Classic Mathematics Books for Lifelong Learners(medium.com)
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Classic Mathematics Books for Lifelong Learners
https://medium.com/however-mathematics/13-classic-mathematics-books-for-lifelong-learners-7ec2759142da
58 comments
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.
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.
> I needed historical context and motivations to plow through those topics and to enjoy math.
Everyone really does! It's a travesty that most books just throw dry definition - theorem - proof at you, when in real world, the concepts were developed in the opposite order, and for a good reason.
Everyone really does! It's a travesty that most books just throw dry definition - theorem - proof at you, when in real world, the concepts were developed in the opposite order, and for a good reason.
I did a rough calculation, and far less than 1% of high school graduates in the US will be introduced to quotient groups at any point. While you might be in the middle majority of the people you interact with every day, you are quite elite compared to the general population, and in no way in any 'middle majority', and if anything you are doing what I was suggesting is necessary to appreciate these topics. If you enjoy reading popularizations on top of that, that's great!
Middle majority among those who pursue STEM in college. Not everyone is or should be in STEM, after all.
> involved with virtually no scientific or mathematical content
You can't say that about this list, on the account of "What is Mathematics" alone.
That's the book that really got me into math (ended up with a PhD in it), and it covers a very wide range of topics from number theory to geometry and topology (and has the best exposition of Calculus I've ever seen).
Also, a lot of branches of math do not require a very long preparation to get into. To get deeply into - yes.
So, perhaps you should narrow your statement to specific books.
You can't say that about this list, on the account of "What is Mathematics" alone.
That's the book that really got me into math (ended up with a PhD in it), and it covers a very wide range of topics from number theory to geometry and topology (and has the best exposition of Calculus I've ever seen).
Also, a lot of branches of math do not require a very long preparation to get into. To get deeply into - yes.
So, perhaps you should narrow your statement to specific books.
I've wonder if we've reached the limits of human knowledge. One has to become extremely specialized, very early on, to get to know anything of significance. Knowing more than one field is almost unheard of, more people probably win the lottery, twice, than can master two fields.
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.
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.
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.
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.
I agree. Some things only stick by repeated application. This becomes more relevant the one older gets.
Depending on your level of math ability you can often learn a lot by reading textbooks and proofs very carefully and closely (my opinion)
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.
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.
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.
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.
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.
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.
You can't hardly learn anything by just reading or watching it.
Any good recommendations on books that are more than pop-fluff but lighter than an academic text?
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.
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.
> 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...
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...
Warning: These books are kind of all over the place as far as technical prerequisites go, but they all satisify being heavier than pop-sci and lighter than a textbook. :)
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Aleksandrov et al - Mathematics: Its content, Methods, ad Meaning. Overview of undergrad curriculum by a number of prominent russian mathematicians.
Levi - The mathematical Mechanic. Uses various physical intuitions and scenarios to solve mathematical problems.
Pask - Magnificent Principia. Tour of newtons principia that includes both newtons geometric language and modern vector calculus version.
Stillwell - Elements of Mathematics. Discusses some important classical problems in the history of mathematics taken from 8 subjects: arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, logic. Wow!
Yandell - The Honors Class: Hilbert's problems and their solvers. Lots of mini biographies with some information describing each of Hilbert's famous problems.
Lawvere & Schanuel- Conceptual Mathematics. Some basic notions of category theory, geared towards teaching you Lawvere's clarification and generalization of the incompleteness theorems.
Cormen - Algorithms Unlocked. Pop Science version of some of the textbook.
Nahin - Inside interesting Integrals. Lots of fun integrations!
Nahin - He also wrote some books about euler and imaginary numbers which ive partially read, they seemed good.
Pierce - An introduction to information theory; symbols signals and noise. sounds and looks like a dover textbook but is a non-fake pop science book.
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Boyer - The History of Calculus and its Conceptual Development. A bit old, and spends too much time on formalism in the 18th and 19th century, there might be something better out now.
Weyl - Philosophy of Mathematics and Natural Science.
Reichenbach - The philosophy of Space and time.
Klein - Elementary mathematics from the advanced viewpoint. these were written at the beginning of the 20th century so are slightly outdated, but kline was extremely forward-looking in his point of view.
Courant & Robbins - What is Mathemaics. Another classic.
Hilbert & Cohn-Vossen - Geometry and the Imagination. The gold standard of popular mathematics writing. A beautiful book.
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Aleksandrov et al - Mathematics: Its content, Methods, ad Meaning. Overview of undergrad curriculum by a number of prominent russian mathematicians.
Levi - The mathematical Mechanic. Uses various physical intuitions and scenarios to solve mathematical problems.
Pask - Magnificent Principia. Tour of newtons principia that includes both newtons geometric language and modern vector calculus version.
Stillwell - Elements of Mathematics. Discusses some important classical problems in the history of mathematics taken from 8 subjects: arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, logic. Wow!
Yandell - The Honors Class: Hilbert's problems and their solvers. Lots of mini biographies with some information describing each of Hilbert's famous problems.
Lawvere & Schanuel- Conceptual Mathematics. Some basic notions of category theory, geared towards teaching you Lawvere's clarification and generalization of the incompleteness theorems.
Cormen - Algorithms Unlocked. Pop Science version of some of the textbook.
Nahin - Inside interesting Integrals. Lots of fun integrations!
Nahin - He also wrote some books about euler and imaginary numbers which ive partially read, they seemed good.
Pierce - An introduction to information theory; symbols signals and noise. sounds and looks like a dover textbook but is a non-fake pop science book.
---------------
Boyer - The History of Calculus and its Conceptual Development. A bit old, and spends too much time on formalism in the 18th and 19th century, there might be something better out now.
Weyl - Philosophy of Mathematics and Natural Science.
Reichenbach - The philosophy of Space and time.
Klein - Elementary mathematics from the advanced viewpoint. these were written at the beginning of the 20th century so are slightly outdated, but kline was extremely forward-looking in his point of view.
Courant & Robbins - What is Mathemaics. Another classic.
Hilbert & Cohn-Vossen - Geometry and the Imagination. The gold standard of popular mathematics writing. A beautiful book.
Regarding Principia, noted astronomer Subramanian Chandrashekhar, had written "Principia for the Common Reader", which has commentary in modern language, while yet using only the same geometric method that Newton used. While I have read only a small portion of it, it was very fascinating.
Somewhat dated but...I really like "A Logical Approach to Discrete Math" by Gries and Schneider. It uses an algebraic approach to introduce logic then works its way into number theory, graphs, and sets. There are a lot of good exercises and the topics are highly relevant for CS.
The content is so valuable that it deserves a year in high school in my opinion, though that is a minority view where I live.
The content is so valuable that it deserves a year in high school in my opinion, though that is a minority view where I live.
I like this book:
https://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-N...
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/
[2] https://pimbook.org/
[1] https://jeremykun.com/
[2] https://pimbook.org/
Man you just hit the nail on the head.
In a similar vein, I love documentaries and I learned much as a child that way. I find that they are way too basic now and often repeat things I've seen/known for years. This is needed for other kids of course, but as an adult the only stuff is really basic or so advanced that it is actual work to make it through (tough after a rough week at work).
One of my favorite examples of finding the middle ground is in dealing with SETI. I love the subject and have seen numerous documentaries on it over the years that cover what they're doing from 50,000 feet. Paul Davies is the lead at SETI and I read his book "The Eerie Silence" and it was really awesome. It perfectly found the middle ground between a documentary which has to be ok for children and a textbook on the subject. Paul Davies is kind of to SETI what Alan Kay is to computing in some ways as someone with deep experience in the field and lots of philosophical thoughts on the subject. His writing is really clear though (I love Kay, but don't have a strong enough computing, historical computing, math, or biology background to keep up with a lot of his writing). Two others that stand out to me are the classic "A Brief History of Time" by Stephen Hawking that blew my mind as a kid and the equally good, but lesser known "Black Holes and Time Warps" by Kip Thorne which made an excellent follow up book that gives a good high level appreciation which would likely be helpful for anyone planning to study those subjects in college at a rigorous mathematical level. Kip's book is also funny. There is a picture of Stephen Hawking delivering him a years subscription to a magazine (I think it might have been lewd in nature) for losing a bet to Kip over whether Cygnus A would have a black hole I think. As a highschool student I wrote to him with some career questions and he was kind enough to respond! Max Tegmark has a book "Our Mathematical Universe" which covers two parts: one on why we might be part of a simulation, and the first part which is like a condensed cosmology lesson from A to Z. He had a big part in several big discoveries and hearing the play by play is gripping. He was apparently a pretty gifted coder, so he got tasked with a lot of the data analysis and therefore he and his first wife were among the first to see the actual results of these big experiments. Code by Petzold is also an amazing tour de force if you want to understand computers.
In a similar vein, I love documentaries and I learned much as a child that way. I find that they are way too basic now and often repeat things I've seen/known for years. This is needed for other kids of course, but as an adult the only stuff is really basic or so advanced that it is actual work to make it through (tough after a rough week at work).
One of my favorite examples of finding the middle ground is in dealing with SETI. I love the subject and have seen numerous documentaries on it over the years that cover what they're doing from 50,000 feet. Paul Davies is the lead at SETI and I read his book "The Eerie Silence" and it was really awesome. It perfectly found the middle ground between a documentary which has to be ok for children and a textbook on the subject. Paul Davies is kind of to SETI what Alan Kay is to computing in some ways as someone with deep experience in the field and lots of philosophical thoughts on the subject. His writing is really clear though (I love Kay, but don't have a strong enough computing, historical computing, math, or biology background to keep up with a lot of his writing). Two others that stand out to me are the classic "A Brief History of Time" by Stephen Hawking that blew my mind as a kid and the equally good, but lesser known "Black Holes and Time Warps" by Kip Thorne which made an excellent follow up book that gives a good high level appreciation which would likely be helpful for anyone planning to study those subjects in college at a rigorous mathematical level. Kip's book is also funny. There is a picture of Stephen Hawking delivering him a years subscription to a magazine (I think it might have been lewd in nature) for losing a bet to Kip over whether Cygnus A would have a black hole I think. As a highschool student I wrote to him with some career questions and he was kind enough to respond! Max Tegmark has a book "Our Mathematical Universe" which covers two parts: one on why we might be part of a simulation, and the first part which is like a condensed cosmology lesson from A to Z. He had a big part in several big discoveries and hearing the play by play is gripping. He was apparently a pretty gifted coder, so he got tasked with a lot of the data analysis and therefore he and his first wife were among the first to see the actual results of these big experiments. Code by Petzold is also an amazing tour de force if you want to understand computers.
What field of mathematics interests you? Or you want some digestable intro to all of them? In that case I would recommend Kevin Houston "How to Think Like a Mathematician".
“An Introduction to the Theory of Numbers” by Hardy and Wright.
The author may have just chosen what's in stock at Amazon, given that all the links are Amazon affiliate links (tag=alikayaspor0b-20).
Would also recommend 'Surreal Numbers' by Don Knut hand 'The Book of Numbers' by Conway and Guy.
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
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
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.
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.
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. :)
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.
Before buying a copy of Prime Obsession, please learn more about the author: 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.
I bought and read his book some time ago, and now I regret supporting him with my purchase.
> 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.
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.
Which group did better than white Europeans?
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.
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.
I can recommend 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.""
"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.""
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.
Any books on how to do proofs, that also includes solutions?
I personally found "How to Prove It: A Structured Approach" by Daniel J. Velleman very helpful.
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!
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!
'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.
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
He included "proofs from the book" which are suppose to be the most elegant proofs.
Proofs from the Book looks superb but does anyone know of another introduction to proofs or compendium of simple proofs suitable for young mathematicians?
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. :)
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. :)
As a primary guide to the subject, I would recommend "The Road to Reality" by Roger Penrose.
Better read Euclid's Elements and then you'll know if you like maths or not.
Why not The Princeton Companion to Mathematics?
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/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
Review at MAA: 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.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
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
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.
"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/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
Review at MAA: 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.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
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
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.
Behind paywall
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.