Ask HN: How to learn mathematical proofs from scratch?

The most basic subject to understand mathematical proofs is Euclidean geometry. There you will learn the basics of proofs and what it means to prove something.

Let’s look at x/a = b/c. You want to show that this equation has the same exact solution set as xc = ab. In order to prove this rigorously you’ll need to prove things about associativity. You’ll also need to prove that a unit isn’t a zero divisor in the real numbers. What we see is that to prove seemingly simple statements requires some machinery and to understand the necessity of this machinery requires mathematical maturity.

But maybe you don’t want to rigorously prove the above. Maybe you just want to understand why it is plausible that this is true. For that, pick up a beginning algebra book and actually read what it says and try to understand it. This is hard to do on your own.

Here’s a plausible explanation for why x/a = b/c has the same solution set as xc=ab. Note that a and c must be nonzero because we can’t divide by zero (this requires proof!). We note that

(x/a) times a

Is the same thing as x times (1/a times a). This is due to associativity. A nonzero number times it’s reciprocal is 1. And 1 times anything is itself. So x/a times a simplifies to x.

So,starting with

x/a = b/c

I can multiply both sides by a. I can do this since a is invertible and multiplying by an invertible element preserves equality (requires proof!). So what I get, after simplifying, is

x = (b/c) times a

I can rearrange things (by associativity) to write this as

x = (ab)/c

Now multiply both sides by c to get (I skipped a step by multiplying and simplifying at the same time)

xc = ab.

Is the formatting wrong on this? Because this isn't an identity.

"Geared to preparing students to make the transition from solving problems to proving theorems, this text teaches them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. To help students construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. "

With that intuition it's simply a matter of slogging through a proof textbook like Velleman's "How to Prove It" until you have the confidence to work through the texts that truly interest you. If you don't feel like a clueless fool you're not trying hard enough. Confusion and self-doubt are sure signs you're finally learning something.

If you're missing some fundamental knowledge of algebra or other high school level math, you should refresh that. You can do so through Khan Academy (https://www.khanacademy.org/math/).

The proof course I took mostly started with examples from number theory to allow students to focus on the mechanics of writing proofs. The course seems to have replaced the book I used (Mathematical Proofs: A Transition to Advanced Mathematics Book by Albert D. Polimeni, Gary Chartrand, and Ping Zhang) with this freely accessible book https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf. This course was a prereq to the analysis, algebra, and other proof based upper level courses in math.

The most important thing is to work the examples in the chapters and solve the exercises in the back of the book. You can usually find lecture notes, problem sets, and assignment solutions on the web these days. If you're having problems with material, someone else probably has, so there is a good chance your question has been asked on Reddit or other math specific forums.

If you're more interested in math applied to computer science, then you can look for a book on discrete math. The only discrete math book I'm familiar with doesn't really make it explicit that you're learning proof techniques the way the above books do, so I can't really recommend it as a first book for self-study.

The sequence of topics covered for a course from Book of Proof by Hammack: - Sets and basic definitions: 1.1 - Logic: 2.1, 2.2, 2.3 - Proofs: 4.1, 4.2, 4.3, 4.4, 4.5 - Logic: 2.4, 2.5, 2.6 - Contrapositive Proof: 5.1, 5.2, 5.3 - Logic: 2.7, 2.8, 2.10 - Proving non-conditional statements: 7.1, 7.2, 7.3, 7.4 - Disproof: 9.1, 9.2, 9.3 - Mathematical Induction: 10.1 and 10.3 - Sets: 1.3, 1.4, 1.5, 1.6, 1.7 - Proofs involving sets: 8.1, 8.2, 8.3 - Sets: 1.2 - Relations: 11.1, 11.2, 11.3, 11.4, 11.5 - Functions: Chapter 12 - Proof by contradiction: Chapter 6 - Cardinality: Chapter 14

With proofs, I also think there are three layers:

1. How do I draw logical conclusions from premises. This is the most straightforward part.

2. What are some of the clever tricks mathematicians use for doing this (e.g. constructing non-intuitive counterexamples, finding equivalences between two seemingly incompatible things, etc). This requires reading other proofs, and is slower, but can be very fun if you like math and find clever proofs beautiful.

3. Finding the right English words and phrases to capture the logic you have in mind (the language used in proofs is not normal English, and it has its own idiosyncrasies and conventions. Like other mathematical notation, it's often specific to particular fields of math and sometimes to a specific author). This also requires reading proofs, I think, and is also where one benefits the most from formal instruction ("how do I say X in my proof?") but I think you can get there on your own with some persistence. It's not Klingon either—proofs are supposed to be readable—but it's a bit like code, maybe, or legalese. If you just try to write nice prose, other mathematicians may find it confusing or non-rigorous.

It can be good to separate the three. 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 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"

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

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.

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).

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

Lots of people are introduced to it in their linear algebra class which is a terrible way to learn it. You want a book like "Proofs and Fundamentals" by Ethan Bloch.

The best one right now is Proofs: A Long Form Textbooks by Jay Cummings. I wholeheartedly recommend it. And this is exactly the one you are looking for.

It is fully intended to teach learners how to write proofs, and not to impress one's peers or get citations.

I loved the book.

I picked it up because although I had a decent curriculum-based Maths education as a Physics undergrad and wrote many proofs, these proofs were always learned in a domain-dependent way. I was clueless about writing a new proof in a new domain.

This book filled a large hole in my life.

And I recommend it.

One other book that is good and should be read if you want to rigorously study Calculus from ground up, and it also teaches you a decent amount of Analysis, too. It's Spivak's Calculus. It's one of the best Math books ever written.

I learned a lot from working through An Infinite Descent into Pure Mathematics by Clive Newstead. It's designed to get someone with minimal math background started with the basics of pure math. Becoming comfortable with proofs happens along the way.

https://infinitedescent.xyz/

Lately, I have also really enjoyed 99 Variations on a Proof by Philip Ording, which is not a textbook. Rather it is an Exercises in Style type book that explores many different ways to express a mathematical proof of the same simple fact. Some of the proofs are whimsical, and others offer genuine insight. If you're looking for something lighter than a textbook that is still interesting and somewhat useful, this book is more approachable.

https://www.amazon.com/99-Variations-Proof-Philip-Ording/dp/...

Journey into Mathematics: An Introduction to Proofs (Dover Books on Mathematics) https://a.co/d/csM8jRd

I seem to remember that by design it didn't require any advanced maths to get started constructing proofs. Good luck!

we can try your example

we want to show that (a^b)^c = a^bc

let's work on the LHS

by the definition of an exponent, we know that a^b is just a * a * ... * a b times, so we can rewrite it as:

(a_0 * a_1 * ... * a_b)^c

by the same definition, we can multiply the quantity inside the parenthesis by itself c times:

(a_0 * a_1...a_b)_0 * (a_0 * a_1 * ... * a_b)_1 * ... * (a_0 * a_1 * ... * a_b)_c

now, use the fact that a^m * a^n = a^(m+n) to consolidate the parenthesis, since each factor has an exponent of 1 we can use simple counting:

(a^b)_0 * (a^b)_1 * ... * (a^b)_c

Repeat the previous step c times, we end up with

a^(b_0 + b_1 + ... + b_c)

which of course is just

a^(bc)

therefore (a^b)^c = a^(bc)

Good luck!