What the picture leaves out

A map as big as the country

In 1893 the English writer Lewis Carroll, author of Alice in Wonderland, published a late novel, Sylvie and Bruno Concluded, in which a stranger describes a country whose mapmakers grew ambitious. First they made a map with a scale of six inches to the mile, then a yard to the mile, and finally a map on the scale of a mile to the mile. The farmers objected that it would cover the whole country and shut out the sunlight, so, the stranger says, the people now use the country itself as its own map, and it does nearly as well.

The joke hides a real puzzle, and a philosopher stated it seriously a few years later. Josiah Royce of Harvard, in The World and the Individual (1899), asked what would happen if someone drew a perfect map of England on a patch of English ground. The map would be part of England, so a perfect map would have to show the map; and that map would be part of England too, so it would need to show a map of the map, and so on without end. A complete picture of a world that contains the picture can never be completed. (Royce meant this seriously: for him a whole that is exactly imaged inside one of its own parts was a model of the universe.) The Argentine writer Jorge Luis Borges told a later version of Carroll’s joke, in a very short story of 1946 that is still in copyright.

Lesson 1 asked whether the total of all things is a thing. This lesson asks a second question about the same totality: can a picture of everything include the one who makes the picture? If I draw a map of the room, I am in the room. But the map does not show me drawing it, unless I add that, and then the map does not show me drawing that. Whoever makes the picture stands one step outside it.

The universe as a condition, not a thing

This course turns the puzzle into a claim about what “universe” means. It says the universe is not a big thing but a field (场), the word as in a person’s commanding presence, 气场: the conditions in which things can show up at all. A stage is not one of the actors, though no actor appears without one. A conversation has a language, a room, a set of rules, none of which says a word. In this reading, the universe is the “where” of everything that is anywhere. If it were itself one more thing, we could ask where it is, and the question would have no end. So it cannot be one of the things it holds, and for the same reason it does not show up among them: it is what makes the showing possible.

This is a philosophical thesis, and it has a respectable ancestor. Kant argued that experience needs a framework that the mind brings to it. Space and time are, in his terms, the two forms of intuition, the way things must be laid out to be perceived at all; and concepts such as cause and effect are categories, the rules by which we connect what we perceive. None of these is a thing we see. They are what seeing needs, and you cannot find them in the picture any more than you can find your own eye in your field of vision. The young Ludwig Wittgenstein said something close in his Tractatus (1921, English 1922): the subject does not belong to the world but is a limit of it, like the eye, which is not part of what it sees.

The next two sections test the thesis on two examples: time and the human being. In each case, an item that seemed to be part of the picture turns out to be part of what makes the picture possible, and trying to put it in produces paradox.

Two kinds of time

In 1687 Isaac Newton began his Principia by defining the terms he needed. Among them: absolute, true, and mathematical time “flows equably without relation to anything external” (in the 1729 English translation). Then he added an admission: “It may be, that there is no such thing as an equable motion, whereby time may be accurately measured.”

He was right to worry. Early clocks used the swing of a pendulum, and Christiaan Huygens, who built the first pendulum clock in 1656, soon saw that the period of a swing is steady only for small swings: at a 30-degree swing it runs about 1.7 percent slower than for a tiny one. The Earth is a clock too, and an unsteady one: tides drag on its spin and slow it, so the day gets longer by about two milliseconds every century, and on top of that the spin speeds up and slows down irregularly from year to year. Between 1972 and 2016, timekeepers added a “leap second” 27 times to keep our atomic clocks in step with the Earth, and in 2022 the world’s time authorities agreed to stop adding them by 2035. Every real clock is judged against some standard of “uniform,” and no real clock is perfectly uniform. So to say any clock is a little fast or slow, you need an idea of perfectly steady time that no clock supplies.

That is the course’s point about Newton, and about Kant, who thought the same: time of this kind is not one more thing we measure but a condition for measuring anything. Kant added that without a continuous, uniform time connecting yesterday to today, there could be no experience at all, since experience is made of connected moments. This sort of time is a condition, not an object.

Einstein’s time

In 1905 Albert Einstein did something that looks, at first, like the opposite. In the first section of his paper on moving bodies, he asked what “at the same time” means for two events in different places, and answered that a definition is only worth having if you can carry it out. His recipe is the one in the activity below. Send a flash of light from A to a mirror at B, and back. If the flash is reflected at B at the instant halfway between leaving and returning, call that instant “simultaneous” with the midpoint of A’s trip. From this operational definition, with a second postulate, that light travels at the same speed for every observer, comes the special theory of relativity: moving clocks run slow, simultaneity depends on the observer, and time becomes something that physics measures and that mass and speed change.

And the clocks agree. In 1971 Joseph Hafele and Richard Keating flew four atomic clocks around the world on commercial airliners and compared them with clocks left at the U.S. Naval Observatory: the traveling clocks lost 59 billionths of a second flying east and gained 273 billionths flying west, within the measurement error of what the theory predicts (a loss of 40 and a gain of 275). The GPS satellites behind your phone’s map carry clocks that tick faster than clocks on the ground by about 38 millionths of a second a day, the net of two relativistic effects, one from the weaker gravity at their height and one from their speed. Uncorrected, that would add up to a position error of about 10 kilometers a day, so engineers set the clocks slightly slow before launch. In 2010 physicists at the U.S. National Institute of Standards and Technology saw an optical clock run measurably faster after it was raised just 33 centimeters (about a foot) above another.

The thesis about time

This course draws a philosophical conclusion from this contrast. Newton’s and Kant’s time is a condition for any measurement, and cannot itself be measured. Einstein’s time is an object of measurement: it varies from place to place and with motion, and clocks show it. It says the two must not be confused. If you read Einstein’s time as if it were Kant’s, you get paradox: if time can be bent by a passing star, how could an experiment done yesterday teach us anything about today? Kant’s time is what guarantees that yesterday’s experiment counts. In this reading, relativity does not replace the earlier time; it moves a different thing, “the speed of light is the same for everyone,” into the role of the unmeasured assumption. We cannot check, in a way that does not already assume it, that light takes the same time to go from A to B as from B to A.

That last point is true, and it is a subtle result that physicists and philosophers have studied for a century. Einstein himself saw it. In the 1905 paper he says the equal-travel-time requirement is established “by definition,” and in his popular book on relativity (1916) he calls it a stipulation that he can make of his own free will, and not a hypothesis about the physical nature of light. Hans Reichenbach, in 1928, wrote it as a free parameter, which is what the slider in the activity above is.

The man in the gallery

The second example is us. The claim is that a human being is two things at once. A person is an item in the world: a body, with a birthday, a family and a bank balance, obeying physics. And a person is the one who makes pictures of the world, who can step back, judge, choose, and change what he does next. We can put the first in the picture. The question is whether the second goes in.

The course asks this with the oldest puzzle of time travel. Suppose you go back to the house where you were born and stand over your own cradle. Can you, in the picture of the world given by physics, shoot the baby? If yes, you destroy your own past and cannot be there; if no, why not, since you are free? The puzzle has a name, the grandfather paradox, and science-fiction readers were already arguing about it in magazine letters in the late 1920s, and René Barjavel’s 1943 French novel Le Voyageur imprudent is often credited as the first novel to use it.

Is time travel physically possible? Not in special relativity, which has no paths that loop back in time. In general relativity, which describes gravity with curved space and time, it is not ruled out at once. In 1949, the year of Einstein’s seventieth birthday, the logician Kurt Gödel, a friend of Einstein’s at the Institute for Advanced Study in Princeton, published a solution of Einstein’s equations, a whole universe that rotates, and in which a traveler can follow a path that loops back into his own past. (Such a path needs rocket power and in practice enormous fuel.) Einstein replied in the volume of essays in his honor, calling Gödel’s work an important contribution to the theory, and asking whether such solutions should be excluded on physical grounds. Gödel’s universe is not ours: it does not expand, and measurements of the cosmic microwave background put any cosmic rotation at levels vastly smaller than his model needs. Other solutions with time loops have since been proposed, such as a spinning cylinder (Tipler, 1974) and wormholes (Morris, Thorne and Yurtsever, 1988). In 1992 Stephen Hawking conjectured that the laws of physics always prevent them, though that is a conjecture with partial results, not a proof.

If loops were possible, what would stop you shooting your grandfather? Physicists give several answers. The most common is that the history must be self-consistent: whatever you do was already part of what happened, and the gun jams. Another is that the shot would start a different branch of history. This course gives a third answer, which is its own: it is the second law of thermodynamics, with its arrow of time, that keeps us out of such paradoxes. The second law does give time its direction, and Einstein’s reply to Gödel says as much: sending a signal is an irreversible process tied to the growth of entropy, and closed time loops would blur “earlier” and “later.” The astronomer Arthur Eddington said the law holds “the supreme position among the laws of Nature.” But whether the second law forbids time loops is not a standard result. The law is statistical, the underlying equations do not point in a direction, and physicists mostly rely on consistency. At most the thermodynamic argument tells against a person with memory, a body and a clock surviving a loop; it is one thinker’s argument, not a consensus.

A gallery with a hole

The favorite picture of the human paradox, in this reading, is Escher’s Print Gallery, made in 1956 by the Dutch artist M. C. Escher. A young man stands in a gallery, looking at a picture of a harbor town. Follow the street along the harbor and you reach a building with an open door, and inside it is the gallery where the young man stands. So the man is looking at a picture that contains him. Escher left a round blank patch at the center of the print, and signed his name there. In 2003 two Dutch mathematicians, Hendrik Lenstra and Bart de Smit, worked out the geometry behind the picture, in which the scene repeats at one 256th of the size on each turn around the center, and used it to fill in the hole.

In this reading, the picture is the portrait of modern science. Science has been drawing the world picture with ever finer detail: galaxies, atoms, genes, neurons. The picture now takes in our bodies, our brains, even our behavior. And yet something has to be outside it, as the gallery needs a viewer: the one who makes the picture, and the standards by which it counts as a good one. The blank patch, in this reading, is where the picture cannot go on.

The course links this to Gödel’s incompleteness theorems, proved in 1931, and the link needs care. The first theorem says that any consistent system of axioms that is rich enough to include arithmetic and that can be listed by rule must contain true statements about numbers that it cannot prove. The common wording, “any complete system is inconsistent,” is the same theorem turned around, and it is correct only if you keep its conditions: the system must be formal, meaning that a rule decides what counts as an axiom, and rich enough to include arithmetic. Some simpler systems, such as arithmetic with addition alone, are complete and consistent. So it is a theorem about formal systems of arithmetic. Using it to say that any picture of the world must have a blind spot is an analogy, and a suggestive one, not a deduction. Philosophers of mathematics have warned about this sort of leap; the Swedish logician Torkel Franzén wrote a whole book on the ways Gödel’s theorem is misused.