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: July 12, 2025
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.
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 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 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))
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. (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 30
“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 33
“invented by” (Newman 54)
Card 35
“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 37
Frege publishes the Begriffsschrift in 1879, “perhaps the most important single work ever written in logic.” (Gray 159)
Card 39
Weil wrote this letter to his sister Simone and 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 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
“shrinks to a point” 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.
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 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” (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 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
“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 52
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 56
The Poisson distribution is just a special case of the binomial distribution. More on this soon!
Card 58
“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)
“every 3-manifold”
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)
“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 originial manifold. (O’Shea 191)
& 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.
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)
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 2002. https://www.ams.org/notices/200503/fea-weil.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)
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)
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