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14
votes
5answers
1k views

What does “formal” mean?

I know the definition of formal power series, power series and polynomials. But what does the adjective "formal" mean? In google English dictionary, does it mean "9. Of or relating to linguistic or ...
13
votes
1answer
312 views

l'Hopital's questionable premise?

Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ ...
13
votes
7answers
1k views

How do we know whether certain mathematical theorems are circular?

There are countless mathematical theorems and lemmata, some of which, obviously, depend on others. My question is: how do we know that, say, Theorem $A_1$- which uses a result proved in Theorem $A_2$ ...
13
votes
7answers
735 views

Is it possible to alternate the law of mathematics?

I am freelance writer. Recently I have been planning a science fiction - just planning, nothing solid yet - and I was wondering would it be possible for some other universes that have different set of ...
13
votes
3answers
5k views

Importance of Linear Algebra

In one of his online lectures Benedict Gross comments that one can never have too much Linear Algebra. Also, looking around it seems like I can find comments to the effect that Linear Algebra has ...
11
votes
2answers
669 views

Why do mathematicians care so much about zeta functions?

Why is it that so many people care so much about zeta functions? Why do people write books and books specifically about the theory of Riemann Zeta functions? What is its purpose? Is it just to ...
11
votes
4answers
2k views

Angle brackets for tuples

I've recently noticed that use of angle brackets for writing tuples, e.g. $\langle x, y \rangle$ instead of the usual round brackets in a few books I've been reading — Lawvere's Sets for Mathematics, ...
11
votes
3answers
2k views

Correct usage of the phrase “In the sequel”? History? Alternatives?

While I feel quite confident that I've inferred the correct meaning of "In the sequel" from context, I've never heard anyone explicitly tell me, so first off, to remove my niggling doubts: What does ...
9
votes
3answers
432 views

How to start learning knot theory?

Knot theory really sounds cool and I'm very interested in it. But I'm wondering what basic knowledge it is required and how I should start learning about it. Thanks
9
votes
4answers
1k views

Why are nets not used more in the teaching of point-set topology?

I just finished working through a proof of Tychonoff's Theorem that uses nets (specifically, as a corollary of the fact that a net in a product space converges iff the projected nets in the components ...
9
votes
3answers
887 views

Best way to set up a wiki for maintaining a structured math journal

Does anyone know of a tool which Can display formulas neatly, preferably like this website without hassle. (Unlike wikipedia with :<math>) Has a wiki like ...
9
votes
4answers
1k views

Publishing elementary proofs of theorems

I'm an undergraduate student and I believe I found another proof of Heron's formula. I have a bunch of questions: I would like to publish this "proof" (I haven't found mistakes yet) in some magazine. ...
9
votes
4answers
1k views

What is the deepest / most interesting known connection between Trigonometry and Statistics?

I'm teaching both at the same time to different classes in high school, so I just wondered about this. Added by OP on 16.May.2011 (Beijing time) I mean Statistics only, without Probability. In ...
8
votes
0answers
111 views

Why are the fundamental theorems of calculus usually associated to the Riemann Integral?

I am writing a "textbook" on Analysis, and I've reached the time I must talk about integrals. I prefer to approach directly the Lebesgue Integral theory. This question is not about the status of this ...
8
votes
2answers
1k views

When can one expect a classical solution of a PDE?

When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer ...
7
votes
4answers
458 views

Mere coincidence? (prime factors) [closed]

Whether some things in mathematics are mere coincidences might keep philosophers busy for 100,000 aeons, but maybe when such a coincidence gets exploited then it's not a "mere" coincidence any more. ...
7
votes
2answers
330 views

self studying advice on analysis

I am trying to learn analysis on my own but there are times when I can't solve the problem or I get the solution wrong after looking it up, but I will only look up the problems online after I am ...
7
votes
6answers
459 views

High School Mathematics

Could you recommend any high school mathematics books that are rigorous and present high school mathematics in a higher level.I just realized that I only memorized a lot of things that I was taught in ...
7
votes
2answers
1k views

The way into set theory

Given that I am going through Munkres's book on topology , I had to give a glance at the topics included in the first chapter like that of Axiom of choice, The maximum principle, the equivalence of ...
7
votes
2answers
241 views

terminology: what is meant if someone writes “calculus of ..”?

This question might be a little soft as it does not have a definite answer, so I hope I do not break the conventions of this forum by posting it here. I have now come across the term "calculus of ...
7
votes
8answers
15k views

Back to basics: What is the fastest way to multiply two digit numbers?

I been playing different math games on my android lately (for example: Math Cruncher). I've noticed that i'm unable to quickly (under 7-8 seconds) multiply two digit numbers (i.e $ 18 * 17$). So my ...
7
votes
2answers
679 views

Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$? [closed]

Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$? Experimenting a bit I also found $\zeta(\frac{8}{3}) \approx e^\frac{1}{4}$, $\zeta(\frac{31}{9}) \approx e^\frac{1}{8}$ and ...
6
votes
2answers
214 views

Zorn's Lemma and Injective Modules

In my study of injective modules over commutative rings, i noticed that Zorn's Lemma is often employed in the proofs. Here are three examples: 1) Baer's Criterion 2) the characterization of injective ...
6
votes
2answers
212 views

Is $(-\infty,\infty)$ a closed **interval**?

Note that we are working in the reals, not the extended reals. Do you understand a closed interval as "an interval that is a closed set" or as "an interval that includes both its endpoints"? If the ...
6
votes
4answers
1k views

Self-learning Mathematics

My question is very elementary, but I hope that you will endure it. I am currently in a CS Master's program and I notice that my skills in mathematics are lacking. My calculus course was 10 years ...
6
votes
2answers
210 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 ...
6
votes
3answers
523 views

Learning about the universe or special/general relativity

I have done a standard course in differential geometry/Riemannian geometry. Am I now able to understand the concepts people talk about when they say things like "spacetime is curved" and when I see ...
5
votes
4answers
362 views

What sequence should I study these topics in?

I will shortly list a series of topics that amount to what is essentially the first two years of an undergraduate degree. I'd like to know what is considered best order in which to study these ...
5
votes
0answers
147 views

Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have ...
5
votes
2answers
166 views

Size Issues in Category Theory

Barr and Wells state in their text Toposes, Triples and Theories (pdf link) It seems that no book on category theory is considered complete without some remarks on its set-theoretic foundations. ...
5
votes
5answers
349 views

Are there more real numbers than we can actually imagine?

I mean, if we could imagine all the real numbers then we could assign each number a finite sentence (or a finite book). Since the set of the finite books is countable then the set of real numbers ...
5
votes
1answer
154 views

What is the minimum required background to understand articles in the nLab?

I am interested in learning more about the nLab categorical perspective on several mathematical subjects such as topology and logic, but found that my understanding of category theory was not ...
5
votes
5answers
753 views

Is mathematics the only language that is not subject of interpretation?

Do you know any other "language" that is used by people except mathematics and is not subject of interpretation? By subject of interpratation I mean e.g. that 1 000 000 people will undertand that 1 + ...
5
votes
2answers
553 views

Advice: How can one prepare for a maths entrance exam? How can one develop mathematical thinking? [closed]

I am Computer science student so pardon me if I am asking this in a wrong place! My inspiration of learning maths is mainly due to algorithms and its feels to me that without mathematics I am not ...
4
votes
2answers
276 views

Books for inequality proofs

I was wondering: what books for proving inequalities are used in universities when studying mathematics (undergraduate)? I know there are lots of books, but I would like to know which ones are ...
4
votes
3answers
363 views

Mental Math Techniques [closed]

What are some interesting mental math techniques that you know? Here's one that I got from my Grandmother who got it from a book: To square a two-digit number (from $26$ to $49$), take the number ...
4
votes
3answers
650 views

Most inspirational mathematical books [closed]

I would like to know which books on mathematics (from university texts to divulgative pop-math books) inspired you the most. My choice is Spivak's Calculus, which is, IMHO one of the most ...
4
votes
1answer
466 views

Tips for finding the Galois Group of a given polynomial

I am currently in an introductory Galois Theory course, and I thought it would be nice to compile a list of standard tricks for finding the Galois Groups of certain polynomials. I am studying from ...
4
votes
2answers
488 views

Homological algebra in PDE

I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me. What is ...
4
votes
3answers
1k views

Help in self-studying mathematics.

Is this a reasonable list for who seek to learn mathematics by "self learning" program? and is it a well sorted list to follow? http://www.math.niu.edu/~rusin/known-math/index/index.html
3
votes
6answers
722 views

Are there broad or powerful theorems of rings that do not involve the familiar numerical operations (+) and (*) in some fundamental way?

I am of, and I would like to retain, a mindset that mathematics does not have to have numbers as the central object of interest. With that in mind, I have done a fair amount of self-study on topics in ...
2
votes
1answer
377 views

Analogy of ideals with Normal subgroups in groups.

I've started with Ideals in ring theory but still not comfortable with the analogy it has with normal subgroups in group theory.Like we can visualize normal subgroups as Is there some good intutive ...
2
votes
1answer
158 views

Understand a weird method of calculus

I see this method of calculus on youtube and my question: is this method valid? How we can understand it? Thanks.
17
votes
5answers
6k views

Is 'no solution' the same as 'undefined'?

Today in class my teacher wrote something along the lines of: $6^x = 0$ And proceed to heed a response from the class. A few people shouted undefined. So the teacher then writes: no solution ...
16
votes
5answers
787 views

Reflections on math education

Why is there such a big difference in math education between The Americas and (Europe and Asia) ? except for a few privileged who have the opportunity to access to math much earlier than the ordinary ...
13
votes
2answers
927 views

Studying mathematics efficiently

I am particularly angry at myself for the last few days. I noticed how inefficiently I work. Here is the general scenario: I decided to study abstract algebra and analysis some days back. I tried ...
13
votes
2answers
895 views

Who is a Math Historian?

In the context of classes, it is very often that discussion on the history of mathematics arises, whether it'd be on who should a lemma be attributed to or a certain event that occurred during the ...
12
votes
2answers
380 views

What was the largest ratio (result size)/(integrand size) you have seen?

Sometimes a definite or indefinite integral of a simple-looking one-liner integrand can give astonishingly huge result. What was the largest ratio of the size of shortest known closed-form result to ...
12
votes
3answers
554 views

Elementary Geometry Nomenclature: why so bad?

A long-ish wall of text, and I apologize. Some background: when I was a first-year university student, my chemistry professor was lecturing and was trying to find the word to describe a shape. A ...
11
votes
2answers
195 views

What's the idea of an action of a group?

I know the formal definition of an action over a set. I'm not asking this. What I'm asking is: what's the intuition of it? It is a way to define an algebra over a set? Since an action can exist in ...