Mathematical proofs

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1. mydriasisFeb 2024
   In my experience, it's because of the order in which math is taught. I didn't _really_ learn about proofs until college, and even then, not until after my first basic calculus course.

   Then, if you wanted an explanation for your example, it'd be a proof. The proof is not hard ( see [0] ), but how can one understand a proof if they've never been taught about proofs?

   This is what I've seen over and over again. If we want folks to understand what they're looking at, the meat is waiting in the fridge for us. It seems to me that instructors don't have the time or inclination to teach these things at all, which is unfortunate because of how foundational it is to really understanding what you're looking at when it comes to math.

   If you're interested in learning how proofs work, and you've missed out on a course like I did, I'd recommend "Mathematical Proofs", by Albert D. Polimeni, Gary Chartrand, and Ping Zhang. You should be able to find a paperback for some bucks.

   Usually mathematics texts are thick and difficult to follow for me, I'm not the brightest, but this one was easy to follow and I worked through the brunt of it in a month of concentrated effort by myself. It really set me up for understanding how proofs are structured and why, and how to reason about them on my own.

   [0] https://tutorial.math.lamar.edu/classes/calci/DerivativeProo...

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2. i_am_a_peasantAug 2023
   There are other books in the literature that mainly focus on how to go about proving things rather than proving specific things.

   Some titles that come up are:

   Mathematical Proofs: A transition to Advanced Mathematics by Chartrand, Polimeni and Zhang.

   Book of Proof: Richard Hammack

   Jay Cumming’s Proofs

   I remember I once had a book that advertised itself as a book on Discrete Mathematics but it turned out to be more in line with the titles above, was a gem, if I find it I'll let you know.

   But it's nice to see I'm not the only guy who gets a kick out of this stuff.

   I wouldn't bother with things from the foundations of mathematics area, like Russel or Frege and whoever else came from that era, back when people still wrote an 800 page treatise on what a "number" even is. A lot of the material coming from that period seems to have been thrown down the drain.

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3. rahimnathwaniMay 2022
   Not an overall answer, but some parts of an answer:

   - Notes on Discrete Mathematics (James Aspnes)

   - Mathematical Proofs (Chartrand, Polimeni et al)

   - TrevTutor videos on YouTube

   - Professor Leonard videos on YouTube

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4. ufopilotFeb 2022
   My very first intro to math was through "Learning to Reason: An Introduction to Logic, Sets, and Relations" by Nancy Rodgers.

   Many books of this kind are usually called "Intro To Discrete Math" because I think discrete math is easier and neater than its continuous counterpart. Most such books consider a sampling of topics from number theory, combinatorics, graph theory and discrete probability. Obviously, they do not even scratch the surface of the subjects they teach as that's not the goal. For example, compare what's in the counting section of a typical intro to discrete math book to an intro to counting proper. For example, my very first discrete math textbook was "Discrete Mathematics with Applications" by Susanna Epp and for counting it was "Combinatorics: A Guided Tour" by David R Mazur. Epp's books covers much, much, much less. Similarly, currently one of the very best intros to elementary number theory is "Number Theory: Step By Step" by Kuldeep Singh. The latter also contains much more material than a typical intro to discrete math.

   Beyond that every author of an into to discrete math likes to include a little taste of their own personal favorites. For example, Susanna Epp has a chapter on algorithm analysis whereas, say, Edward Scheinerman talks a little bit about group theory in his book "Mathematics: A Discrete Introduction". Yet other authors include chapters on geometry, vectors spaces, generating functions etc.

   I noticed (almost) every math/CS adjacent thread on HN gets mention of Big O notation. IMO, the best place to learn that is a textbook on discrete math. For example, Susanna Epp has a whole chapter dedicated to this topic in her intro to discrete math book. Consider [0] from Epp's book. Since it's a math book it's very careful and thorough meaning if you don't understand something in that example reading the material that precedes will clarify everything; just a matter of backtracking far enough. Since it's an extremely introductory book, it's also very patient. I personally struggled with the motivation for defining total derivative through little-o notation.

   There are a ton more textbooks on discrete math like, say, " Essential Discrete Mathematics for Computer Science" by Lewis & Zax or "Discrete Mathematics: An Open Introduction" by Oscar Levin.

   Some intro books also attempt to include a tiny bit of real analysis and general topology. Topics include ways of using least upper bound, epsilon-delta arguments, open/closed sets, a little on metric spaces etc. Obviously, such books don't get carried away by ditching epsilon-delta/balls arguments in favor of more general open sets to define continuity, for example. One such book that comes to mind is "Mathematical Proofs: A Transition to Advanced Mathematics" by Chartrand, Polimeni, Zhang.

   I think the best book in this introductory genre is "Book Of Proof" by Richard Hammack. It's freely available. I see the latest edition added a few techniques for proving convergence and continuity on the real line: epsilon-delta and epsilon-N stuff.

   I call the books mentioned above "Intro To Math 1". After these books one can get started with more in depth books on the subjects considered in the aforementioned books or pick up an intro to real analysis, topology, abstract algebra, whatever.

   One of the most effective textbooks on elementary math analysis is "Principles of Mathematical Analysis" by Rudin. But one will have a real tough time with it even after studying the aforementioned books. But there are fantastic preparatory books like "Linear Algebra: Step by Step" by Kuldeep Singh which one needs because Rudin considers Euclidean Spaces proper, not just the reals. Other helpful books include "Yet Another Introduction to Analysis" by Bryant, "How to Think About Analysis" by Lara Alcock, "A First Course in Mathematical Analysis" by Brannan, "Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings" etc. None of these books are perfect. For example, none of them (but that of Cummings) include the necessary topological concepts. Also, even though the book by Bryant is chatty, his exercises can be tough. Case in point, he expects you to prove that rationals are incomplete in the very first chapter. To his credit, he gives a lot of hints and every one of his exercises have detailed solutions. When I first saw Alcock's book I got the impression that her book was an excuse to explain how we think about the convergence in reals. The rest of the book felt like a filler, even though the book is worth it for that chapter alone. It's good to look into many different books because the authors make different choices. For example, Cummings proves Bolzano-Weierstrass theorem through the fact that every sequence has a monotone subsequence whereas Brannan does it by way of bisection argument. Other authors might make yet other choices. In math the more techniques you have in your tool chest, the easier it is to attack problems. If you solve a problem one way, but the solution given is different it's worth memorizing the alternative solution as it's another technique that will come in handy in a different situation.

   Ok, I didn't say anything about abstract algebra and other subjects, but this post is already turning into a novel.

   After making very first and wobbly steps into math through the intro to discrete math books, the other path is to study what I call "Intro To Math 2". There are books like "Mathematical Concepts" by Jost or (especially) "Essential Mathematics for Undergraduates" by Giossi. Wanted to say more about this or the other topics above. Maybe next time.

   Please, excuse the grammar and inconsistent notation (& naming convention) as it is very late here.

   [0] https://ibb.co/Q81b96R

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5. cg30eNov 2021
   For sure. I'll list some books for introduction to proofs, abstract algebra, real analysis, topology and category theory. These are not comprehensive, just listing books off the top of my head. I'll definitely be leaving off personal favorites other people have. You'll like some better than others. Some of these are beginner books and some are more advanced. A good tutor can help you get through the more advanced books. I tried to list the most beginner friendly book first in the list under each subject. Then the more advanced books later in the list.

   Introduction to Proofs:

   Just pick one of these that speaks to you the most. All three are good.

   Discrete Mathematics with Applications - Epp

   Discrete Mathematics and Its Applications - Rosen

   Mathematical Proofs: A Transition to Advanced Mathematics - Chartrand, et al.

   Abstract Algebra:

   How to Think about Abstract Algebra - Alcock

   Abstract Algebra - Pinter

   Abstract Algebra: A First Course - Saracino

   Algebra - Artin

   Abstract Algebra - Herstein

   Abstract Algebra - Dummit & Foote

   Linear Algebra:

   Maybe an engineering based book first if you haven't seen linear algebra in a while (e.g. Strang or Linear Algebra: Step by Step by Singh).

   Then:

   Linear Algebra - Friedberg, et al

   Linear Algebra Done Right - Axler

   Linear Algebra - Hoffman & Kunze

   Real Analysis:

   How to Think About Analysis - Alcock

   Understanding Analysis - Abbott

   Tao's Analysis text

   Principles of Mathematical Analysis - Rudin

   Topology:

   Topology - Munkres

   Topology A Categorical Approach - Tai-Danae Bradley, Tyler Bryson, and John Terilla

   Check out this list:

   http://pi.math.cornell.edu/~hatcher/Other/topologybooks.pdf for others.

   Category theory:

   Categories and Toposes: Visualized and Explained - Southwell

   Conceptual Mathematics: A First Introduction to Categories - Lawvere

   Category Theory for Programmers - Milewski (if you like functional programming)

   Programming with Categories - Fong, Milewski, Spivak (if you like functional programming)

   Category Theory in Context - Riehl

   There are a few others by Spivak which you may like.

   If you don't know category theory whatsoever then I like Southwell the best (pair them up with his youtube videos). Eugenia Cheng also has a nice set of lecture videos.

   If you already know math pretty well, then Riehl is a favorite.

   Hope that helps!

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6. whatever1078Sep 2021
   You need to transition to learning math.

   One possible starting point is Book of Proof by Richard Hammach[0]. It's free. It will show you the rudiments of how math is actually done and thought of. Another good choice is Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang. These books have no prerequisites(good to go if you know how to do arithmetic with fractions/decimals; not that these are strictly necessary, rather that's about the level of comfort with math one should posses). Search on Amazon and you'll see there are a ridiculous number of books on this subject of all varieties of ways of presentation. Almost every week a new book pops up, it seems. Note, intro-to-proofs books are not only for novices, though. There are kinds written for advanced undergrads and beginning grad students as well. For example: Mathematical Concepts by Jugen Jost. This speaks to the wealth of the kinds of math books available.

   Concurrently and/or after that you can study introduction to any of linear algebra, discrete math, group theory, analysis of reals, combinatorics, number theory, graph theory, algorithms, probability, category theory...whatever. Each of these are very deep subjects and there exist hundreds(if not 1000s of books from the past, current and future) of every imaginable level, depth and presentational quirk. Again check Amazon for titles. Check libge*n for actual files.

   The following intro books are so easy anyone and they momma can read these with utmost ease:

   Discrete Math by Susanna Epp

   How to Think about Analysis by Lara Alcock

   Linear Algebra: Step by Step by Kuldeep Singh

   Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings

   A Tour Through Graph Theory by Karin Saoub

   The above mentioned books are a few examples of elementary books that you can get started with right this minute.

   Note and remember the wealth of possible books, though. For example, consider the subject of math analysis.

   We can break it into three general parts: real analysis, complex analysis, functional analysis. Each can be further taken apart into measure theory, vector spaces, topology etc. Further, topology alone branches out into point-set topology, algebraic topology, differential topology etc. Each of these fly under many different names. Say, vector spaces is a huge part of linear algebra. For the beginners, it's usually the finite dimensional vector spaces (aka the title of a famous linear algebra book) version whereas the infinite version will take you into func analysis. Vector spaces are also usually included as a chapter in books on abstract algebra as they are an algebraic structure just like groups, rings, modules. But if there is linear algebra, there must be non-linear algebra too. Well, yes. Roughly, it corresponds to what's called algebraic geometry. It never ends.

   The point is every topic in math has loads of books dedicated to it. In turn every topic has a (sub)^{n}topic and there are tons of books on those. Each of the books come in elementary, quirky, armchair enthusiast, average undegrad student, middling grad student, kickazz graduate math aficionado, researcher, undergrad researcher wanna-be etc levels. There are also olden books, classics, those just published (say, at most 5 years ago) and up-and-coming ones. Also, each term in math has bagillion synonyms. So multiply the number of books above by n. I am not even gonna mention books written in different languages and lecture notes that are available for free online. A lot of published book often spend years as a free set of lecture notes before ending up on Amazon as a newly published book. There are also Olympiad books, proof compilations, capstone compilations, intro-to-research books, those that prove a single theorem and in the process go through tons of disparate math etc Take advantage of all of this.

   [0] https://www.people.vcu.edu/~rhammack/BookOfProof/

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