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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?2012-04-26
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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.2012-04-26
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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.2012-04-26
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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?2012-04-26
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    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.2012-04-26

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