# 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

# Art Songs

$\begin{array}{l l} \text{Parlo d'amor vegliando,} & \text{Whatsoe'er I am doing,} \\ \text{Parlo d'amor sognando,} & \text{Whatsoe'er I'm pursuing,} \\ \text{All' acqua, all' ombra, ai monti,} & \text{In sunshine or in showers,} \\ \text{Ai fiori, all' erbe, ai fonti, ...} & \text{At home or midst the flowers, ...} \\ \text{All' ecco---all' aria---a venti,} & \text{I sigh---I pant---I languish,} \\ \text{Che il suon de vani accenti, ...} & \text{In bliss that throbs like anguish, ...} \\ \end{array}$

Sung by the pubescent, flirtatious Cherubino (he’s played by a soprano), these words are notable for their arresting meter and rhyme, the heavenly beauty of their melody, and, well, their humor. Mozart’s The Marriage of Figaro—recently performed fantastically by the Peabody Opera Theater—presents such an explosive combination of theatricality, musicality, and hilarity as to make Mozart come across as a supernatural genius.

Only later did I learn that Mozart did not work alone [1]. The Felix Krull-esque French man-of-the-world Pierre Beaumarchais originally wrote the French play Le Mariage de Figaro in 1784; only then did Lorenzo Da Ponte, an equally fascinating Italian librettist, translate the play into Italian, excise a tirade against inherited nobility (thus making the play acceptable to the censors), and set certain of its passages to meter and rhyme. These developments, finally, prepared the way for Mozart to set music to the entire work, which premiered in 1786. The libretto’s rhymed passages became the opera’s arias.

This realization, in fact, placed Mozart into a long tradition within classical music. Continue reading

# Swiss French, Swiss German

Ferdinand de Saussure was a profound linguistic thinker of the early 1900s. During a legendary series of lectures given at University of Geneva, de Saussure, a French-speaking Swiss, introduced to the world many ideas which have since become fundamental — even “self-evident” — within the discipline of linguistics. De Saussure suggested, for example, that the historical and etymological emphases of his day failed to recognize as the central object of linguistics the instantaneous internal structure of a language, to which prior evolutionary contingencies are irrelevant. A language’s internal structure, in fact, exists moreover independently of the writing system it uses, of the concrete sounds of its phonetic system, and even of its words. It consists entirely, de Saussure argued, of an abstract system of so-called signs — each linking an idea to a sound — which subsist only through the network of relationships among them and persist only through the coordinated ativity of a linguistic community. “It is because the linguistic sign is arbitrary that it knows no other law than that of tradition,” de Saussure famously wrote, “and because it is founded upon tradition that it can be arbitrary.” [1, p. 74]

De Saussure’s facility with historical linguistics was, to his credit, uncanny. Discussing the unfortunate common tendency to confuse historical (diachronic) with instantaneous (synchronic) linguistics, for example, he writes:

In order to explain Greek phuktós, it might be supposed that it suffices to point out that in Greek g and kh become k before a voiceless consonant, and to state this fact in terms of synchronic correspondences such as phugeînphuktóslékhosléktron, etc. But then we come up against cases like tríkhesthriksí, where a complication occurs in the form of a ‘change’ from t to th… [1, p. 96]

De Saussure proceeds like this effortlessly, citing detailed examples variously from Sanskrit [1, p. 2], Latin [1, p. 95], Old High German [1, p. 83], Anglo-Saxon [1, p. 83], early Slavonic [1, p. 86], and, of course, French [1, pp. 31, 69, 85, 95, 104, 106, …].

De Saussure’s true genius, however, was evident perhaps most of all in his novel theory of signs, which emphasized the social, conventional, and ultimately arbitrary nature of linguistic systems — and which eventually produced the field of semiotics.

This theory was validated during my recent trip to Switzerland. Continue reading

# The Philosopher

“I know dusk / And dawn, rising like a multitude of doves. / What men have only thought they’d seen, I’ve seen,” [1, p. 89] writes the late-1800s French poet Arthur Rimbaud, in a quote elevated by his latest translators, Jeremy Harding and John Sturrock, to their edition’s back cover.

It is telling that Harding and Sturrock choose this quote. These scholars characterize Rimbaud’s style as one infused by “a disordering of all the senses” (Rimbaud’s words) “often with the aid of alcohol and drugs” (their words). Rimbaud was only 15 when he abandoned his native Charleville for Paris, where he was arrested for not paying his train fare. By 17, he had been taken in more stably by Paul Verlaine, a young leader of Paris’ so-called Parnassian school of poets, whose wife and in-laws Rimbaud then taunted by demanding the removal of a picture from one of their walls and by “vandalizing an ivory Christ” [1, p. xxv], and, later, through the developing homosexuality of his relationship with Verlaine. Under the protection of another poet, Théodore de Banville, Rimbaud allegedly “slept in his boots, smashed the china and sold the furniture” [1, p. xxvi], and also stripped in front of an open window and threw his clothes onto the roof. Rimbaud later joined the avant-garde Zutistes’ circle, who “convened for regular drinking sessions in a hotel overlooking the Boulevard St Michel” [1, pp. xxvi-xxvii]. By 21, famously, Rimbaud had abandoned poetry forever, travelling around Europe and eventually settling as a colonial trader in Africa. He died at 37.

Rimbaud’s “disordering”, Harding and Sturrock write, was most of all one of the self itself, which in his poetry “is wilfully distended and distressed, offering the maximum surface area to which unusual information… can adhere” [1, p. xxiv].

This “meme” used to be visible in the library’s basement floor outside the darkened, locked door of a graduate student carrell.

Just how Rimbaud’s poetry achieves its striking character is perhaps too subtle to write down. Surprising insight, though, might be gained through the eliminative materialist ideas of the modern philosopher Paul Churchland. Continue reading

# A Mathematical Daydream

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

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