Where Wisdom Is Buried

Every formula is borrowed, taken from people who composed them in their own laboratories and signed them with their own names; the question, however, for whose sake the whole architecture of pointers, graphs, and archive was set in motion in the first place, is mine, and it sounds plain and engineerly: where, exactly, does wisdom live.

A literal question, without rhetoric — the way a programmer would ask about a variable: at what address. And if wisdom has no address, then perhaps it doesn’t exist either, and there’s only the habit of calling, after the fact, wise whatever happened, after the fact, to work.

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At what address.

The list of places in which wisdom is usually sought is short to the point of indecency.

In books. The first thing that comes to a philologist’s mind. You go to the library, take the right volume off the shelf, everything is written in it. Any book without a reader, however, is a stack of bindings, and that stack grows wise only when somebody walks up and picks it up; while the shelf stands by itself, there is as much wisdom in the shelf as in the brick wall behind it.

In a teacher. You go to a teacher, you sit across from him, you listen — and after two hours you come out with the sense that wisdom, apparently, slipped past somewhere nearby. The strange thing is only that for every student who came out of the same room, wisdom slipped past in a different place, and these places cannot be compared with each other. The same teacher transmitted ten different wisdoms to ten students — which says, among other things, that wisdom, apparently, isn’t in him.

In experience. “Age brings wisdom,” as they say. In practice, however, two people with one and the same experience extract from it opposite things: one decides he must never do that again, the other — that next time he just needs to be more aggressive. One experience, two wisdoms — which means wisdom isn’t in experience either, but in some separate mechanism that pulls something out of experience.

In the brain. The most honest of the answers: inside the skull, in three pounds of gray matter. The trouble is that the brain is substrate, and cutting it open in hopes of hitting wisdom is a pursuit whose name is already indecent. No matter how many brains neuroanatomists have opened, no separate compartment for wisdom in cortex, hippocampus, or cerebellum has ever turned up.

Four addresses — and at each, empty. Something is off with the question itself.

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Infinite, with nothing to measure with.

This strangeness, as often happens, isn’t mine but borrowed — one that algebraic quantum field theory ran into first, somewhere in the middle of the last century.

It runs as follows. Take any finite region of spacetime — a cube the size of an apple, say — and write down carefully all the quantum operators that, in principle, live in it. You get a mathematical object called an algebra of observables. For such an algebra, by construction, you’d expect a measure: “how many operators are in it,” “how much information,” “how much entropy.” This measure, in standard theory, is called the trace, and it works roughly like a graduated measuring cup: feed in an operator, get out a number.

For the algebra of observables of a quantum region, an unpleasant thing turns out: it has no trace. Not in the sense of “not found yet”; in the sense of principally does not exist, and cannot exist, because of the structure of the algebra. Technically it’s called a Type III von Neumann algebra, and its main property is this: Type III has no trace, — which means entropy is undefined, no normalized state exists, the notion of “how much” is not, on this algebra, defined at all. Infinitely many operators in the algebra, nothing to measure them with; no scale, period.

To feel this on your fingers. Infinity by itself does not yet mean the absence of a measure. The natural numbers also have infinitely many elements — but they can be counted (one, two, three, …), and such an infinity at least has some “how many.” Type III is a special infinity, which doesn’t even admit that kind of count.

Two old puzzles illustrate it. The first — the continuum on an interval: take the segment from zero to one and notice that, by points, there are as many real numbers in it as on the entire number line; to ask in which of the two segments there are more points is simply forbidden, because their cardinality is the same — the entire “number world” fits between zero and one. The second — Hilbert’s hotel: picture a hotel with an infinite number of rooms, fully occupied, — one guest per room. A new arrival shows up. The clerk announces that the resident of room one moves to room two, of room two to room three, and so on by one: room one frees up, the hotel is “full” again. By the same maneuver an entire bus of guests is accommodated, an infinite fleet of buses, — any countable number of new residents the hotel takes on, remaining “full.” No “how many are occupied” exists in this hotel, — only rearrangements of pointers from room to room. The same happens with the Type III algebra: there are infinitely many operators, and “more” or “less” among them is not defined.

The consequence is unpleasant: the world in quantum field theory, by itself, is unmeasurable. The word unmeasurable here means that the world has no scale at all until something else is attached to it that brings that scale with it.

And for decades it remained unclear what exactly had to be attached: they tried the apparatus, then boundary conditions, then coordinate environment, — every time something was missing, so that the scale stubbornly failed to appear, and the algebra remained without a trace.