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20

This shouldn't be surprising. There are only so many letters that we can easily access. $\Bbb{P,Q,R}$ can denote a forcing notion, and have nothing to do with probability, the rationals or the reals. $\pi$ denotes a real number, the ratio between a diameter and circumference of a circle, but in many places it also denotes a projection map, or an ...


8

A quick example that came to mind: consider $\Bbb R^n$ with the euclidean metric and $\Omega \subset \Bbb R^n$ a non-empty open and connected set. Take $p \in \Omega$. Then $\Omega \setminus \{p\}$ is also connected..... except when $n = 1 $.


7

I think this is an excellent question, and I would hope no one would down vote it. There was a time when mathematics had a similar goal. There were several issues found in the late 19th century which demonstrated that a naive approach to mathematics led to contradictions. For example, the set of all sets that do not contain themselves is a contradictory ...


6

That there exist transcendental numbers. This was fist shown by Liouville, who proved that Liouville's number: $$\sum_{i=0}^\infty10^{-i!}$$ is transcendental. The "modern" proof would be due to Cantor: There are countably many algebraic numbers and uncountably many reals. Therefore there exists a transcendental number. Proving that Liouville's number ...


6

Any of the following might mean "such that": $: \ ; \ , \ / \ | \ s.t.$ I don't know if there are others.


5

The book you are reading is excellent, and you should enjoy it and get a good deal of satisfaction. However, unless you have truly upper-echelon talent and are planning to spend the bulk of your time for a couple of years immersed in what has already been done and in building your skills, you are very unlikely to "discover something novel about prime ...


5

This sum-of-squares theorem of Fermat may qualify as an example: An odd prime $p$ is expressible as the sum of squares $x^2+y^2$ if and only if $p\equiv 1 \text{ mod } 4$. You can read this Wikipedia article (as of the most recent update to this answer) to see the difference in mental effort in the original proof by Euler, as opposed to a modern ...


5

Expanding my comment: The vector space $\mathbb{R}$ can be equipped with an order relation $\le$ such that $ (\mathbb{R},\le)$ is a total ordered vector space with the Archimedean property. Tis is not possible in $\mathbb{R}^2$: here the lexicographic order is total but not Archimedean. Another property: Rotations in $\mathbb{R}^2$ commutes, but in ...


5

In my opinion, mathematics is an art, not a science. So, this question is as relevant as asking "Is there a fundamental goal of painting, or literature, or movies?" There are many goals, some personal, some societal, some financial, some, some, some. I know I love doing math (the limited amount that I can do), and hope to continue doing it for quite a ...


5

You might enjoy The Art of Problem Solving. Here is a link: http://www.artofproblemsolving.com/


4

Usually, in elementary number theory and the like, the symbol "$\mid$", which is typeset by \mid, means "divides." That is, $2\mid 4$ means "$2$ divides $4$," whereas something like $2\not\mid 3$ would mean "$2$ does not divide $3$." The difference between : and | in set theory is really a nominal one. The symbol | is used in set-builder notation. The linked ...


4

This extension has been studied and is known as product calculus. Look here for product integral.


4

The question of proof equivalence is quite an old one! In fact, David Hilbert considered adding it (or a similar one) to his celebrated list of open problems, but finally decided to leave it out, so it is sometimes referred to as Hilbert's 24th problem. There is a rather well-established field investigating proof equivalence, though definitely no clear ...


4

Let $\mathcal C$ be a category. A Cartesian square is a commutative square$\require{AMScd}$ \begin{CD} A @>a>> B\\@VVbV @VVcV\\ C @>d>> D \end{CD} that is universal with respect to all commutative squares having the same bottom right hand corner. This means that if we have any commutative square of the form \begin{CD} F @>f>> ...


3

Kurt Heegner showed the Stark-Heegner theorem. At that time, he wasn't connected to any university, in fact, no one looked to his proof until Stark showed the same result. He wasn't young, but Yitang Zhang wasn't either.


3

You don't need to have an extensive background to understand the fundamental ideas of Knot Theory. I would suggest reading The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots by Colin C. Adams. It's an excellent introduction to the subject and it will give you a starting point for further studies. From the Preface: My hope is ...


3

While I've never used it myself, Kahn Academy is supposed to be very good for teaching yourself mathematics. It goes into some quite advanced topics as well so it should keep you covered for a while.


3

There's no royal road to doing well in math. Basically, just work your ass off, and things will come to you in due time. Don't give up hope; just keep working that ass off until there's only a nub.


3

I don't think your characterization of "the" fundamental goals of physics is accurate. At any given time, there are unsolved problems in physics, and there are unsolved problems in mathematics, and these problems maybe catch the interest of a lot of people, or maybe a smaller number of people, or maybe just a few people. I think physics and mathematics are ...


3

The idea of compact came from sequences. You can see how one might come up with the name 'compact' to describe a set where you can't have an infinite set of points that are a minimum fixed positive distance from each other; that would not be a 'compact' set. Finding a cluster point gives you way to find a limit, and that's the importance of a 'compact' set. ...


3

All nontrivial rotations in $n$ dimensions are about a single $(n-2)$-dimensional axis. This holds in dimensions 2 and 3, but fails in dimensions 4 and higher. (Of course, in $1$ dimension, there are no nontrivial rotations.)


3

I would say that computing the Fourier coefficients of a tamed function is a triviality today even at an engineering math 101 level. Ph. Davis and R. Hersh tell the long and painful story of Fourier series. I quote from their book: "Fourier didn't know Euler had already done this, so he did it over. And Fourier, like Bernoulli and Euler before him, ...


3

In the nineteenth century expressing the antiderivative of an elementary function as an elementary function was an open problem. Nowadays, Risch algorithm, which can be run on machines, decides whether such operation can be done and, if so, yields a version of the correct result. I cannot speak for past mathematicians, but I think this is a useful tool.


2

Physics provides a lot of examples of this type which can be translated into mathematical statements. One of my favorites comes from the history of the Ising model. Let us quote Wikipedia: In his 1924 PhD thesis, Ising solved the model for the 1D case. In one dimension, the solution admits no phase transition. On the basis of this result, he ...


2

Suppose you run a numerical analysis program, and it outputs $$0.5323$$ What does that even mean? Does it mean: $$x = 0.5323$$ $$0.5323 \le x < 0.5324$$ $$0.53225 \le x < 0.53235$$ $$\text{Probability}\left(\frac{|x - 0.5323|}{0.5323} < 0.01\right) > 95 \%$$ The meaning of a numerical program still must be expressed and proven ...


2

Have you discussed this with your teachers and parents? If not, bring it up! Depending on your school and location, you may try seeking: A school teacher who may volunteer to coach you. Ask the head of your school's math department for guidance. A school math club. These clubs often have a teacher sponsor and aim to expand your knowledge by training for ...


2

The $p$-norm is a generalization of the standard Euclidean norm for vectors. It is defined as: $${\parallel\mathbf{x}\parallel}_p = \left(\sum_{i = 1}^{n} \left|x_i\right|^p\right)^{1/p}$$ This is for $p > 0$. What about $p = 0$? Well, maybe we can call that the $0$-norm. The $0$-norm, is very useful in measuring the "sparsity" of a vector. A vector is ...


2

Galois theory is a branch of abstract algebra that gives a connection between field theory and group theory, by reducing field theoretic problems to group theoretic problems. It started out by using permutation groups to give a description of how various roots of a polynomial equation are related, but nowadays, Galois theory has expanded to involve ...


2

I had this problem. Of course, I wanted to solve all the problems on my own, but my time was limited by the other work I had to do. And besides, sometimes you know you're just stuck and have no idea how to proceed. My rule was to ask for help after $X$ amount of time had passed since my last piece of progress on the problem. The exact amount of time ...


2

Re: cographs, here is one point of view. Write $G$ for the category with two objects $0, 1$ and two morphisms $l, r : 0 \rightrightarrows 1$. Then a graph internal to a category $C$ is a contravariant functor $G^{op} \to C$, and dually a cograph internal to a category $C$ is a covariant functor $G \to C$. $G$ is analogous to but simpler than the simplex ...



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