The point

Paul Klee drew a squiggle and called it “an active line on a walk.” I was so floored by this that I wrote “Draw a Line. Now Take it on a Walk (and maybe bring a poop bag)” in my notes. Then I drew a line and I also drew a poop bag. That was the start of this idea. This piece is not about lines or about Paul Klee, though I did grab his Pedagogical Sketchbook this morning just in case.

POINT

Euclid calls a point “that which has no part.” But it has a little part. And you’ve just seen it. Twelve times! Not counting the caption. Which just saying that added two more dots and now this one added three. How long can I get stuck in this loop? A long time.

In physics, point particles are leptons, quarks and gauge bosons. We can safely say everything in our universe that ever was or ever will be is some combination of these points and their antimatter partners.

Sir Thomas L. Heath, definitive translator of Euclid, footnotes that you could translate Euclid’s sentence as “A point is that which is indivisible into parts.”

Aristotle called a point indivisible too. But he wrestled with this definition and started beefing with Plato because if a point is indivisible, is infinitely small, how do you place points together to make something divisible — a line? He concluded that a line is not made up of points. A point is made up of now, a piece of time so small it is not part of time at all. The line only forms once the point moves.

Points caused chaos and disruption in 300 BC when the Greeks realized that you could not make a line with √2 number of points. Lines and points informed their entire worldview so this was a cataclysmic discovery, what Jeremy Gray calls “the erosion of the concept of number.” 

Wassily Kandinsky Interruption

Kandinsky made the first works of art that made no sense whatsoever. He is probably the reason people at museums rub their chins and go “hm” even though they are totally confused. He is probably the reason others go “my kid could draw that” even though they can’t. (His 1911 Impression III (Concert) is abstract, but you can maybe make out that this is a concert. Not so his next one: Picture with a Circle. It’s supposed to represent a musical score, but this isn’t obvious at all.) Kandinsky was also a writer. In the three sentences below, he moves the point from the end, where it belongs, to the middle, where it’s weird, to an illogical place near the start. Kandinsky is working hard to trying to strip the point of meaning:

Today I am going to the cinema.

Today I am going. To the cinema.

Today I. Am going to the cinema.

Then he “divorces” (his word) the point completely from its sentence. It rolls off and hangs underneath it.

If the size of the point itself, and of the empty space surrounding it are increased, the sound of the writing becomes diminished, and the sound of the point gains in clarity and stength (Fig 1). (Kandinsky 1928)

The sound of the point! His “Fig 1” is a giant dot and it’s probably the most straightforward thing Kandinsky has ever drawn. The point is now free:

Nonetheless, the point has been wrenched free from its habitual state and thus prepares itself for the leap from one world into another, in which it is emancipated from the tyranny of the practical-purposive, in which it begins to live as an independent entity, and where its subordination has inner purpose. This is the world of painting. (Kandinsky 1928)

In grade school the point is a point of confusion, i.e. in moments like “0.125 = 1/8”: the point and line separating two numbers — the point doing the heavy lifting, if we’re being honest — in two (seemingly different) sets of ways. Kids actually have a hard time believing these two things are related at all.

A similar finding is reported by Neumann (2001) who noticed that seventh graders had difficulties accepting that there could be a fraction between 0.3 and 0.6 (Vamvakoussi and Vosniadou, 2010)

What is the point?

I don’t know.

Square rooting the frog

Theories come and theories go. The frog remains. — Jean Rostand

Analyzing humor, as E.B. White famously said, is like dissecting a frog; few people are interested and the frog dies of it. — Teddy Wayne

FROG

The other day I came across Bob Mankoff’s 2013 TED talk “Anatomy of a New Yorker Cartoon.” Mankoff was a legendary and longtime New Yorker cartoon editor.

Mankoff basically says that humor mashes up two things that don’t go together. This is the theory of benign violation.

The Benign Violation Theory (image from Peter McGraw’s Humor Research Lab)

This means that if something is benign, it’s not funny. If something is a violation, it’s not funny. But a benign violation — a harmless subversion of what we expect — is funny.

The square root of a cow is a benign violation. This was a revelation to me because I was trying to be ridiculous, but not in a way that was grounded in any scientific reality. But when we see the words “find the square root of” we’re primed to expect a number or a variable to follow. A cow is the violation. (What’s the malignant violation of a square root!)

My reasoning for a prompt like this is that it flattens the concept of square root. Since nobody has taken the square root of a cow, there is no fear of getting it wrong. A fourth grader and a Field’s Medalist and a parent that’s forgotten everything from school can discuss the square root of a cow and everyone will sound equally dumb. When you divorce math from the strictures of “right” and “wrong” you are free to play with a mathematical idea. Maybe you will even understand it better.

But then look at this too!

…students who were taught class material with humor retained more of the class learnings, scoring 11 percent higher on their final exams. (Aaker & Bagdonis 2021)

Here is a string of benign violations at various levels. I’ll keep adding to this list. Some are on the cards and some aren’t:

Are parallel lines lonely?

Should the fraction bar be a squiggle?

What’s the derivative of a dinosaur?

Can you flush a four-dimensional toilet?

What’s half a cookie times half a duck?

What’s a potato to the zebra power?

Find the Euler number of a lamp!

Would you ride the tangent curve roller coaster?

These are not quite benign violations but still verbal math exercises that are fun to explore:

DARK: Turn off the lights and describe an object using only its geometric properties. This forces you to use words like line, plane, circle and sphere. These are called manifolds in topology!

PET: Hold a pen or pencil to a piece of paper. Now take this line on a walk! (This was inspired by Paul Klee and referenced in the caption on this blog post.)

DOTS: Try to draw dots that are randomly far apart! This is an exploration of hyperuniformity.

Grothendieck, Flaubert and the search for the perfect word.

Thus are we constantly led to “invent” the language best suited to ever more finely express the intimate structure of mathematical things — Alexander Grothendieck

If I put 'blue' after 'stones,' it's because 'blue' is le mot juste, believe me — Gustave Flaubert

LE MOT JUSTE

(this piece is incomplete!)

So here is Gustave Flaubert, famous for obsessing over the perfect word. One can imagine the stress of translating Madame Bovary to English. Merde alors! Lydia Davis did it. She won a prize for her translation too.

In rewriting, he would watch out for poor assonances, bad repetitions of sounds and of words (especially qui and que, which he occasionally underlined and apologized for even in his letters) — Zola remarks that “often a single letter exasperated him.” (Davis 2010)

What to say of Alexander Grothendieck, mathematician?

There is no need to introduce Alexander Grothendieck to mathematicians: he is one of the great scientists of the twentieth century. (Cartier 2014)

From here I will bounce between Flaubert and Grothendieck using words from their own writings

Unfortunately, the role played by projective spaces in all this still seems rather excessive. I feel like looking into whether one doesn’t get something for “regular arithmetic varieties” which are “complete” (i.e. obtained by gluing together spectra of regular rings). But to start with, do you have any idea of what complete really means in this context? (Grothendieck)

My winter is to pass in complete solitude, good way of making life run along rapidly. (Flaubert)

Cartier, Pierre. A Country Known Only by Name. https://inference-review.com/article/a-country-known-only-by-name

McKenzie, Aimee L. The George Sand Gustave Flaubert Letters. London: Duckworth & Co. (1922) https://ia601603.us.archive.org/29/items/georgesandgustav00sandiala/georgesandgustav00sandiala.pdf

Grothendiek, Alexander. Récoltes et Semailles. Note: The 1900-page R&S has not been fully translated to English, but Saad Slaoui, a PhD student at UT Austin, is knee-deep (202 pages) into one. See here: https://web.ma.utexas.edu/users/slaoui/notes/recoltes_et_semailles.pdf

Pater, Walter. Style. Aesthetic Criticism and Polemic. (1889) https://aestheticperceptioncognition.se/pdf/pater-on-style.pdf

Why writers should read math proofs

It occurred to me the other day that there is a lot of great writing in math. There are sentences with rhythm and sometimes sentences that are funny! But this is about words.

In Pete Wells’s New York Times review of Eleven Madison Park, he writes:

In tonight’s performance, the role of the duck will be played by a beet, doing things no root vegetable should be asked to do. (Wells 2021)

The line struck like a bolt me when I saw the performance of the word “yoga” in Alexander Grothendieck’s 1,900 page Récoltes et Semailles (a book nobody has ever finished, not for lack of trying):

The deepest (to my eyes) of these twelve themes, are the notion of motives, and the closely related yoga of nonabelian algebraic geometry, and Galois-Teichmüller theory. (Grothendieck 1986)

Here yoga is in a scene with nonabelian algebraic geometry, of all things! Where else were familiar words doing stints in unfamiliar roles? I needed to nose around.

In this monograph, conventional subjects — symmetric group, Lie algebras (and, to a lesser extent, continuous Lie groups) — are presented in a somewhat unconventional way, in a flavor of diagrammatic notation that I refer to as “birdtracks.” (Cvitanović 2008)

All this time, birdtracks have been used as a way to simplify index notation.

A little depiction of birdtrack notation. Stefan Keppeler. “Birdtracks for SU(N)” SciPost Phys. Lect. Notes 3 (2018) · published 27 September 2018. Note that with a space (“bird tracks”), you are back to using footprints in the sand to track if you have a webbed, totipalmate, or zygodactyl bird.

Let’s keep looking (emphasis mine):

When working with a Coxeter group, one is sooner or later faced with problems concerning the combinatorics of reduced words. When do two such words represent the same group element? (Bjorner 2000)

In this case it is natural to make a surgery by cutting open the neck and gluing small caps to each of the boundaries, and then to continue running the Ricci flow. (Perelman 2024)

Just as the Dirac gamma matrices lead to spinor reps of SO(n), the Grassmann valued γµ give rise to Sp(n) reps, which we shall call spinsters. (Cvitanović 2008)

Yoga, birdtracks, words, surgery, spinsters! You know who loves this kind of thing? Writers. Writers are word watchers and get a serious thrill when a fellow writer deploys a word in an unexpected way. Lydia Davis writes glowingly about Samuel Beckett’s spin on the word “dint”:

There was his precise and sonorous use of the Anglo-Saxon vocuablary — especially, in this example, the way he gives a familiar word like dint a fresh life by using it in an unfamilar way. (Davis 2019)

Francine Prose writes about F. Scott Fitzgerald’s unexpected use of “deferential” to describe a row of palm trees. Donald Barthelme paid close attention to what Edward Gorey did with “it” in the sentence “Last night it did not seem as if today it would be raining”:

The meeting of the “it” which did not seem and the “it” which would be raining is inspired. (Barthelme 1997)

It’s equally thrilling to find these words a reader. I can’t think of anything more prosaic than a paragraph. Francine Prose calls it a “literary respiration.” Sloane Crossley imagines Dorothy Parker’s “delight in filleting a book.” InTrout Fishing In America, Richard Brautigan mistakes a woman for a trout stream:

I remember mistaking an old woman for a trout stream in Vermont, and I had to beg her pardon. “Excuse me,” I said. “I thought you were a trout stream.” (Brautigan 2010)

William Vollman made “flypaper” a verb. William Faulkner called an empty road a postulate.

Isn’t it magical that mathematicians and writers are playing with words in similar ways?

IRRATIONAL MANNINGFALTIGKEIT

The link between words and math is ancient and storied. For over 2500 years, mathematicians have had to use the vocabulary of their time to give “text-clothes” to the inexplicably abstract contours of their world. Take irrational. This word burst onto the scene, uninvited, because nobody could measure the diagonal of a square. In 300 BC, everything could be neatly expressed as the ratio of integers or line segments. But the length of the square’s diagonal was √2, a number that did not exist. This was profoundly disturbing. Here is Lucio Russo writing about this episode in The Forgotten Revolution:

If we discover that a scientific theory is contradictory, it’s no big deal: we change theories. But what can we do if we discover, or think we have discovered, a contradiction in reality itself? (Russo 2000)

The event exploded the concept of a number. 100 oxen were sacrificed. It’s rumored someone died. “Irrationalis” is a calque of the Greek ἀλόγος: √2 is called irrational because there is simply no ratio.

And now we have this word.

In 1854, Bernhard Riemann gave a lecture on space that would change the world (and give Einstein a geometric scaffold for relativity). Riemann called a set of points Mannigfaltigkeit which became manifold. In 1891, Georg Cantor, inventor of set theory, called his set of points Mannigfaltigkeit too (“Uber eine elementare Frage der Manningfaltigkeitslehre”). His Mannigfaltigkeit became “Menge” and Riemannian manifolds remain.

***

I was at the library in middle school and for some reason checked out Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. I didn’t know Fermat (and pronounced the “t” in his name well into my 30s) but the book was short and “ancient mathematical problem” sounded promising.

Well this book blew my mind. The idea of Andrew Wiles sitting at a desk doing a math problem — a math problem! — for seven years was the most diabolical thing I’d ever heard. I couldn’t get it out of my head. (Turns out Grigori Perelman did basically the same thing ten years later with the Poincaré Conjecture.) These people were either a variety of bored beyond all Earthly comprehension or math was more interesting than The Establishment was leading on. I was leaning towards the latter because this world was bursting with words that set fire to my imagination. Here’s Wiles:

Using this, we complete the proof that all semistable elliptic curves are modular. (Wiles 1995)

Semistable! Elliptic! Modular! There are so many more! Rings, unramified cohomology classes — I mean who has ever seen “ramify” conjugated this way? — isomophism, homomorphism.

I had a vague grasp of Wiles’s proof at 13. Not because I was a math genius but because the guy who wrote that book, Amir Aczel, was a damn good writer. Intermission time.

INTERMISSION

If being a math genius is this pink dot: .

I’m having coffee on a faraway galaxy where, not only do I fail to see the dot, but the concept of “dot” doesn’t exist because light from this blog won’t reach me until the universe has exploded. That’s how far away from being a math genius I am.

INTERMISSION FINISHED

All of these words were interesting. This world was interesting. But not everyone wants to plod through grad school to get there. Not everyone can! Besides, what if you don’t even like math? What if you never learned math past 4th grade? Does that mean you should be totally shut out from all of this stuff?

No it absolutely does not.

Five days ago, I attended — well, crashed — a talk at the Courant Institute. I say crashed because this security guard was not messing around. That’s when the speaker, a mathematician at the Institute of Advanced Study at Princeton, came out to use the bathroom and I flagged him down. This wonderful soul was greatly confused but happy to let me in. I now know you have to be on a list to attend the Oscars, the MET Gala, and math seminars at Courant.

While this speaker, a bona fide math genius, made some very salient points and proved an important new result, I’m sure, you can see from the exclamation point at the bottom that my biggest thrill came from seeing “blow-down” on a math slide. Here it is in the paper:

The proof of Theorem 1.2 is analogous, but the contradicting sequence is obtained by blowing down (M4 , g). Indeed, the blow-down of a manifold with Euclidean volume growth is a metric cone and, again by [10], the cross-section of the latter is orientable. (Brena 2024)

I thought about “blow-down” all the way home. “Blow-down” was the most exciting thing I’d seen since Faulkner called an empty road a postulate.

This post is for writers and word lovers. If you hate math, gorge on a hidden secret: its words.

Barthelme, Donald. Not-Knowing: The Essays and Interviews. Counterpoint. (1997)

Bjorner, Anders & Brenti, Francesco. Combinatorics of Coxeter Groups. Springer. (2000)

Brautigan, Richard. Trout Fishing in America. First Mariner Books. (2010)

Brena, Camillo et. al. “Lower Ricci Curvature Bounds and the Orientability of Spaces.” arXiv:2412.19288

Cvitanović, Predrag. Group Theory: Birdtracks, Lie's, and Exceptional Groups. Princeton University Press (2008)

Davis, Lydia. Essays One. Picador. (2019)

Grothendieck Circle, Chapter 3. https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Mathematics/chap3.pdf

Grothendieck, Alexander. Récoltes et Semailles. (1986)

Parker, Dorothy. Constant Reader: The New Yorker Columns 1927-28. Forward by Sloane Crosley. McNally Editions. (2024)

Perelman, Grisha. “The entropy formula for the Ricci flow and its geometric applications” https://arxiv.org/abs/math/0211159v1

Prose, Francine. Reading Like A Writer. Harper Perennial. (2006)

Serre, Jean-Pierre & Colmez, Pierre. Grothendieck-Serre Correspondence (English and French Edition). American Mathematical Society (2022) https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Letters/GS.pdf

Russo, Lucio. The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn. Springer (2000)

Vollman, William. You Bright & Risen Angels. Penguin Group. (1987)

Wells, Pete. “Eleven Madison Park Explores the Plant Kingdom’s Uncanny Valley.” New York Times. (2021) https://www.nytimes.com/2021/09/28/dining/eleven-madison-park-restaurant-review-plant-based.html

Wiles, Andrew John. Modular elliptic curves and Fermat’s Last Theorem. Annals of Mathematics, 141 (1995), 443-551. http://www.scienzamedia.uniroma2.it/~eal/Wiles-Fermat.pdf

Williams, Gilda. How to Write About Contemporary Art. Williams writes about about how art theorist Boris Groys suggests that words give art “protective text-clothes.”

Why write it?

One of the properties of language is its ability to generate sentences that have never been heard before. — Donald Barthelme

I really do not know that anything has ever been more exciting than diagramming sentences. — Gertrude Stein

I’m interested in the genre of the sentence,

The genre that’s always overlooked. — Verlyn Klinkenborg

Well that’s a lot of decorative quotes about sentences.

For some time, I have been obsessing over the mythical divide between word people and math people. And I’m sorry to say I’m partial to one side, the word people, because they live in a hushed fear that they are a little bit broken.

I want us to write the longest sentence because the sentence is the principle obsession of word people.* But if this sentence is mathematical, we can bridge the mythical divide. At least this is my idea.

Now there is already a literary form in math: the word problem. But we all know from homework problems about socks that it’s the worst literary form in the world.

A collaborative record. An audacious public arts project. But why?

Math is awesome but it also scares a lot of people including me. (Did you know that, in one study, a participant — a graduate student! — broke into tears at the idea of 46+18? Math anxiety is real and starts as young as 5.)

This is why I am so committed to bringing people to math with words and without the stress of time pressure and computation.

Which got me thinking about word problems, the most groan-inducing dreck in all of math. Can we breathe new life into the word problem?

We can! We’re going to write the longest one — together. Word by word, clause by clause. I’d like to imagine that diagramming this monstrous sentence would give Gertrude Stein chills.* Stanley Fish writes that “Flaubert’s famous search for the “mot juste” was not a search for words that glow alone, but for words so precisely placed that in combination with other words, also precisely placed, they carve out a shape in space and time.” 

How thrilling to carve a gigantic, meandering, syntactically complex word problem in space and time. Anyone can participate and because of this, everyone is invited to belong to math in a very public and personal way.

The word problem will be displayed in New York City. Location to come.

Please check the submission page for updates and to add your words!

As a personal exercise to see how we would slot all of our submissions together into one winding sentence here is my attempt to blurb this page into a 195-word sentence. Of course ours will be much longer:

So yes, math is the grand scaffold of our universe, of its clockwork orbits and particulate interactions, math with its ability to describe with baffling accuracy — sometimes to fifteen digits or more — the behavior of pollen grains or black holes or electrons; but it scares people too, including me, including several people in a study that approached problems like 46+18 with visible distress, nervous laughter and trembling palms, one participant bursting into tears, the anxiety an aching spiral of dread and embarrassment that sets in as young as 5, sending a chill worse than any haunted house really, out of which blooms decades of math avoidance: all this because of a pervasive notion that math is for numbers people and not words people, this notion persisting until now, right now, when we will weave the longest word problem together, word by word, clause by clause, a deluge of nouns adjectives and verbs tasked with computation, all of us contributing parcels — Chomsky’s kernels, Flaubert’s “mot juste” — in such a way that this weird and wild sentence, a real working math problem, carves a shape in space and time for all to see. 

* In Lydia Davis’s book Essays One, she has an eight-page piece called “Revising One Sentence.”

Cy Twombly and the art of mistakes

There is something exhilarating about Cy Twombly’s art. His strokes are whimsical and spontaneous. He isn’t stressed about perfection. This is absolutely not the way we do math.

With math, we are chock full of stress, bound by perfection and terrified of looking stupid. We are so ashamed and stressed we erase mistakes and scrap work — even when the scrap work is right! But here’s Cy Twombly:

I use paint as an eraser. If I don’t like something, I just paint it out. (Jacobus 2016)

So casual. So devil may care. Cy Twombly celebrated his mistakes.

When I look at a Twombly canvas, I don’t see mistakes. I see passion and evolution of thought. Look at the gorgeous drop and slop of Untitled (Gaeta):

Cy Twombly, Untitled (Gaeta) 1993 © Cy Twombly Foundation.

Can you imagine throwing paint at hard math problems? MoMA would call. And the Guggenheim. Larry Gagosian, if we’re lucky. And since we’d like a bidding war, we’d make more and more mistakes. Splish, sploosh, splash. The irony is we’d become better and better at math:

One of the most interesting findings from research on the brain to emerge over recent years is something that I try to communicate as widely as I can. We now know that when students make a mistake in maths, their brain grows, synapses fire, and connections are made This finding tells us that we want students to make mistakes in maths class and that students should not view mistakes as learning failures but as learning achievements. But students everywhere feel terrible when they make a mistake. They think it means they are not a “maths person.” (Boaler 2009)

Mistakes actually help our brains grow! (Moser 2011)

I have a couple of cards about this. So we can all get going making world-famous works of mess ups.

Boaler, Jo. The Elephant in the Classroom. Helping Children Learn and Love Maths. Souvenir Press, an imprint of Profile Books Ltd. (2009)

Jacobus, Mary. Reading Cy Twombly. Poetry in Paint. Princeton University Press. (2016)

Moser, J. S. et. al. Mind Your Errors: Evidence for a Neural Mechanism Linking Growth Mind-Set to Adaptive Post-Error Adjustments. (2011)

Everything Bagel Notes & Sources

These are notes and sources for the EVERYTHING BAGEL deck. Please note that this page will make minimal sense without the cards. But I am continually updating it (chaotically and out of order) so feel free to check in!

If you have the deck, just hit "Ctrl+F" (or ⌘+F on a Mac) and type “Card 6” (or whatever you need) to jump to the relevant notes. This list is getting quite unwieldy!

A little of this, a little of that. Coming very very soon!

Card Notes

Last updated: November 25, 2025

Intro Card

I mentioned Greeks and the square root of two but made sure not to say “ancient Greeks,” a giant tarp that blurs real distinctions between classical Socrates-era philosophizing and the far more rigorous and deductive Hellenistic-era science and math that began around 300 BC. Pythagoras and his crew were aware of the incommensurability of the side of a square and its diagonal but lacked the deductive reasoning (it simply wasn’t invented yet) to say much else about it. See Russo (citations below) for more. This book blew my socks off.

Card 1

“Nathaniel Johnston” (Johnston 2009)

You can still read his 2009 blog post here! Johnston consulted the Online Encyclopedia of Integer Sequences (OEIS) and found that 11630 was the first number that didn’t appear anywhere in the database. How clever is that! (Note: He calls this number “uninteresting” and not boring.)

There’s a neat book called Those Fascinating Numbers by Jean-Marie De Koninck that goes through almost every number and lists something interesting about it. It’s a fun book to have around! I have a copy and the first number that doesn’t have an entry is 95.

Card 2

I never would’ve thought this topic, this completely elementary topic we learn about in 5th or 6th grade, would turn out to be such a rabbit hole.

There is a quote I wanted to include by Roger Penrose. He writes: “What might it mean to say there are minus three cows in a field?” (Penrose 63) We take so many math concepts for granted that we gloss over, or forget, how baffling these ideas once were. (And then wonder why people don’t understand that -4 x -2 = 8!) The idea that Augustus De Morgan, the great pioneer of logic and computer science, was still unsure about negative numbers in 1843 is, to me, a testament to this.

Information about De Morgan plus Liu Hui and Bhaskaracharya are revealed in an incredible paper called What’s so Baffling About Negative Numbers? – a Cross-Cultural Comparison by the genius David Mumford. Honestly I think it’s telling that after reading a hundred things, the most insightful, comprehensive and thoughtful work about this little sixth grade topic came from a Fields Medalist, MacArthur winner and Putnam fellow. See the paper here.

“as early as 1000 BC” (Mumford)

“as late as 1843!” (Mumford)

Card 3

“definitely true or definitely false”

The study of logic begins with statements. A statement is a sentence or mathematical expression that is either definitely true or definitely false. (Hammock 2018)

Card 4

"The ancient Greeks ran into” there’s a rumor that irrational numbers were so upsetting someone was drowned over this! For more on the history, see The World of Mathematics Vol 1, James R. Newman.

“Eudoxus of Cnidus” (Newman 525) Almost everything we know about Eudoxus comes by way of Euclid. Book V of The Elements, which concerns the theory of ratios (what I called Eudoxus’s workaround to irrationals), is almost entirely inspired by (or written by?) Eudoxus.

“Richard Dedekind” (Newman 525)

This is one of those topics that’s so big it felt impossible to write something meaningful in 90 words or less. Many cool things were left out, like the Babylonian tablet YBC 7289 which contains an estimate of the square root of 2 to six decimal places! There is so much to say about Richard Dedekind, the legendary mathematician who came up with his theory of irrationals while teaching at a technical high school. (Newman 527)

Also, it’s hard to overstate how unsettling irrational numbers were — and for how long. Here’s E.T. Bell in 1937:

As not one of these three roots can be extracted exactly, no matter to how many decimal places the computation is carried, it is clear that the verification by multiplication as just described will never be complete. The whole human race toiling incessantly through all of its existence could never prove in this way that √3 x √2 = √6. (Newman 526)

Card 5

DISCLAIMER: Card 5 has a very long unfocused and rambling note because this topic is enormous and very interesting!

A postulate is a statement we can assume is true without proof. Euclid starts The Elements with 23 definitions of things like circles, points and lines. He then writes five postulates — statements he can safely and logically gather from these definitions without proving them. Here are the first four postulates (Euclid/Thomas L. Heath, The thirteen books of Euclid’s Elements, 135):

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any centre and distance.

4. That all right angles are equal to one another.

Here is the fifth:

5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side which are the angles less than the two right angles.

There is an easier-to-understand restatement of the fifth postulate called Playfair’s Postulate:

Through a given point only one parallel can be drawn to a give straight line. (Heath 1956)

Back to Euclid. Right away, something is different. This postulate is long and not at all obvious. But Euclid included it as if it was! Was it obvious to him? Did he add it on a wing and a prayer?

It’s a very strange statement. It’s a blot. Because it’s a leap of faith unlike all the other postulates. (Gray 2009)

When we consider the countless successive attempts made through more than twenty centuries to prove the Postulate, many of them by geometers of ability, we cannot but admire the genius of the man who concluded that such a hypothesis, which he found necessary to the validity of his whole system of geometry, was really indemonstrable. (Heath 202)

It turns out the parallel postulate is true in Euclidean geometry (the flat paper geometry we learn in school) but false in other geometries. These other geometries were discovered by Gauss, Bolyai, Riemann and Lobachevsky, not in that order. I have to add a note here by Roger Penrose:

It is, however, the conventional standpoint (somewhat unfair, in my opinion) to deny [Heinrich] Lambert the honor of having first constructed non-Euclidean geometry, and to consider that (about half a century later) the first person to have come to a clear acceptance of a fully consistent geometry, distinct from that of Euclid, in which the parallel postulate is false, was the great mathematician Carl Friedrich Gauss. (Penrose 2005)

“over 1000 books” is from page 59 of a terrific book called The Poincare Conjecture by Donal O’Shea (this book also mentions the second most-read book and Lincoln).

“bouts of insanity” people the world over really went insane over the parallel postulate and I wish I had more room on the card to elaborate. János Bolyai was one of them. János’s dad, Farkas Bolyai, was a genius. (He was friends with freakin’ Gauss.) Farkas decided hey — let me make my son János an even bigger genius! (János knew calculus, analytical mechanics and several languages by 13. This is also in O’Shea.)

Farkas, the dad, worked on the parallel postulate with Gauss but it was János who got really obsessed. Here’s what Farkas wrote to his son:

I implore you to make no attempt to master the theory of parallels; you will spend all your time on it…Do not try…either by the means you mentioned or any other means…I passed all through the cheerless blackness of this night and buried in it every ray of light, every joy in life. For God’s sake, I beseech you, give it up. Fear it no less than sensual passions, because it too may take all your time, deprive you of your health, peace of mind and happiness in life. (O’Shea 69)

A quick note about Euclid: It’s true that very little is known about him. Lucio Russo, in (the mind-blowing book) The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had To Be Reborn writes:

Euclidean geometry has remained throughout the centuries the framework for basic mathematical teaching. But Euclid himself has been taken out of history. In his case the mechanism is opposite the one used for Archimedes: instead of being depicted in legend and in anecdotes, he is offered to us without any historical context, laying down “Euclidean geometry” as if it were something that had always been there at mankind’s disposal. If you are not convinced of this, try asking your friends what century Euclid lived in. Very few will answer correctly in spite of having studied Euclidean geometry for several years. (Russo 7)

Lastly, If you want a masterpiece treatment of the parallel postulate in the broader context of logic, read Logicomix, An Epic Search For Truth. The parallel postulate is on page 70.

Card 6

Not to pile on trapezoids more, but triangles are even mentioned in Plato’s Timaeus in relation to the human body:

[Consider] the young constitution of the whole animal, which has the triangles of the elements new…Since the triangles coming in from the outside, which make up food and drink, are older and weaker than its own triangles, it overpowers them and cuts them up with its new triangles, making the animal grow by nourishing it with many similar elements. (Russo 38)

Also, when Archimedes was trying to find the area of a parabola bounded by a line (Quadrature of the Parabola), he did so by adding up the areas of infinitely many triangles:

This example makes it clear why Hellenistic mathematicians laid out with great care such simple theories as that of triangles, presented in the Elements: they were useful tools for tackling even problems whose original statements had no connection whatsoever with the auxiliary theory. (Russo 51)

Triangle tidbit: One way to find the area of a triangle is through its base. The first use of “base” in Euclid’s Elements was in Book 1 Proposition 4:

Here we have the word base used for the first time in the Elements. Proclus explains it as meaning (1), when no side of a triangle has been mentioned before, the side “which is on a level with the sight” and (2), when two sides have already have already been mentioned, the third side. (Heath 1956)

Proclus wrote the commentary for Euclid’s Elements.

Card 10

“Kepler dedicated” (Newman 124)

“took a voyage”

“My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto Astronomy, viz. the Logarithms.” — Henry Briggs (Newman 124)

“was wishing he’d hurry up”

The Danish astronomer looked for an early publication of the logarithmic tables; but it was long before they were completed. Napier, in fact, was slow but sure. (Newman 123)

Card 11

We return to David Mumford’s paper (everyone should read this and he deserves all of his awards) for an example of Girolamo Cardano trying to figure out a real-world example of negative square roots:

The dowry of Francis’ wife is 100 aurei more than Francis’ own property, and the square (?) of the dowry is 400 more than the square of his property. Find the dowry and the property (Mumford)

Mathematicians thought, then, that imaginaries, though apparently uninterpretable and even self-contradictory,  must have a logic. So they were used with a faith that was almost firm and was only justified much later. Mathematicians indicated their growing security in the use of √-1 by writing “i” instead of “√-1” and calling it “the complex unity,” thus denying, by implication, that there is anything really imaginary of impossible or absurd about it. (Newman 30)

Card 15

This card mentions bona fide math genius Terrance Tao. He keeps a blog here that everyone should read.

Euclid proved that primes are infinite in Book IX Proposition 20 of The Elements. Note that he does this “without ever dealing directly with infinity by reducing the problem to the study of finite numbers.” (Russo 2000)

Card 17

There is a quote I wish I had room to add on this card and it’s by Tom Sherrington, author of the great teacherhead blog and also of the book The Learning Rainforest: Great Teaching in Real Classrooms. On one of his Great Lessons posts, he writes:

I’d suggest that the most important diagram in the universe is the number line.  The key to good numeracy is a strong mental model of numbers in sequence and scale. I’ve often found that people with weak numeracy skills have a poor foundation at this basic level.  Before we get into complex operations, just having a really good feel for number is vital.  Having an intuition that 0.6 is less than 2/3 or that 3/4 is bigger than 0.7 – and so on come – from a good visual map of numbers in scale and sequence. (Tom Sherrington, Great Lessons 6: Explaining, 2013

I agree with him and really wish I had space on the card to include that quote!

Card 19

Frege published his Begriffsschrift in 1879, “perhaps the most important single work ever written in logic.” (Gray 159)

Card 28

A fun bit about large numbers: Immanual Kant argued that all you need for proofs is your intuition. Frege said something like how do we know that 123,456,789 + 987,654,321 = 1,111,111,110? Not by counting dots! You can’t use your intuition for things like really big numbers. You need rules. (Gray 81).

What Archimedes did trying to count the number of grains of sand that would fit in the universe is nothing short of jaw-dropping.

Card 29

“if a quantity is increased or decreased by an infinitesimal, [it] is neither increased or decreased” Johann Bernoulli (Thompson, 22)

“was pilloried far and wide for a long time” Bishop George Berkeley, 1734: “And what are these same evanescent incremenents? They are neither finite quantities, nor quantities infinitely small nor yet nothing.” (Thompson 21)

Bertrand Russell (1903) called them “mathematically useless.” Charles Pierce “strongly disagreed” but “was almost alone in his day in siding with Leibniz, who believed that infinitesimals were as real and as legitimate as imaginary numbers.” (Thompson, 23)

There’s evidence Zeno of Elia was aware of infinitesimals in 500 BC! Let’s say you’re crossing a street. You walk halfway across the street and pause. Then you walk half the distance that’s left and pause. Etc. Etc. The size of your steps is getting really really small right? Also: Will you ever cross the street? That’s Zeno’s Paradox.

Card 32

The integral sign being invented by Gottfield Wilhelm von Leibniz is from Newman 54.

Card 34

“here, color is to do everything” (van Gogh 86)

“in a word, looking at the picture” Van Gogh enclosed a lovely black and white drawing of his bedroom. The colored-in version would one day be world famous! (van Gogh 86)

Card 38

The footnote is from a letter from André Weil to his sister Simone from Bonne-Nouvelle Prison — a military prison — in Rouen, March 1940:

One would be totally obstructed if there were not a bridge between the two.And just as God defeats the devil: this bridge exists; it is the theory of the field of algebraic functions over a finite field of constants (that is to say, a finite number of elements: also said to be a Galois field,, or earlier "Galois imaginaries" because Galois first defined them and studied them; they are the algebraic extensions of a field with p elements formed by the numbers 0, 1, 2, .. . , p- 1 where one calculates with, them modulo p, p = prime number). They appear already in Dedekind. (Krieger 2005)

He also said some pretty complimentary things about another person we know — Riemann!

I am surely one of the most knowledgeable persons about this subject; mainly because I had the good fortune (in 1923) to learn it directly from Riemann’s memoir, which is one of the greatest pieces of mathematics that has ever been written; there is not a single word in it that is not of consequence. (Krieger, 2005)

Card 42

David Hilbert: “I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, we are a university, not a bathing establishment.” (Woit, 43)

“stop by later for more food, drinks and discussion” Noether would host events at her apartment where students and professors could come to relax and chat about math over food, desserts and wine. (On Emmy Noether and Her Algebraic Works, Deborah Radford, 9)

Card 43

“shrinks to a point” Note that the sphere is a “three-sphere” and both the torus and sphere are examples of “three-manifolds.”

Card 44

Cantor’s set theory was the first big move to strip intuition from logic and make it cold, consistent and formal. Set theory faced infinity head-on (Gauss did not want to face infinity head-on) and made math axiomatic from the ground up. (Goodbye intuition! This distressed Poincaré who viewed the new agenda as a move to math a soulless machine that spits out answers. In the based-on-reality but fictionalized Logicomix, Poincaré is depicted as saying “[Hilbert] wants a machine to feed it axioms and make theorems, like one where a pig enters the one side and the sausages come out from the other!”)

Russell’s paradox about sets that contain themselves turned this whole project upside down. Probably Poincaré was pleased. (A distraught Frege wrote an addendum in his Grundgesetze basically saying Russell collapsed one of his laws. See his letter to Russell here.) Cantor it seems took it somewhat well because his “set of all sets” was now impossible.

In modern ZFC set theory, there is no set that contains itself.

Here is a neat quote about Cantor’s proof by Hans Hahn:

The essence of Cantor’s proof is that no comprehensive counting procedure can be devise for the entire set of real numbers, nor even for one of its proper subsets, such as al the real numbers lying between 0 and 1. By various ingenious methods certain infinite sets such as all rational fractions ir a agebracic numbers can be paried off with the natural numbers; every attenmpt, however, to construct a formula for counting the all-inclusive set of real numbers is invaribaly frustrated. No matter what counting scheme is adopted it can be shown that some of the real numbers in the set so considered remain uncounted, which is to say that the scheme fails. It follows that an infinite set for which no counting method can be devised in noncountable, in other words nondenumerably infinite. (Newman 1597)

Card 45

“talk nobody cared about” Riemann’s talk at the University of Göttingen wasn’t meant to be a big ordeal because the habilitation is a ho-hum requirement for a German teaching position. But one person in attendance was paying close attention: Gauss.

“one of the greatest moments in the history of science” (The Poincare Conjecture, Donal O’Shea, 74)

More praise:

The speech completely recast three thousand years of geometry, and did so in plain German with almost no mathematical notation. (O’Shea 75)

Riemann’s geometry was the key to solving the puzzle Einstein had been wrestling with all those years. (Yau 31)

…Bernhard Riemann, who was widely recognized as the most original mathematician of the mid-nineteenth century... (Gray 18)

“I still can’t see how he thought of it” (Livio, 169)

“a terrible mess” this was said by Einstein’s friend, the geometer Marcel Grossman. The full quote: “a terrible mess which physicists should not be involved with.” (Yau, 31)

“general relativity is born” note that special relativity is another geometric framework: Minkowskian.

Card 46

"modern-day Euclid” (Roberts 2003)

“what shape is that?" (Roberts 2005)

“It seems worth while…” (Coxeter 1998)

Card 47

On hard problems fostering creativity and joy:

When students are invited to ask a harder question, they often light up, totally engaged by the opportunity to use their own thinking and creativity. (Mindset Mathematics, Jo Boaler)

Card 48

You might have a fun time seeing the Bourbaki group mentioned in the storied 294-page Grothendieck-Serre Correspondence, a torrent of letters of sent between Alexander Grothendieck and Jean-Pierre Serre. Grothendieck is considered one of the greatest geniuses in the history of math.

Before leaving for a Bourbaki congress, I will try to answer the torrent of questions you asked in your last letter. — Serre

In no6, I have marked two passages with a “?” sign in the margin, to indicate that if you feel that such unhatched chickens have no place in a Bourbaki talk, then you can simply delete them. — Grothendieck

You must think me a terrible correspondent for not having answered your letter sooner, but I have just come back from the Bourbaki meeting, and I had loads of things to do.

We would really like you to come to the Bourbaki meeting in October, if possible (and ditto for the others, of course! I don’t remember exactly what the program is to be (in any case, there will be a reading of my draft on filtered rings etc.), and I don’t think you will find it particularly interesting. But one is not in Bourbaki for fun, as Dieudonn´e never stops repeating… — Serre

etc.

Card 50

“delivered his talk” The anecdote about Grigori Perelman is from Donal O’Shea’s amazing book The Poincare Conjecture:

As at MIT, everyone in the room, young and old, except the reporters, realized that what they were hearing was the culmination of over a century of the greatest flowering of matheiacical thought in our species’ history. The lecture demanded close attention, leaving little space for stray throughts. (O’Shea 3)

How did Perelman feel about the possibility of winning that kind of money? As it dawned on them that he did not care, they changed their approach and wrote stories about a reclusive Russian making a big math discovery, and speculated that he would reject the prize. (O’Shea 3)

Card 51

This didn’t fit on the card but here are some very nice words about Arthur Cayley courtesy of James R. Newman:

Cayley brought mathematical glory to Cambridge, second only to that of Newton, and the fertility of his suggestions, in geometry and algebra, continues to influence the whole range that is now studied at home and abroad. To this versatility Cayley added a Gauss-like care and industry. (Newman 164)

Card 54

Apollonius of Perga discovered all of this in around 200 BC (!). See his eight-volume treatise Conics.

Card 55

The Poisson distribution is just a special case of the binomial distribution. More on this soon!

Card 58

“cutting and glueing”

There is a problem in The Knot Book that demonstrates the syntax of math surgery:

Cut S open along C, obtaining two copies of C in the cut open S. Glue discs to each of the new curves, where each disk is parallel to the disk bounded by C in F. (Adams 103)

“to classify different knots” This is mentioned in an absolutely lovely paper by Jonathan Marty:

Despite the rather mechanical and unintuitive nature of the surgery operation, it has a wide variety of applications. Physicists use it to study topology change under events that “cut” spacetime. Knot theorists use it to classify knots by breaking down their the Seifert surfaces associated with them. (Marty 1)

“Dehn surgery”

In 1910, Dehn and Heegaard published a famous paper that used Dehn surgery to produce an infinite series of three-manifolds that were homology spheres. (O’Shea 141)

The 1910 paper is interesting for a number of other reasons. It showed that there was a connection between homology spheres and non-Euclidean geometry. It also investigated some connections between the theory of knots and three-manifolds. (O’Shea 142)

“every 3-manifold”

It has been known for over 30 years that every closed connected orientable 3- manifold is obtained by surgery on a link in S^3. (Lackenby, 1997)

“A Seifert surface is”

Given a knot K, a Seifert surface for K is an orientable surface with one boundary component such that the boundary component of the surface of the knot is K. We have just described one way to obtain a Seifert surface for a knot. However, there may be other Seifert surfaces for the same knot. (Adams 99)

“Ricci flow with surgery”

Ricci flows typically develop singularities within finite time. Perelman’s groundbreaking work allowed for the classification of these singularities and introduced the concept of Ricci flow with surgery. (Yike 2024)

As Hamilton’s colleague Shing-Tung Yau of Harvard University pointed out, these necks mark the spots where mathematicians should perform the “surgery” Thurston’s conjecture requires. (Mackenzie 2003)

Now consider the Ricci flow and let the manifold evolve in accordance with it. If the manifold is simply connected (that is, if it is such that every loop can be shrunk to a point) , then Perelman proves that the Ricci flow, after perhaps some harmless surgeries, will eventually smooth out the extremes of curvature, giving a manifold with constant positive curvature homeomorphic to the original manifold. (O’Shea 191)

“all the possible shapes”

Poincare was particularly concerned with three-dimensional manifolds. These modeled possible shapes that our universe might have. (O’Shea 132)

& Sources

Adams, Colin C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society (2004)

Boaler, Jo. Mathematical Mindsets. Jossey-Bass. (2016)

Coxeter, H.S.M. “Whence does an ellipse look like a circle?” C. R. Math. Rep. Acad. Sci. Canada Vol. 20 (4) 1998, pp. 124–127. https://mathreports.ca/article/whence-does-an-ellipse-look-like-a-circle/

De Koninck, Jean-Marie. Those Fascinating Numbers. American Mathematical Society. (2009)

Euclid/Thomas L. Heath. The Thirteen Books of the Elements, Vol 1. Dover. (1956)

Gray, Jeremy. Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton University Press. (2009)

Gillings, Richard J. Mathematics in the Time of the Pharoahs. Cambridge, Mass. M.I.T. Press. (1972)

Hammack, Richard. Book of Proof, Third Edition. Published by Richard Hammack (2018)

Johnston, Nathaniel. “11630 is the First Uninteresting Number.” https://www.nathanieljohnston.com/2009/06/11630-is-the-first-uninteresting-number/ (2009)

Krieger, Martin H. A 1940 Letter of André Weil on Analogy in Mathematics. Translated by Martin H. Krieger. Notices of the American Mathematical Society, Volume 52, Number 3. March 2005. https://www.ams.org/notices/200503/fea-weil.pdf

Lackenby, Marc. Dehn Surgery on Knots in 3-Manifolds. Journal of the American Mathematical Society. Volume 10, Number 4, October 1997, Pages 835–864 S 0894-0347(97)00241-5. https://www.ams.org/journals/jams/1997-10-04/S0894-0347-97-00241-5/S0894-0347-97-00241-5.pdf

Livio, Mario. The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. Simon & Schuster. (2006)

Mackenzie, Dana. “Mathematics World Abuzz Over Possible Poincaré Proof.” Science Vol 300 18 April 2003. https://www.science.org/doi/10.1126/science.300.5618.417

Marty, Jonathan. Surgery Theory. December 27, 2021. https://www.uvm.edu/~smillere/TProjects/JMarty21f.pdf

Mumford, David. “What’s so Baffling About Negative Numbers? — a Cross-Cultural Comparison.” In: Seshadri, C.S. (eds) Studies in the History of Indian Mathematics. Culture and History of Mathematics. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-49-1_6 (2010).

Newman, James R. Volume 1: The World of Mathematics. Simon and Schuster. (1956)

Krieger, Martin H. A 1940 Letter of Andre Weil on Analogy in Mathematics. Notices of the AMS. Vol 52 No 3 (2005). https://www.ams.org/notices/200503/200503FullIssue.pdf

O’Shea, Donal. The Poincare Conjecture: In Search of the Shape of the Universe. Walker & Company. (2007)

Penrose, Roger. The Road to Reality: A Complete Guide to the Laws of the Universe. New York, A. A. Knopf (2005)

Roberts, Siobhan. King of Infinite Space. Walker Publishing Company. (2006)

Roberts, Siobhan. “Donald Coxeter: The Man Who Saved Geometry” Siobhan Roberts. Toronto Life, January 2003.

Russo, Lucio. The Forgotten Revolution: How Science Was Born in 300 BC and Why it Had to Be Reborn. Springer (2000)

Serre, Jean-Pierre & Colmez, Pierre. Grothendieck-Serre Correspondence (English and French Edition). American Mathematical Society (2022) https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Letters/GS.pdf

Sherrington, Tom. "Great Lessons 6: Explaining” https://teacherhead.com/2013/02/13/great-lessons-6-explaining/ (2013)

Thompson, Silvanus P. and Gardner, Martin. Calculus Made Easy. St. Martin’s Press. (1998)

van Gogh, Vincent. The Complete Letters of Vincent Van Gogh, Volume III. Bulfinch Press. (2000)

Woit, Peter. Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law. Basic Books. (2007)

Yau, Shing-Tung and Nadis, Steve. The Shape of Inner Space: String Theory and the Geometry of the Universe’s Hidden Dimensions. Basic Books. (2010)

Yike, He. Ricci Flow With Surgery. The University of Chicago Department of Mathematics. (2024) https://math.uchicago.edu/~may/REU2024/REUPapers/He,Yike.pdf

Zee, A. Quantum Field Theory As Simply As Possible. Princeton University Press. (2023)

Anatomy of a blurb

(Math + Nora) ÷ Lydia

That’s my formula and I’m sticking to it.

The back of each card contains a doodle and a blurb. The doodle takes up space so the blurb has to be tight. Around 65 to 85 words tight.

There isn’t room for flimflam. (Though the flimflam is funny.) With so little space, and so much to say, there was only one place to turn: Lydia Davis. Her story In a House Besieged is 65 words. They Take Turns Using a Word They Like — one of my favorites — is 11.

She is a genius. There is no better person to try (and fail and try and fail) to copy.

Explaining the monster group was the first time I had a legitimate crisis. The guiding mission of these cards is to write them in a way that is accessible to everyone, regardless of math experience. A principle I won’t compromise on. Of group theory, mathematician Cassius Keyser wrote that he could not, in one hour, “give you anything like an extensive knowledge of it, nor facility in its technique, nor a sense of its intricacy and proportions…” Oh great!

Still, Cassius Keyser was helpful. And James R. Newman. And Erica Klarreich, who wrote this great big wonderful piece. See also Columbia math professor Peter Woit, who writes the brilliant Not Even Wrong blog (named after the beyond brilliant Not Even Wrong book). But groups are still gnarly to weed wack.

Back to Lydia. And really, why bother being consistent with tense? Since I can’t hypnotize us both and somehow telepathically borrow her brain, the next best thing is her green book Essays One which contains invaluable glimpses into her brain like Revising One Sentence and Commentary on One Very Short Story (“In a House Besieged”). These pieces are, thank God, more than 10 words long.

Anyway this was all still hard. But I started to gather ideas from here, quotes from there, and foof them like a floral arrangement around mathematical anchors (invariants, groups, transformations, the boy genius Évariste Galois) that I definitely wanted to include.

The missing item is Nora, of course. The late Nora Ephron. One of the greatest sentence writers of all time and my flimflam idol. For this card, I think this sacred ingredient is still missing. Though Nora did say this:

I can't stand writers who quote people saying very mundane lines like 'I was born in 1934.' You read something like that and wonder to yourself why did he quote that, it doesn't take you anywhere or show you anything the writer couldn't have done himself in a more interesting way. (Ephron 2015)

Fortunately the topic of groups contained some great, un-mundane lines. Here’s James R. Newman in Volume 3 of The World of Mathematics (1534):

The term group was first used in a technical sense by the French mathematician Évariste Galois in 1830. He wrote his brilliant paper on the subject at the age of twenty, the night before he was killed in a stupid duel.

So maybe I am halfway there. “Stupid duel” is funny (though Galois’s story is tragic. He is so interesting he spawned his own card.)

Which makes this a working blurb. It comes in at 82 words. Is Hermann Weyl important? I think so. Weyl, a mathematician, wrote a groundbreaking physics book that was so math-y physicists struggled to understand it. (Apparently Wolfgang Pauli called the whole field die Gruppenpest — the plague of group theory.)

A potential swap for Hermann Weyl is the aforementioned mathematician Cassius Keyser. There’s a question that haunted him “a good deal from time to time in recent years” and which he’s “not yet prepared to answer confidently.” Is mind a group? This finite mind group would include abstract things like feeling, believing, seeing, tasting, hoping, etc.

That is interesting right? It’s also 17 words.

Anyway here’s the working blurb:

Groups were invented by Évariste Galois “the night before he was killed in a stupid duel.” At 20. Groups are collections of things that meet a few crucial conditions. They can be finite or infinite. You can transform groups and see what changes. But what’s more interesting is what doesn’t! (The invariants). Lie (“LEE”) groups are a big part of quantum mechanics. In The Group Concept, Cassius Keyser wonders if the mind is a group. The monster is a very big group!*

82 words! Did we make mathematicians proud and also Lydia Davis? Let’s hope.

Let’s hydraulic press this all into one blurb!

*808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

Should triangles wear top hats?

Like all great fashion epiphanies, I had this one while watching E! Live From the Red Carpet. Why don’t triangles wear hats? There are so many to choose from! I had to change things.

So I did what anyone would do. I sent an email blast to triangles, even the scalene ones, inviting them to a neighborhood happy hour. I picked triangles because historically they are the most responsive to emails. At least this is what Pythagoras wrote in his diary.

Two days later, the triangles showed up.

I could tell they were thrilled, so I got straight to the point.

“I am going to get straight to the point. Your fashion sense is nonexistent and you should all wear hats. Preferably top hats for the alliteration.” I pointed to a box of hats.

“Imagine a world” — I could feel the excitement building inside of me — “where triangles walk runways in Paris or get 12-page editorial spreads in Vogue!”

The triangles blinked. In math, ideas of great significance take time to sink in.

Then I heard a voice. “How will people know what the angle measures are if everything is obstructed by a hat?”

I said that in the grand scheme of things, the measure of an angle in one corner of a triangle never really mattered to anyone.

Wow. Chills!

I think this could solve the global math crisis. I really do

Extensive Bibliography

I didn’t do this alone. These cards were built with love on the backs of brilliant people. Here are their names and works:

• Aaker, Jennifer & Bagdonis, Naomi. Humor, Seriously. Random House (2021)

• Abeles, Vicki. Beyond Measure. Simon & Schuster. (2015)

• Adams, Colin C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society (2004)

• Ashcraft, M.H. “Math anxiety: Personal, educational, and cognitive consequences.” Current Directions in Psychological Science 11 (5) pp. 181-185 (2002)

• Angrist, Noam et at. “How to Improve Education Outcomes Most Efficiently? A Comparison of 150 Interventions Using the New Learning-Adjusted Years of Schooling Metric.” World Bank Group. Education Global Practice & Development Research Group. October 2020. Policy Research Working Paper 9450

• Ball, Deborah Loewenberg. “Magical Hopes: Manipulatives and the Reform of Math Education.” American Educator (1992)

• Barton, Craig. How I Wish I’d Taught Maths. John Catt Educational Ltd. (2018)

• Bingham, T. & Rodriguez, R.C. “Understanding Fractions Begins with Literacy.” Texas Association for Literacy Education Yearbook Vol 6: Discover the Heart of Literacy ISSN: 237400590 online. (2019)

• Boaler, Jo. Fluency. “Without Fear: Research Evidence on the Best Ways to Learn Math Facts.” youcubed at Stanford University (2015)

• Boaler, Jo. Mathematical Mindsets. Jossey-Bass. (2016)

• Boaler, Jo. The Elephant in the Classroom. Souvenir Press (2015)

• Burnett-Bradshaw, Camille S. “From Functions As Process to Functions as Object: A Review of Reification and Encapsulation.” A qualifying paper for the Doctor of Philosophy. Tufts (2007)

• Christian, Brian; Griffiths, Tom. Algorithms To Live By: The Computer Science of Human Decisions.” Picador (2016)

• Dweck, Carol. Mindset. Ballantine Books. (2006)

• Euclid. The Thirteen Books of the Elements, Vol 1. Dover. (1956)

• Gillings, Richard J. Mathematics in the Time of the Pharoahs. Cambridge, Mass. M.I.T. Press. (1972)

• Hammack, Richard. Book of Proof, Third Edition. Published by Richard Hammack (2018)

• Kaplan, Robert; Kaplan, Ellen. Out of the Labyrinth: Setting Mathematics Free. Oxford University Press. (2007)

• Klee, Paul. Pedagogical Sketchbook. Faber & Faber Limited. (1953)

• Liljedahl, Peter. Building Thinking Classrooms.

• Lockhart, Paul. Measurement. Harvard University Press. (2012)

• Lin, Thomas. The Prime Number Conspiracy. The Simons Foundation (2018)

• Lomborg, Bjorn. Best Things First. Copenhagen Consensus Center. (2023)

• Newman, James R. Volume 1: The World of Mathematics. Simon and Schuster. (1956)

• Nuthall, Graham. The Hidden Lives of Learners. Nzcer Press. (2007)

• Penrose, Roger. The Road to Reality: A Complete Guide to the Laws of the Universe. New York, A. A. Knopf (2005)

• Pettis, Christy Rae. Preservice Elementary Teachers’ Understandings of the Connections Among Decimals, Fractions, and the Set of Rational Numbers: A Descriptive Case Study. A Dissertation Submitted to the Faculty of University of Minnesota. (2015)

• Ramirez, Gerardo et al. Math Anxiety, Working Memory, and Math Achievement in Early Elementary School, Journal of Cognition and Development, 14:2, 187-202. (2013)

• Roberts, Siobhan. King of Infinite Space. Walker Publishing Company. (2006)

• Russo, Lucio. The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn. Springer-Verlag (2004)

• Sherrington, Tom. The Learning Rainforest. John Catt Educational Ltd. (2017)

• Steward, Ian. Taming the Infinite

• Thompson, Silvanus P. & Martin Gardner. Calculus Made Easy. St. Martin’s Press. (1998)

• Unesco Institite for Statistics. “More Than One-Half of Children and Adolescents Are Not Learning Worldwide.” UIS Fact Sheet No. 46. September 2017

• Vamvakoussi, X. & Vosniadou, S. How many decimals are there between two fractions? Aspects of secondary school students’ reasoning about rational numbers and their notation. Cognition & Instruction, 28(2), 181-209. (2010)

• Wallace, David Foster. Everything and More: A Compact History of Infinity. W. W. Norton & Company. (2003)

• Willingham, Daniel T. Why Don’t Students Like School? Jossey-Bass. (2021)

• Zeki, Semir et. al. (2014) The Experience of Mathematical Beauty and its Neural Correlates. Frontiers in Human Neuroscience. Vol 8 Article 68

A Global Mission

The UN has big goals — 17 of them across 169 targets. Written in 2016, these Sustainable Development Goals (SDGs) tackle issues like tuberculosis and clean water and are meant to be achieved by 2030.

The fourth goal (SDG 4) is that by 2030, all girls and boys have access to quality education. This sounds ambitious and vague, but there are targets and indicators for progress. One (4.1.1) is that everyone hits something called a Minimum Proficiency Level (MPL) for their grade level.

SDG 4: Ensure inclusive and equitable quality education and promote lifelong learning opportunities for all

What’s that mean? Let’s look at math. For grades 2 and 3, the MPL means you can name shapes. You can identify patterns. You can, within reason, divide, double and compare numbers under 100. By grade 5, you can skip count, use decimals, tell time, and read a bar graph. By grade 8, you can use exponents and do simple probability. (Unesco Institute for Statistics, 2022)

The Minimum Proficiency Levels are reachable. Unfortunately, six out of ten kids are not there in reading and math. Among the poorer countries, that’s 364 million students. (Lomborg 61) This goal is behind and on track to be fulfilled by 2056 — if that.

In his book Best Things First, Bjorn Lomborg states that SDG4 is achievable — in fact, the second most achievable one. One way to do this is to teach at the right level, using either high-tech methods (tablets and computers) or low-tech methods (classroom shuffling so you spend a portion of the day in a class at your level). These are tested methods that work.

But will kids even consume problems at their level if they don’t want to? Will it translate to substantive learning if you are clamped and resistant to the whole subject? It’s important to whet an appetite for math — perhaps by lowering the temperature and reducing the anxiety surrounding the subject.

Students are constantly on their guard against being conned into being interested.

— Nuthall, 2007

Why Humor Matters.

For some, math is a peregrine falcon, spreading its wings and leading us to vistas of wild geometries and telescoping series and probability problems without socks. For others, math is a subway pigeon with explosive diarrhea that leads us nowhere and follows us everywhere we go.

Math anxiety can creep up any time. But a lot of it happens at school, when being fast and accurate is rewarded. Math tests with a time component are important (Barton 2018), but the pressure to be fast and accurate is stressful.

You can find math anxiety in first and second graders and even kids as young as 5. (Ramirez 2013) This can snowball, creating a lifelong creepy-crawling feeling that you are not a math person and maybe not even smart.

College-aged participants in one lab study were asked to solve elementary problems like 46+18. Many showed signs of distress including nervous laughter and trembling palms. A few asked if this was going to reflect on their intelligence. One even burst into tears. (Ashcraft 2002).

In 2023, half of New York City students in grades 3 to 8 were not proficient in math. In 2022, scores on the National Assessment for Educational Progress showed their biggest drop in math for grades 4 and 8 since 1990.

It’s not even close to the whole problem. But I think some of this stems from people having an air of anxiety and dislike around math.

Mathematics is a beautiful subject, with ideas and connections that can inspire all students. But too often it is taught as a performance subject, the role of which, for many, is to separate students into those with the math gene and those without.  — Mindset Mathematics

So we are going to retool this whole thing. Because I am one of those people with pigeon droppings on my head and I’ve discovered math is for everybody. It is mind-boggling and beautiful and you don’t have to do a single ounce of calculation to enjoy thinking and talking about it.

Find the funny.

Humor helps us learn. I thought nothing of this until I real papers like Avner Ziv’s Teaching and Learning with Humor: Experiment and Replication. It starts by saying there are 50 other papers praising humor in education! In this paper, humor was injected into a statistics class and the students scored 11% higher on their final exam.

This is why Duolingo writes funny sentences on purposes. Laughing actually helps us solve creative problems. (Isen 1987) Best of all, laughing and smiling feel good. So why not melt away math anxiety by associating math with that feeling?

The goal of SHEESH is to make it funny. It’s not possible to find the square root* of a cow. But by wiping out the biggest contributors to math anxiety, you can really think about a square root. You can make this simple low-floor high-ceiling prompt as deep as you’d like.

And maybe, just maybe, you can find the square root of a cow.

*Psst: It’s also totally okay to not know what a square root is. The topics are wide-ranging and everything is defined on the back!

Building engaging prompts that are “low floor, high ceiling.”*

In her phenomenal book Mathematical Mindsets, Stanford professor Dr. Jo Boaler asks what makes a good math question. There are several characteristics. One, she says, is that the question is “low floor, high ceiling.” (Boaler, 62)

A question’s floor is its entrance point. Who can understand it? A question has a high ceiling if you can keep exploring the problem as it leaps and grows in complexity.

Fermat’s Last Theorem has a low floor. The question can be understood by a high school algebra student and takes up less than an inch of space. It has an extremely high ceiling. Proving the question took Andrew Wiles 7 years and 129 pages and involved a modularity theorem for semistable elliptic curves.

I went to work writing funny zero calculation prompts that were “extremely low floor and possibly high ceiling with a mezzanine area for dawdlers.” Because the mission is to make math funny, especially to those with math anxiety, everyone should be able to engage with the prompts regardless of age (within reason) or mathematical ability.

Let’s walk through two example cards from the picture above.

Make the world’s skinniest triangle

Kids as young as 3 can identify shapes and kids as young as 4 can badly draw them. (Quinn, 65) This is a low floor.

Triangles also represent curvature. If a triangle has more than 180 degrees, the surface it’s on has positive curvature. Less than 180 and the surface has negative curvature. We are now sort of exploring the Riemann curvature tensor. I will call this high ceiling.

Everything in between makes you play with the definition of a triangle. This vast middle is the mezzanine, where most of us regular folks live.

This prompt invites anyone to play around with triangles. No pretense, no calculation, no pressure.

What’s half a cookie times half a duck?

Fractions. They’ve been causing headaches since the Egyptians. Here’s a quote from Ian Stewart’s Taming the Infinite:

Fractions caused the Egyptians severe headaches. (Stewart 2008)

Here is another from Mathematics in the Time of the Pharaohs:

When the Egyptian scribe needed to compute with fractions, he was confronted with many difficulties arising from the restrictions of his notation. (Gillings 1972)

Perhaps this card should us ask to give the Egyptians some Girl Scout cookies for all that stress.

Fractions are weird. Kids learn that multiplying two numbers makes a bigger number. But multiplying two fractions makes a smaller fraction! This is jarring and something we gloss over as adults.

Teaching fractions is inherently challenging because the operations are counterintuitive to what students already know about whole numbers. — Ni & Zhou, 2005

This prompt is low floor because a duck-cookie crayon/marker/whatever mash is accessible to everyone. (The worse the drawing the better.) It is high ceiling because we can bring ourselves back to the Egyptians and ask what is really going on when we multiply fractions. The mezzanine for this card serves milk and cookies.

This leads to a question though. Instead of cookies and ducks…

Why not just write math questions on the cards?

There are numerous incredible resources out there for those that are math-inclined, math educators or both. This deck was lovingly crafted for those, young and old, who see flashcards and math games and puzzles and recoil a bit into their shell.

Sometimes that math-phobe or math-hater is a student. Other times it’s a parent. The truth is there is usually someone in a student’s orbit who has expressed outright dislike of math.

I have come to the realization that, as their maths teacher, I am possibly the only positive mathematical role model in many of my students’ lives. Sadly, students are likely to be surrounded by math-haters or math-avoiders. — Craig Barton, How I Wish I’d Taught Maths

There are no math questions on the cards because math questions exist all over the place. For the math-phobes, math haters and math avoiders: This is my present to you.

*Boaler, Jo. Mathematical Mindsets. Jossey-Bass. (2016)