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I can second "Proofs from the Book", but would advise against it unless you're already comfortable with the basics of linear algebra and perhaps some analysis.

I can second "Proofs from the Book", but would advise against it unless you're already comfortable with the basics of linear algebra and perhaps some analysis. For a total beginner there is no better choice than Steward and Tall's "The Foundations of Mathematics", an incredibly readable guide which takes you from high-school calculus through a good portion of intro analysis and algebra. Reading this and doing the exercises was enough to get me through my first year of real math classes. There is no praise great enough for this book, and no sufficient recommendation I could give. With that under your belt, if you'd like a "real textbook" I enjoyed Axler's "Linear Algebra Done Right". It has great exercises and should get you used to proofs done in the textbook style (though considering the high quality it may well not prepare you for the bleak world of lesser options).

tines 4y ago
Does The Foundations of Mathematics focus on performing calculations, or does it have proofs as well? I feel like I'm in a hard place because I'm not a total beginner; I've taken calculus I and II, some discrete math, and I'm fine with the basics of algebraic manipulations. But I need something to bridge the gap between calculation and proving theorems. As I mentioned in the original post, Linear Algebra Done Right is above my head at this point, I can't even do all the exercises at the end of the first section.
- ravi-delia 4y ago
  Not only does it have proofs, they're the subject of the book! You are exactly the target audience, someone who knows math but wants to make the leap to proofs. There is essentially no calculation, it starts with a lovely introduction to logical quantifiers and sets and moves on from there. By the end you'll have constructed the Reals and experimented with everything from vector spaces to the hyperreals to the basics of galois theory. LADR is definitely a good pick for after you're comfortable with proofs. They really aren't especially hard, but I remember how tricky it was until I actually learned how to start and where to go.

tines 4y ago

Does The Foundations of Mathematics focus on performing calculations, or does it have proofs as well? I feel like I'm in a hard place because I'm not a total beginner; I've taken calculus I and II, some discrete math, and I'm fine with the basics of algebraic manipulations. But I need something to bridge the gap between calculation and proving theorems. As I mentioned in the original post, Linear Algebra Done Right is above my head at this point, I can't even do all the exercises at the end of the first section.

ravi-delia 4y ago
Not only does it have proofs, they're the subject of the book! You are exactly the target audience, someone who knows math but wants to make the leap to proofs. There is essentially no calculation, it starts with a lovely introduction to logical quantifiers and sets and moves on from there. By the end you'll have constructed the Reals and experimented with everything from vector spaces to the hyperreals to the basics of galois theory. LADR is definitely a good pick for after you're comfortable with proofs. They really aren't especially hard, but I remember how tricky it was until I actually learned how to start and where to go.