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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?

#21

Post by dragon96 »

Regardless of which text you choose, I recommend writing proofs (full sentences and all) for the solutions that were harder to come by, or you feel difficulty expressing clearly. And if you can, get some feedback on the proofs. (Happy to take a look at a few if you DM me)

The process of writing will hopefully help you:

  - build awareness of when your arguments are not airtight or when you make false assumptions
  - modularize your thinking 
  - become more fluent with logical "vocabulary"
As for books, my personal favorites were Problem Solving Strategies (Engel) and Art and Craft of Problem Solving (Zeitz). They're both really approachable, have plenty of examples, and will give you a different perspective on what math can be about.

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

#23

Post by chc4 »

I had a lot of fun going through incredible.pm/ which is gamified writing set theory proofs. Following along with Software Foundations[0] also teaches you how to use the Coq proof assistant and write proofs in it, which I found entertaining. I think of proof solving as manipulating symbols from A to B, with a proof that each intermediate transformation is valid and sound.

0: https://softwarefoundations.cis.upenn.edu/

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

#24

Post by rofo1 »

Spend time on the mechanics (understanding and writing out full proofs) and spend some time on the intuition (why is this important, examples and counter-examples).

If you keep doing this, you'll build up a whole library of examples and counter-examples for various statements and you'll get a feeling of how you can approach any problem (of similar difficulty).

Start with any Real analysis I and Algebra I book, and try to understand every part of it - don't skim chapters, just work at your own pace. Your pace will improve over time, that's guaranteed.

And keep doing the same thing - work out the proofs, exercises, examples and counter-examples. Re-read old ones, etc.

A lot of math knowledge is really perfect or near-perfect understanding of the basic principles.

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

#26

Post by ookdatnog »

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 it was interesting and relevant to your question.

(Here's also a talk by the professor about his rationale for using Lean: https://youtu.be/Dp-mQ3HxgDE).

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

#27

Post by ghufran_syed »

This book has a creative commons license and is good for beginners:

https://www.people.vcu.edu/~rhammack/BookOfProof/

It’s important to get feedback on your proofs, so it’s worth learning to use Lyx which is a LaTeX editor - once you know some LaTeX, you can post questions on math.stackexchange.com and people are usually glad to help

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

#29

Post by JacobiX »

For me the book that changed my perspective on proofs was : Proofs from THE BOOK by Martin Aigner and Günter M. Ziegler. I always thought that proofs are just means to an end, but this book showed me that proofs can be more interesting than the end result. The book begins with six proofs of the infinitude of the primes, and you can read it in any order. It contains some of the most elegant proofs of mathematical theorems.

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

#30

Post by mintyDijon »

greymalik wrote: post #9
bitxbitxbitcoin wrote: post #4

Earlier quoted context omitted.

+1 for Euclidian geometry. Once you “prove” that you can find angle B and C knowing angle A it’s a pretty eye opening experience. This is why this is/was? emphasized in middle school geometry.

Any particular books you can recommend?

Euclids Elements
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