Rabbinic Mathematics

יַּ֥עַשׂ אֶת־הַיָּ֖ם מוּצָ֑ק עֶ֣שֶׂר בָּ֠אַמָּה מִשְּׂפָת֨וֹ עַד־שְׂפָת֜וֹ עָגֹ֣ל׀ סָבִ֗יב וְחָמֵ֤שׁ בָּֽאַמָּה֙ קוֹמָת֔וֹ ׳וּקְוֵה׳ ״וְקָו֙״ שְׁלֹשִׁ֣ים בָּֽאַמָּ֔ה יָסֹ֥ב אֹת֖וֹ סָבִֽיב׃
מלכים א 7:23

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.
I Kings 7:23

This Hebrew Bible passage from I Kings—along with a similar one from II Chronicles—forms the biblical basis for Talmudic scholar Matityahu Hacohen Munk’s suggestion that “some of the geometrical rules did not hold in King Solomon’s temple,” a heavenly ‘‘world of truth’’ beyond our own, mathematical historians Tsaban and Garber write [1].

What’s so heavenly about the Molten Sea, a putative basin created by King Solomon in the ancient Temple of Jerusalem for ritual ablution? And why do the Rabbis Johanan and Papa discuss it extensively in the Babylonian Talmud, bickering in particular about its brim—“[as thin as] the flower of a lily… a handbreadth thick… wrought like the brim of a cup” [2, Eruvin 14a:29-31]?

The simple answer is that this particular snippet of the Word of God contains an oddity, asserting that this circular basin’s circumference is thrice its diameter—or that the geometrical constant π, rather than an irrational number, with an infinite and unpredictable decimal expansion, is in fact rational, and indeed an integer—the number 3, to be exact. Continue reading

A Mathematical Daydream

This article is part of a series on Complex Algebraic Geometry. See also:
1. The Hardest Conjecture; 2. The Valley; 3. A Mathematical Daydream

“Those limits in certain functor categories that cannot be computed pointwise? They don’t actually exist,” one of us declared authoritatively. We, all math grad students, sat on an apartment floor, in a circle, with cards scattered over the floor.

Blank looks abounded. “Meaning, there don’t actually exist monic natural transformations some of whose components are not monomorphisms,” he clarified.

We had invented a variant of a popular card game called “spies vs. revolutionaries” – we called it “students vs. professors” – in which, in particular, each new round was heralded by its leader’s presentation of an established mathematical truth that he or she had decided we were to overturn for good. “The Banach-Tarski paradox is still true, but requires using at best six pieces, not five,” another student later suggested. “2 isn’t actually a prime,” one student blustered, citing the integer’s pathological character in many number-theoretic environments.

Soon it was my turn. “The Hodge Conjecture is false,” I fibbed, “and a counter-example is provided in my paper.”

A good laugh and a general readiness to proceed with the game cut my monologue short. Ridiculous as it was, though, I was ready to continue. The matter was one to which I had given some thought. Indeed, my work presents an interesting testing ground for a few of the ideas surrounding the Hodge Conjecture, and in particular seems to invite a heuristic argument whereby it could be used to furnish a counter-example to the conjecture. I’ll explain this mathematical daydream, and how it can be ultimately debunked. Continue reading

The Valley

This article is part of a series on Complex Algebraic Geometry. See also:
1. The Hardest Conjecture; 2. The Valley; 3. A Mathematical Daydream

“I see a massive valley before me. The only question is… Should I cross it?”

This is how, about three years ago, and living in Moscow, I posed to a friend the dilemma of whether I should attend graduate school for math.

The analogy held up well. I often envisioned my progress through the program in terms of the same imposing valley. “The brambles are getting so thick, at this depth, that the light which once flowed liberally from the surface is becoming damp and attenuated,” I once thought, as I entered the later stages of the notorious first-year program in algebra and analysis. “I’m so far from charted land that I’m encountering species wholly unknown to the outside world,” I thought later, during my second year – as I first began to encounter Hodge Theory – envisioning passing into a dark forest where marvelous, strange creatures chirped and glowed.

Once in while I’d perceive myself on an unexpected trail towards a promising discovery. “It’s as if I’ve glimpsed light on the other side much earlier than I had anticipated,” I’d think.

The analogy seems to have lost some of its power now. I’ve made my way to the other side, sure. But much of the valley remains unexplored – by me or by anyone else. Continue reading

The Hardest Conjecture

This article is part of a series on Complex Algebraic Geometry. See also:
1. The Hardest Conjecture; 2. The Valley; 3. A Mathematical Daydream

Among the so-called Millenium Prize Problems – seven notoriously difficult mathematical problems, each open for decades, and each now carrying, courtesy of the Clay Mathematics Institute, a million-dollar prize – Kieth J. Devlin places the Hodge Conjecture last. “[A]n author should delay as long as possible introducing anything is likely to make his reader give up in despair,” Devlin writes in his book, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of our Time. “There is no… path even to the problem’s front door.”

The Hodge Conjecture fascinated me even before I began studying mathematics. Its sheer inscrutability surely played a role. I sensed, behind the incomprehensible words and symbols of its Wikipedia page, a bafflingly deep, and coherent, world. I had to understand.

The problem’s formidability was only partly to blame. This world – which I envisioned – was not just expansive, but beautiful. I perceived, there, something like Dante’s “music of the heavenly spheres”.

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French artist Gustave Doré’s depictions of Dante’s Paradise.

My graduate study has taken me into a field of math relatively close to that which the Hodge Conjecture occupies. As I’ve explored the rich foothills of this towering mountain, its peak has become even more stunning and mysterious.

I will try to explain the conjecture to this blog’s lay readers. Continue reading

Game of Theorems

This article is part of a series on The Structure of Theorems. See also:
1. Theorems’ Almanack; 2. The Greatest Theorem; 3. Game of Theorems

Few (currently) practicing mathematicians – in my experience – deign to concern themselves with issues surrounding set theory and the foundations of mathematics. In these areas reside the very definitions upon which the rest of our discipline rests; in any case, our discipline proceeds nonetheless, despite its practitioners’ regrettable ignorance. Mathematicians are pragmatic people. In 1949, Bourbaki – perhaps apprehending a subtle need to defend itself – titled a paper Foundations of Mathematics for the Working Mathematician [1].

This state of affairs, unfortunate as it is, explains the surprise and intrigue I often feel when I take time to explore foundational issues. The definition of the so-called Axiom of Determinacy particularly struck me. This set-theoretic axiom is formulated in terms of a certain type of two-player game – an infinite sequential game, in fact, in which two players take turns playing integers, leading ultimately to a sequence of integers of infinite length. The axiom (indeed, it’s something we might choose to suppose) states that every such game – that is, every choice of a victory set, a distinguished collection of possible infinite sequences whose members define the winning outcomes for the first player – is determined, in the sense that one player or the other in the game has a dominant strategy.

I’ll explain these terms below. The important thing, here, is that this mathematical property is defined by the existence of dominant strategies for a certain class of two-player games. This intrusion of an apparently economic, or game-theoretic, notion – that of the two-player game – into mathematics surprised me.

This intrusion, in retrospect, should have been less than surprising. Continue reading

Conway’s Game of Coinage

It was a long time before I talked to Professor Emeritus John Boardman. I’d seen him walking in the halls before, sure. Wisps of white hair ran across the top of his high forehead, and he walked with a stoop. Sometimes his eyes appeared pried too far open, as if he were struggling to look ahead despite the downward incline of his head. He had a habit of incessantly clearing his throat, even as he walked, and even as he sat in his office; his office was adjacent to the math help room and I often looked up from an undergrad’s work only to notice the familiar sound again. I’d never seen anyone else talk to him, either.

This changed one evening as we all sat down for dinner in a sparsely patronzied restaurant after an invited speaker’s seminar talk. There were about six or seven of us at the table, and the others quickly became distracted in conversation. Professor Boardman, alone at the end of the table, sat to my left.

The first thing I noticed was his sharp, almost Russell-esque British accent. Continue reading

Lesson Time

This article is part of a series entitled Everyday Game Theory. See also:
1. The Escalator’s Dilemma; 2. Electoral College; 3. Passing Curiosity; 4. Lesson Time

This is (a slightly modified version of) a text message exchange which recently occurred between my violin teacher and me.

  1. Teacher: “Can we meet today instead of tomorrow?”
  2. Me: “That’d be great!”
  3. Teacher: “Cool, see you this afternoon.”
  4. Me: “Ok.”

It would not have been acceptable for me to fail to respond to my teacher’s message (1). If I didn’t respond, my teacher would have no way to know whether I ever received her message – and, hence, whether to come today or tomorrow.

Neither would it have been ok, for that matter, for my teacher to let the conversation end at message (2). Until I receive her confirmation (3), I can in no way be sure whether she has seen or acknowledged my message (2). In other words, with her message (3) unsaid, it could remain the case, for all I know, that my teacher, as yet unaware of my response (2), imagines me unaware of (1) and still intent to come tomorrow.

Even after I received my teacher’s message (3), though, it was important for me to send the further message (4). After all, until she receives my message (4), my teacher may well imagine me unaware of her message (3). In that situation — her thought process might go — I would, unaware of her confirmation (3), be liable to suspect her unaware of my response (2), and hence unsure of my receipt of (1), and so liable to come tomorrow.

Why doesn’t this continue? Continue reading