The title of the question already says it all but I would like to add that I would really like the book to be more about geometric algebra than its applications : it should contain theorems' proofs. Just adding that I have never taken a course on geometric algebra. I'm a 2nd year engineering student, so a "beginner" book style will be very good!!! Also mentioning what would be the prerequisites for mastering the branch is appreciated. Thanks.
[Math] Good introductory book on geometric algebra
book-recommendationclifford-algebrasgeometric-algebrasgeometryreference-request
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Are you looking for a book on Hilbert axiomatic systems in particular, or just a book on proof theory more generally? There are many different kinds of proof systems, apart from the axiomatic Hilbert-style systems. In particular, proof theory benefits most from the sequent calculus presentation. If you're only interested in Hilbert systems, then I don't know of any resources that focus exclusively on that. If you're just interested in proof theory, there's an array of books that would suit most of your needs.
The thing about proof theory books is that, most of the time, they focus mostly on two logics: classical and intuitionistic. Those, it seems, have been the most influential in developing the course of proof theory. Still, to understand proof theory for other logics, you need to understand some of the basics for proof theory in classical and intuitionistic logic first. Negri and von Plato's Structural Proof Theory is a good place to start. Takeuti's Proof Theory develops proof theory from a different angle, focusing on arithmetic and consistency proofs. For something a little faster paced, look at Troelstra and Schwichtenberg's Basic Proof Theory. All of these are written post-2000.
If you've got some understanding of sequent calculi, then the book you're looking for is definitely Greg Restall's An Introduction to Substructural Logics. This is a fantastic book (one of my favorites actually), and I keep going back to it again and again. Only the first 7 chapters talk about proof theory, but he talks about proof systems with fewer connectives, nonclassical proof systems, modal proof systems, etc., and gives plenty of good examples. The cool thing about the substructural viewpoint is that it's an amazingly elegant way to characterize a lot of nonclassical and many-valued logics in one fell swoop. Again though, to fully appreciate it, you should be sure you know some sequent calculus before reading this book.
From the Chicago Undergraduate Mathematics Bibliography:
Gelfand/Shen, Algebra
Gelfand/Glagoleva/Shnol, Functions and graphs
Gelfand/Glagoleva/Kirillov, The method of coordinates
These three little white books come from the Soviet correspondence school in mathematics, run by I. M. Gelfand for interested people of all ages in the further reaches of the USSR. Rather than trying to be artificially "down-to-earth" in the way Americans do, Gelfand simply assumes that you can understand the mathematics as it's done (and avoids the formal complexities mathematicians are inured to). YSP and SESAME give these out by the carload to their students, who mostly love them. TMoC is notable for its intriguing four-axis scheme for making flat graphs of $\Bbb R^4$. Overall a fresh, inspiring look at topics we take for granted, and a good thing to recommend to bright younger students or friends (or parents!)
Cohen, Precalculus with unit circle trigonometry
[Rebecca Virnig] I used this book in high school and absolutely loved it. It's very skimpy on proofs, and really should not be used for that sort of insight. However, in terms of understanding how to apply various mathematical concepts it's wonderful. It has a large number of graphs, examples, and easy reference tables. It covers all the algebra, trig, and cartesian geometry that any good high school math sequence should deal with. I have used it for years as a reference book (e.g., what exactly is Cramer's rule again...) Solutions to a number of the problems are in the back, and the problems are not entirely applications.
Best Answer
The classic reference is David Hestenes' New Foundations for Classical Mechanics which is by one of the early developers of geometric algebra.
You may find it easier to learn geometric algebra from Geometric Algebra for Physicists by Doran and Lasenby though (I certainly did). The link is to a sample version of chapter 1.
A reference that I've never looked at is Geometric Algebra for Computer Science which details the geometric algebra approach to computer graphics, robotics and computer vision.
As for prerequisites - certainly some familiarity with linear algebra. For the 'geometric calculus' component a first course in multivariable calculus would be sufficient. Since the big developments in geometric algebra in the 1980s were by physicists, many of the examples tend to be physically motivated (spacetime algebras, relativistic electrodynamics etc) and a passing familiarity with (special) relativity, rigid body dynamics and electromagnetism would be useful (though certainly not essential).