Book of Proof
This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra.
Stats for nerds
54recommend
86mentions
66people
2012–2026years
81% recommend19% neutral
Mentions per year
What Hacker News says
- This is absolutely not a helpful list for someone who wants to prove things in lean, and just seems like weird gate-keeping. In particular, Proof Trees are completely irrelevant, incompleteness and compactness are only relevant if you are proving things in those specific fields and the only thing you need to know about constructive vs intuitionistic vs classical logic in lean is if you want to use the law of the excluded middle or some non-constructive proofs and tactics, lean won’t force that on you, so you sometimes need to “open classical”. (And this is covered in “Mathematics in Lean” and “Theorem Proving in Lean” at the appropriate place).
Knowing that the Curry-Howard correspondence exists is important if you care about the CS magic that makes lean work and to understand how term mode and tactic mode relate to each other but again it’s really not necessary to understand the correspondence itself to use lean as a proof assistant. (And this is covered extensively in “Theorem proving in Lean” if that’s your jam).
Whatever mathematical background you have is obviously helpful and will widen the scope of what you can do, but you don’t need to learn a huge amount of foundational mathematics to get your hands dirty in lean. For example, “The Mechanics of Proof” by Heather Macbeth was written as a course for 1st year undergrads so only assumes high school maths knowledge. Here’s a list of learning resources that the lean prover community recommends https://leanprover-community.github.io/learn.html
One thing I would add is if you want to learn about proof writing in general there are a lot of good resources out there including “The Book of Proof” which is free online. https://rcsnyder.github.io/open-frontier-curriculum/07-resou...
I personally really enjoyed Jay Cummings’ “Proof: A long-form mathematics textbook” which is on that list as it provides lovely little intros to various areas of mathematics along the way. I have done every exercise in that book and had a lot of fun in the process.
But these aren’t things you necessarily need to do before getting started in lean.
- First step into math is to learn how math people think. Here below is an intro textbook meant to do just that[0]. It's a high quality, free introduction. You don't need to know much past 6th/7th grade level math which amounts to knowing how to do arithmetic with fractions and decimals and maybe an ability to in/equalities in one variable. I will also add a couple of google links that might be more relevant to what you might be trying to do[1][2].
[0] Book of Proof by Richard Hammack https://www.people.vcu.edu/~rhammack/BookOfProof/
[1] Linear Algebra: Step by Step by Kuldeep Singh https://www.google.com/books/edition/Linear_Algebra/BJNoAgAA...
[2] Number Theory: Step by Step by Kuldeep Singh https://www.google.com/books/edition/Number_Theory/YZr9DwAAQ...
[3] Undergraduate Analysis: A Working Textbook by Aisling McCluskey and Brian McMaster
https://books.google.com/books?id=7U9HtAEACAAJ&printsec=copy...
There's even more relevant book for you[4]
[4] Foundations of Applied Mathematics by Jeffrey Humpherys, Tyler J. Jarvis, Emily J. Evans
https://books.google.com/books?id=CEc3DwAAQBAJ&pg=PR3&source...
- Learning grade school math won’t really help you understand these things. Yes, you will learn to think numerically and practice applying opaque algorithms, but you’d be better off starting with basic set theory and logic and learning “real” math. Book of Proof is one of my personal favorites. Then you can move onto some Real Analysis while brushing up on Calculus, then maybe consider formal probability starting to learn the foundations
- For math, the first step is to know how to prove math statements. Usually, it's learned together with the subject at hand(analysis, algebra), but you can come prepared. Book Of Proof by Hammack(might've misspelled that) is a good one. It's free online. Very similar to working out at the gym when you engage in a sports discipline whether thats boxing or gymnastics.
- I would recommend Book of Proof: https://www.people.vcu.edu/~rhammack/BookOfProof/
We used this in my Discrete Mathematics class (MATH 2001 @ CU Boulder) (it is a pre-requisite for most math classes). The section about truth tables did overlap a bit with my philosophy class (PHIL 1440 Critical Thinking)
- There is a whole "grammar of proofs and basic intro to higher mathematics" style course nowadays.
Book of Proof has a free PDF version. It covers the basic styles of proof (necessary to learn but still not super advanced).
Other texts of this sort include Jay Cummings long form texts (I myself haven't worked through or even exaimed these but they seem popular.
Related
Mathematical proofs
17 recommend · 23
2017–2024
Linear Algebra
9 recommend · 11
How to prove it
78 recommend · 129
How to Think About Analysis
15 recommend · 16
Learning to Reason
6 recommend · 6
Proofs from the book
28 recommend · 60
A transition to advanced mathematics
15 recommend · 20
