Card 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!
A little of this, a little of that. Coming very very soon!
Card Notes
Last updated: June 11, 2025
Card 1
“Nathaniel Johnston” 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
Information about Liu Hui, Bhaskaracharya and Augustus De Morgan were 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.
“minus three cows” (Penrose 63)
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. (The Book of Proof, Third Edition, Richard Hammock)
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.
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.
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. (Jeremy Gray, Plato’s Ghost)
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. (The Road To Reality, Roger Penrose, 44)
“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 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).
Archimedes’ system of octads is described in Newman’s World of Mathematics, Volume 1 (page 418)
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. (van Gogh 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! (van Gogh 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.” (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 44
“This loop can “slide” off a sphere.” What’s actually meant is that this loop can shrink to a point. Also note that the sphere is a “three-sphere” and both the torus and sphere are examples of “three-manifolds.”
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.
Card 52
This didn’t fit on the card but here are some very nice words about Arthur Cayley courtesy of James R. Newman’s 1956 World of Mathematics Volume 1 (page 164):
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.
Card 60
The Poisson distribution is just a special case of the binomial distribution. More on this soon!
& Sources
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)
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)
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)
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)
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)