Many students begin learning proofs in Analysis 1 or Linear Algebra. The most pedagogically friendly book I've seen for this is Terence Tao's Analysis 1.
Ask HN: How to learn mathematical proofs from scratch?
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Re: Ask HN: How to learn mathematical proofs from scratch?
#12How to Prove It by Daniel J. Velleman.
"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. "
Re: Ask HN: How to learn mathematical proofs from scratch?
#13Undergrad discrete mathematics and symbolic logic made proof writing click for me. You have a set of things known to be true as handed down from on high, you have a set of operations to transform those true things into equivalent statements, you then go about the work of using those transformations to connect true statements together into a massive graph and extend that graph through speculation if possible.
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.
Re: Ask HN: How to learn mathematical proofs from scratch?
#14Math undergrad student here. As another comment has already mentioned we really start learning proofs in Analysis I (the introductory Linear Algebra course in my school wasn't proof heavy) but before that at least in my school we take Intro to math I and Intro to math II, I can't recommend you a book but the topics that are covered are usually basic set theory, rules of logical inference and induction. For Analysis I my prof used baby Rudin however I don't think it's a good book for self study (I'm a bit biased here since I'm really having a hard time with baby Rudin). Two books that are somewhat a more gentle introduction to Analysis are Kenneth A. Ross' Elementary Analysis and Abbott's Understanding Analysis.
Re: Ask HN: How to learn mathematical proofs from scratch?
#15Which texts have you been looking at that are too advanced for you?
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
Re: Ask HN: How to learn mathematical proofs from scratch?
#16Not 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 exceptionally good book for self-study, though)
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
Re: Ask HN: How to learn mathematical proofs from scratch?
#17Re: Ask HN: How to learn mathematical proofs from scratch?
#18[deleted]
Re: Ask HN: How to learn mathematical proofs from scratch?
#19How to Prove It by Daniel J. Velleman. "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…
Came to recommend this. This book was used in my undergrad discrete math class. I really enjoyed it and it paid dividends as I finished my math degree.
Re: Ask HN: How to learn mathematical proofs from scratch?
#20Not 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 of computation, discrete math, introduction to higher math and then some upper div cs courses in ai/ml.
what you say here rings absolutely true to me. as someone coming in with a long time background in coding and computers, i always found myself wanting to apply the same unambiguousness and precision that one uses to express computations in a programming language in mathematical proofs. this was a huge stumbling block for me! proofs are written in shorthand (!) by humans for other humans who have the same base knowledge, obvious things are omitted. once you have that knowledge it makes sense, but before that it seems like giant leaps are being taken without a rigorous line of reasoning between them.
to answer op's question: "an introduction to mathematical reasoning" by peter eccles was helpful for me. it's basically an expanded version of what you'll find at the start of many intro cs/math courses. another option, if sets, number theory and such are confusing to you is the computability angle (cs theory, computation). personally i found this material a lot easier to reason about which then imputed confidence, which is really the magic ingredient for good proofs.