Dividing a number by zero doesn’t produce an infinitely large number as an answer. The reason is that division is defined as the inverse of multiplication; if you divide by zero, and then multiply by zero, you should regain the number you started with. However, multiplying infinity by zero produces only zero, not any other number. There is nothing which can be multiplied by zero to produce a nonzero result; therefore, the result of a division by zero is literally “undefined.”

“Any hypothetical path would require more time to traverse than the one actually taken. In other words, the route that the light ray takes is always the fastest possible one. That’s Fermat’s principle of least time.”

“You’re used to thinking of refraction in terms of cause and effect: reaching the water’s surface is the cause, and the change in direction is the effect. But Fermat’s principle sounds weird because it describes light’s behavior in goal-oriented terms. It sounds like a commandment to a light beam: ‘Thou shalt minimize or maximize the time taken to reach thy destination.’ ” I considered it. “Go on.” “It’s an old question in the philosophy of physics. People have been talking about it since Fermat first formulated it in the 1600s; Planck wrote volumes about it. The thing is, while the common formulation of physical laws is causal, a variational principle like Fermat’s is purposive, almost teleological.”

The idea of thinking in a linguistic yet non-phonological mode always intrigued me. I had a friend born of Deaf parents; he grew up using American Sign Language, and he told me that he often thought in ASL instead of English. I used to wonder what it was like to have one’s thoughts be manually coded, to reason using an inner pair of hands instead of an inner voice. With Heptapod B, I was experiencing something just as foreign: my thoughts were becoming graphically coded. There were trancelike moments during the day when my thoughts weren’t expressed with my internal voice; instead, I saw semagrams with my mind’s eye, sprouting like frost on a windowpane.

Consider the phenomenon of light hitting water at one angle, and traveling through it at a different angle. Explain it by saying that a difference in the index of refraction caused the light to change direction, and one saw the world as humans saw it. Explain it by saying that light minimized the time needed to travel to its destination, and one saw the world as the heptapods saw it. Two very different interpretations. The physical universe was a language with a perfectly ambiguous grammar. Every physical event was an utterance that could be parsed in two entirely different ways, one causal and the other teleological, both valid, neither one disqualifiable no matter how much context was available.