits (signifying the ...
1
vote
1answer
301 views

Differential-geometry textbook with solved problems

I'm looking for a textbook in differential geometry which inside has exercises with (at least) final answers. Since it's my first course in differential geometry it doesn't have to cover material (we ...
6
votes
2answers
247 views

Why is there apparently a consensus on the P = NP question?

So through my years of education I have heard a lot about the famous $\mathrm{P}=\mathrm{NP}$ problem. I have seen that a significant number of mathematicians believe that this result is false (and ...
2
votes
1answer
42 views

Difference between contradiction and paradox?

In multivalued logic one can distinguish at contradictions (of the type $P\wedge\neg P$) and paradoxes (of type $P\leftrightarrow \neg P$). How about in mathematics? Does the appearance of ...
2
votes
0answers
43 views

Polynomial decomposition

I've just recently learned about the neat algorithm that, given a polynomial $f$ finds (non linear) polynomials $h,g$ such that $$f = g \circ h \quad (1),$$ or decides that there are no such ...
2
votes
3answers
305 views

Existence of numbers such as $\pi^{-1}$

For my non-mathematics students (this particular class are computing), I would define $\displaystyle \frac{1}{n}$ for $n\in\mathbb{N}$ as the solution of the equation $$nx=1,$$ and then ...
5
votes
0answers
96 views

Can we consider a hypergeometric function as a closed-form?

Let's say a calculus problem like an integral or a series has a solution that inevitably involving a hypergeometric function. It turns out that hypergeometric function cannot be expressed in term of ...
6
votes
4answers
284 views

Combinatorial group theory books

I would please like some recommendations for an introductory level book on combinatorial group theory, by which I mean a group theory book which places emphasis on generators and relations and free ...
-1
votes
1answer
92 views

Most general mathematical framework

One can think of the same mathematical object in many different ways. For example take $\mathbb{R}$. One can think of this as (assume necessary hypotheses and so on) As a group. As a one ...
1
vote
1answer
113 views

Are the quaternions obsolete in pure mathematics?

I remember I read an article saying that "The quaternions $\Bbb{H}$ are obsolete in pure mathematics since the theory of vectors has been developed enough, however it is useful in computer science". ...
0
votes
1answer
87 views

2 x 2 matrix game and expected value

I have found the optimal strategy for the row player and column player. How do I find the expected value of the game for the row player and determine whether the game is favourable to the row player ...
2
votes
3answers
133 views

Prove that $S= \{ (x,y) : x^2 - y^2 <1 \}$ is open in $\mathbb{R}^2$

Prove that $S= \{ (x,y) : x^2 - y^2 < 1 \}$ is open in $\mathbb{R}^2.$ The question itself is rather easy and trivial by observing the $S$ in $\mathbb{R}^2$ geometrically. But if we are ...
0
votes
1answer
73 views

Discuss the following graphs(Differential Equations)

So I have a differential equations midterm coming up soon, and in my last exam I messed the graphing question up. It was very similar to the one I am posting. All the questions said was "Discuss the ...
6
votes
4answers
237 views

Software, techniques and tricks of experimental mathematics to conjecture possible closed forms

It often happens that people conjecture possible closed forms of integrals, series, and so on starting from a numerical value calculated to very high precision. What are the techniques, tricks, ...
0
votes
0answers
85 views

Does there exist some kind of irreversible transfomations on maths?

I know that this kind of transformation by itself without control can lead to contradiction because its value changes depending on the state of the function where you do the transformation. Anyway I ...
5
votes
0answers
125 views

From algebraic master degree to algebraic geomery Phd

I am a foreign master student in algebra at the final year. I'm familar with categorical algebra and have interesting in algebraic geomery and number theory. I have learned some knowledge about scheme ...
32
votes
5answers
2k views

“Stick it to the man!” Mathematical discoveries that resulted in persecution.

As the old story goes, Pythagoras and his followers were adamant that all numbers were rational, until Hippasus came along and proved that $\sqrt{2}$ (the length of the diagonal of the unit square) is ...
3
votes
1answer
52 views

Recommendation on setting the reference axis for mathematical objects

(I don't know what the title should be for this post, please change it if you have a better title. Also tags) In many situations, there arises cases that one mathematical structure embeds into ...
3
votes
4answers
333 views

Theorems in number theory whose first proofs were long and difficult

What are the examples of important theorems of number theory that has been shown to have surprisingly simple proofs though their first demonstration wasn't at all simple enough. Now simple proof is an ...
2
votes
1answer
215 views

Algebraic approach to analysis

Can topics and foundations of real analysis be interpreted and profitably explained in terms of abstract algebraic structures? If so, what papers or books (accessible to undergraduate students) ...
2
votes
1answer
174 views

How can we draw a Blaschke $3$ ellipse?

Today I read the article Ellipses and Finite Blaschke Products by Ulrich Daepp, Pamela Gorkin, and Raymond Mortini. In there they have proved very nice geometric results about per-images of Blaschke ...
6
votes
3answers
272 views

Big list of books on counterexamples and other clever observations in different topics

This question is related to Counterexample Math Books, but I'm looking for books in areas which aren't covered there (for example, number theory). In addition, books that focus on clever ...
1
vote
1answer
46 views

Fixed-point theorem restriction in numerical analysis

The Banach fixed-point theorem states that if $f:[a,b]\to [a,b]$ is $\lambda$-Lipschitz where $\lambda\in[0,1)$ is such that satisfies $|f(x)-f(y)|\leq \lambda |x-y|$ for every $x,y\in [a,b]$ (I'm ...
21
votes
6answers
3k views

Why do some mathematical ideas seem counter-intuitive?

Suppose you play the following game: There's a certain buy-in, and at every turn you flip a coin. If anytime you flip a tail, you lose the game and leave with your winnings. If you flip a head on the ...
2
votes
3answers
180 views

Undergrad level presentation on homological algebra and some related topics

I'm a TA of an introductory course about modules, category theory and homological algebra and the students have to do a 2 hour long presentation as a final exam. There's one student who really likes ...
2
votes
1answer
58 views

Intuition behind an algebraic identity

This problem Compute $$\sqrt{\sqrt{44\cdot 45 \cdot 46 \cdot 47+1}-44}$$ has a nice solution that relies on the identity $$n(n+1)(n+2)(n+3) +1 = \left(n^2 + 3n + 1\right)^2$$ a word form of ...
1
vote
1answer
59 views

What best explains the 'perspective' effect in this image?

I found this image and I would like to replicate the effect algorithmically. How would one describe this distortion effect mathematically? The rate of change of scale seems very familiar, and the ...
1
vote
2answers
85 views

Is it possible to not have irrational numbers?

(Math noob question): Is there a base that can be used like binary that produces no irrational numbers or numbers with an infinite amount of one number after the decimal (don't know the name)? I feel ...
3
votes
1answer
126 views

Intuitive understanding of logarithms

I know logarithms are supposed to be the inverse of exponential functions, and while this makes sense, it seems to me that a more intuitive and significant property is $$\log (ab) = \log(a)+\log(b)$$ ...
2
votes
0answers
360 views

Learning Roadmap to Mathematical Physics

Currently, I am a graduate student specializing in algebraic geometry. On the other hand, I have also become extremely interested in the mathematical physics. However, I am not sure what steps I ...