Card Notes & Sources

PLEASE NOTE: This page will make absolutely no sense without the deck. But I am continually updating it (chaotically and out of order) with notes and citations so feel free to check in!

Last updated: April 11, 2025

Card 1

“Nathaniel Johnston” you can still read his 2009 blog post here!

“very original thinking” he used 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.)

Card 5

The line in italics is actually a simplified version of the parallel postulate called Playfair’s Postulate (after mathematician John Playfair). It’s easier to read and generally preferred to the clunkier line Euclid wrote:

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. (Euclid/Thomas L. Heath, The thirteen books of Euclid’s Elements. page 135)

The part about “1000+ books” is from page 59 of a beyond terrific book called The Poincare Conjecture by Donal O’Shea (this book also mentions Lincoln and The Elements being the second most read book). And but this isn’t even the whole story because those 1000+ books were only written between 1607 to 1880! The parallel postulate is a rabbit hole worth exploring. If you want a masterpiece comic book treatment of this, read Logicomix, An Epic Search For Truth. The parallel postulate is on page 70.

“semi-obvious” is a tricky compromise I landed on after confusion (to me anyway) about whether Euclid found this parallel line thing obvious or not. It helps to know the other 4 postulates are really obvious. They’re sentences like “That to all right angles are equal to one another” or “To draw a straight line from any point to any point.” (Euclid/Heath)

“It’s a very strange statement. It’s a blot. Because it’s a leap of faith unlike all the other postulates.” That’s Jeremy Gray, mathematical historian and author of Plato’s Ghost, an excellent academic work that I return to over and over again. But then Heath himself says:

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. (Euclid/Heath 202)

So I don’t know.

“People go mad” man this is wild and I wish I had more room on the card to elaborate. People really became obsessed with the parallel postulate. Farkas Bolyai, the dad, was already a genius. (He was friends with freakin’ Gauss.) So he 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 tinkered with 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)

The line “Bolyai did not give it up” on card 53 is a reference to the fact that he ignored his dad, kept obsessing, and invented a whole new branch of geometry. As one does.

A quick note about Euclid (he just goes by that, like Madonna): It’s true that very little is known about him. It’s sad. 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)

But you all know since it’s on the first line of the card!

Card 29

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. (Plato’s Ghost, Jeremy Gray, 81).

Card 30

“if a quantity is increased or decreased by an infinitesimal, [it] is neither increased or decreased” Johann Bernoulli (Calculus Made Easy, Sylvanus 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 35

“here, color is to do everything” this is in a letter by Vincent Van Gogh to his brother Theo. (The Complete Letters of Vincent Van Gogh, Volume III, page 86)

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

Card 37

Frege publishes the Begriffsschrift in 1879, “perhaps the most important single work ever written in logic.” (Plato’s Ghost, Jeremy Gray)

Card 43

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.” (Not Even Wrong, Peter 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 45

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.

Card 46

“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” (The Equation That Could Not Be Solved, Mario 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.” (The Shape of Inner Space, Shing-Tung Yau, 31)

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

Card 47

"modern-day Euclid” this was referenced in “Donald Coxeter: The Man Who Saved Geometry, “ Siobhan Roberts. Toronto Life, January 2003.

“what shape is that?" (King of Infinite Space, Siobhan Roberts)

“It seems worth while…” Whence Does an Ellipse Look Like a Circle,” readable here!

Card 48

“There’s joy and creativity in drumming up hard problems”

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 51

The anecdote about Grigory Perelman is from Donal O’Shea’s amazing book The Poincare Conjecture.