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Ask HN: Where to Read Proofs?

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Ask HN: Where to Read Proofs?

#1
I'm attempting to self-study mathematics, and I'm having a very hard time with proofs. My formal maths education goes up to calculus II. I've never taken linear algebra, for example, and trying to learn it with a textbook like Linear Algebra Done Right is very difficult. I have a terrible time even making it through the first set of exercises once proofs come into the picture.

I've heard many many times that you can only learn math by doing it, which is certainly true, and is akin to saying that you can only learn a language by attempting to speak it. But to begin to learn to speak well, one must hear tons and tons of speech. Similarly, to begin to learn to write well, one must read tons and tons of writings.

Are there any resources for people who just want to read proofs? Preferably well-commented ones suited for beginners like myself who are trying more to get a feel for proof as an activity rather than trying to learn any particular branch of mathematics through them (at this point).

Re: Ask HN: Where to Read Proofs?

#2
"Reading" in mathematics is usually about doing problem sets and working through proofs step by step by yourself.

If you try to read math like a novel your brain will just go "zip...zip" and jump over important things. You really have to make math your own.

A very interesting case is

http://www.takayaiwamoto.com/Pythagorean_Theorem/Pythagorean...

because there are so many ways to do it. This book has an insane number of proofs of it

https://www.amazon.com/exec/obidos/ISBN=0873530365/ctksoftwa...

Re: Ask HN: Where to Read Proofs?

#4
I haven't read it, but I have heard good things about "Proofs from THE BOOK":

https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK

Also, to get you into the right mood, I highly recommend "Fermats Last Theorem", which is light on mathemtics but quite interesting nontheless:

https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem_(book)

and this article:

http://projectwordsworth.com/the-paradox-of-the-proof/

Re: Ask HN: Where to Read Proofs?

#6

I haven't read it, but I have heard good things about "Proofs from THE BOOK": https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK Also, to get you into the right mood, I highly recommend "Fermats Last Theorem", which is light on mathemtics but quite interesting nontheless: https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem_(book) and this article: http://projectwordsworth.com/the-paradox-of-the-proof/

I feel like I need something between Proofs from the Book and Fermat's Last Theorem. PftB is too hard, but I'm a bit beyond popular math books.

Re: Ask HN: Where to Read Proofs?

#7
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).

Re: Ask HN: Where to Read Proofs?

#8
Your approach will not work. Either continue attempting proofs or give up. It's fine to read the answers after you've tried.

> I've heard many many times that you can only learn math by doing it, which is certainly true

yes

> and is akin to saying that you can only learn a language by...

no. this is you evading the main point.

Taking a graded class with homework can help. So can finding an elementary book on a subject that interests you (topology, combinatorics, algebra, ...). Linear is dry, that may be your issue.

Re: Ask HN: Where to Read Proofs?

#9

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 exe…

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.
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