A Gentle Introduction to the Art of Mathematics


314 points | by Keegs 12 days ago


  • rramadass 11 days ago
    Seems very student-friendly and accessible which is good. A quick browse shows that it covers Set Theory, Logic and Proofs i.e. the absolute basics for "Modern Mathematics". One of the biggest difficulties for the student i.e "Mathematical Notations" (which are merely a shorthand language) are explained from the ground up which is great.
  • auggierose 11 days ago
    At the very start of the book:

    > It has been said that “God invented the integers, all else is the work of Man.” This is a mistranslation. The term “integers” should actually be “whole numbers.” The concepts of zero and negative values seem (to many people) to be unnatural constructs.

    It is not a mistranslation. In German "ganze zahlen" actually means "integers". It is just if you translate it word for word, "ganze" -> "whole", "zahl" -> "number", that you get whole numbers, which would be a mistranslation, because in English whole numbers mean 0, 1, 2, ...

    • tromp 11 days ago
      > in English whole numbers mean 0, 1, 2, ...

      According to Wikipedia [1], the term is ambiguous. Its talk page [2] has plenty discussion about it though.

      [1] https://en.wikipedia.org/wiki/Whole_number

      [2] https://en.wikipedia.org/wiki/Talk:Whole_number

      • zelphirkalt 11 days ago
        In German though "Ganze Zahlen" is not ambiguous. At least in my entire life I have not seen any other understanding of it and every child at school learns about them and that they include 0 and negative numbers, in contrast to natural numbers ("Natürliche Zahlen").
        • freilanzer 11 days ago
          As a German, this is exactly what whole numbers means - numbers such as -1, 0, 1. Natural numbers are whole numbers > 0.
          • ahf8Aithaex7Nai 10 days ago
            Every mathematical introductory text that excludes zero from the natural numbers contains an unwieldy additional notation like N_0 in the very next sentence. Why the hell should the number that is both used in the first Peano axiom and is the additive neutral element of the natural numbers not be included in the natural numbers? I've never understood why so many people insist on this, relying on pseudo-anthropological reasoning or something. Zero is at least two and a half thousand years old. You could just as easily claim that the natural numbers end with the number 10 because humans don't have any more fingers.
        • jacobolus 11 days ago
          Okay, but what about in the 1880s?
          • weinzierl 11 days ago
            I think, historically the term "Ganze Zahl" (a whole or entire = integer number) was always used in contrast to "Gebrochene Zahl", meaning broken or fractured number.

            Negative numbers are not broken, so they have always been considered whole. For example, Leonhard Euler wrote in his "Vollständige Anleitung zur Algebra" from 1767:

            "Alle diese Zahlen, so wohl positive als negative, führen den bekannten Nahmen der gantzen Zahlen, welche also entweder größer oder kleiner sind als nichts. Man nennt dieselbe gantze Zahlen um sie von den gebrochenen, und noch vielerley andern Zahlen, wovon unten gehandelt werden wird, zu unterscheiden."


      • auggierose 11 days ago
        In German, it is not and never was ambiguous, so it is not a mistranslation. What the meaning of "Whole number" in English is, is not really relevant, except for explaining the mistake in the book.
        • Someone 11 days ago
          > In German, it is not and never was ambiguous

          I’m not sure about that ”and never was”. Mathematicians used to have fairly loose definitions for all kinds of things and different fields sometimes have incompatible definitions for a term, so I think it’s not impossible “ganzzahlig” was used ambiguously for ℤ or ℕ (in- or excluding zero) for a while in some corners.

          When was the phrase “natural number” even invented? The best I can find is https://jeff560.tripod.com/n.html (via https://mathoverflow.net/questions/379699/origin-of-phrase-n...) which says “Chuquet (1484) used the term progression naturelle for the sequence 1, 2, 3, 4, etc.”

          There may well have been a time where “ganzzahlig” existed but “Natürliche Zahl” didn’t yet.

      • zorked 11 days ago
        English is being weird by having three terms then?

        Inteiros - (...-2, -1, 0, 1, 2...)

        Naturais - (1, 2, 3...)

        • noobr 10 days ago
          Olha, primeira vez que vejo algo em português aqui!
      • n4r9 11 days ago
        Huh, I never knew there was an ambiguity. To me it makes total sense to include negative integers. "Whole" means "without a fractional part".
        • lupire 11 days ago
          Is i a whole number?
          • auggierose 11 days ago
            Yes and No.

            No, it is not even a real number, but every whole number surely is a real number.

            Yes, it is a "Gaußsche ganze Zahl".

    • yboris 11 days ago
      I find that philosophically, "all math is discovered" is a better way to think of mathematics.

      E.g - it's always been true, that given a set of axioms A, B, and C, the claim X is true. Humans discover these conclusions when they explore implications of taking different assumptions to be true.

    • lupire 11 days ago
      Mistranslation or not, conceptually "natural numbers Z+" or "whole numbers Z*" are a more fitting claim.

      Peano would go further: God invented 0 and +1, and all the rest are the work of people.

      • dotancohen 11 days ago
        In software development it is often said that there are only three relevant numbers: 0, 1 and infinity. Everything else is contrived.

        Certainly, nature follows a very similar pattern. For limited values of infinity ))

    • falcor84 11 days ago
      I can't seem to find a reference for it now, but a math professor once told us that some early cultures didn't even have a proper word/symbol for the number "one", and the act of counting didn't kick in until there were two items.
      • jacobolus 11 days ago
        Greek geometers (e.g. in Euclid's Elements, Book 7) define a "unit" to be 1 and "numbers" (arithmoi) to start from 2. https://mathcs.clarku.edu/~djoyce/java/elements/bookVII/book...

        But in practice there are a bunch of propositions/proofs where 1 is treated as a number just like any other.

        • lupire 11 days ago
          log(1)=0, so it's unavoidable to consider 1 a something at least slightly different from the other numbers, as much as 0 is.
  • danielvaughn 11 days ago
    I was prepared to hate this book as much as I've hated any other "introduction" to math, since I've found that mathematicians generally suck at introducing or explaining things. But the first little section that breaks down set notation into plain english was fantastic. That's exactly the kind of thing I need personally, as notation is extremely off-putting and confusing for me.
    • ChainOfFools 11 days ago
      > I've found that mathematicians generally suck at introducing or explaining things.

      and then there's Grant Sanderson

      • wyager 11 days ago
        I don't think it's much of an exaggeration to describe Sanderson as a contemporary Feynman, at least from the pedagogy side of things.
        • Nifty3929 11 days ago
          Perhaps on pedagogy, or at least nice, intuitive, layperson (but technically accurate) explanations of things. And plenty of fully-rigorous explanations of some things too.

          I'm a big fan of Sanderson.

          But I don't think we can compare Sanderson to Feynman on raw intelligence or contributions to the state of the art in math/science.

        • ChainOfFools 11 days ago
          And Feynman himself was characterized by his advisor in a recommendation letter as "another Dirac, but this time human."[0]

          [0] paraphrasing from memory, might have been a different supervisory figure.

  • magpi3 11 days ago
    It's been hugged to death perhaps. The site is also hosted on github here:


    • leadingthenet 11 days ago
      In fact, this seems to link to an even newer version of the text.
  • Keegs 12 days ago
    The quality of writing in this book is fantastic. If you're studying math or have a recreational interest I really recommend it.
    • __rito__ 11 days ago
      What other resources would you recommended for learning Math that is a bit more advanced?

      It needs to be suitable for someone studying alone.

    • dr_kiszonka 11 days ago
      Thanks for sharing here! Do you know of any textbooks in this spirit that cover other topics in math?
  • thewanderer1983 11 days ago
    Fearless Symmetry and Elliptic Tales are fascinating, accessible and fun introductions to the art of mathematics.
    • UncleOxidant 11 days ago
      > Fearless Symmetry

      I happened to pick up a copy of Fearless Symmetry at a garage sale recently. Was wondering if it was worthwhile (kind of hard to go wrong for $1), thanks for the recommendation.

    • Koshkin 11 days ago
      But (as usual) reviews on Amazon offer a wide spectrum of opinion on these.
  • __rito__ 11 days ago
    This looks good. And I have already covered these things.

    What are some other resources for learning more Math that are very approachable when studying alone?

    • tarkin2 11 days ago
      Khan academy will teach you all of pre-k, high-school, pre-calc, calculus and infinite series pretty well. mathsisfun.com is a great resource if that's ever lacking. High school physics on khan is good to give you a practical application of the concepts if they seem too abstract.

      I found Strang's youtube lectures the best for linear algebra along with 'vector maths for 3d graphic'. Khan's linear algebra course seems like more a collection of practical things you can do with linear algebra rather than a coherent cource but it's still a great resource.

      Khan academy's multi-variable calculus is good enough to give you a reasonable grasp, but I'd call it a good complement rather than sole resource.

      I've tended to go through the cycle of watching the lectures, then doing the exercises, going forward, going back after a few months to ensure I understand the whys and not just the hows, and then writing my own notes. Machine Learning et al is a great practical application of all this maths.

      • __rito__ 10 days ago
        I have a Bachelor in Physics, and a Master’s in CS.

        I am not really looking for the basics. I am looking for advanced material that can be handled without the help of a teacher. The book should be written in that manner.

        As I have a good background, I can read AI papers and get the math directly or study some and get that.

        But whenever I start to study something, more often than not, the book is written in a dry manner and cannot hold my interests.

    • rramadass 10 days ago
      "Approachability" might vary somewhat but you might find the following useful;

      * For an introduction (and a reference) to various areas of Modern Mathematics that one didn't even know existed, The Princeton Companion to Mathematics and The Princeton Companion to Applied Mathematics are a must.

      * All the Math You Missed: (But Need to Know for Graduate School) by Thomas Garrity - A survey and a good adjunct to a textbook.

      * Mathematics: Its Content, Methods and Meaning by Kolmogorov et al. - Classic text from the great Russian Mathematicians.

      * Methods of Mathematics Applied to Calculus, Probability, and Statistics by Richard Hamming - Unique text from the great Richard Hamming (also checkout his other books).

      There is plenty more of course, specifically; checkout "Dover Publications" texts, many of which are classics and affordable.

      • __rito__ 10 days ago
        Yeah, thanks for the recommendation.

        I already know the basics, and I want to learn more.

        I cam handle hard, but the textbooks need to be written in a manner that doesn't require the help of a teacher or a classroom.

        • rramadass 10 days ago
          None of the above are "basics", they range from undergraduate to research level topics, in particular; the two Princeton Companions are a must have.



          PS: In an earlier HN thread, somebody had highly recommended the 4-vol Foundations of Applied Mathematics developed for Brigham Young University’s Applied and Computational Mathematics degree program for beginning graduate and advanced undergraduate students. I have not browsed/read them yet but they are on my "future acquisition and study" list. They seem great and well worth looking into - https://foundations-of-applied-mathematics.github.io/

          • __rito__ 10 days ago
            Thanks for the original recommendations, and these two. I have heard about the Princeton Review books. I have not read them yet.

            Among your original recommendations, I have heard about "All the Maths you Missed". And I have read several chapters of the one by Kolmogorov et al. It's a fantastic book. It lays the landscape really well, discusses things, and covers the breadth of the field rather than the depth of any particular field. I find the writing style really good. It could be seen as a reference book for people who already know the stuff, or are looking to know about new stuff, but want to have ideas about what those stuff might be.

    • WillAdams 11 days ago
      My context is CNC programming, but I've found the Make: Geometry, Make: Trigonometry, and Make: Calculus books excellent.
  • xk_id 11 days ago
    Would be nice if we could see experienced mathematicians endorse this, please. It would help to decide whether to put time into it. At first glance it looks really nice.
    • vouaobrasil 11 days ago
      I'm a mathematician (PhD in numnber theory...) and I took a look. It's a basic textbook about some concepts like functions, relations, and basic proof techniques. It seems okay. I was expecting something different from the title, though. It's not too different from many basic books introducing such concepts but obviously a lot of effort was put into it.

      Personally, I think if you want an introduction to the "art" of mathematics, it would be a lot better to pick up a more idiosyncratic book that doesn't aim to cover the basics of the standard curriculum in a textbook-style way, which in my opinion is rather tedious. That could either be a more high-level book like Ian Stewart's "From Here to Infinity" or one of Raymond Smullyan's fun texts on logic.

      Or for a more basic book, something like "The Mathematical Universe" by William W. Dunham is a much more interesting introduction to the "art" of mathematics than a textbook-style intro.

      • skhunted 11 days ago
        I’m ABD in math for discolosure purposes. I strongly disagree with your recommendations if the purpose is to get an introduction to higher math. The book in question is much, much better for introducing one to higher math than any of the books you recommended.

        Smullyan’s books are great but one isn’t going to go from Smullyan to abstract algebra, point set topology, or real analysis.

        • vouaobrasil 11 days ago
          Meh, well different strokes for different folks I guess. I got into higher math while reading Ian Stewart's book in grade school but I guess some people are going to want to go the standard way.

          My problem with the book we are discussing is that it seems rather prosaic -- it doesn't really give a sense of the true reason to practice math: the asking of interesting questions and creating new universes. It's just the same old stuff that we're taught because it's a convention.

          • skhunted 11 days ago
            The language comes first then the applications. There’s a reason the order of topics evolved the way it has.
            • vouaobrasil 11 days ago
              Well, I'm just going by empirical evidence: what has worked for me and what has worked for many of my fellow colleauges that have done actual research in mathematics.
              • skhunted 11 days ago
                Before one can do advanced mathematics (I have too have done research in math) one needs to learn the basics. Reading Smullyan did not in any way help you learn advanced mathematics. It may have helped you to get motivated to learn math and want to learn advanced math but it didn’t help you accomplish this. There’s a reason just about every mathematics department teaches classes on how to do proofs and on basic set theory but almost none of them teach from Smullyan’s book.

                The overwhelming empirical evidence is that having a course on proofs and basic set theory is much better preparation for advanced mathematics than reading Smulyan’s recreational math books. I guess you’d rather your students read Martin Gardner and then do Fraliegh’s Abstract Algebra book. No one does it that way but go with your so called empirical evidence.

                Having a Ph.D. in math ought to have taught you to reason better than to use “actual research” as part of your reasoning when discussing learning topics that are not cutting edge. One doesn’t need to have done research in mathematics to know about Gorenstein rings or projective dimension or other such stuff. It also has nothing to do with teaching basic math.

                I could be wrong in my opinion but attack it on its merits without using superfluous things like “actual research” when research has nothing to do with the topic.

                • rramadass 11 days ago
                  Well said! You are absolutely right and "vouaobrasil" is wrong.

                  As a person interested in self-learning Mathematics, i have read and amassed a lot of "popular mathematics" books by authors like Ian Stewart, W.W.Sawyer, E.T.Bell, George Gamow etc. all of which were great motivators but none of which taught me the basics of "Modern Mathematics" which i could only get from Textbooks. The quality of Textbooks are of course all over the map and so i am always on the lookout for the simplest, clearest and yet rigourous explanations available. The book under discussion seems to check all such boxes for a beginning student.

            • lupire 11 days ago
              It's weird that everyone from Euclid to Gauss disagrees with you.

              The language is a baby compared to the applications.

              • skhunted 11 days ago
                Do you have any evidence that Euclid disagreed with me? Elements starts off with a list of postulates, definitons, and common notions. Then he proceeds to proposition 1. He does not motivate why one would want to come up with proposition 1 or why would should care about it. Have you read Elements?

                How do you propose one do applications of point set topology without knowing about sets and mappings? Before the applications one must know the language. We don’t teach the quadratic formula and solving simple velocity problems before teaching students how do the basics of manipulating algebraic expressions.

                One must first be a baby before being an adult.

      • seanhunter 11 days ago
        The purpose of the book as stated in the preface is to be used for a course in proof writing ie it is a sort of bridge to higher maths which the author defines as "defining axiomatic systems and proving statements within them" vs "elementary maths" which he defines as "solving problems".

        So I think the idea behind the title is get students to see this as the gateway to the good stuff as opposed to a lot of proof texts which might be seen as irrelevant.

      • beryilma 11 days ago
        One the same subject and as accessible, I love the two books from Jay Cummings: Proofs and Real Analysis. Each just $16 on Amazon. It is a joy to read these books and try some of the exercises. I wish PDF versions were also available...

        For those interested: https://webpages.csus.edu/jay.cummings/Books.html

    • nicf 11 days ago
      I'm a mathematician, currently working as a private tutor for adults who want to learn proof-based math. I had a quick glance through this book and it seems to me like a pretty nice version of this "intro to proofs" sort of book. This is a topic that's done well in a lot of different books, though, so if you really want to dig into this topic I'd maybe recommend looking at a couple different ones and finding the one that agrees with you the most.

      Right now I have a student working on this material and we're using "How to Prove It: A Structured Approach" by Daniel Velleman, which so far I'm finding decent. Some others I've seen (but that I haven't looked at in as much detail) are "Proofs: A Long-Form Mathematics Textbook" by Jay Cummings and "Book of Proof" by Richard Hammack.

      • nyrikki 11 days ago
        As someone who tutors adults, can you suggest a more digestible book for abstract algebra?

        While I was motivated, I used one of the typical college books. For me Abstract Algebra is what opened a lot of doors for me... but I am simply using applied math.

        That moving away from proofs being magical across sub-topics is what I would like to share with some co-workers who are unwilling to buy a textbook and answer key.

        As I didn't even mind Spivik for calc, my radar is way off for making suggestions to most people.

        • tzs 11 days ago
          Pinter, "A Book of Abstract Algebra", is very nice. It's rigorous but not too terse. It divides the material into many small chapters with many exercises. Chapters are mostly around 10+/-3 pages with about 40-60% of that being text and the rest exercises.

          The exercises for each chapter are split into several sections each section covering a different aspect of the chapter's material. Sometimes there is a section of exercises applying the material to some interesting area.

          For example, the chapter on groups of permutations has 6 pages of text, then 5 pages of exercises divided into 9 sections. Those sections are: computing elements in S6 (5 problems), examples of groups of permutations (4 problems), groups of permutations in R (4 problems), a cyclic group of permutations (4 problems), a subgroup of SR (4 problems), symmetries of geometric figures (4 problems), symmetries of polynomials (4 problems), properties of permutations of a set A (4 problems), and algebra of kinship structures which consists of 9 problems covering how anthropologists have applied groups of permutations to describe kinship systems in primitive societies.

          There are answers in the back for a decent number of the exercises.

          It's a Dover republication so is not too hard on the wallet. List price is $30 at Dover but its around $20 on Amazon.

          The combination of short chapters and lots of exercises make it easier than most textbooks to fit into a busy adult schedule.

          • nyrikki 10 days ago
            Thank you, ordered it.
        • sn9 11 days ago
          Abstract Algebra: A Student Friendly Approach by Dos Reis and Dos Reis [0] is like The Little Schemer but if it was a first course in abstract algebra.

          [0] https://www.amazon.com/gp/product/1539436071?psc=1

        • nicf 11 days ago
          I assume you're talking about an algebra book for self-study? Gallian's "Contemporary Abstract Algebra" is a common suggestion for a more accessible algebra book, and people also sometimes suggest Fraleigh's "A First Course in Abstract Algebra", but I can really only speak to what it's like to work on this stuff with a teacher --- since my students are by definition not self-studying the things I'm working on with them, my suggestions might be of limited use!

          In general, I think self-studying proof-based math can certainly be done if someone's motivated enough, but it's pretty hard and takes a lot of work, especially if you're still getting used to the skill of reading and writing proofs. It's very valuable to be able to have a person available to evaluate the proofs you're writing, and I've definitely seen a few people who came to me thinking they'd mastered proof-writing on their own and were kind of mistaken about that. (I've definitely also seen people who really did learn this skill pretty well without help! It varies a lot.)

          • nyrikki 11 days ago
            Thanks for the suggestion. I personally used Dummit and Foote's book and found it useful, but like early Calculus with Spivak, it seems most people prefer clear and concise over slightly more comprehensive and rigorous while still being introductions.

            With self study I prefer a bit more breadth to make up for the realities of needing to self study which often ends up with deep but not wide understanding of topics.

            Having a brother who had a PHD in complex analysis to bother probably helped with self-learning. That is the only option when you are on-call for decades at a time as higher math courses are/were always in person.

            But hopefully someone will figure out a business model to help people who need to grow and adapt.

            Thanks again for the suggestions, I have ordered both books to add to my lending library.

      • sn9 11 days ago
        The 3rd edition of Velleman's book has an online supplement that uses Lean to work through the book so you can get feedback about your proofs.
      • __rito__ 11 days ago
        I have read many parts of Cummings'es book, and I can vouch for its quality.

        It's the ideal book for learning proofs if you are self-learning.

    • mcguire 11 days ago
      Putting on my Computer Scientist hat (and getting all of the tentacles out of my face so I can see), GIAM seems to be a good, if a little verbose[1], introduction to formal logic and basic set theory. Those are fundamental to CS and this is a heck of a lot better than many of the intros to those topics that I have seen in curricula without a dedicated class.
  • drBonkers 11 days ago
    Are there prints for sale?
  • VinLucero 11 days ago
    Been looking for this!