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Ask HN: Where to Read Proofs?
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).
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- PaulHoule 4y ago"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_Theorem.html 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/ctksoftwareincA/ https://www.amazon.com/exec/obidos/ISBN=0873530365/ctksoftwa...
- Errorcod3 4y agoI just took a course on this and purchased multiple books for it. My favorite to work though because it is to the point was Book of Poof by Richard Hammack https://www.amazon.com/Book-Proof-Richard-Hammack https://www.amazon.com/Book-Proof-Richard-Hammack
- kratom_sandwich 4y agoI haven't read it, but I have heard good things about "Proofs from THE BOOK": https://en.wikipedia.org/wiki/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) https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem_(book) and this article: http://projectwordsworth.com/the-paradox-of-the-proof/ http://projectwordsworth.com/the-paradox-of-the-proof/
- tines 4y agoI 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.
- anon27182818 4y agoPerhaps the ProofWiki will be useful for your efforts: https://proofwiki.org/wiki/Main_Page https://proofwiki.org/wiki/Main_Page
- tines 4y agoI've seen this, but the proofs I've read there seem targeted toward people with a level of mathematical maturity I don't possess yet.
- TheProof 4y agoI am involved in Proofwiki and I am not mathematician, so I had to learn from scratch what it means to prove things. In one of Numberphile videos Terence Tao said that there are 3 levels of mathematical maturity. On the first level people simply use math language like any other language without paying attention to details, so their arguments quickly become sloppy and ill-defined. On the second level people study maths as a network of definitions and theorems, i.e. people need to understand what are the inputs and outputs of definitions and theorems. This becomes tedious very quickly, because every single logical step you make that is not an identity must be supported by a definition or a theorem. However, after some training people reach the third level where they understand, that some of the arguments can be omitted, so they appear to speak like people on the first level, but now they know what and why they are omitting. I guess you are trying to transition from the first to the second level. The way I learnt to prove things was to take a book, look at its solutions and try to understand what makes a "proof" a proof. This meant that for each sentence I had to write down which previous theorem allows one to connect arguments in neighboring sentences. Now, it really depends on a book you start with. Some very standard books are actually terrible references for proofs, and finding the right one involves luck. I did not use any actual book about proofs, because no matter how nice a proof for number theory is, if I am working with functional analysis, I need to study proof structure for functional analysis including tiniest lemmas that I could only learn by studying this stuff from scratch. In your case, I would say that Linear Algebra Done Right is an ok book, but you need to do some extra work. Namely, since you do not yet have a working knowledge needed for proofs, you need to make something like a dictionary, i.e. for a given chapter you need to write out all the definitions and theorems and clearly state what are their inputs and outputs. Once you have this list, given an assumption of an exercise now you can start branching out by connecting all these inputs and outputs. This is the reason why I joined Proofwiki - I use it as a giant dictionary to store such details. To a classically trained mathematician this may sound like overkill - proofs tend to become 2,3 or 4 times longer than what you find in books, but this is what it takes. With time your working knowledge will grow, you will start noticing repetitions in the proofs and you will stop using the same lemma for the n-th time. At this point you will be able to say that you own the second level, at least in some tiny area of maths. If making the dictionary I presented above is still too complicated, I may provide you some more direct help not constrained by this forum.
- ravi-delia 4y agoI 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 agoDoes 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 agoNot 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.
- kcl 4y agoYour 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.
- ravi-delia 4y agoI don't think that's fair. Basic proofs are fairly rote, but if you don't know where to start it can be challenging to replicate them. You sort of begin by learning to translate definitions into algebra, and only branching out from there. Until you get a sense for which tools to reach for when, you could bang your head against that wall for a while without making any real progress (and pick up some terrible habits in the process).
- Jtsummers 4y agoI have not read it, but have often seen this one recommended: https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-dp-1108439535/dp/1108439535/ref=mt_other?_encoding=UTF8&me=&qid= https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-... How to Prove It: A Structured Approach by Velleman. New edition came out in 2019. It appears to be aimed at your level, and pricewise isn't too bad.
- markus_zhang 4y agoI think an introductory course in Analysis, or a Number theory course, could be an entry course for proofs.
- mb7733 4y agoMost math textbooks are proofs, with lots of explanations, and then exercises that ask you to prove related theorems using similar methods. So really any math text appropriate to your level will work.
- maxminminmax 4y agohttps://www.amazon.com/Nuts-Bolts-Proofs-Introduction-Mathematical/dp/0123822173 https://www.amazon.com/Nuts-Bolts-Proofs-Introduction-Mathem... https://www.amazon.com/Art-Craft-Problem-Solving/dp/0471789011 https://www.amazon.com/Art-Craft-Problem-Solving/dp/04717890...
- capitalsigma 4y agoTake a real, formal class. You really can't self-study in my experience. I highly recommend the Harvard Extension School math 23 sequence.
- tines 4y agoIt's unfortunate that each course is over a thousand USD.
- capitalsigma 4y agoIt's a serious undergrad class with very accessible TAs, lots of resources. My experience there was orders of magnitude better than taking courses at my local public university. For proof-writing you really need an experienced human to be reviewing your work because the mistakes can be very subtle -- you'd never catch them by yourself.
- tines 4y agoI totally agree, I just don't have that kind of money right now, hence my question about reading proofs.
- rg111 4y agoYou should read Jay Cumming's "Proofs: A Long-Form Math Textbook" [0]. It is very well-written. And it actually tries to teach you rather than documenting math knowledge or impressing peers. I highly recommend it. [0]: https://www.amazon.com/gp/aw/d/B08T8JCVF1 https://www.amazon.com/gp/aw/d/B08T8JCVF1