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One thing that has captured my interest lately is the depth of mathematics. I see questions and answers discussed on Math.SE with so much different notations, terms which seem quite alien to me. Yet, I understand many things and have successfully applied them in many fields, from physics to computer science (and there to computer graphics and 3D interactive applications).

Yet, I find myself baffled by all the terms being thrown around here since I'm self-taught. I would really like to grasp things with more rigour like it is expressed in many answers (and even questions), but I don't know where to start. Isomorphisms, automorphisms, intricate notions of probability theory, measures, non-commutative rings, automorphism groups... Sheesh!

It really seems overwhelming, but is this really the case? If someone could someone point me in the direction of good books, I'd be more than willing to go through things I already have a solid understanding ( I hope ) in order to capture the formal spirit of modern mathematics.

Is such rigour compatible with intuitive understanding? I really appreciate a firm "feel" for any subject. And then when I "learn" to trust it, I can generalize it further.

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This seems much too broad for any hope of a useful answer. Is there any specific area of math you'd like to start with? –  Zev Chonoles Apr 26 '12 at 22:12
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I agree with Zev; you can't hope to learn this stuff all at once. It will take time and for now this is far too broad of a question. –  Qiaochu Yuan Apr 26 '12 at 22:33
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It is, in fact, overwhelming. Moreover, it never gets better. I'm overwhelmed every day. And I know every word on your list, but only because I'm old. You'll find that every time you cross a word off your list, you'll add two. –  user29743 Apr 26 '12 at 22:35
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I recommend trying to master something small. Most of the items on your list would be defined in a first course in algebra. Maybe buy Dummit and Foote's textbook on the subject, and try to work through some of it? –  user29743 Apr 26 '12 at 22:35
    
Mathematics has its own specialist language for concepts which mathematicians have found to be useful - as a conceptual subject, on the whole, with the process of abstraction recognised as progress (!), the specialist language of mathematics is extensive. This specialist language makes sense in its various appropriate contexts - and the best ones clarify rather than complicating, once they are properly understood. The language is learned by study of specific areas, which is more often productive than not. –  Mark Bennet Apr 26 '12 at 22:45
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4 Answers

I do not believe there are, or could be, any books that do what you ask.

Mathematics is an extremely broad and deep subject. Most likely, the words you describe are not simply esoteric labels for things you already "have a solid understanding of," but rather words for increasingly difficult abstractions. Learning these abstractions well enough to get a "feel" for them, as you put it, is certainly possible, but takes a concerted effort over months or years. In particular, if you are primarily interested in the sort of applications you mention in your question, it is unlikely to be worth your while.

On the other hand, sometimes these words do come up in describing things that could be explained in terms you would understand. Perhaps you could think of them a bit like highways: to someone who knows how to drive, highways are a great shortcut. To someone on a bicycle, they are deservedly intimidating. In the mathematical world, however, it's more like there are several dozen different kinds of highways; each kind of highway requires a different kind of car; and each kind of car takes months or years to learn how to drive. Sometimes mathematicians in different subject areas will be unable to understand each other's descriptions of the same phenomenon, because they are taking different "highways."

If you do want to attempt "learn to drive," I would not recommend starting with Wikipedia: it may be a useful resource, but it's more like a car manual than an instructor. You may want to start with the first couple of chapters of either of the two books Algebra (Michael Artin) or Abstract Algebra (Dummit and Foote). These should be enough to give you a "feel" for the sort of abstract thinking that is required.

For the record, here is a rough list of where the specific words you mention might be defined:

Isomorphism: this is ubiquitous in mathematics, but is (arguably) most naturally introduced in abstract algebra. See one of the two books I mention above.

Automorphism, Automorphism of groups: Again, these are algebraic notions, as is the notion of "group"--which, incidentally, means something quite different from a "set."

Probability theory: I don't really know. This one is outside my expertise.

Measures: I recommend Royden, Real Analysis. Another often-used (more difficult) option is Rudin, Real and Complex Analysis. Alternatively, a textbook on theoretical probability (of which I don't know any offhand) should discuss and define measures in a different context.

Non-commutative rings: This is again an algebraic notion. Of the two sources I mentioned, Dummit and Foote says much more about non-commutative rings than Artin, but in both cases, the topic does not show up in the first few chapters.

One final source: There's a short book by Curtis called Abstract Linear Algebra that introduces most of the algebraic words you bring up. I would not recommend attempting to study this book without a knowledgable mentor, but it does have the advantage of brevity.

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Start with Wikipedia: Isomorphism, etc.. Much of it is excellent, though often poorly presented, though some parts are awful.

Or read a book like The Princeton Companion to Mathematics with contents and sample chapters available here.

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Trying to "learn mathematics" is like trying to "learn science".

Now which branch of science would that be? Because, you know, there are scientists who study subatomic particles, and there are scientists who study galaxies. There are scientists who study the properties of physical materials, and scientists who study radio waves. There are scientists who study how the human brain works, and others who study the evolutionary history that brought about the human brain.

Attempting to "study science" is like attempting to study all of human knowledge about the physical world (not to mention aspects of the mental worlds as well).

A lot of people fail to grasp this, but mathematics is a similarly vast subject. That's what makes it so damned interesting! :-D

So, yeah, before you can meaningfully "learn mathematics", you need to decide which parts of it are interesting to you. (E.g., I find set theory utterly boring, but group theory fascinates me. Everybody is different, of course...)

HTH.

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I, too, am a Mathematical Mutt (i.e., no papers.) And,as @countinghaus has pointed out, it is almost impossible to learn everything.

Mathematics is all about the patterns and the best way to learn about a specific pattern is to chase references. Find a book that describes the pattern, then find the books in the references, and then find the books in those references, etc. If there are no circular references, you will eventually know everything about that pattern.

Of course when I started using this method in high school over 50 years ago, it involved real books. The internet makes it much simpler. Try Wikipedia, then the references on the Wikipedia page, etc.

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