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> somebody knows a book about quantum field theory that is actually mathematically rigorous

I'm not sure that this really exists. You could maybe try Peter Woit's notes, but I wouldn't stick to a mathematician's take. I've never read Steven Weinberg, but I would trust him over pretty much anyone else. In any case, you may be underestimating the difficulty of physics even given a mathematics and physics background. Consider giving Sakurai or possibly Ballentine a thorough read before delving into quantum field theory. Asher Peres has a great book too. Chances are, you haven't really considered what, mathematically, the momentum and energy operators are actually doing. Respectively, they are generators of the groups of spatial translations/rotations (depending on if you are considering linear or angular momentum) or of time translation. This is pretty important to understand clearly, and I think it's worth appreciating the physical intuition before delving too deeply into the math involved.

One of my favourite books is Matrix Analysis, by Rajendra Bhatia. I think it's a crying shame that most (all?) undergraduate curricula do not cover the calculus of matrices (as opposed to the algebra of matrices). I think it's the logical conclusion of the sequence Single Variable Calculus --> Multivariable Calculus --> Vector Calculus. In particular, one should be aware that smooth functions of a diagonalizable matrix are equivalent up to a basis change to a smooth function of the eigenvalues of that matrix. This is a consequence of the Cayley-Hamilton theorem. But then you have to worry about the nastiness of errors in specifying either matrix elements or eigenvalues. There are lots of thorny but fascinating issues to consider here. This is, to me, the real foundation of quantum mechanics. All the junk about observables needs to be appreciated in context of the ability of measurement devices to respond to the eigenvalues of a Hamiltonian.

I think it's better to keep a focussed and small list of things to read. If you have some kind of electronic reading device, you'd be better advised to put PDFs of good books/notes/articles rather than carrying a bunch of paper. But if you're in Mozambique and therefore unlikely to have reliable power or internet (never been, so I could be wrong), I think you are better advised to pick one book and work through it diligently. I'd strongly recommend Hartshorne's Algebraic Geometry for this, but that's a pretty herculean effort. Algebraic Geometry is nice, though, because it requires every aspect of mathematical thought and is beautiful to boot.

A suggestion that is not so directly related to the ones you have given: Donald Knuth's The Art of Computer Programming. It could be the most important book of the twentieth century.

As with Michael Spivak's Calculus, Apostol's two-volume Calculus is much, much more proof-centric than your introduction to calculus has been until now. That will make the material challenging but really rewarding, too.

Since you've had three semesters of calculus, I'm confident you likely have the relevant calculus background. I'd be more interested, though, in your background in reading, understanding, and writing proofs. Have you taken any such proof-based courses? In many American university curricula, for example, this is often introduced in a class like discrete mathematics. You'll likely have to be comfortable with the following, for example:

• set and function notation
  
  
• basic results in set theory, including unions, intersections, collections of subsets, and possibly countably and uncountably infinite sets
  
  
• basic results and concepts with functions, including the image and preimage/inverse image of a function
  
  
• basic ideas about sequences and subsequences
  
  
• mathematical induction
  
  
• familiarity with logical quantifiers
  
  This is just off the top of my head, but don't worry if the above list seems intimidating. What you don't already know, but will need, should be included in the text itself.
  
  Self-study is great, and I applaud your ambition in choosing this text. From my experience, you'll likely be even more successful if you can find someone to join you in self-study, especially if you don't have a teacher or professor to guide you through what will inevitably include very new material. If you can't find someone local with whom you can study together in person, then the internet may be a good way to find a fellow study partner. If you know (or are learning) LaTeX, then Overleaf or similar tools can be really useful for sharing math that's actually legible.
  
  I'd add one other remark: if memory serves, Apostol introduces integration before differentiation, something that I believe is uncommon. Since you're already familiar with both integration and differentiation, that will likely matter less for you.
  
  Good luck with your project!

Thank you for the recommendations, I'll definitely add How to Prove it to my reading list, and Spivak will be read someday for sure. It's a mathematical classic as far as I can tell.

I'm not so sure that rigorous proof is strictly required to appreciate beautiful solutions to problems. I think of mathematics much the terms outlined in chapter 1 and 12 of Mac Lane (1985) Mathematics: Form and Function where it is explained how human cultural activities give rise to certain ideas, which are formalized into neat structures which can be then serve as a source of new ideas to improve the cultural activities.

This back and forth of activity, ideas and formalisms is the best way I know of to make cultural progress. It involves art, philosophy and mathematics. It's a damn pity that our society has divided these labors into seemingly unrelated activities when they are practiced at their finest in tandem.

So a book like Bigg's Quite Right: The Story of Mathematics, Measurement and Money may not be a proof based text but it gives one a huge appreciated for the beauty of modern notation, units of measurement, algorithmic simplicity and computational resources compared to those the ancients and medieval people had to work with, for example. This is an appreciation for the beauty of certain mathematical practices that is lost when teaching children numeracy devoid of an exploration the cultural contexts our current methods evolved from.

Actually, in cultural context, it seems absurd that we spend as much time as we do teaching young children paper and pencil arithmetic and so little time on spreadsheets and other computational methods, but I digress.

RPCV checking in. This is a good idea... you're going to have a lot of downtime and it's a great opportunity to read all the things you've wanted to but haven't yet found the time for. That could mean math, or languages, or just old novels.

When I was learning functional analysis, if found this book by Bollobas to be incredibly helpful. Of course, the only real analysis reference you need is Baby Rudin, but if you want to learn measure theory you may want his Real & Complex Analysis instead.

For texts on the other subjects, take a look at this list. You should be able to find anything you need there.

If you have any questions about Peace Corps, feel free to PM me. Good luck!

I bought graph paper the summer before I started taking math classes after Calc II, and I haven't opened the package yet. I had Number Theory, Calc III, Vector Calc, Abstract Algebra I and II, a Problem Solving course and Geometry and found no need for graph paper. I saw it for a dollar and thought it might be useful. If there's anything that needs a graph, I find that a sketch works just fine.
I'm kinda weird when it comes organization. Everything needs to be exactly where I want it (I'm one of those people). So what I do is separate the binder into sections with these exact Binder Tabs. I use this Binder. You may want to try a 2" binder though. The only reason I don't is that I don't want to carry the 2" binder around.
So there are 8 sections in the binder. My last section is for extra paper. The other 7 cover 3-4 classes. I use 2 sections for a math class: one for the notes and one for the homework and other problems. I date every sheet to make finding things easier. With the remaining section after 3 math classes, I will put a random humanity or philosophy class. They only need one section. Economics took only one section for me, but I had a really easy professor. I've never taken a programming course (until this upcoming fall), but most of my friends are computer science majors (why I'm on Reddit), and I've never seen any of them take notes for any computer science classes.
Upper level math courses are fun with the right professors. A horrible subject becomes fun with a good professor; a great subject becomes dreadful with a bitch. You'll need patience to succeed. It gets really frustrating sometimes.
Also, I'd like to add that one of my teachers took notes on a tablet instead of the whiteboard, and she posted them online. The notes looked nice and I'd recommend the tablet if you feel like spending the money. I don't like using tablets, and there are many hundreds of things I'd rather do with $300, so I just stick to paper.

My pleasure :\^) It's hard to say what a local community college would have, since courses seem to vary a lot from school to school. The best thing you could find would probably be a class on something like "Set Theory" or "Mathematical Thinking" (those usually tend to touch on subjects like this without being pathologically rigorous), but a course in Discrete Math could do the trick, since you often talk about counting which leads naturally to countable vs uncountable sets. If you really want to learn the hardcore math, a course in Real Analysis is what you want. And if you don't know where to begin or are too busy, I can't recommend this book enough: http://www.amazon.com/Everything-More-Compact-History-Infinity/dp/0393339289. It's DFW so you know it's good ;)

I'm actually an undergrad studying Computer Science and Math but yes, I plan to end up a teacher after some other sort of career. Feel free to PM me if you have any more questions.

hey there the bridges conference is about your research topic. Here is a really cute video displaying some of the pieces, which there are descriptions of on the site.

This youtube channel also has a lot of other maths inspired art such as this sculpture and a cute little video on symmetry in music.

Good luck with your project!

e: also thirding the mc escher suggestion :)

e2: also if you're interested here is an accessible book (pdf)on symmetry in mathematics, which as you can imagine, ends up being a relevant topic for thinking about art.

I think the most important part of being able to see beauty in mathematics is transitioning to texts which are based on proofs rather than application. A side effect of gaining the ability to read and write proofs is that you're forced to deeply understand the theory of the math you're learning, as well as actively using your intuition to solve problems, rather than dry route calculations found in most application based textbooks. Based on what you've written, I feel you may enjoy taking this path.

Along these lines, you could start of with Book of Proof (free) or How to Prove It. From there, I would recommend starting off with a lighter proof based text, like Calculus by Spivak, Linear Algebra Done Right by Axler, or Pinter's book as you mentioned. Doing any intro proofs book plus another book at the level I mentioned here would have you well prepared to read any standard book at the undergraduate level (Analysis, Algebra, Topology, etc).

Here's an easy read that I liked: Concepts of Modern Mathematics by Ian Stewart. It gives a pretty broad overview. And you can't beat the price of those Dover paperbacks.

You may also be interested in a more thorough exploration of the history of the subject. Try History of Mathematics by Carl Boyer.

AH HA, one of the few times I will link a dover book in good heart!

http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178

Pinter offers a fine introduction to abstract algebra.

We used the Dover textbook by Pinter. It's my favorite math textbook ever, the writing was just so clear, and even entertaining and funny. We had a good professor too.

One of my favorite recent mathematics books - and one that offers a nice continuum between 'pure' mathematics and a specific application of it, as well as a nice spread of mathematical sophistication from pop math to some research-level depth, is The Symmetries Of Things by John Conway, Heidi Burgiel and Chaim Goodman-Strauss. It's an exploration of 'discrete' symmetries of the plane and of space - and of the tilings, polyhedra, etc. that they give rise to - as well as an introduction to some aspects of Coxeter groups and a (slightly out-of-place) chapter on the number of finite groups of various orders. I can highly recommend all of Conway's writing, but this is perhaps the finest instance available right now.

Try Naive Set Theory by Paul Halmos. I think it's aimed at undergraduates, so the content is a bit dense, but the style and tone is very conversational and engaging. I thoroughly recommend it.

Don't give her books just on pure math, as an undergrad in math, one of the most fascinating books I ever read was a biography of the master mathematician Leonard Euler (pronounced Oiler) http://www.amazon.com/Euler-Master-Dolciani-Mathematical-Expositions/dp/0883853280/ref=sr_1_2?s=books&ie=UTF8&qid=1450373882&sr=1-2&keywords=leonard+euler he was one of the giants in the field, overcoming the loss of multiple families members, disease, his sight and hearing, and yet was still a level of brilliant that is marveled even by today's standards.

There is a resource list on math stack for undergraduate and graduate level books inmathematics. There are similar pages like this but this has a few good ones.

​

Typical math undergrad curriculum goes: Calc 1-3, Diff eq, proofs, real analysis, linear algebra, abstract algebra, complex analysis and topology along with some electives.


i see high school students try to take on too much frequently. I'd look at real analysis andlinear algebra.

Not exactly what you are looking for, but I would recommend the classic:

How to solve it, by George Polya

https://www.amazon.com/How-Solve-Aspect-Mathematical-Method/dp/4871878309/ref=pd_lpo_sbs_14_t_1?_encoding=UTF8&psc=1&refRID=JK14NS0GEZBXYG7DA5CK

You're talking about Love and Math ( https://www.amazon.com/Love-Math-Heart-Hidden-Reality/dp/0465050743)

There is a fine book by Willian Dunham called "Euler: The Master of Us All." Take a look at amazon's preview to see if you will be comfortable with the level of the mathematics.

This webpage has a solid list of recommended textbooks: https://mathblog.com/mathematics-books/

For Linear Algebra, Linear Algebra Done Right (3rd Ed.).

This book is full of proofs you can work through. It could keep you busy for quite a while and it's considered a standard for analysis.

https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X

Good advice, but I'd add that if you do revisit calc get an intro to analysis textbook to understand how we derived the rules that calc uses. For instance, an integral is not defined as an antiderivative, that had to be proven.
Edit: My class used Principles of Mathematical Analysis by Rudin. It requires little to no initial knowledge and essentially builds multivariable calculus from the ground up.