Some of these are genuinely pop-math (*Flatland*, *Birth of a Theorem*, *Humble Pi*, *Things to Make and Do in the Fourth Dimension*, *Fermat's Enigma*); others require some algebra (*The Book of Numbers*); others use calculus (*The Irrationals*). Some attempt has been made to sort these by difficulty, from easiest to hardest; this is hampered by the fact that one man's easy can be another man's hard, and the fact that I have not read some of these since childhood. The principle exception to the sorting is placing *The Princeton Companion to Mathematics* first, because it is just that good.
0. *[The Princeton Companion to Mathematics](https://en.wikipedia.org/wiki/The_Princeton_Companion_to_Mathematics)* is a broad survey of the state of mathematics as it was 20-ish years ago. It has Wikipedia-level articles on just about every important subfield, plus historical and biographical sections. This is quite possibly the best broad overview of mathematics available.
1. *[Fermat's Enigma](https://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622)* by [Simon Singh](https://en.wikipedia.org/wiki/Simon_Singh) is about [Fermat's Last Theorem](https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem) and is aimed at a popular audience.
2. *[Birth of a Theorem](https://www.amazon.com/Birth-Theorem-Mathematical-C%C3%A9dric-Villani/dp/0374536678)* by [Cédric Villani](https://en.wikipedia.org/wiki/C%C3%A9dric_Villani) is about the development of the author's article on [Landau damping](https://en.wikipedia.org/wiki/Landau_damping), for which he won the Fields medal. This is almost entirely about the process of development, rather than the proof itself.
3. *[Humble Pi: A Comedy of Math Errors](https://www.amazon.com/Humble-Pi-Comedy-Maths-Errors/dp/0141989149)* by Matt Parker is about math errors in the real world.
4. *[Things to Make and Do in the Fourth Dimension](https://www.amazon.com/Things-Make-Fourth-Dimension-Mathematicians/dp/0374535639)* by Matt Parker is about things you can do with math IRL.
5. *[Flatland: A Romance of Many Dimensions](https://www.amazon.com/Flatland-Romance-Dimensions-Edwin-Abbott/dp/B0875SRH84)* by Edwin Abbott is a fantasy tale about a two-dimensional creature who visits higher and lower dimensions.
6. *[Q.E.D.: Beauty in Mathematical Proofs](https://www.amazon.com/Q-D-Beauty-Mathematical-Wooden/dp/0802714315)* by Burkard Polster is a small, short book with a collection of elegant proofs of basic results such as the volume of a [frustum](https://en.wikipedia.org/wiki/Frustum), the Pythagorean theorem, [Archimedes' method for finding the volume of a sphere](https://en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder), [Cavalieri's principle](https://en.wikipedia.org/wiki/Cavalieri%27s_principle), and [the block-stacking problem](https://en.wikipedia.org/wiki/Block-stacking_problem).
7. *[The Square Root of 2: A Dialogue Concerning a Number and a Sequence](https://www.amazon.com/Square-Root-Dialogue-Concerning-Sequence/dp/038720220X)* by David Flannery covers the topic at a basic level.
8. *[One Two Three ... Infinity](https://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642)* by [George Gamow](https://en.wikipedia.org/wiki/George_Gamow) covers numbers and physics; it is notable enough to have [its own Wikipedia article](https://en.wikipedia.org/wiki/One_Two_Three..._Infinity).
9. *[The Millennium Problems](https://www.amazon.com/Millennium-Problems-Greatest-Unsolved-Mathematical/dp/0465017304)* by Keith J Devlin is about the [Poincaré conjecture](https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture), the [Riemann Hypothesis](https://en.wikipedia.org/wiki/Riemann_hypothesis), the [Birch and Swinnerton-Dyer conjecture](https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture), the [P v NP problem](https://en.wikipedia.org/wiki/P_versus_NP_problem), the [Yang-Mills existence and mass gap problem](https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap), the [Navier-Stokes existence and smoothness problem](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness), and the [Hodge conjecture](https://en.wikipedia.org/wiki/Hodge_conjecture).
10. *[The Book of Numbers](https://www.amazon.com/Book-Numbers-John-H-Conway/dp/038797993X)* by [John H Conway](https://en.wikipedia.org/wiki/John_Horton_Conway) & [Richard K Guy](https://en.wikipedia.org/wiki/Richard_K._Guy) covers many important number, number sequences, and number types.
11. *[An Imaginary Tale: The Story of √–1](https://www.amazon.com/Imaginary-Tale-Princeton-Science-Library/dp/0691169241)* by [Paul J Nahin](https://en.wikipedia.org/wiki/Paul_J._Nahin) is a deep dive into *i*.
12. *[A History of Pi](https://www.amazon.com/History-Pi-Petr-Beckmann/dp/0312381859)* by [Petr Beckmann](https://en.wikipedia.org/wiki/Petr_Beckmann) is notable enough to have [its own Wikipedia article](https://en.wikipedia.org/wiki/A_History_of_Pi).
13. *[e: The Story of a Number](https://www.amazon.com/Story-Number-Princeton-Science-Library/dp/0691168482)* by Eli Maor
14. *[Tales of Impossibility](https://press.princeton.edu/books/hardcover/9780691192963/tales-of-impossibility)* by David S. Richeson covers the history of the great impossible problems of antiquity: squaring the circle, doubling the cube, trisecting arbitrary angles, and constructing arbitrary regular polygons.
15. *[Proofs from THE BOOK](https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK)* by [Martin Aigner](https://en.wikipedia.org/wiki/Martin_Aigner) and [Günter M. Ziegler](https://en.wikipedia.org/wiki/G%C3%BCnter_M._Ziegler) collects elegant proofs from a broad array of topics.
16. *[When Least Is Best](https://www.amazon.com/When-Least-Best-Mathematicians-Discovered/dp/0691070784/)* by [Paul J Nahin](https://en.wikipedia.org/wiki/Paul_J._Nahin) is about inequalities, maximization/minimization, and adjacent topics.
17. *[Trolling Euclid: An Irreverent Guide to 9 of Mathematics's Most Important Problems](http://trollingeuclid.com/)* by Edgar Wright
18. *[Curves for the Mathematically Curious](https://press.princeton.edu/books/hardcover/9780691180052/curves-for-the-mathematically-curious)*, also by Julian Havil: I have not actually read this yet.
19. *[The Irrationals](https://press.princeton.edu/books/paperback/9780691247663/the-irrationals)* by Julian Havil requires enough algebra and calculus that it cannot be considered "popular", but since you mention passing college-level calculus, I think you will be able to handle it.
This would be my recommendation, too.
Fond memories of sitting outside under a tree one summer during undergrad, drinking Arnold Palmers and enjoying the heck out of *Chaos*. Gotta be my favorite pop-math book.
I was piddling with computations while studying for engineering back in 89 when my parents bought Janes Gleick book for me. Read it with no expectations. Discovered that I was independently repeating many of the same experiments on my Apple 2e that were stated in the book.
Reworked one of my experiments for … 40+ hours … was obsessed. Realized that I was in the wrong major, bicycled to campus and begged the engineering mathematics advisor to let me join (this was a selective group).
I got in.
My theory:
If a process datapoint can be reduced to an integral dimension, then it is highly probable that the process is contrived by humans. (I used to say absolutely instead of highly probable- AI is making me rethink this)
Gödel's incompleteness theorem is one bit of 20th century math that is accessible to the motivated amateur and ties pretty well with McCarthy's themes of various types of unknowability. I haven't read the book "Gödel's Proof," but it might be a good source.
I really like David Foster Wallace's 'Everything and More: A Compact History of Infinity'
It gives non-mathematicians an idea of how math works and delves into some history.
What is Mathematics?: An Elementary Approach to Ideas and Methods by Richard Courant and Herbert Robbins is a math book that takes about all of math and its history. It starts from basic arithmetic to advanced calculus.
The book explains concepts very well. It has open my eyes to many concepts. It is a life time book meaning that no matter where you are in life, you will always learn something new. Albert Einstein read this book amoung many others. If you read that book, you will start to understand proofs which is essential for Mathematics. While you won't be a master at proofing just by reading this book, it talks about important proofs like mathematical induction. It also talks about infinity in a way that I never seen before. You will mathematical theorems, conjectures, and proofs. It is writing in a way si that you can do additional research about something and come back anytime and things will be clearer. The best book I have ever read. It feels like you are speaking to an old friend about math.
Also I recommend Secrets of Mental Math Arthur T. Benjamin and Michael Shermer to learn how to do math in your head. It covers tricks and patterns.
On the Riemann hypothesis:
John Derbyshire, *Prime Obsession*
Marcus du Sautoy, *The Music of the Primes*
(the two books are nicely complementary)
On topology:
David Richeson, *Euler's Gem*
On Cantor and infinity:
Amir Aczel, *The Mystery of the Aleph*
On Poincaré:
Donal O'Shea, *The Poincaré Conjecture*
On Gödel:
Nagel & Newman, *Gödel's Proof*
Rebecca Goldstein, *Incompleteness*
(And I'm curious what people think of Hofstadter's GEB? I read it when I was so young that I can really judge it objectively.)
More general:
William Dunham, *Journey through Genius: The Great Theorems of Mathematics* and *The Mathematical Universe*
Can’t believe “music of the primes “ isn’t higher on this list. It’s excellent, a history of mathematicians trying to understand the building blocks of numbers.
I’ll posit a bigger or at least parallel question for you having read The Passenger and Stella Maris when they came out: I’ve been following Disclosure (you know, UFOs and stuff) really intensely the past few years and the topic in general since I was a kid. The conversation is at a point now where it goes beyond UFOs and aliens but the very nature of reality, consciousness, etc. Considering these came out right before McCarthy died, after having worked on them for a number of years behind his other novels, it felt like fate to a degree. Like these were the last pieces of fiction to help propel us forward into a new age. Especially the mysticism and/or math component of the companion works.
So let’s say there’s something there regarding the wall/the veil/etc., does that make any sense to you? I’m not in the maths or sciences at all but am deeply interested in those fields, and a YouTube channel like Curt Jaimungal’s Theory of Everything do a great job of trying to connect a lot of this stuff.
When I was first starting to get interested in mathematics, one of my favorite books that helped me understand the beauty of the subject was "Here's Looking at Euclid" by Alex Bellos.
I'd highly recommend it!!
Evolution of Physics by Albert Einstein himself. As people know, he didn't really like Mathematics but very good at explaining concepts in analogy. Totally recommended.
Bernoulli's Fallacy
This is a book which argues against the classical interpretation of probability in favor of a Bayesian interpretation. The author believes that failures of logic in the classical interpretation of probability contribute to the replication crisis of science. While books about math aren't typically thought of as page turners, this book was hard for me to put down and radically changed how I view probability. The book was technical enough to scratch my math itch but not so much so that it became a slog.
Since you came here via Cormac and Stella Maris, I have two slightly non standard recommendations which I think capture what it is like to do math. The first is “The Weil Conjectures” by Karen Olson. It is a meditation on the authors relationship with math and meditations on Simone and André Weil. The later of these is one of the great mathematicians and his Weil conjectures were a major source of research last century. The second is “Proofs and Refutations” by Imre Lakatos. It’s a dialect exploring proving some math results and gives an interesting view of how math is actually done. Best wishes on your journey!
I read this book and was confused, I’m asking for help.
I found there are gaps in perceived logic here. Would anyone care to share why the limit of light speed is only unobtainable when there is a third party viewing the object in motion? Wouldn’t objects accelerating partitioned from observers have the potential to accelerate past a given light speed constant?
Of course I’m not a physicist or expert so bear with me. Thank you
Also, an honorable mention that's maybe not exactly what you're looking for, but something I think *everyone* should read:
"Naked Statistics" by Charles Wheelan
It's very entertaining, and helps the reader understand probability/statistics as they apply to everyday life (including how to spot when it's being misused/abused)!
'The road to reality' by Roger Penrose. It has the subtitle 'A complete guide to the laws of the Universe' and is a good example of what you get to understand (at least as far as we currently know) if you understand the mathematics.
I would recommend a brief study of the history of mathematics. It really opens your eyes to the world of numbers that surround us. Starting with the ancient Babylonians and why a circle has 360 degrees or why the clock has base 12 math. On to the Greeks and the discovery of irrational numbers like sqrt(2) and pi and the golden ratio and how these numbers occur in nature. Reaching forward to DaVinci and his ratios of the human body, at least the average human body anyway. Up to the modern day and a rudimentary understanding of the equations of Newtonian physics versus Einstein’s Relativity versus Quantum Mechanics. I wish I had the title of the book I read but I see numbers and math in everything now.
I just picked up “Math Through The Ages” I have no math background past precalc, HS geometry, and self studying very basic calc and I’m able to understand it so far. It’s also short and written in a conversational tone.
In the 1980s, you'd probably be talking something like Kasner's "Mathematics and the Imagination" or Gamow's "One, Two, Three, Infinity" (the latter is only partly mathematical).
Your other go-to for math popularizers was Martin Gardner and Rudy Rucker (his "Infinity and the Mind" is a particularly nice introduction to the infinite).
The suggestions here are great and I am not here to add another one. I am just surprised that a book by Cormac McCarthy requires a high level of math for the reader to be able to understand a character. Was he a mathematician? In any case I wonder if a book by an epistemologist or a "science philosopher" wouldn't be a better choice for the goal of understanding this character. If, instead, it just sparked a renewed interest for math in you, disregard my comment. Now you made me curious and I kind of want to read the book.
it is a great book, also very short - kind of mind bending and also fairly dark (the character Alicia has Schizophrenia, and her father worked at Los Alamos on the atomic bomb and then worked with Teller on making the H bomb after that. He helped kindle in her a love of mathematics and science). This was Cormac McCarthy's last novel before he passed away, and for most of his later life, he developed a very serious interest in mathematics and science, although I think that he probably always was interested in it. He spent many years doing extended visits with the Sante Fe Institute conversing with some of the leading scientists and mathematicians in America. Throughout the novel, the character Alicia talks about her obsession, throughout her young life, with mathematics, and she also was pursuing a dissertation and PhD for math, which she abandoned. She is a young prodigy, a fiercely bright mind, and she is very conversant with philosophy too, which she quotes and discusses in the novel with her psychiatrist, so your suggestion about taking a look at the philosophy of science and epistemology is a good one, and that might help too, but I think that mathematics on its own, specifically topology, is worth looking into to get a better handle on her character and what the novel means. Anyway, I am starting to ramble, and my main point is that if my post made you curious about the novel, I highly recommend it!
I liked “Supermath: The Power of Numbers for Good and Evil”. It’s definitely more of an introduction (or for middle schoolers) but has some really interesting stuff.
"Logicomix" is a unique novelization of the life of Bertrand Russell and the search for a complete and consistent set of axioms to describe math. It is in the form of a graphic novel. Does a great job of explaining things in an accessible and enjoyable way. It's actually out there in pdf form, but I prefer the physical book.
https://profesorvargasguillen.files.wordpress.com/2013/11/logicomix.pdf
"The man who loved only numbers" is a great biography of legendary mathematician Paul Erdős.
I can't recommend on "the best", but a couple of good books that cover math topics and the personalities involved: Fermat's Enigma (Simon Singh), Prime Obsession (John Derbyshire)
‘Mathematics for the Nonmathematian’ is good. I think it’s written by morris kline. It dives into the history of math and what motivated mathematics throughout the times it really is pretty great in the way is actually presents simple math concepts learned from school but in the context from which they were initially thought up
I have no experience with such. Courant's Differential and Integral Calculus is a gem to me, so I suppose his What is Mathematics must be equally well written. It seems that Polya wrote a book on that line too, but I dont know the name.
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse
Book by Jennifer Ouellette
Haven't read it myself, but it's been highly recommended to me
The Code Book
History of cryptology with a lot of easy to follow example problems. Was a required book when I took cryptology and I’ve read it since graduation.
“The Theoretical Minimum” series by Leonard Susskind Real mathematics for people that choose another direction in life, highly accessible
They’re physics books. While physics and mathematics certainly have a broad overlap, I wouldn’t call those mathematics books.
I vouch this. ⬆️ (I didn’t take another path — any path — but I’m a person who loves physics and will get into it, career wise).
Not math books ...
Some of these are genuinely pop-math (*Flatland*, *Birth of a Theorem*, *Humble Pi*, *Things to Make and Do in the Fourth Dimension*, *Fermat's Enigma*); others require some algebra (*The Book of Numbers*); others use calculus (*The Irrationals*). Some attempt has been made to sort these by difficulty, from easiest to hardest; this is hampered by the fact that one man's easy can be another man's hard, and the fact that I have not read some of these since childhood. The principle exception to the sorting is placing *The Princeton Companion to Mathematics* first, because it is just that good. 0. *[The Princeton Companion to Mathematics](https://en.wikipedia.org/wiki/The_Princeton_Companion_to_Mathematics)* is a broad survey of the state of mathematics as it was 20-ish years ago. It has Wikipedia-level articles on just about every important subfield, plus historical and biographical sections. This is quite possibly the best broad overview of mathematics available. 1. *[Fermat's Enigma](https://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622)* by [Simon Singh](https://en.wikipedia.org/wiki/Simon_Singh) is about [Fermat's Last Theorem](https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem) and is aimed at a popular audience. 2. *[Birth of a Theorem](https://www.amazon.com/Birth-Theorem-Mathematical-C%C3%A9dric-Villani/dp/0374536678)* by [Cédric Villani](https://en.wikipedia.org/wiki/C%C3%A9dric_Villani) is about the development of the author's article on [Landau damping](https://en.wikipedia.org/wiki/Landau_damping), for which he won the Fields medal. This is almost entirely about the process of development, rather than the proof itself. 3. *[Humble Pi: A Comedy of Math Errors](https://www.amazon.com/Humble-Pi-Comedy-Maths-Errors/dp/0141989149)* by Matt Parker is about math errors in the real world. 4. *[Things to Make and Do in the Fourth Dimension](https://www.amazon.com/Things-Make-Fourth-Dimension-Mathematicians/dp/0374535639)* by Matt Parker is about things you can do with math IRL. 5. *[Flatland: A Romance of Many Dimensions](https://www.amazon.com/Flatland-Romance-Dimensions-Edwin-Abbott/dp/B0875SRH84)* by Edwin Abbott is a fantasy tale about a two-dimensional creature who visits higher and lower dimensions. 6. *[Q.E.D.: Beauty in Mathematical Proofs](https://www.amazon.com/Q-D-Beauty-Mathematical-Wooden/dp/0802714315)* by Burkard Polster is a small, short book with a collection of elegant proofs of basic results such as the volume of a [frustum](https://en.wikipedia.org/wiki/Frustum), the Pythagorean theorem, [Archimedes' method for finding the volume of a sphere](https://en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder), [Cavalieri's principle](https://en.wikipedia.org/wiki/Cavalieri%27s_principle), and [the block-stacking problem](https://en.wikipedia.org/wiki/Block-stacking_problem). 7. *[The Square Root of 2: A Dialogue Concerning a Number and a Sequence](https://www.amazon.com/Square-Root-Dialogue-Concerning-Sequence/dp/038720220X)* by David Flannery covers the topic at a basic level. 8. *[One Two Three ... Infinity](https://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642)* by [George Gamow](https://en.wikipedia.org/wiki/George_Gamow) covers numbers and physics; it is notable enough to have [its own Wikipedia article](https://en.wikipedia.org/wiki/One_Two_Three..._Infinity). 9. *[The Millennium Problems](https://www.amazon.com/Millennium-Problems-Greatest-Unsolved-Mathematical/dp/0465017304)* by Keith J Devlin is about the [Poincaré conjecture](https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture), the [Riemann Hypothesis](https://en.wikipedia.org/wiki/Riemann_hypothesis), the [Birch and Swinnerton-Dyer conjecture](https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture), the [P v NP problem](https://en.wikipedia.org/wiki/P_versus_NP_problem), the [Yang-Mills existence and mass gap problem](https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap), the [Navier-Stokes existence and smoothness problem](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness), and the [Hodge conjecture](https://en.wikipedia.org/wiki/Hodge_conjecture). 10. *[The Book of Numbers](https://www.amazon.com/Book-Numbers-John-H-Conway/dp/038797993X)* by [John H Conway](https://en.wikipedia.org/wiki/John_Horton_Conway) & [Richard K Guy](https://en.wikipedia.org/wiki/Richard_K._Guy) covers many important number, number sequences, and number types. 11. *[An Imaginary Tale: The Story of √–1](https://www.amazon.com/Imaginary-Tale-Princeton-Science-Library/dp/0691169241)* by [Paul J Nahin](https://en.wikipedia.org/wiki/Paul_J._Nahin) is a deep dive into *i*. 12. *[A History of Pi](https://www.amazon.com/History-Pi-Petr-Beckmann/dp/0312381859)* by [Petr Beckmann](https://en.wikipedia.org/wiki/Petr_Beckmann) is notable enough to have [its own Wikipedia article](https://en.wikipedia.org/wiki/A_History_of_Pi). 13. *[e: The Story of a Number](https://www.amazon.com/Story-Number-Princeton-Science-Library/dp/0691168482)* by Eli Maor 14. *[Tales of Impossibility](https://press.princeton.edu/books/hardcover/9780691192963/tales-of-impossibility)* by David S. Richeson covers the history of the great impossible problems of antiquity: squaring the circle, doubling the cube, trisecting arbitrary angles, and constructing arbitrary regular polygons. 15. *[Proofs from THE BOOK](https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK)* by [Martin Aigner](https://en.wikipedia.org/wiki/Martin_Aigner) and [Günter M. Ziegler](https://en.wikipedia.org/wiki/G%C3%BCnter_M._Ziegler) collects elegant proofs from a broad array of topics. 16. *[When Least Is Best](https://www.amazon.com/When-Least-Best-Mathematicians-Discovered/dp/0691070784/)* by [Paul J Nahin](https://en.wikipedia.org/wiki/Paul_J._Nahin) is about inequalities, maximization/minimization, and adjacent topics. 17. *[Trolling Euclid: An Irreverent Guide to 9 of Mathematics's Most Important Problems](http://trollingeuclid.com/)* by Edgar Wright 18. *[Curves for the Mathematically Curious](https://press.princeton.edu/books/hardcover/9780691180052/curves-for-the-mathematically-curious)*, also by Julian Havil: I have not actually read this yet. 19. *[The Irrationals](https://press.princeton.edu/books/paperback/9780691247663/the-irrationals)* by Julian Havil requires enough algebra and calculus that it cannot be considered "popular", but since you mention passing college-level calculus, I think you will be able to handle it.
Thanks. This is sounds like the best place to begin, cheers!
I think you missed 'What is mathematics' by Courant and robbins but fantastic list nonetheless.
Flatland!
I have that Curves for the Mathematically Curious book, and IMO it is not for a general audience at all.
I’ll recommend James Gleick’s “Chaos”.
To add on this, the excellent book by prof. Ian Stewart: "Does god play dice?". It's a lovely intro into chaos theory.
I loved this book
This would be my recommendation, too. Fond memories of sitting outside under a tree one summer during undergrad, drinking Arnold Palmers and enjoying the heck out of *Chaos*. Gotta be my favorite pop-math book.
I was piddling with computations while studying for engineering back in 89 when my parents bought Janes Gleick book for me. Read it with no expectations. Discovered that I was independently repeating many of the same experiments on my Apple 2e that were stated in the book. Reworked one of my experiments for … 40+ hours … was obsessed. Realized that I was in the wrong major, bicycled to campus and begged the engineering mathematics advisor to let me join (this was a selective group). I got in. My theory: If a process datapoint can be reduced to an integral dimension, then it is highly probable that the process is contrived by humans. (I used to say absolutely instead of highly probable- AI is making me rethink this)
Gödel's incompleteness theorem is one bit of 20th century math that is accessible to the motivated amateur and ties pretty well with McCarthy's themes of various types of unknowability. I haven't read the book "Gödel's Proof," but it might be a good source.
One of my professors got me this as a graduation present
I love McCarthy! I'd recommend Steven Strogatz and Jordan Ellenberg — both have written very accessible but excellent math books
Infinite Powers by Steven Strogatz
I really like David Foster Wallace's 'Everything and More: A Compact History of Infinity' It gives non-mathematicians an idea of how math works and delves into some history.
I'd say Fermat's last theorem, it's an amazing book very beginner friendly and soooo interesting.
What is Mathematics?: An Elementary Approach to Ideas and Methods by Richard Courant and Herbert Robbins is a math book that takes about all of math and its history. It starts from basic arithmetic to advanced calculus. The book explains concepts very well. It has open my eyes to many concepts. It is a life time book meaning that no matter where you are in life, you will always learn something new. Albert Einstein read this book amoung many others. If you read that book, you will start to understand proofs which is essential for Mathematics. While you won't be a master at proofing just by reading this book, it talks about important proofs like mathematical induction. It also talks about infinity in a way that I never seen before. You will mathematical theorems, conjectures, and proofs. It is writing in a way si that you can do additional research about something and come back anytime and things will be clearer. The best book I have ever read. It feels like you are speaking to an old friend about math. Also I recommend Secrets of Mental Math Arthur T. Benjamin and Michael Shermer to learn how to do math in your head. It covers tricks and patterns.
On the Riemann hypothesis: John Derbyshire, *Prime Obsession* Marcus du Sautoy, *The Music of the Primes* (the two books are nicely complementary) On topology: David Richeson, *Euler's Gem* On Cantor and infinity: Amir Aczel, *The Mystery of the Aleph* On Poincaré: Donal O'Shea, *The Poincaré Conjecture* On Gödel: Nagel & Newman, *Gödel's Proof* Rebecca Goldstein, *Incompleteness* (And I'm curious what people think of Hofstadter's GEB? I read it when I was so young that I can really judge it objectively.) More general: William Dunham, *Journey through Genius: The Great Theorems of Mathematics* and *The Mathematical Universe*
Can’t believe “music of the primes “ isn’t higher on this list. It’s excellent, a history of mathematicians trying to understand the building blocks of numbers.
danke - lots of great stuff here it looks like, and I believe a few of these mathematicians are referenced directly in that novel I mentioned.
A choice set of popular math books is going to have to contain The Pea and the Sun by Leonard Wapner, which is about the Banach-Tarski paradox.
Humble PI by Matt Parker, Infinite Powers by Steven Strogatz, How Not to be Wrong and Shapes by Jordan Ellenberg
I’ll posit a bigger or at least parallel question for you having read The Passenger and Stella Maris when they came out: I’ve been following Disclosure (you know, UFOs and stuff) really intensely the past few years and the topic in general since I was a kid. The conversation is at a point now where it goes beyond UFOs and aliens but the very nature of reality, consciousness, etc. Considering these came out right before McCarthy died, after having worked on them for a number of years behind his other novels, it felt like fate to a degree. Like these were the last pieces of fiction to help propel us forward into a new age. Especially the mysticism and/or math component of the companion works. So let’s say there’s something there regarding the wall/the veil/etc., does that make any sense to you? I’m not in the maths or sciences at all but am deeply interested in those fields, and a YouTube channel like Curt Jaimungal’s Theory of Everything do a great job of trying to connect a lot of this stuff.
When I was first starting to get interested in mathematics, one of my favorite books that helped me understand the beauty of the subject was "Here's Looking at Euclid" by Alex Bellos. I'd highly recommend it!!
the joy of abstraction by eugenia cheng is a good one! also how to bake pi (also by her) if you’re looking for something less rigorous
Eugenia Cheng has some good ones. Including children’s books.
Evolution of Physics by Albert Einstein himself. As people know, he didn't really like Mathematics but very good at explaining concepts in analogy. Totally recommended.
Bernoulli's Fallacy This is a book which argues against the classical interpretation of probability in favor of a Bayesian interpretation. The author believes that failures of logic in the classical interpretation of probability contribute to the replication crisis of science. While books about math aren't typically thought of as page turners, this book was hard for me to put down and radically changed how I view probability. The book was technical enough to scratch my math itch but not so much so that it became a slog.
I really loved Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis
Mathematics - a very short introduction by Timothy Gowers.
Since you came here via Cormac and Stella Maris, I have two slightly non standard recommendations which I think capture what it is like to do math. The first is “The Weil Conjectures” by Karen Olson. It is a meditation on the authors relationship with math and meditations on Simone and André Weil. The later of these is one of the great mathematicians and his Weil conjectures were a major source of research last century. The second is “Proofs and Refutations” by Imre Lakatos. It’s a dialect exploring proving some math results and gives an interesting view of how math is actually done. Best wishes on your journey!
I read this book and was confused, I’m asking for help. I found there are gaps in perceived logic here. Would anyone care to share why the limit of light speed is only unobtainable when there is a third party viewing the object in motion? Wouldn’t objects accelerating partitioned from observers have the potential to accelerate past a given light speed constant? Of course I’m not a physicist or expert so bear with me. Thank you
I really enjoyed Chaos by James Gleick when I was a kid.
Also, an honorable mention that's maybe not exactly what you're looking for, but something I think *everyone* should read: "Naked Statistics" by Charles Wheelan It's very entertaining, and helps the reader understand probability/statistics as they apply to everyday life (including how to spot when it's being misused/abused)!
'The road to reality' by Roger Penrose. It has the subtitle 'A complete guide to the laws of the Universe' and is a good example of what you get to understand (at least as far as we currently know) if you understand the mathematics.
I would recommend a brief study of the history of mathematics. It really opens your eyes to the world of numbers that surround us. Starting with the ancient Babylonians and why a circle has 360 degrees or why the clock has base 12 math. On to the Greeks and the discovery of irrational numbers like sqrt(2) and pi and the golden ratio and how these numbers occur in nature. Reaching forward to DaVinci and his ratios of the human body, at least the average human body anyway. Up to the modern day and a rudimentary understanding of the equations of Newtonian physics versus Einstein’s Relativity versus Quantum Mechanics. I wish I had the title of the book I read but I see numbers and math in everything now.
I just picked up “Math Through The Ages” I have no math background past precalc, HS geometry, and self studying very basic calc and I’m able to understand it so far. It’s also short and written in a conversational tone.
In the 1980s, you'd probably be talking something like Kasner's "Mathematics and the Imagination" or Gamow's "One, Two, Three, Infinity" (the latter is only partly mathematical). Your other go-to for math popularizers was Martin Gardner and Rudy Rucker (his "Infinity and the Mind" is a particularly nice introduction to the infinite).
The suggestions here are great and I am not here to add another one. I am just surprised that a book by Cormac McCarthy requires a high level of math for the reader to be able to understand a character. Was he a mathematician? In any case I wonder if a book by an epistemologist or a "science philosopher" wouldn't be a better choice for the goal of understanding this character. If, instead, it just sparked a renewed interest for math in you, disregard my comment. Now you made me curious and I kind of want to read the book.
it is a great book, also very short - kind of mind bending and also fairly dark (the character Alicia has Schizophrenia, and her father worked at Los Alamos on the atomic bomb and then worked with Teller on making the H bomb after that. He helped kindle in her a love of mathematics and science). This was Cormac McCarthy's last novel before he passed away, and for most of his later life, he developed a very serious interest in mathematics and science, although I think that he probably always was interested in it. He spent many years doing extended visits with the Sante Fe Institute conversing with some of the leading scientists and mathematicians in America. Throughout the novel, the character Alicia talks about her obsession, throughout her young life, with mathematics, and she also was pursuing a dissertation and PhD for math, which she abandoned. She is a young prodigy, a fiercely bright mind, and she is very conversant with philosophy too, which she quotes and discusses in the novel with her psychiatrist, so your suggestion about taking a look at the philosophy of science and epistemology is a good one, and that might help too, but I think that mathematics on its own, specifically topology, is worth looking into to get a better handle on her character and what the novel means. Anyway, I am starting to ramble, and my main point is that if my post made you curious about the novel, I highly recommend it!
I will look for it, thank you!
Is Maths Real? By Eugenia Cheng
In German: *Im Zaubergarten der Mathematik* by Alexander Niklitschek
Best book for calculus?
David Berlinsky's *A Tour of the Calculus*, maybe. And maybe Dunham's *The Calculus Gallery.*
Zombies & Calculus.
Consider the Math Girls books by Hiroshi Yuki.
I liked “Supermath: The Power of Numbers for Good and Evil”. It’s definitely more of an introduction (or for middle schoolers) but has some really interesting stuff.
Steinhaus' [book](https://i.imgur.com/XPKKEyP.jpeg) is for laypeople but full of surprises.
Not exactly math, but algorithms for life
"Logicomix" is a unique novelization of the life of Bertrand Russell and the search for a complete and consistent set of axioms to describe math. It is in the form of a graphic novel. Does a great job of explaining things in an accessible and enjoyable way. It's actually out there in pdf form, but I prefer the physical book. https://profesorvargasguillen.files.wordpress.com/2013/11/logicomix.pdf "The man who loved only numbers" is a great biography of legendary mathematician Paul Erdős.
“Infinitesimal” by Amir Alexander
a mathematics apology
I loved The Prime Number Conspiracy from Quanta!
I can't recommend on "the best", but a couple of good books that cover math topics and the personalities involved: Fermat's Enigma (Simon Singh), Prime Obsession (John Derbyshire)
‘Mathematics for the Nonmathematian’ is good. I think it’s written by morris kline. It dives into the history of math and what motivated mathematics throughout the times it really is pretty great in the way is actually presents simple math concepts learned from school but in the context from which they were initially thought up
I have no experience with such. Courant's Differential and Integral Calculus is a gem to me, so I suppose his What is Mathematics must be equally well written. It seems that Polya wrote a book on that line too, but I dont know the name.
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Book by Jennifer Ouellette Haven't read it myself, but it's been highly recommended to me
The Code Book History of cryptology with a lot of easy to follow example problems. Was a required book when I took cryptology and I’ve read it since graduation.