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14
votes
3answers
2k views

Explaining what real math is to a high school student

I think, after reading through some of the questions here and their answers, that there are many people here who share my opinion on high school mathematics that it's quite different from "real ...
5
votes
4answers
4k views

What is more elementary than: Introduction to Stochastic Processes by Lawler

I have trouble to reading this book! What book is more elementary/preliminary than this book: Introduction to Stochastic Processes by Lawler
2
votes
1answer
116 views

Finding his own path [closed]

I am a graduate student in major mathematics and now the time has come to choose a "specialization" (I have to choose 3 between 4 subjects). I like algebra a lot and I am also interested in ...
4
votes
3answers
228 views

Theory of Equations and Undergraduate Mathematics

I was browsing through a book entitled: Field Theory and Its Classical Problems by Charles Hadlock, one of the Carus Mathematical Monographs titles. In his preface he says the following: In ...
3
votes
2answers
161 views

The extent of chaos

In chaotic systems the typical situation is that at a low level trajectories of points are wild, but overall there is a nice statistical description of the system. For example, consider the ...
18
votes
1answer
618 views

How do you manage your “pedanticism”?

After I took my first analysis course and learned how to be truly "pedantic," I have always been having a dilemma of balancing myself between being "detailed" and "intuitive". I would like to ask how ...
2
votes
2answers
268 views

Texts that define the derivative as the “Anti-integral”…?

Every text that I read starts by defining differentiation then integration... but does anyone know if there is one that goes the other way? Also is there any harm in taking this approach.... to me, ...
4
votes
1answer
354 views

why algebraic structures?

According to wikipedia, an algebraic structure is an arbitrary set with one or more finitary operations defined on it. From a model theory perspective, I understand this definition as: structure with ...
4
votes
2answers
101 views

Why saying that “$x$ is an indeterminate real number” is misleading?

I'm reading: Behnke's Fundamentals of Mathematics, Vol.1 On page 23, he says: In order to indicate that a variable $x$ has the real numbers for its range, mathematicians often say that $x$ is ...
4
votes
1answer
299 views

proof that $0 = \infty$

I have constructed a proof of $0 = \infty$ that I know is incorrect, although I'm not quite sure why. it goes like this: $$0 = 0 = (1-1) + (1-1) + (1-1) + ...$$ but it is also true that $0. ...
3
votes
0answers
108 views

A few questions about nonabelian cohomology of finite groups.

I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
2
votes
1answer
111 views

How bad can perturbing a functional equation really make things?

A long time ago, I found occassion to find solutions to a functional equation of the following form $f(x-y) = f(x) - f(y) + \delta $ with $\delta \in \mathbb{R}.$ Using the same exact techniques as ...
5
votes
1answer
375 views

Academic failure [closed]

I'm an undergrad student majoring in maths in my next semester. Up until the current semester I was able to get very good grades in almost every course and I had a grade average which I think would be ...
13
votes
1answer
997 views

What is the expected mathematical repertoire of a Ph.D. program applicant?

I am an undergraduate (currently a sophomore) studying to prepare for applying to a Ph.D. program in mathematics. I have thus far structured my course selection upon the advice of a friend I met ...
4
votes
3answers
923 views

How can One Generalize a Result?

Suppose we are trying to generalize some result, for example, a well known theorem or some theory. The generalization must improve the result in some way. What are the best tactics to do this? Let me ...
114
votes
33answers
13k views

Can you provide me historical examples of pure mathematics becoming “useful”?

I'm trying to think/know about something but I don't know if my basis premise is plausible, here we go. Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because ...
1
vote
1answer
170 views

Intuition Regarding Homotopic Spaces

I am just starting to do some algebraic topology (very basic stuff) so have obviously just been introduced to the notion of homotopys, contractible spaces and homotopic spaces. It's this last one that ...
3
votes
1answer
92 views

Presentation, Reduction and Generalization in Mathematics: The Case of Linear Algebra

Apologies for the grandiose title, but it is motivated by a serious consideration. Linear algebra, LA hereafter, is an enormously interesting area of mathematics. What's more, it is fairly ...
11
votes
2answers
1k views

Is there a collection of alternative mathematical notation? (Semi-soft Question)

I'm interested in alternative systems of notation for mathematics. I've often heard how mathematical notation is illogical, inconsistent, filled with grandfather clauses that serve no purpose, and ...
36
votes
1answer
1k views

Counterexample Math Books

I have been able to find several counterexample books in some math areas. For example: $\bullet$ Counterexamples in Analysis, Bernard R. Gelbaum, John M. H. Olmsted $\bullet$ Counterexamples in ...
6
votes
0answers
122 views

Proof that there is no algebraic proof of Fundemental Theorem of Algebra [duplicate]

Possible Duplicate: Is there a purely algebraic proof of the Fundamental Theorem of Algebra? I have heard many times that there is no algebraic proof of the fundamental theorem of algebra ...
91
votes
7answers
8k views

How to read a book in mathematics?

How is it that you read a mathematics book? Do you keep a notebook of definitions? What about theorems? Do you do all the exercises? Focus on or ignore the proofs? I have been reading Munkres, Artin, ...
70
votes
11answers
10k views

Results that came out of nowhere.

Most big results in mathematics are built on years and years of groundwork by the author and other mathematicians, such as Wiles' proof of FLT or the classification of finite simple groups. ...
7
votes
1answer
587 views

Is a sustained,lifelong dedication & commitment to Mathematics worth every bit of it? [closed]

It may not be the kind of question I am supposed to ask here.I apologize if I violate any terms and conditions.But I really need an answer to it.I am in my mid 20s and just another average guy.But for ...
3
votes
2answers
275 views

Why are only the first four alternating groups are non-simple?

I know asking for intuition in math is a generally flawed approach, but can anyone give any reason why only the first four alternating groups are non-simple?
41
votes
8answers
5k views

How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards?

I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by ...
5
votes
2answers
277 views

How does one best balance learning from a “problem based book” with supplementary material?

We all know that when learning math, one has to do more than just simply read - one must try to solve problems and work actively with the material. Many books try to force the reader to participate ...
1
vote
1answer
70 views

Interesting verification of functoriality

Functors and morphisms of functors (aka natural transformations) have become powerful tools in all areas of pure (and meanwhile also applied) mathematics. There are lots of nontrivial constructions of ...
1
vote
1answer
139 views

Book recommendation request for geometric bodies (cube, pyramid, prism etc.)

Can anyone recommend books that deal with geometric bodies (cube, pyramid, prism etc.)? I haven't been able to find any.
8
votes
3answers
783 views

Some intuitive vision on abstract algebra

I'm currently on my second year of Mathematics and first one I'm learning abstract algebra. So far it's being the most interesting subject I've seen, but I have a problem with some intuitive visions ...
4
votes
1answer
989 views

Complex Variable vs Real Analysis 1

I took Real Analysis 1 last semester, and it was challenging, but not as bad as I thought it would be. I am considering taking Function of a Complex Variable this semester, but I am torn. I don't know ...
3
votes
3answers
216 views

Groups and Rings course suggestion

I am a physics undergrad, looking to explore pure maths. I apologize if this question is not appropriate for MathSE, but I couldn't resist posting it. Feel free to close it down. I haven't taken ...
-1
votes
1answer
151 views

What does it mean for a random variable to describe an experiment?

I often hear the expression, random variable (or sequence of rd's) to describes an experiment (or sequence of experiments. But that sound totally unrigorous to me: So we're given a mapping $X:\Omega ...
18
votes
2answers
486 views

A structural proof that $ax=xa$ forms a monoid

During the discussion on this problem I found the following simple observation: If $M$ is a monoid and $a \in M$ then $\{x: ax = xa\}$ is a submonoid. This is trivial to prove by checking ...
12
votes
1answer
492 views

Are there independent research groups?

I wanted to ask if someone knows how to find independent researchers or research groups. I mean, people interested in doing Math in a research level that are not related to universities neither ...
16
votes
2answers
5k views

How to go about studying chaos theory/dynamical systems/fluid dynamics in grad school with a physics background?

I'm turning to you as I find myself in need of help as to how to best go about studying something related to chaos theory/dynamical systems/fluid dynamics in postgraduate school. I'm currently in my ...
4
votes
3answers
3k views

Does the set of all sets that contain themselves contain itself?

We always hear about the paradox of the set of all sets that don't contain themselves and whether it contains itself or not. What about the set of all sets that do contain themselves? Is that an ...
5
votes
4answers
236 views

I need a lot of questions for mathematics. Algebra to calculus so that I learn by solving.

One huge problem I have with learning mathematics is that I have not got enough problems to solve, with answers. Is there a resource that I can get hundreds of mathematical questions, small questions, ...
8
votes
2answers
497 views

Learning category theory before abstract algebra

I'm reading this excellent pdf http://homepages.math.uic.edu/~acamer4/aluffi.pdf which is an algebra book, beginning with category theory and then use it for groups, rings,... My question is : Is it ...
1
vote
3answers
597 views

Looking for philosophical subject for my Bachelor Thesis

In may 2013 I have to write a Bachelor Thesis for my bachelor Mathematics. I prefer to choose a subject which involves philosophy. At the same time I have the feeling that my university wants me to ...
0
votes
1answer
139 views

Examples of dictionaries between two distinct fields of mathematics (or between “differents” structures of math).

I'd like to meet explicit examples of dictionaries between two distinct fields of Mathematics (or between two "different" structures of Mathematics). I'm not interested in the usual sense dictionary ...
1
vote
0answers
472 views

Suggestion of Books for MCQ

I'm preparing for a competitive exam for which I need to practice multiple choice questions (both of single and multiple correct type) on the following topics: Linear Algebra; Abstract Algebra; ...
3
votes
1answer
170 views

Prize for textbook aesthetics

When browsing through Alain Connes' textbook on Noncommutative Geometry, whose illustrations must have been conceived as true works of love, I was wondering if there is a recognized prize for ...
1
vote
2answers
239 views

Cantor's intersection theorem and Baire Category Theorem

From an old post in math stackexchange, I read a comment which goes as follows " I like to think of Baire Category Theorem as spiced up version of Cantor's Intersection Theorem". My question -----is ...
9
votes
5answers
2k views

Mathematics, Philosophy and writing.

Do you know of any famous mathematicians who were also philosophers? I have heard of Descartes, Plato and Leibniz. Are there other good examples, especially more modern examples? Also welcome are ...
7
votes
1answer
740 views

Liouville's proof of the existence of transcendental numbers

The existence of transcendental numbers can be shown easily by considering the cardinality of the set of solutions to polynomials with integer cofficents and the cardinality of the real numbers. It ...
0
votes
1answer
130 views

Infinite versus unendlich and double-negation

The German term for infinite is unendlich, which transliterates as non-ending, or non-finite. This is just word-play but from a constructive point of view, is the shift from a negative to a positive ...
14
votes
2answers
398 views

The double factorial notation

The double factorial is defined as $$n!! = \begin{cases} n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1 = \dfrac{(n+1)!}{2^{(n+1)/2}((n+1)/2)!} & \text{ If $n \in \mathbb{Z}^+$, is odd}\\ n \cdot ...
5
votes
1answer
355 views

Galois Theory Texts

What is a very comprehensive text regarding Galois Theory? I'm about to take a course in Galois Theory this spring, and I usually like to complement my course texts with something more rigorous, ...
3
votes
3answers
295 views

How to do diagram chasing effectively?

I am trying to teach myself some homological algebra, and the book I am using is Aluffi's wonderful Algebra: Chapter 0, which introduces homology at the end of chapter 3. I have spent a lot of time ...