Combinatorial Grammar

This article is part of a series entitled Language Games. See also:
1. Wild Grammar; 2. Combinatorial Grammar; 3. Pragmatic Grammar

“All doors will not open.” — operator, Amtrak Northeast Regional train, Charlottesville

In language, scope ambiguities are one of the trickiest parts of semantic theory. Semantic meaning is famously said to be determined compositionally: the meaning of a larger sentence is determined by the meanings of its smaller parts, as well as by the way these smaller parts are assembled into a whole. Even so, there can be interactions between these parts, in the sense that certain words can exert control over other words. When one word influences how another is interpreted, it is said to hold that word in its scope.

For example:

Every king admires himself. [2]

In this situation, the reflexive pronoun himself is given meaning by the separate noun king, which holds himself in its scope.

Let’s consider another example:

Puck didn’t solve one problem. [2]

What does this mean? It actually depends on how scope is assigned. It’s ambiguous. Continue reading


What Mathematical Theorems Do

Gauss’s theorem of quadratic reciprocity “like none other has left its mark on the development of algebraic number theory,” [1], writes Jürgen Neukirch, in his celebrated Algebraic Number Theory. Davenport calls it “one of the most famous theorems in the whole of the theory of numbers.” [2] Gauss himself calls it the fundamental theorem in his Disquitiones Arithmeticae, and privately he referred to it as The Golden Theorem. [3] Gauss discovered the law at the age of 19.

Gauss offered seven proofs of the theorem during his lifetime. Each relied on very different techniques. [2] Even after the emergence of various proofs, Davenport writes, “[t]he desire to find what lies behind the law has been an important factor in the work of many mathematicians, and has led to far-reaching discoveries.” [2] Quadratic reciprocity has undergone several successive sweeping generalizations — see the reciprocity laws of Eisenstein, Kummer, and Hilbert — culminating in Artin’s reciprocity law and even the titanic modern Langlands program.

Why didn’t the work stop after Gauss’s first proof? Gauss, Artin and Langlands weren’t after proof. They were after understanding. Continue reading

The World-Builders

This article is part of a series entitled The Unlimited Mind. See also:
1. On Memory; 2. The Genius Within; 3. The World-Builders

I’ve been fascinated with expertise since childhood. And it started over the chessboard. My dad would beat me—swiftly, crushingly, and above all, effortlessly—time and time again. He understood lines and positions in a way that I just couldn’t, and, as it seemed to me, would never be able to. My first question at the end of most games was: “where did I go wrong?”


Chess has served as a popular topic of study for those seeking to understand expertise.

Almost more unnerving than my dad’s ability was the fact that there were people out there who could, just as easily, beat him. “In college in Russia, I played a classmate of mine, who was a master,” he told me once. “I would think all night about my move, and then the next day in class, he’d move right away. Still, he beat me easily.”

Thus my interest in expertise was born. It seemed that some just had some sort of divine gift, which beckoned them onto a higher plane of understanding. For me to attempt to reach those heights would be futile. I could only watch in awe from below.

As I grew older, my skills improved. My games with my dad grew stricter and cleaner, until, one day, I beat him. In time, whether I won or lost, I was always able to give him a fair fight. I came to appreciate chess as an incredibly rich and rewarding game.

But my view of expertise—now that I had a taste of it—had lost a bit of its sparkle. Continue reading

Heavenly Host

This article is part of a series on Italian Renaissance Literature. See also:

  1. Bocaccio: Youthful Idyllic Escape
  2. Petrarch: The Troubled Wanderer
  3. Dante: Heavenly Host

Dante’s 1320 Divine Comedy is perhaps most known for its Inferno — the first among three parts — in which Dante’s self-styled Pilgrim is led, by Virgil, through Hell’s concentric rings. Each ring contains souls given a particular penance suited to a particular sin. These crime-punishment pairs range from the comical to the poignant. In the fourth circle of Hell, two screaming groups — the Prodigal (“why hoard?”) and the Miserly (“why waste?”) — roll massive weights towards each other, until the balls smash into one another and the groups turn around and start once again. [1] In the barren forest of suicides — those who commit suicide are deprived even of their body in Hell — the Pilgrim, puzzled, breaks a twig off a nearby tree. To his horror, blood begins to ooze from the broken branch. A voice emerges: “Why are you tearing me?” [1]

Here, though, I’ll study not Dante’s Inferno or Purgatory but rather his Paradise. In Paradise, Dante is led by Beatrice — the heavenly instantiation of his earthly love, whom he was to meet only twice in his life, for the first time at the age of nine — through Heaven’s concentric spheres. Dante follows Beatrice through a world of unimaginable religious splendor — towards a place where “where joy becomes one with eternity.” (Paradise X.148) [1]

A depiction of angels by Italian Renaissance painter Melozzo da Forlì. [6]

Depictions of angels by the Italian Renaissance painter Melozzo da Forlì. [6]

Deeply woven alongside this depiction of heavenly beauty, however, is Dante’s depiction of Beatrice’s female beauty, which, in fact, “becomes more radiant with every step / of the eternal palace that we climb” (Paradise XXI.7-9) [1]. The religious and romantic are inseparable. This adroit act by Dante gives Paradise much of its power. It also informs deep philosophical questions about the nature of religious love. Continue reading

Passing Curiosity

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

We were on I-80 Eastbound somewhere around Nebraska. It had just gotten dark. The large semi-trucks on the road – usually the highway’s peaceful, lumbering mammoths – seemed to be turning aggressive.

I moved to the left lane, preparing to pass a slower semi truck looming ahead on my right. Suddenly the red and yellow lights of a large truck appeared in my rearview. The truck was approaching behind me, at high speed, and pressuring me into the right lane.

Not one for confrontation – and, let’s face it, this was a Ford Taurus – I acquiesced, laying on the brakes and ducking behind the first truck, almost as if for cover. The newcomer blew past both of us.

“Man,” I said to Josh, shaking my head. It was clear to us that the speeding truck’s behavior had been unfriendly. But it wasn’t quite clear why.

A more peaceful moment on the highway.

A more peaceful moment on the highway.

Driving across the country – Josh and I just made the trip from Portland to Baltimore – one has time to think about many things. The above event, along with countless others, got us pondering how highway passing works. Passing behavior on the interstate, in fact, follows predictable and consistent patterns, and these patterns are somewhat complex. We concocted a game-theoretic account of passing behavior. Continue reading