Darren Glass, editor of the book review section of the American Mathematical Monthly, writes,

For this month’s column, I [Glass] thought that, rather than provide an in-depth review of a new monograph, I would ask a number of members of our community about some of the “mathematical comfort food” that they have turned to or that they might recommend people seek out.

A couple days ago I shared my recommendation, Proofs and Refutations.

Here are the books and other products recommended by the math teachers who were asked to contribute to this review:

Number Theory, by Edit Gyarmati, Paul Turán, and Róbert Freud

The Symmetries of Things by John H. Conway, Heidi Burgiel, and Chaim Goodman- Strauss

Black Mathematicians and Their Works, edited by Virginia K. Newell, Joella H. Gipson, L. Waldo Rich, and Beauregard Stubblefield

The movies Hidden Figures and October Sky

The article Computing Machinery and Intelligence, by Alan Turing, and the book Alan Turing: The Enigma, by Andrew Hodges

Origami Journey: Into the Fascinating World of Geometric Origami, by Dasa Severova.

Crocheting Adventures With Hyperbolic Planes, by Daina Taimina.

The game Prime Climb, by Dan Finkel and friends

An Imaginary Tale: The Story of √−1, by Paul Nahin

A Topological Picturebook, by George Francis

Proofs and Refutations by Imre Lakatos.

These were not what I would’ve expected! I thought my suggestion of Lakatos was a little bit offbeat, but the others are even more surprising.

If you’d asked me what a bunch of mathematicians would’ve selected as their mathematical comfort food, I’d’ve thought the list would have included some of these:

Mathematical Snapshots, by Hugo Steinhaus

Just about anything by Martin Gardner

Maybe something by Smullyan too

Can one hear the shape of a drum, by Mark Kac

How long is the coast of Britain, by Benoit Mandelbrot.

I’d add Eric Temple Bell’s Men of Mathematics but maybe that’s too old-fashioned to get a recommendation.

I don’t have any problem with the items in the first list above; they’re just not what first comes to mind when I think of mathematical comfort food. But it could be that the others were just like me and were trying to avoid the obvious choices. Maybe next month the journal can run a set of reviews, Canonical Mathematical Comfort Food.

The Art of Mathematics, by Jerry P. King, and Love and Math, by Edward Frenkel.

Thanks for listing for those great resources.

Bell’s Men of Mathematics is a great read!

I am most certainly NOT a mathematician as anything beyond differential calculus makes my brain hurt.

I do, however, get great pleasure from reading about the application of mathematics to the natural world. And the actions of humanity.

Men of Mathematics is a tempting, but to Clifford Truesdell’s quite is worth recalling:

“…[Bell] was admired for his science fiction and his Men of Mathematics. I was shocked when, just a few years later, Walter Pitts told me the latter was nothing but a string of Hollywood scenarios; my own subsequent study of the sources has shown me that Pitts was right, and I now find the contents of that still popular book to be little more than rehashes enlivened by nasty gossip and banal or indecent fancy.”

Came in to say “Just about anything by Martin Gardner” but you already said it.

Jordan:

Yeah, I was thinking that too, but I checked out a couple Gardner books from the library the other day and flipped through them . . . they were ok but not as fun to read as I was expecting. Maybe just because I read them before, or maybe because I wasn’t ready to put in the effort like I was in the old days, discovering amazing nuggets in back issues of Scientific American, building hexaflexagons out of paper, going to the store and buying a go set so I could try out the Game of Life for myself, and so on. This is comfort reading in a sense, but reading that takes a lot of work to really enjoy. I remember once collecting a bunch of matchboxes and using them to build a tic-tac-toe learning machine that Gardner had described in one of his columns. Just reading the columns passively isn’t as fun as I was imagining.

Julian Havil’s “Gamma: Exploring Euler’s Constant” was a beautiful read.

Does fiction count? I really enjoyed “The Housekeeper and the Professor” by Yoko Ogawa.

https://ibaracaldo.files.wordpress.com/2013/06/ogawa-yoko-the-housekeeper-and-the-professor.pdf

James Newman’s 4 volume The World of Mathematics is dated now, but I still pull a volume out every couple of years and browse, happily.

Hodge’s Turing touched me quite profoundly as a youngster. Thanks for the reminder that I should revisit it!

Thirty years later it’s still my model of a biography that does justice to both the intellectual and emotional side of its subject.

Single Digits, by Marc Chamberland, is kinda fun

The Code Book, by Simon Singh, is terrific

Infinity: Beyond the beyond the beyond, by Lillian R. Lieber. Stumbled across this book in a secondhand book store. Fantastic and fun introduction to Cantor

I can’t think of anything that currently I could call “mathematical comfort food”. But here are some of the things that warrant that label at earlier ages. They are things that I thought about and/or otherwise “did”

When I was very very young (five? six? seven?) I enjoyed my older siblings telling me what a “google” and a “googleplex” were. We wrote out all the digits of a google on our kitchen blackboard — it look several lines.

At around age 9, I liked writing out Fibonacci and similar sequences. I didn’t know that name at the time. I just liked to write 1, then 2 under I, then add to get 3, then add the 2 and 3 to get 5, etc. –and do the same process starting with two other numbers.

At around age 12, I liked calculating square roots by hand. Then a friend of mine wondered if there was an algorithm for calculating cube roots. Our teacher (who sponsored the math club) didn’t know, but my friend found an algorithm in a very old encyclopedia that had belonged to her grandparents. So we had some fun calculating cube roots.

I think it was in high school that I enjoying making hexaflexagons and other 3-D mathematical models. I also enjoyed geometric proofs and algebraic derivations, and the ideas (e.g. limits, asymptotes, derivatives and their uses) in calculus.

In college, I enjoyed doing proofs, and seeing or reading especially elegant ones. I also enjoyed solving mathematical problems.

In graduate school, I enjoyed doing problems more than listening to lectures, although some lectures were very good. And I enjoyed working on the problems that made up my Ph.D. thesis. I wasn’t just that I worked so much on a particular problem, but that sometimes I generated my own problems to solve.

After getting my Ph.D., I also enjoyed working on conjectures, and sometimes proving them (and sometimes finding counterexamples, and sometimes generating new conjectures). But as I did more and more teaching, I began to enjoy it more and more — especially when a student “got it”, and enjoyed thinking about how to teach in a way that helped students “get it” rather than just “follow rules that were taught”.

Hexaflexagon! I did not know that word, although I’ve seen examples of them. Indeed, kids (mostly girls, but not exclusively) liked to make them in junior high school.

Phil:

Those fortune-teller things were similar to hexaflexagons, but not quite the same thing.

I don’t have any particular books that come to mind (Maybe Godel, Escher, Bach, if that counts? I haven’t read it in decades though). I like to do mathematical problems as a “comfort food,” especially ones that require reasoning and proof more than calculation. I never participated in mathematics competitions (which I kind of regret), but during vacations I’ll sometimes look up the latest Putnam problems and try to do them. I also sometimes watch YouTube videos on mathematics—I highly recommend 3Blue1Brown’s channel in particular.

Speaking of channels, Michael Penn’s videos are great fun. (Mostly math olympiad problems and the like. The problems are hairy enough that they function as good examples of particular techniques, e.g. using modularity to solve number theory problems. If you remember thinking Why did I have to study that in calculus class, you’ll find some answers here.)

https://www.youtube.com/channel/UC6jM0RFkr4eSkzT5Gx0HOAw

Yeah that looks great too. Actually, there’s so much really good mathematics content on YouTube that it can be kind of a time sink.

I’m still not quite sure what “mathematical comfort food” means. If it’s something [mostly] accessible to a general audience that conveys some of the flavor or beauty of math, then maybe Conway/Berlekamp/Guy “Winning Ways for your mathematical plays” with all sorts of games, serious mathematical analysis and pun-filled writing twisted together, something that you might pull off the shelf and reread for pleasure on a lonely winter night, which seems like comfort food to me.

I also want to make a plug for Ron Aharoni’s “Mathematics, Poetry, and Beauty” — though I have to qualify it a little as the math examples may be too familiar for a mathematical audience (the usual collatz conjecture, irrationality of sqrt(2), etc), yet his descriptions might be too condensed for someone for whom they are new. But it’s readable and engaging and I really like his approach in trying to convey the beauty of math to a general audience.

I remember really liking “The pleasure of Counting” (Körner) but I can no longer find my copy, it may have been lent to someone.

“Math from Three to Seven” by Alexander Zvonkin is the story of a math circle for pre-schoolers that Zvonkin ran from his apartment in Moscow for his son (Dmitri Zvonkine) and friends in the early 1980s. Very readable, here and there the day-by-day diary aspect may bog down, but there are some beautiful moments, too.:

“…The boys talked about their recent visit to the zoo where they were shown monkeys. I interrupted, telling them that they were not shown monkeys, they were shown *to* monkeys. This naturally provoked a heated protest, but they did not immediately find the right argument.

“We were looking at them.”

“You were looking at them — big deal! They were also looking at you.”

Their second argument was more solid: “We can walk wherever we want to, and the monkeys can’t.”

But I still refuted them with, “No, you can’t go inside the cage. And the monkeys can’t go outside the cage. There are bars separating you, and you walk wherever you want on one side of them and the monkeys do the same on the other side.”

We were arguing this way for some time until Dima exclaimed with delight, as if he had caught me red-handed, “Hey, Dad! We’re doing math again!”

Ted:

I looked up that book, and . . . it’s wonderful! I am just now overcome with sadness that nobody did this for

mewhen I was four years old. And also this makes me feel like such a bad teacher, as I’ve been finding it so difficult to engage students in this way when everything’s on zoom and we can’t work together face to face.I suppose it’s true — seeing the 3rd and 4th graders coming out of his classroom chattering excitedly in the pre-covid days, I’ve often wished I could be a third grader in Ilya Zakherevich’s math circle. And I *know* high-performing high school kids today have access to so much online enrichment [and not just in math] that was unimaginable in the late 1970s.

Still, it’s hard to think of oneself as underpriviledged for growing up in suburban DC 40-odd years ago.

Ian Hacking’s “Emergence of probability” is entertaining and informative (if sometimes a bit much). Maybe not comfort food, but food for thought.

Proofs without Words – Roger B. Nelson

Sphereland – Dionys Burger

The Fourth Dimension – Rudy Rucker

Fifty Challenging Problems in Probability – Frederick Mosteller

Aah – – yes, I can consider Nelson’s “Proofs without Words” books* as comfort food!

* I’ve got volumes I and II, but I see on the web that there’s a volume III — and his page https://sites.google.com/a/lclark.edu/nelsen/ has some other titles that are enticing; in particular:

Charming Proofs

Math Made Visual

Nuggets of Number Theory: A Visual Approach, and

A Cornucopia of Quadrilaterals

“Sphereland” (which I’ve never read) makes me think of the excellent book “Flatland”, by Edwin Abbott. How has it not been mentioned already? What a great book.

This is a fun conversation. It makes me think we should have a blog called Statistical Modeling, Causal Inference, and Social Science where we just talk about fun nerd stuff all year long.

Yes, I was going to mention Frederick Mosteller’s Fifty Challenging Problems in Probability; it’s right here on my desk, and I was first introduced to it in childhood. A delightful, challenging little book.

Steinhaus’s Mathematical Snapshots made me a mathematician. I built models then and still do. I can trace jalmost every piece of “real research” I did over the years to pictures there: fair division, weighing problems, Apollonian circles, zonohedra, slicing cubes, stellated dodecahedra.

Ethan:

That reminds me that I plan to write a book called Statistics Stories where I just tell all the statistics stories I know, in abbreviated form. Not all the stories, I guess. Just all the stories that have happened to me. But maybe for completeness I’ll list the famous stories I know that haven’t happened to me but are still worth mentioning. Between my own stories and all the others, I guess I’d have a few hundred?

You can call it “That reminds me…”

+1

Please do write the bok of Statistics Stories. More scholars should do that as they get older. Lots of us collect great stories that we tell in class to illustrate points, and more of us should write them down.

I’m thinking of writing something in the style of William Riker’s The Art of Political Manipulation, with some of my own stories of campus or local politics and some from history. Stories are a good way of getting deep ideas “into the bones” of the listener (things like the Sports Illustrated Effect for Regression to the Mean, or that air force pilots story about it).

“I’d add Eric Temple Bell’s Men of Mathematics but maybe that’s too old-fashioned to get a recommendation.”

It has many inaccuracies as well.

David:

For sure. But nobody expects comfort food to be nutritious!

I strongly recommend looking at

Prime Numbers and the Riemann Hypothesisby Mazur and Stein.https://www.amazon.com/Prime-Numbers-Riemann-Hypothesis-Barry-dp-1107101921/dp/1107101921/ref=mt_other?_encoding=UTF8&me=&qid=1613340578

Bob76

I used to like Joel Smoller’s “Shock Waves and Reaction-Diffusion Equations” — I’d pick that up when I was feeling low. He quotes the Beatles in the epigraph: “I get by with some help from my friends”. I still keep that book close, even though I cannot make any sense of anything in it anymore. And Smoller’s gone too, dammit.

A recent book that I really enjoyed was David Acheson’s THE WONDER BOOK OF GEOMETRY. An easy read, it reminds you how doing mathematics can be a case of looking at a problem the right way. Unfortunately, I am unable to enjoy Bell’s MEN OF MATHEMATICS as it has a reputation for often being highly inaccurate.

“Mathematics Under the Microscope” by Alexandre Borovik

Hermann Weyl’s “Symmetry”.

Solved and Unsolved Problems in Number Theory, by Dan Shanks. A “a charming, unconventional, provocative, and fascinating book on elementary number theory”. It’s an in-depth mathematical presentation of the area, giving proofs in depth and also interesting discussion of the idea of proofs, theorems, conjectures, and lots of related topics. There’s a PDF here:

https://mathematicalolympiads.files.wordpress.com/2012/08/solved-and-unsolved-problems-in-number-theory-daniel-shanks.pdf

_Paradoxes_ by Sainsbury

_Godel’s Proof_ by Nagel and Newman

_Interpreting Probability_ by Howie is contending for inclusion too

A great 19th century book is W.K. Clifford’s “Common Sense of the Exact Sciences”. I found it very useful when when I was a high school math teacher 40 years ago.

Love and Math: The Heart of Hidden Reality by Edward Frenkel