# 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

# 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

# A Fragile Truth

My four-week psychiatry rotation at Western State Hospital landed smack in the middle of peak general election season. And, oddly enough, these two experiences have yielded remarkable similarities. In both cases, I have been forced to entertain various versions of the truth.

Many of the patients here at Western State are psychotic. Our known and stated goal, then, is to return these patients to reality-based thinking. Only then might they qualify for discharge. This exercise has presented philosophical challenges. Certainly, sometimes, our job is easy. One of our patients, who signs her forms as Michelle Obama Prince Harry Elizabeth Queen Zealand, communicates with Russia, Germany, Berlin, Jerusalem, East Germany, West Germany, South Germany, and Russia, by radio, television, and satellite, including the satellite in the backyard of her palace, which she built, and in which we currently reside. Another patient, though, gave me pause. Continue reading

# I Have a Unique Coffee Mug

I’ll attempt to explore a few of the subtleties surrounding the word unique in English.

I’ve drunk too many of these, and this post is how you can tell.

I’ll begin with an exploration of words like the same and different. Sentences containing these words will prove a fertile initial testing ground.

First, an observation. Consider the sentences:

1. Every coffee mug in my department is brittle.
2. Every coffee mug in my department is the same.

Though these sentences appear structurally similar, their predicates are deceptively different. Continue reading

# Lesson Time

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

# Relativistic Linguistics

A study in linguistic relativity.

English – as, it would seem, all human languages – assumes a Newtonian conception of space and time. These languages, and in particular their tense systems, postulate, in addition to three spatial dimensions, a single temporal dimension. These dimensions equip space-time with universally valid coordinates. The time dimension in particular assigns – or so we imagine – to every event a unique time value, valid for all observers. By comparing these time values, we introduce such notions as simultaneity and precedence; these emerge in language through grammatical tense, as well as through certain prepositions.

In the Newtonian model, all relations of simultaneity and precedence (which might hold between any two space-time events) hold independently of observer. The notion of duration is also well defined and consistent across observers. Finally, geometric notions such as length and angle are independent of observer. The Newtonian framework closely approximates physical reality when all observers travel at speeds well below the speed of light.

Questions such as the following could empirically test whether English presupposes Newtonian physics:

1. Ben and Josh are twins, born on the same day in the same hospital. Ben knows that the supernova of the red supergiant KSN2011d became visible to Earth before his twenty-fourth birthday. Can Josh necessarily say the same? (No.)
2. Was Ben really born first? (Yes – even under relativity – because our births were very close spatially. Sorry Josh.)
3. Have you ever traveled near the speed of light – in this life, or throughout your evolutionary history? (No. Hence the Newtonian trappings of natural language.)

A Newtonian language ill befits communication between observers moving at speeds close to the speed of light. Continue reading

# Buried Treasure

A week ago, I took the USMLE step 1, an 8-hour Goliath of a test. As daunting of a prospect as that is, the study process was much more extensive. I studied for twelve hours a day for six weeks.

I had intended to spend those six weeks memorizing a whole lot of facts. But I eventually, I found that I wasn’t just learning facts; I was learning structures. This took a lot of the drudgery away, since the latter are quite a bit more fun to study.

Now that boards are over, I find myself stepping away from medicine and looking towards other fields. Do other fields, like medicine, produce elaborate structures from the underlying facts and principles? Must they? Are some resultant structures better than others? In medicine, there often is a right answer (especially on boards). Is the same true of other fields? Where, if at all, does the rubber meet the road? Continue reading