Mathematics has a problem.
Well, lots of problems, actually. (My favorite is the Kakeya Conjecture.) But I’m talking about a culture problem, a communication problem: the gap between mathematics as written and mathematics as practiced.
Mathematical work is full of loop-de-loops and dead ends. It’s multi-modal, multi-player, multi-dimensional, multicolored.
But mathematical writing is like a stone monument from a lost civilization. Opaque. Mystifying. Mathematical texts loom like obelisks in the desert, their meanings hard to trace, their origins almost impossible to imagine.
I propose a solution: the quotation.
Quotations have a mysterious dual property: highly contextual, yet utterly decontextualized.
A quotation, by nature, is a thing that someone said. It comes from a specific voice at a specific moment. That makes it personal, subjective. It has what mathematics often lacks: an actual human heartbeat.
Yet at the same time, a quotation becomes a quotation when it is lifted from that context. When you quote someone, you pass the microphone to someone who isn’t there, to say something that belongs to a different conversation. Decontextualization is what gives quotations their air of authority and universality–or at least, their quality of definiteness.
And so what better way than quotation to crystallize all the aspects of mathematics that are missing in our usual forms of discourse?
Which brings me to the context for these decontextualized thoughts.
A few days ago, at MathFest 2024 in Indianapolis, I ran a session titled Quotable Mathematics. Over a hundred mathematicians gathered in a ballroom; I pointed them towards a pile of pens and a stack of some 300 quotations; then I said, in so many words, “go to town.”
They voiced approval (✔) and disapproval (X). They agreed and disagreed. They annotated and arranged. And, I hope, they assembled the raw material for speaking what has too long remained unspoken. I hope that they found quotations able to name and crystallize some of those unnamed vapors without which we’d be unable to breathe mathematics at all.
Some originally non-mathematical quotes (like this line, which I lifted from an Anna Sfard paper) found favor among the mathematicians:
Other witticisms incurred witty rejoinders of their own:
Sometimes a beloved speaker — like Bill Dunham, who just hours before had delivered a fabulous lecture on century-old Bryn Mawr entrance exams — drew surprisingly sharp disagreement. (Apparently we are over the 19th-century obsession with rigor, and have circled back around to intuition as foundational to mathematics.)
Some lines attracted silent nods of approval:
While others set off whole complex dialogues:
Not all the quotes went over as well as I imagined they would. One of historian Michael Barany’s favorite lines elicited raised eyebrows. (Perhaps we were not a crowd of pork-eaters.)
Meanwhile, an Edward Tufte favorite of mine, lifted from its original context on data visualization, no longer rang true.
On one table, the mathematicians arranged a pile of Rising Stars. Some happened to come from world-famous dead philosophers, but of course, the speakers are not the rising stars; it’s the words spoken.
One particular rising star was Nick Trefethen, a numerical analyst whose book of index-card musings I’d recently enjoyed, and from which I’d farmed a few favorite comments.
Some quotes stood out as particularly controversial–or not even controversial, just universally reviled, like Alfred Adler’s venomous assessments of mathematical research careers.
But the most fun we had was in arranging the great grid of wit and insight. With masking tape (borrowed from the fabulous Dave Richeson) I marked out two axes: wit on the x-axis, and insight on the y-axis. Then we arranged quotes in their appropriate positions, like a kind of real-world, real-time xkcd comic.
Here’s Dave attempting to document the results, while I settle for the easier task of documenting Dave’s documentation:
Unsurprisingly, a line from Terry Tao landed in the coveted top-right corner:
Although some folks felt that a similar sentiment had been even more felicitously expressed by Jordan Ellenberg:
I enjoyed the conversations arising from this line. (The point, I think, is that the incompleteness theorems mean that mechanistic processes can never encompass all of mathematical truth — so any mental process that can do so must be more than simply mechanistic.)
And even as this blog post stretches toward infinite length, I can barely do justice to the quotations and the commentaries. Here are a few more I saved from the recycling bin:
As the session hummed along toward its finish, many eager participants asked me things like “What’s your takeaway?” or “What are you hoping to achieve from this?” or “Do your actions on this planet ever have a purpose, Ben? Be honest.”
I told them the truth: I just wanted to see more fun quotations in circulation, and to accelerate the process of mathematical culture-building.
But I would like to extend this project beyond those delightful 90 minutes in Indianapolis. There are already some excellent compilations of mathematical quotations out there. I hope to create my own — ideally, a living and continually updated version, into which I and others can pour interesting quotes as we find them (with an eye toward greater diversity, in every sense of that word). If you’d like to join me as a partner in that project, reach out — I could use a co-creator with better Python skills (my own for loops being quite feeble).
Either way, I invite you to keep saying witty and insightful things, and — better yet — to quote such things when your colleagues say them. Speech fades; quotation is forever.
I’ve loved “collecting” quotations since I was a kid. Yes, Ben, yes! Here’s one: “The order of operations is morally wrong because it turns humans into robots.” From Minute Physics: https://youtu.be/y9h1oqv21Vs?si=M_wApqsIhVwX0Ahh
Ha, that’s nice!
My own preferred replacement for PEMDAS is to just say: “More powerful operations take priority.” Hence, in 5+4*3^2, exponentiation (most powerful) happens first, giving us 5+4*9; then, multiplication (next most powerful), giving us 5+36; then, addition (least powerful), giving us 41.
Still a bit robot-like, in the sense that it requires us all to parse the same symbols in the same way, but that level of robot-icism seems unavoidable!
I teach a homeschool class and I have a weekly mathematical quote on their assignment page. I’ll be adding some of these to the rotation. Wish I had been there!
Ah, cool — I’ll need to check out your rotation if/when I get my database up and running!
Love your logical page, Ben! Been following your blog posts since in graduation, and I am doing PhD now! Best regards from India (not polis sadly).
Thank you for reading, and good luck with the PhD!
If that’s any comfort, the one about the pig makes perfect sense to Italian ears 🙂
I had the pleasure of participating in your Quotable Mathematics session at MathFest last week (thx!), and I was hoping to come across a quote or two from the French philosopher Simone Weil – who also happened to be the sibling of French mathematician André Weil; she surfed on into mathematics frequently in her own writings, and I’ve includes a few of her quotes within my own math’centric essays.
Were there any from her in the ~300 quotes(!) you provided for our review, and I just missed it?
This is one of my faves of hers: “Harmony is proportion. It is also the unity of contraries.”
That’s nice! I don’t think I had any of either Weil sibling in the stack, although I did have at least one from Karen Olsson’s fabulous book on them (and I quote that book, The Weil Conjectures, a bunch of times in my own forthcoming book Math for English Majors).
That’s great to hear that you reference Olsson’s book in your upcoming release. I agree that her book is a delight to read while also being super informative – a total publishing win/win.
I sourced that Weil quote from a book that’s a collection of her writings.
David Berlinski isn’t exactly the most popular fellow around, but he is a bit of a wordsmith, and I’ve always enjoyed this longish thought-twisting bit from his volume, “The King of Infinite Space”:
“Like any other mathematician, Euclid took a good deal for granted that he never noticed. In order to say anything at all, we must suppose the world stable enough so that some things stay the same, even as other things change. This idea of general stability is self-referential. In order to express what it says, one must assume what it means. Euclid expressed himself in Greek; I am writing in English. Neither Euclid’s Greek nor my English says of itself that it is Greek or English. It is hardly helpful to be told that a book is written in English if one must also be told that written in English is written in English. Whatever the language, its identification is a part of the background. This particular background must necessarily remain in the back, any effort to move it forward leading to an infinite regress, assurances requiring assurances in turn. These examples suggest what is at work in any attempt to describe once and for all the beliefs ‘on which all men base their proofs.’ It suggests something about the ever-receding landscape of demonstration and so ratifies the fact that even the most impeccable of proofs is an artifact.”
Nice. I have quibbles with Berlinski’s pedagogy, but he never lacks for poetry!
Great idea, Ben!
Wish I’d been there too…
and BTW – your books are phenomenal and original – I’m currently reading through them one by one… Love them!
Not exactly pithy, but I keep this quote nearby when writing papers:
“In the remaining sections of this paper we briefly discuss various occurrences of the stability and pinching phenomena in differential geometry. The results we present are, for the most part, not new and we do not provide the detailed proofs. (These can be found in the papers cited in our list of references). What may be new and interesting for non-experts is an exposition of the stability/pinching philosophy which lies behind the basic results and methods in the field and which is very rarely (if ever) presented in print. (This common and unfortunate fact of the lack of an adequate presentation of basic ideas and motivations of almost any mathematical theory is, probably, due to the binary nature of mathematical perception: either you have no inkling of an idea or, once you have understood it, this very idea appears so embarrassingly obvious that you feel reluctant to say it aloud; moreover, once your mind switches from the state of darkness to the light, all memory of the dark state is erased and it becomes impossible to conceive the existence of another mind for which [the] idea appears non-obvious.)” –Mikhail Gromov, “Stability and Pinching” pp. 64-65.
Ooh, that’s great! The parenthetical is separable from the rest, and is a very nice meditation on mathematical communication.