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I am an undergraduate engineering physics student since 1 1/2 years back. I've taken classes in real analysis, complex analysis and linear algebra.

When one tries to read about math on Wikipedia the articles talk about mathematical structures, mathematical objects and so on. I'd like to know more about this, the "foundations of mathematics".

I want to know if there are any appropriate books, given my background, that would be possible to use for self-study?

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By a mathematical structure, do you mean groups, rings, fields, vector spaces, finite geometries, division rings, Modules and so on? –  user21436 Feb 10 '12 at 20:11
    
Start with any good textbook on university algebra (see prior questions). Supplement that with Shafarevich's very beautiful book Basic notions of algebra - which bursts at the seams with so much motivation and insight (and has many examples that will be of interest to a physicist). –  Math Gems Feb 10 '12 at 20:16
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See also: Saunders Mac Lane: Mathematics: Form and Function –  Math Gems Feb 10 '12 at 20:24
    
For context, you might also appreciate a historical book like the one by John Stillwell. Also, plus-one for the Saunders Mac Lane book. –  yep Feb 10 '12 at 21:25
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What do folks here think of Basic Concepts of Mathematics by Elias Zakon? It's free online: trillia.com/zakon1.html –  Ben Crowell Mar 11 '12 at 22:22

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A classic text that used to be (and still may be) a standard undergraduate text for courses in the foundations of mathematics is Introduction to the Foundations of Mathematics by Raymond Louis Wilder. You can find this book in virtually every U.S. college/university library, and apparently (from what I've just read online) Wilder's book is being reprinted by Dover Publications. I think this book would be a very good fit for what you're looking for and your background. (I've had a copy of the 1965 edition for nearly 30 years, so I'm somewhat familiar with it.)

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Here is a BAMS review by O. Frink. It's not clear if the OP is asking about such traditional "foundations" or about structuralism, universal algebra, category theory, etc. –  Math Gems Feb 10 '12 at 21:28
    
I recently discovered that a digital copy of the 1965 edition of Wilder's Introduction to the Foundations of Mathematics is freely available (and apparently legally available) on the internet. –  Dave L. Renfro Oct 10 at 14:03

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