8 ms·

Classic Mathematics Books for Lifelong Learners

191 points by furcyd 8y ago

dankle 8y ago
Behind paywall
billfruit 8y ago
Many of the list are pop-maths rather than fundamental texts. I do see many in software field recommend Chrystal:Algebra an Elementary Textbook. I also find Don Knuth, 'Concrete Mathematics' a interesting book for software people.
- commandlinefan 8y ago
  I came to a realization years ago that when it comes to math/science/computers, if I just read about it, I’m not really learning anything. I don’t learn unless I’m participating. Math books without _lots_ of exercises probably don’t really impart that much, although they can be a fun distraction if you need something to read on an airplane.
  - arvinsim 8y ago
    I agree. Some things only stick by repeated application. This becomes more relevant the one older gets.
  - hnuser355 8y ago
    Depending on your level of math ability you can often learn a lot by reading textbooks and proofs very carefully and closely (my opinion)
    - whatshisface 8y ago
      It depends on your definition of learning. If by learning you mean remembering how one author defined and proved various things then sure. If by learning you mean gaining the ability to do what the author did except for new problems, then at some point you will have to grapple with the fact that coming up with new ideas is a very different mental exercise from recalling things you've read. A supporting factor in the necessity of exercises for learning to actually do the math is that authors often hide important technical tricks in the exercises. Of course, a sufficiently intelligent person could learn anything by reading tea leaves and then deriving on their own whatever it was they wanted to learn, so in a sense you can learn math by doing anything.
    - throwawaymath 8y ago
      I disagree. Generally speaking students aren't able to teach themselves mathematics on their own until they hit the graduate level. That's not a hard rule, it just depends on mathematical maturity. But everyone I know who is able to teach themselves math without lectures or a professor still does the exercises in the textbook. Math isn't a spectator sport, it's really important to do active learning.
    - The_suffocated 8y ago
      This depends on the textbooks. It is very, very hard to find a textbook that is (a) well written, (b) suitable for self-study and (c) affordable.
      - hnuser355 8y ago
        Idk, I tried to self study the first two chapters of a very hated textbook (Principles of Mathematical Analysis). If you’d have asked me if I understood the material I would have said no. But by the time I took the actual class on that material I made high As on all the exams and could call out mistakes my professor made on metric sets with counter examples.
  - jordigh 8y ago
    In my country we have a saying: la matemática entra por la mano, no por los ojos -- mathematics is absorbed with your hands, not with your eyes.
  - kiba 8y ago
    You can't hardly learn anything by just reading or watching it.
- mymythisisthis 8y ago
  Any good recommendations on books that are more than pop-fluff but lighter than an academic text?
  - technofire 8y ago
    Jeremy Kun, who writes the "Math ∩ Programming" blog,[1] recently published a book, "A Programmer's Introduction to Mathematics,"[2] that seems to leverage the conceptual knowledge programmers already have to help them learn serious math. There also are sections in the book on math culture which seems interesting, something I haven't seen elsewhere. [1] https://jeremykun.com/ https://jeremykun.com/ [2] https://pimbook.org/ https://pimbook.org/
  - tyler- 8y ago
    I like this book: https://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-Numbers/dp/0691138710 https://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-N...
  - bgutierrez 8y ago
    I really liked Introduction to Graph Theory by Trudeau. No upper level math required, but it got rigorous and fun at times. Mostly spatial reasoning. David Foster Wallace wrote Everything and More which included a lot of math history, while driving into transfinite numbers and set theory. Kindle typesetting is generally terrible for math, but was completely readable in this case. The forward by Neal Stephenson really sets the mood.
    - a_wild_dandan 8y ago
      > David Foster Wallace wrote Everything and More This is far and away my favorite math book. The way DFW can seamlessly transition from discussing a serious mathematical concept, to the nature of insanity, to the philosophy of abstraction all within a few pages is astounding. He somehow manages to do it while maintaining a conversational, two-folks-at-a-bar tone, teaching me dozens of new word, and being insightful/funny. Like an English professor sneaked into the math dept. I didn't know it was available on Kindle. Guess I'm buying it again...
- okl 8y ago
  Would also recommend 'Surreal Numbers' by Don Knut hand 'The Book of Numbers' by Conway and Guy.
- masonic 8y ago
  The author may have just chosen what's in stock at Amazon, given that all the links are Amazon affiliate links (tag=alikayaspor0b-20).
jeff76 8y ago
Any books on how to do proofs, that also includes solutions?
- akinsketch 8y ago
  I personally found "How to Prove It: A Structured Approach" by Daniel J. Velleman very helpful.
- waynecochran 8y ago
  He included "proofs from the book" which are suppose to be the most elegant proofs.
- okl 8y ago
  'Surreal Numbers' by Donald E. Knuth is a ca. 100-page booklet that develops rules for surreal numbers with proofs. It's colorful but brief.
- pps 8y ago
  https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf
- rguzman 8y ago
  How to Solve It by Polya.
  - emgee_1 8y ago
    This.
- hnuser355 8y ago
  I used the book “A Transition to Advanced Mathematics” in undergraduate and I loved it. Many but not all exercises have answers in the back
- graycat 8y ago
  Sorta, kinda supposed to learn to "do proofs" by reading books that are done with theorems and proofs and have exercises that ask for proofs. Commonly the first start on proofs, in either the books or the exercises, is high school plane geometry. The proofs there are commonly in a rigid format which is traditional and maybe good for a start. Later, the good proofs are still fully precise but without the rigid format. Then high school second year algebra may have some proofs; trigonometry pretty much does; solid geometry does; analytic geometry mostly does. Then first calculus commonly goes a little light on the proofs. After I went through all that, I took a course in college abstract algebra, and (1) it was all proofs and, best of all, (2) the prof read some of the homework and gave little notes on how to do proofs. E.g., he said to use "since" instead of "because". That was about all I needed in one on one tutoring in how to do proofs. Do calculus with all the proofs later in pure treatments of advanced calculus, e.g., W. Rudin, Principles of Mathematical Analysis. As I read proofs by some good authors, especially, Nering, Halmos, Rudin, Fleming, Spivak, Coddington, Tukey, Neveu, but more, I refined how I wrote proofs. One step I took was, in a proof, when use one of the clearly stated assumptions in the statement of the theorem, mention that are using the assumption, that is, so that can be more sure are actually using the assumption, and where and for what, for yourself and any readers. If don't use the assumption, then either have (A) a more general result and maybe something exciting or (B) an error, and in practice more likely (B)! Later in a course in measure theory, a prof corrected some of my homework: In measure theory, are working with infinity so much that have to be careful not to, say, subtract one possibly infinite quantity from another one; that is, have to be a little careful slightly to refine some algebraic manipulation skills learned before. It's all quite doable, but if get that far in math then should learn this little point of being careful. When you have the actual ideas necessary for a proof, the above is about all you need -- or so it seems to me, and I have an applied math Ph.D. and have published papers with theorems and proofs. For thinking of the ideas needed for an especially challenging textbook exercise or, and mostly for, research, i.e., a new theorem and its proof, commonly my approach is (1) have good enough understanding of the likely main prerequisite material, techniques, and tools, (2) think intuitively, e.g., build little intuitive models, (3) with the little models, or actual reasoning if can, do some fast thought experiments, working quickly, with a lot of intuition and not very precisely, something like "If A is true, then it looks like B is also true, but, gee, B sounds like asking too much, so maybe A isn't true.". I.e., trying to prove things that likely aren't true is at best a long shot! Can continue by looking at what does/does not hold in simple cases; if can find a counterexample in a simple case, then can give up on a generalization being true! If the result does hold in a simple case and have a proof it does, then can look at the proof and start to guess where the simple proof would fail in the more general case. Uh, just because one proof of a simple case doesn't generalize does NOT mean that the more general case doesn't hold! Be careful about that! There start to back off a little, ascend to maybe 2000 feet up, and start to look at the larger picture and what the important parts, assumptions, techniques, are there. Then with this larger view, get some hints for a proof for that more general situation. If something works in some simple cases, then start to guess, test, understand where generalizations might or do fail. Use more such simple, intuitive, meta stuff, maybe from what I outlined here or what you can dream up and try. From such guessing around, may start to have some idea what is true and false and why. Then can set out to do a real proof. If are still stuck, then try to back off and guess, maybe, what assumption actually holds that so far I have not noticed and not exploited. Or, a polished, correct proof is very precise; coming up with that proof might have used a lot of intuition, guessing, etc. For winning the Abel Prize, f'get about my advice!
frostburg 8y ago
The Courant-Robbins book is good but I wouldn't actually suggest it for beginners, it's harder than necessary in many topics. Concrete mathematics is great but not something you could suggest to, for example, a biologist or medical doctor that wants a deeper understanding of some aspect of mathematics. It's (explicitly) meant for computer science.
- kayuri 8y ago
  I've read it in high school and was truly fascinated with which elegance and simplicity the book approached some of the most complex math problems. Of course it just scratched the surface but partially because of this book I chose the Math Department. And our school teacher recommended Courant-Robbins to all the students in a class, many of which actually joined Math Dept too. So, it might be not just me. :)
mbrudd 8y ago
Before buying a copy of Prime Obsession, please learn more about the author: https://en.wikipedia.org/wiki/John_Derbyshire https://en.wikipedia.org/wiki/John_Derbyshire I bought and read his book some time ago, and now I regret supporting him with my purchase.
- yters 8y ago
  > There have of course been some blots on the record, but I don't see how it can be denied that net-net, white Europeans have made a better job of running fair and stable societies than has any other group. If colonialism, WW2 and USSR are 'blots' he might have to reexamine his definition of 'blot'. White supremacists hold themselves up as people willing to speak difficult truths. But their logic tends to be pretty question begging. It's like atheists who claim religion is the root of violence, all the while ignoring the genocidal atheists regimes of the 20th and 21st centuries.
  - p1esk 8y ago
    Which group did better than white Europeans?
    - yters 8y ago
      The groups that didn't start global wars or genocidal regimes that wiped out hundreds of millions of people? Another odd thing is that these white supremacists act like Judeo-Christian culture (which is what they are really praising) sprang fully formed from the Caucasian race. However, the white Europeans were just warring tribes until they were colonized by the Romans. And, Judeo-Christian culture was founded by Jews (Jesus was a Jew), whom the white supremacists tend to dislike. It'd be interesting if they could name some significant aspect of Judeo-Christian culture that came entirely from white Europeans. White supremacists come off as bumbling ethnocentric historical revisionists.
      - OpenBSD-supreme 8y ago
        >white supremist this >white supremist that Why is no other racial pro-group labelled "supremist" while white/ethnic Euro interest groups are? Also, Christianity has been an anchor around Europe's neck for millennia; it's about time a scientific secular system of thought puts it to bed permanently. One can be pro Euro and also be secular, we each evolved as separate groups with local adaptations. One isn't necessarily "supreme" anymore than a poodle is supreme over a chihuahua. It is also not illogical to have a preference for one's own over outsider groups, otherwise why is anti-racism drilled so hard in school and childrens' pop media? (I can't think of a future more deprived of diversity than the "inevitable" mocha-colored masses of a millennia from now, all sharing the same global culture.)
molteanu 8y ago
Better read Euclid's Elements and then you'll know if you like maths or not.
winchling 8y ago
Proofs from the Book looks superb but does anyone know of another introduction to proofs or compendium of simple proofs suitable for young mathematicians?
- dragon96 8y ago
  I'm a big fan of Art and Craft of Problem Solving by Zeitz. The main downside is that there isn't an official solution manual, so checking your work takes a little more work. But one of the main upsides is that there isn't an official solution manual, so you actually have to work through the problems. :)
sonabinu 8y ago
I would recommend "Fermat's Enigma" by Simon Singh. It's is book that goes into the heart of how mathematics is sometimes elusive to the most sincere pursuit and at the same time yields itself to a creative mind that chases it with passion.
soVeryTired 8y ago
For anyone with an undergrad maths background, I'd recommend 'Fearless Symmetry' and 'Elliptic Tales'. They would be tough going for someone without a maths background, but they give a view to the forefront of some active areas of research in maths.
raintrees 8y ago
Likely listed elsewhere, from the article for convenience: Zero: The Biography of a Dangerous Idea - Charles Seife Measurement - Paul Lockhart Prelude to Mathematics - W. W. Sawyer Proofs from The Book - Aigner and Ziegler The Joy of x - Steven Strogatz Things to Make and Do in the Fourth Dimension - Matt Parker What is Mathematics? - Courant and Robbins A History of PI - Petr Beckmann An Imaginary Tale - Paul Nahin e: The Story of a Number - Eli Maor Imagining Numbers - Barry Mazur Journey Through Genius - William Dunham Prime Obsession - John Derbyshire
tzs 8y ago
Some suggestions, aiming for books that will teach you some interesting math rather than teach you about some interesting math like most of the books on that list do (except for "Proofs From the Book" and "What is Mathematics?", which would be on my list below if they weren't already in the submitted list). The following range all over the place in prerequisites, from things you could probably do with just middle school algebra to things that probably need early college level. "Challenging Mathematical Problems with Elementary Solutions" by Yaglom and Yaglom, Volume 1 and 2. Volume 1 contains 100 problems from probability and combinatorics. Volume 2 contains 74 problems from a variety of areas including points and lines, lattices of points in the plane, topology, convex polygons, distribution of objects, nondecimal counting, theory of primes. Complete solutions are included for each problems, as well as hints. Available from Dover so relatively inexpensive but good quality. Here are the Dover links, but of course they are available from Amazon and the other usual places. I'm linking to Dover because that will have the most complete description. http://store.doverpublications.com/0486655369.html http://store.doverpublications.com/0486655369.html http://store.doverpublications.com/0486655377.html http://store.doverpublications.com/0486655377.html ---------------------------------------- "Three Pearls of Number Theory" by Khinchin. One of Khinchin's former students was seriously wounded in WWII, and to pass the time during his long recovery in the hospital he wrote to his old professor and asked if he had anything mathematical to study to pass the time. Khinchin wrote back with three problems in elementary number theory that had recently been solved by people who were not a "great number theorist". Khinchin gave his former student the proofs along with guidance, examples, clarifications, and notes to help understand them. Dover link: http://store.doverpublications.com/0486400263.html http://store.doverpublications.com/0486400263.html Review at MAA: https://www.maa.org/press/maa-reviews/three-pearls-of-number-theory https://www.maa.org/press/maa-reviews/three-pearls-of-number... ---------------------------------------- "The Enjoyment of Math" by Rademacher and Toeplitz. The MAA review has a good summary: > This is a serious math book that has minimal prerequisites: geometry and college algebra, but no trig or calculus. It contains 28 largely independent chapters that solve a variety of famous and difficult math problems, mostly in the areas of plane geometry and number theory. The problems include: Fermat’s last theorem for exponent 4, unique factorization in number fields, a number of geometrical maximization problems including several versions of the isoperimetric problem, some transfinite numbers, the 5-color map coloring theorem, and the arithmetic mean - geometric mean inequality. There’s no analysis per se in the book, but several topics depend on the analytic ideas of continuity and variation. > This book was first published in German in 1930 and in English in 1957 as The Enjoyment of Mathematics, and is still in print today in both languages. This implies that there is still an audience for it, but it is hard to imagine exactly what this audience is. The book was developed out of a series of public lectures and was intended as a “popular math” book. While it is very clear and well-written, the reasoning in all the chapters is very intricate (especially in the geometric problems), and the book is much more difficult than anything that appears in popular math books being written today. It’s also too difficult for a math appreciation text. The modern (2000) Preface to the German edition suggests that the book is suited for bright high-school students who are hungry for learning, and maybe this is its real audience today https://www.maa.org/press/maa-reviews/the-enjoyment-of-math https://www.maa.org/press/maa-reviews/the-enjoyment-of-math https://www.amazon.com/gp/product/B07DMWX5FC/ https://www.amazon.com/gp/product/B07DMWX5FC/ ---------------------------------------- Anneli Lax New Mathematical Library is a whole series of books described thusly at the AMS site: > Featuring fresh approaches and broad coverage of topics especially suitable for high school and the first two years of college, the volumes in this series are an excellent source of enrichment material for teachers and students. Good mathematical reading with lively exposition. https://bookstore.ams.org/nml https://bookstore.ams.org/nml I read "Ingenuity in Mathematics" by Honsberger in high school and it was good. Kind of like "The Enjoyment of Mathematics" but a lot easier. A lot of books in this series can be good stepping stones to more advances books. For example, Olds "Continued Fractions" could be a reasonable read before then reading Khinchin's "Continued Fractions". The latter is available from Dover and is about 1/3 the price of the Olds book, so personally I'd start with Khinchin, and if it turns out a simpler intro is needed then I'd get Olds. This is a good point to toss in a note about Dover. They like to take older books, often out of print, get the rights to them, and publish a relatively inexpensive but high quality paperback edition. The difficulty level ranges from classic elementary intro texts to advanced material for practicing mathematicians. (And not just math...they do this for physics, chemistry, and various other fields of science and engineering). If you are interested on some math topic and want a book on it, it is usually a good idea to have a look at the Dover catalog to see if they have something about that at the level you are looking for. ---------------------------------------- "A Book of Abstract Algebra" by Pinter, available as a Dover edition. http://store.doverpublications.com/0486474178.html http://store.doverpublications.com/0486474178.html The usual undergraduate abstract algebra stuff: groups, rings, fields, the impossibility of the classic Greek duplicating the cube and trisecting the angle problem, Galois theory and solvability by radicals. What sets this book apart is that although it is rigorous and proves nearly everything, it takes things in smaller steps than a lot of other books, and has a lot of well chosen exercises that further cement the material, often by applying it to some interesting practical area. The exercises are grouped into sections, each of which focuses on a particular concept from the chapter, or develops and proves interesting things. One or two exercises from each of these sections usually has a solution given. Only about $12 at Amazon. If you haven't done much proof-based math before this could be a good first proof-based book.
ltbarcly3 8y ago
These sorts of books can be interesting as a sort of 'history of science', but I find they are very misleading, and can't help but end up as exaggerated dramatizations of a handful of the personalities involved with virtually no scientific or mathematical content. I understand that this is probably the best the authors can do, since actual content would require years of academic preparation and near full time study on the audience's part to be able to start to approach most of these topics, but that should be a huge clue that the essence can't really be boiled down. If any of the core ideas of these subjects were accessible in any significant way via just reading a book with no real prerequisites or preparation, people wouldn't have to spend 4 years of full time study just to get to the point that some of them are considered to be prepared to start to study them in a serious way. There may be some value to science in that these popularizations increase support for science funding by creating 'fans of science', the people that read them are no better off or more educated than if they had just read a romance novel or a western. TLDR; you aren't learning anything when you read these books, other than a exaggerated biography (with largely invented stories of conflict and drama) of some of the scientists.
- g9yuayon 8y ago
  Did you read the book What Is Mathematics? It is certainly not "an exaggerated biography for some of the scientists". And the mentioned books, at least some of them, serve different purposes: they aim to inspire, to motivate, and to offer historical context and intuition. The last is especially important, as they show people how abstract concepts emerged from historically concrete endeavors. When it comes to learning math, you can't take an elite's view. Not everyone is born Bourbaki dudes or Galois or people like them. Ordinary people like me don't just fall in love with maths. I was certainly not interested in number theory as I thought it was too fundamental for me to spend serious time on. And I'm still not. I was certainly puzzled on why my professors introduced the concept of functional in linear algebra or quotient groups in algebra or lattice equations in program analysis or category theory in model checking or probability space in probability. After all, all I wanted was to learn how to model the world to be a better programmer. And I was not able to grasp the abstractions without serious effort. I needed historical context and motivations to plow through those topics and to enjoy math. Yes, the elites will love and excel at math for no particular reason. Yet it is the middle majority like me who will greatly benefit from the mentioned books and biographies and what not.
jim_bailie 8y ago
As a primary guide to the subject, I would recommend "The Road to Reality" by Roger Penrose.
turingspiritfly 8y ago
Why not The Princeton Companion to Mathematics?
amai 8y ago
I can recommend https://en.wikipedia.org/wiki/God_Created_the_Integers https://en.wikipedia.org/wiki/God_Created_the_Integers "God Created the Integers: The Mathematical Breakthroughs That Changed History is an anthology, edited by Stephen Hawking, of "excerpts from thirty-one of the most important works in the history of mathematics.""

191 points by furcyd 8y ago

Behind paywall

Many of the list are pop-maths rather than fundamental texts. I do see many in software field recommend Chrystal:Algebra an Elementary Textbook. I also find Don Knuth, 'Concrete Mathematics' a interesting book for software people.

commandlinefan 8y ago
I came to a realization years ago that when it comes to math/science/computers, if I just read about it, I’m not really learning anything. I don’t learn unless I’m participating. Math books without _lots_ of exercises probably don’t really impart that much, although they can be a fun distraction if you need something to read on an airplane.
- arvinsim 8y ago
  I agree. Some things only stick by repeated application. This becomes more relevant the one older gets.
- hnuser355 8y ago
  Depending on your level of math ability you can often learn a lot by reading textbooks and proofs very carefully and closely (my opinion)
  - whatshisface 8y ago
    It depends on your definition of learning. If by learning you mean remembering how one author defined and proved various things then sure. If by learning you mean gaining the ability to do what the author did except for new problems, then at some point you will have to grapple with the fact that coming up with new ideas is a very different mental exercise from recalling things you've read. A supporting factor in the necessity of exercises for learning to actually do the math is that authors often hide important technical tricks in the exercises. Of course, a sufficiently intelligent person could learn anything by reading tea leaves and then deriving on their own whatever it was they wanted to learn, so in a sense you can learn math by doing anything.
  - throwawaymath 8y ago
    I disagree. Generally speaking students aren't able to teach themselves mathematics on their own until they hit the graduate level. That's not a hard rule, it just depends on mathematical maturity. But everyone I know who is able to teach themselves math without lectures or a professor still does the exercises in the textbook. Math isn't a spectator sport, it's really important to do active learning.
  - The_suffocated 8y ago
    This depends on the textbooks. It is very, very hard to find a textbook that is (a) well written, (b) suitable for self-study and (c) affordable.
    - hnuser355 8y ago
      Idk, I tried to self study the first two chapters of a very hated textbook (Principles of Mathematical Analysis). If you’d have asked me if I understood the material I would have said no. But by the time I took the actual class on that material I made high As on all the exams and could call out mistakes my professor made on metric sets with counter examples.
- jordigh 8y ago
  In my country we have a saying: la matemática entra por la mano, no por los ojos -- mathematics is absorbed with your hands, not with your eyes.
- kiba 8y ago
  You can't hardly learn anything by just reading or watching it.
mymythisisthis 8y ago
Any good recommendations on books that are more than pop-fluff but lighter than an academic text?
- technofire 8y ago
  Jeremy Kun, who writes the "Math ∩ Programming" blog,[1] recently published a book, "A Programmer's Introduction to Mathematics,"[2] that seems to leverage the conceptual knowledge programmers already have to help them learn serious math. There also are sections in the book on math culture which seems interesting, something I haven't seen elsewhere. [1] https://jeremykun.com/ https://jeremykun.com/ [2] https://pimbook.org/ https://pimbook.org/
- tyler- 8y ago
  I like this book: https://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-Numbers/dp/0691138710 https://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-N...
- bgutierrez 8y ago
  I really liked Introduction to Graph Theory by Trudeau. No upper level math required, but it got rigorous and fun at times. Mostly spatial reasoning. David Foster Wallace wrote Everything and More which included a lot of math history, while driving into transfinite numbers and set theory. Kindle typesetting is generally terrible for math, but was completely readable in this case. The forward by Neal Stephenson really sets the mood.
  - a_wild_dandan 8y ago
    > David Foster Wallace wrote Everything and More This is far and away my favorite math book. The way DFW can seamlessly transition from discussing a serious mathematical concept, to the nature of insanity, to the philosophy of abstraction all within a few pages is astounding. He somehow manages to do it while maintaining a conversational, two-folks-at-a-bar tone, teaching me dozens of new word, and being insightful/funny. Like an English professor sneaked into the math dept. I didn't know it was available on Kindle. Guess I'm buying it again...
okl 8y ago
Would also recommend 'Surreal Numbers' by Don Knut hand 'The Book of Numbers' by Conway and Guy.
masonic 8y ago
The author may have just chosen what's in stock at Amazon, given that all the links are Amazon affiliate links (tag=alikayaspor0b-20).

jeff76 8y ago

Any books on how to do proofs, that also includes solutions?

akinsketch 8y ago
I personally found "How to Prove It: A Structured Approach" by Daniel J. Velleman very helpful.
waynecochran 8y ago
He included "proofs from the book" which are suppose to be the most elegant proofs.
okl 8y ago
'Surreal Numbers' by Donald E. Knuth is a ca. 100-page booklet that develops rules for surreal numbers with proofs. It's colorful but brief.
pps 8y ago
https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf https://www.people.vcu.edu/~rhammack/BookOfProof/Main.pdf
rguzman 8y ago
How to Solve It by Polya.
- emgee_1 8y ago
  This.
hnuser355 8y ago
I used the book “A Transition to Advanced Mathematics” in undergraduate and I loved it. Many but not all exercises have answers in the back
graycat 8y ago
Sorta, kinda supposed to learn to "do proofs" by reading books that are done with theorems and proofs and have exercises that ask for proofs. Commonly the first start on proofs, in either the books or the exercises, is high school plane geometry. The proofs there are commonly in a rigid format which is traditional and maybe good for a start. Later, the good proofs are still fully precise but without the rigid format. Then high school second year algebra may have some proofs; trigonometry pretty much does; solid geometry does; analytic geometry mostly does. Then first calculus commonly goes a little light on the proofs. After I went through all that, I took a course in college abstract algebra, and (1) it was all proofs and, best of all, (2) the prof read some of the homework and gave little notes on how to do proofs. E.g., he said to use "since" instead of "because". That was about all I needed in one on one tutoring in how to do proofs. Do calculus with all the proofs later in pure treatments of advanced calculus, e.g., W. Rudin, Principles of Mathematical Analysis. As I read proofs by some good authors, especially, Nering, Halmos, Rudin, Fleming, Spivak, Coddington, Tukey, Neveu, but more, I refined how I wrote proofs. One step I took was, in a proof, when use one of the clearly stated assumptions in the statement of the theorem, mention that are using the assumption, that is, so that can be more sure are actually using the assumption, and where and for what, for yourself and any readers. If don't use the assumption, then either have (A) a more general result and maybe something exciting or (B) an error, and in practice more likely (B)! Later in a course in measure theory, a prof corrected some of my homework: In measure theory, are working with infinity so much that have to be careful not to, say, subtract one possibly infinite quantity from another one; that is, have to be a little careful slightly to refine some algebraic manipulation skills learned before. It's all quite doable, but if get that far in math then should learn this little point of being careful. When you have the actual ideas necessary for a proof, the above is about all you need -- or so it seems to me, and I have an applied math Ph.D. and have published papers with theorems and proofs. For thinking of the ideas needed for an especially challenging textbook exercise or, and mostly for, research, i.e., a new theorem and its proof, commonly my approach is (1) have good enough understanding of the likely main prerequisite material, techniques, and tools, (2) think intuitively, e.g., build little intuitive models, (3) with the little models, or actual reasoning if can, do some fast thought experiments, working quickly, with a lot of intuition and not very precisely, something like "If A is true, then it looks like B is also true, but, gee, B sounds like asking too much, so maybe A isn't true.". I.e., trying to prove things that likely aren't true is at best a long shot! Can continue by looking at what does/does not hold in simple cases; if can find a counterexample in a simple case, then can give up on a generalization being true! If the result does hold in a simple case and have a proof it does, then can look at the proof and start to guess where the simple proof would fail in the more general case. Uh, just because one proof of a simple case doesn't generalize does NOT mean that the more general case doesn't hold! Be careful about that! There start to back off a little, ascend to maybe 2000 feet up, and start to look at the larger picture and what the important parts, assumptions, techniques, are there. Then with this larger view, get some hints for a proof for that more general situation. If something works in some simple cases, then start to guess, test, understand where generalizations might or do fail. Use more such simple, intuitive, meta stuff, maybe from what I outlined here or what you can dream up and try. From such guessing around, may start to have some idea what is true and false and why. Then can set out to do a real proof. If are still stuck, then try to back off and guess, maybe, what assumption actually holds that so far I have not noticed and not exploited. Or, a polished, correct proof is very precise; coming up with that proof might have used a lot of intuition, guessing, etc. For winning the Abel Prize, f'get about my advice!

frostburg 8y ago

The Courant-Robbins book is good but I wouldn't actually suggest it for beginners, it's harder than necessary in many topics. Concrete mathematics is great but not something you could suggest to, for example, a biologist or medical doctor that wants a deeper understanding of some aspect of mathematics. It's (explicitly) meant for computer science.

kayuri 8y ago
I've read it in high school and was truly fascinated with which elegance and simplicity the book approached some of the most complex math problems. Of course it just scratched the surface but partially because of this book I chose the Math Department. And our school teacher recommended Courant-Robbins to all the students in a class, many of which actually joined Math Dept too. So, it might be not just me. :)

mbrudd 8y ago

Before buying a copy of Prime Obsession, please learn more about the author: https://en.wikipedia.org/wiki/John_Derbyshire https://en.wikipedia.org/wiki/John_Derbyshire I bought and read his book some time ago, and now I regret supporting him with my purchase.

yters 8y ago
> There have of course been some blots on the record, but I don't see how it can be denied that net-net, white Europeans have made a better job of running fair and stable societies than has any other group. If colonialism, WW2 and USSR are 'blots' he might have to reexamine his definition of 'blot'. White supremacists hold themselves up as people willing to speak difficult truths. But their logic tends to be pretty question begging. It's like atheists who claim religion is the root of violence, all the while ignoring the genocidal atheists regimes of the 20th and 21st centuries.
- p1esk 8y ago
  Which group did better than white Europeans?
  - yters 8y ago
    The groups that didn't start global wars or genocidal regimes that wiped out hundreds of millions of people? Another odd thing is that these white supremacists act like Judeo-Christian culture (which is what they are really praising) sprang fully formed from the Caucasian race. However, the white Europeans were just warring tribes until they were colonized by the Romans. And, Judeo-Christian culture was founded by Jews (Jesus was a Jew), whom the white supremacists tend to dislike. It'd be interesting if they could name some significant aspect of Judeo-Christian culture that came entirely from white Europeans. White supremacists come off as bumbling ethnocentric historical revisionists.
    - OpenBSD-supreme 8y ago
      >white supremist this >white supremist that Why is no other racial pro-group labelled "supremist" while white/ethnic Euro interest groups are? Also, Christianity has been an anchor around Europe's neck for millennia; it's about time a scientific secular system of thought puts it to bed permanently. One can be pro Euro and also be secular, we each evolved as separate groups with local adaptations. One isn't necessarily "supreme" anymore than a poodle is supreme over a chihuahua. It is also not illogical to have a preference for one's own over outsider groups, otherwise why is anti-racism drilled so hard in school and childrens' pop media? (I can't think of a future more deprived of diversity than the "inevitable" mocha-colored masses of a millennia from now, all sharing the same global culture.)

molteanu 8y ago

Better read Euclid's Elements and then you'll know if you like maths or not.

winchling 8y ago

Proofs from the Book looks superb but does anyone know of another introduction to proofs or compendium of simple proofs suitable for young mathematicians?

dragon96 8y ago
I'm a big fan of Art and Craft of Problem Solving by Zeitz. The main downside is that there isn't an official solution manual, so checking your work takes a little more work. But one of the main upsides is that there isn't an official solution manual, so you actually have to work through the problems. :)

sonabinu 8y ago

I would recommend "Fermat's Enigma" by Simon Singh. It's is book that goes into the heart of how mathematics is sometimes elusive to the most sincere pursuit and at the same time yields itself to a creative mind that chases it with passion.

soVeryTired 8y ago

For anyone with an undergrad maths background, I'd recommend 'Fearless Symmetry' and 'Elliptic Tales'. They would be tough going for someone without a maths background, but they give a view to the forefront of some active areas of research in maths.

raintrees 8y ago

Likely listed elsewhere, from the article for convenience: Zero: The Biography of a Dangerous Idea - Charles Seife Measurement - Paul Lockhart Prelude to Mathematics - W. W. Sawyer Proofs from The Book - Aigner and Ziegler The Joy of x - Steven Strogatz Things to Make and Do in the Fourth Dimension - Matt Parker What is Mathematics? - Courant and Robbins A History of PI - Petr Beckmann An Imaginary Tale - Paul Nahin e: The Story of a Number - Eli Maor Imagining Numbers - Barry Mazur Journey Through Genius - William Dunham Prime Obsession - John Derbyshire

tzs 8y ago

Some suggestions, aiming for books that will teach you some interesting math rather than teach you about some interesting math like most of the books on that list do (except for "Proofs From the Book" and "What is Mathematics?", which would be on my list below if they weren't already in the submitted list). The following range all over the place in prerequisites, from things you could probably do with just middle school algebra to things that probably need early college level. "Challenging Mathematical Problems with Elementary Solutions" by Yaglom and Yaglom, Volume 1 and 2. Volume 1 contains 100 problems from probability and combinatorics. Volume 2 contains 74 problems from a variety of areas including points and lines, lattices of points in the plane, topology, convex polygons, distribution of objects, nondecimal counting, theory of primes. Complete solutions are included for each problems, as well as hints. Available from Dover so relatively inexpensive but good quality. Here are the Dover links, but of course they are available from Amazon and the other usual places. I'm linking to Dover because that will have the most complete description. http://store.doverpublications.com/0486655369.html http://store.doverpublications.com/0486655369.html http://store.doverpublications.com/0486655377.html http://store.doverpublications.com/0486655377.html ---------------------------------------- "Three Pearls of Number Theory" by Khinchin. One of Khinchin's former students was seriously wounded in WWII, and to pass the time during his long recovery in the hospital he wrote to his old professor and asked if he had anything mathematical to study to pass the time. Khinchin wrote back with three problems in elementary number theory that had recently been solved by people who were not a "great number theorist". Khinchin gave his former student the proofs along with guidance, examples, clarifications, and notes to help understand them. Dover link: http://store.doverpublications.com/0486400263.html http://store.doverpublications.com/0486400263.html Review at MAA: https://www.maa.org/press/maa-reviews/three-pearls-of-number-theory https://www.maa.org/press/maa-reviews/three-pearls-of-number... ---------------------------------------- "The Enjoyment of Math" by Rademacher and Toeplitz. The MAA review has a good summary: > This is a serious math book that has minimal prerequisites: geometry and college algebra, but no trig or calculus. It contains 28 largely independent chapters that solve a variety of famous and difficult math problems, mostly in the areas of plane geometry and number theory. The problems include: Fermat’s last theorem for exponent 4, unique factorization in number fields, a number of geometrical maximization problems including several versions of the isoperimetric problem, some transfinite numbers, the 5-color map coloring theorem, and the arithmetic mean - geometric mean inequality. There’s no analysis per se in the book, but several topics depend on the analytic ideas of continuity and variation. > This book was first published in German in 1930 and in English in 1957 as The Enjoyment of Mathematics, and is still in print today in both languages. This implies that there is still an audience for it, but it is hard to imagine exactly what this audience is. The book was developed out of a series of public lectures and was intended as a “popular math” book. While it is very clear and well-written, the reasoning in all the chapters is very intricate (especially in the geometric problems), and the book is much more difficult than anything that appears in popular math books being written today. It’s also too difficult for a math appreciation text. The modern (2000) Preface to the German edition suggests that the book is suited for bright high-school students who are hungry for learning, and maybe this is its real audience today https://www.maa.org/press/maa-reviews/the-enjoyment-of-math https://www.maa.org/press/maa-reviews/the-enjoyment-of-math https://www.amazon.com/gp/product/B07DMWX5FC/ https://www.amazon.com/gp/product/B07DMWX5FC/ ---------------------------------------- Anneli Lax New Mathematical Library is a whole series of books described thusly at the AMS site: > Featuring fresh approaches and broad coverage of topics especially suitable for high school and the first two years of college, the volumes in this series are an excellent source of enrichment material for teachers and students. Good mathematical reading with lively exposition. https://bookstore.ams.org/nml https://bookstore.ams.org/nml I read "Ingenuity in Mathematics" by Honsberger in high school and it was good. Kind of like "The Enjoyment of Mathematics" but a lot easier. A lot of books in this series can be good stepping stones to more advances books. For example, Olds "Continued Fractions" could be a reasonable read before then reading Khinchin's "Continued Fractions". The latter is available from Dover and is about 1/3 the price of the Olds book, so personally I'd start with Khinchin, and if it turns out a simpler intro is needed then I'd get Olds. This is a good point to toss in a note about Dover. They like to take older books, often out of print, get the rights to them, and publish a relatively inexpensive but high quality paperback edition. The difficulty level ranges from classic elementary intro texts to advanced material for practicing mathematicians. (And not just math...they do this for physics, chemistry, and various other fields of science and engineering). If you are interested on some math topic and want a book on it, it is usually a good idea to have a look at the Dover catalog to see if they have something about that at the level you are looking for. ---------------------------------------- "A Book of Abstract Algebra" by Pinter, available as a Dover edition. http://store.doverpublications.com/0486474178.html http://store.doverpublications.com/0486474178.html The usual undergraduate abstract algebra stuff: groups, rings, fields, the impossibility of the classic Greek duplicating the cube and trisecting the angle problem, Galois theory and solvability by radicals. What sets this book apart is that although it is rigorous and proves nearly everything, it takes things in smaller steps than a lot of other books, and has a lot of well chosen exercises that further cement the material, often by applying it to some interesting practical area. The exercises are grouped into sections, each of which focuses on a particular concept from the chapter, or develops and proves interesting things. One or two exercises from each of these sections usually has a solution given. Only about $12 at Amazon. If you haven't done much proof-based math before this could be a good first proof-based book.

ltbarcly3 8y ago

These sorts of books can be interesting as a sort of 'history of science', but I find they are very misleading, and can't help but end up as exaggerated dramatizations of a handful of the personalities involved with virtually no scientific or mathematical content. I understand that this is probably the best the authors can do, since actual content would require years of academic preparation and near full time study on the audience's part to be able to start to approach most of these topics, but that should be a huge clue that the essence can't really be boiled down. If any of the core ideas of these subjects were accessible in any significant way via just reading a book with no real prerequisites or preparation, people wouldn't have to spend 4 years of full time study just to get to the point that some of them are considered to be prepared to start to study them in a serious way. There may be some value to science in that these popularizations increase support for science funding by creating 'fans of science', the people that read them are no better off or more educated than if they had just read a romance novel or a western. TLDR; you aren't learning anything when you read these books, other than a exaggerated biography (with largely invented stories of conflict and drama) of some of the scientists.

g9yuayon 8y ago
Did you read the book What Is Mathematics? It is certainly not "an exaggerated biography for some of the scientists". And the mentioned books, at least some of them, serve different purposes: they aim to inspire, to motivate, and to offer historical context and intuition. The last is especially important, as they show people how abstract concepts emerged from historically concrete endeavors. When it comes to learning math, you can't take an elite's view. Not everyone is born Bourbaki dudes or Galois or people like them. Ordinary people like me don't just fall in love with maths. I was certainly not interested in number theory as I thought it was too fundamental for me to spend serious time on. And I'm still not. I was certainly puzzled on why my professors introduced the concept of functional in linear algebra or quotient groups in algebra or lattice equations in program analysis or category theory in model checking or probability space in probability. After all, all I wanted was to learn how to model the world to be a better programmer. And I was not able to grasp the abstractions without serious effort. I needed historical context and motivations to plow through those topics and to enjoy math. Yes, the elites will love and excel at math for no particular reason. Yet it is the middle majority like me who will greatly benefit from the mentioned books and biographies and what not.

jim_bailie 8y ago

As a primary guide to the subject, I would recommend "The Road to Reality" by Roger Penrose.

turingspiritfly 8y ago

Why not The Princeton Companion to Mathematics?

amai 8y ago

I can recommend https://en.wikipedia.org/wiki/God_Created_the_Integers https://en.wikipedia.org/wiki/God_Created_the_Integers "God Created the Integers: The Mathematical Breakthroughs That Changed History is an anthology, edited by Stephen Hawking, of "excerpts from thirty-one of the most important works in the history of mathematics.""