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Ask HN: How to learn mathematical proofs from scratch?

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Re: Ask HN: How to learn mathematical proofs from scratch?

#42

Post by suture »

greymalik wrote: post #8
suture wrote: post #5

Earlier quoted context omitted.

Do you want something at the level of beginning algebra? Or something more advanced?

I know how to do beginning algebra and geometry but I don’t necessarily know why the various rote techniques work. So beginner as far as intuition. But also beginner for technique for things like linear algebra and calculus that I never learned at all.

There is a book called “Number, Shape, & Symmetry”. You can download it at z-lib.org. It’s a book that will give you the flavor of mathematics and prove some of the basic algebraic properties. It does not require calculus but will require a desire to understand. It’s an art form to read mathematics and understand. I recommend the book along with using a tutor or math.stackexchange.com.

Good luck!

Re: Ask HN: How to learn mathematical proofs from scratch?

#43

Post by spekcular »

You need a good tutor. I suggest finding a math major at a local university to teach you.

The problem is that, from what I can infer from your description, you don't have the fundamental skills necessary to self-tech effectively. You could try using some of the resources mentioned by other commenters, but chances are this process will much more tedious than if you had a mentor, and you'll probably come to believe various incorrect things that you'll have to unlearn later. Real-time feedback and correction would be more effective.

Also, I disagree with some of the advice given here. (Suggesting resources on Coq and ZFC to someone asking how exponents work? Really?) Tread carefully, and prefer the recommendations of people who have experience teaching high school students and undergraduates.

Re: Ask HN: How to learn mathematical proofs from scratch?

#44

Post by patrakov »

a-dub wrote: post #20
msteffen wrote: post #16

Not quite the question you asked, but "Classic Set Theory for Guided Independent Study" by Derek Goldrei is a great self-study intro to ZFC, which is the formal foundation of any other math you'll read about. I think the early chapters are simple enough that they'd make good practice for reading and writing proofs (though I didn't encounter it until I already had some experience doing that. I still think it's an exce…

> Specifically, when learning a new field of math, the proofs sometimes don't feel rigorous to me right away, but once I get used to the the basics and the linguistic conventions, I'm more able to fill in the holes in my head the most complicated proofs i ever dabbled in were things like proofs of convergence for algorithms, but i encountered various types of proofs in several intro courses: computability and theory…

I think that if you want to go away from "proofs written in shorthand by humans for other humans", then you are talking about formal verification of proofs. So, you are talking about something like making your proofs understandable to Coq or Isabelle, or maybe reading how others did it: you can start from http://us.metamath.org/

Re: Ask HN: How to learn mathematical proofs from scratch?

#45

Post by dorchadas »

magnio wrote: post #39

I am in a math-learning Discord server led by a PhD guy, where everyone works through math books recommended by him, solves the exercises and posts the solutions to get checked. Of course, you can always ask questions. All of the math is proof-based, so we start with books that teach just that: proof writing, basic logic and set theory. Then you can branch out and learn what you like. Each person goes at their own pa…

I'm also in this server. It's honestly the single best place, with the one caveat that you must be serious about it. I've gone through proofs, basic algebra, basic analysis, linear algebra, metric spaces, and am currently doing deeper work on Ring Theory, Statistics and Set Theory, all under the PhD's guidance, and it's literally worth its weight in gold.

Re: Ask HN: How to learn mathematical proofs from scratch?

#46

Post by eterps »

thealistra wrote: post #31

For me Coq - the automated proof assistant was it. You can consider it a game where you try to prove something. And it is green and says okay when you are correct. If you just do it on paper you are never sure about if you are correct.

So, what would be a gentle introduction to Coq? Any recommendations?

Re: Ask HN: How to learn mathematical proofs from scratch?

#48

Post by martincmartin »

ookdatnog wrote: post #26

At Imperial College London there's an intro to proofs course taught with interactive exercises supported by a proof assistant, Lean. The exercises (and the proof assistant) are freely available online at https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g... . Disclaimer: I didn't fully work through this game, I never studied at ICL and I can't vouch for its effectiveness, I simply heard about it and thought…

This might not be the best introduction for someone who hasn't proved "a = b iff a - c = b - c", and e.g. may not be familiar with "if and only if," or the fact that P => Q is considered true if P is false, even if Q is also false.
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