Time becomes a number

The last of the magicians

On 13 and 14 July 1936, in a London auction room, Sotheby’s sold 329 lots of old manuscripts: what remained of Isaac Newton’s papers. Newton died in 1727 without a will, and the papers passed through his niece’s husband, John Conduitt, to the Wallop family, the Earls of Portsmouth. In 1872 the fifth Earl gave the scientific papers to Cambridge University; the rest, on theology, prophecy, chronology, and alchemy, went back to the family and were sold in 1936 for just over 9,000 pounds. The economist John Maynard Keynes, alerted late to the sale, spent years buying what he could, and he wrote that he managed to reassemble about half. They were not quite the secret hoard of legend: Newton’s Chronology of Ancient Kingdoms Amended (1728) and his commentary on Daniel and Revelation (1733) were printed soon after his death. What the sale revealed was how much more there was.

Historians estimate that Newton wrote perhaps a million words on alchemy and several million more on religion; William Newman puts his religious papers at about four million. Keynes’s response was an essay, “Newton, the Man,” written for the celebrations of the tercentenary of Newton’s birth, held in July 1946. Keynes died that April, and his brother Geoffrey read the essay for him. Newton, Keynes wrote, was “not the first of the age of reason” but “the last of the magicians,” a mind that saw the world with the eyes of the Babylonians. (Course 2 told how Keynes bought the papers.) The question here is narrower: what do they say about time?

This course argues, boldly, that Newton was at heart a thinker of cycles: that he believed the world, like every cycle, must end and begin again, and so spent his later years recovering the wisdom of the ancients, since looking back and looking forward come to the same thing. Part of that is documented. Newton was sure that the ancients had known deep truths, and that a pure original wisdom had been corrupted and could be recovered; his chronology tried to shorten the dates of early Greek history (Course 20 mentions it in a line).

But the cyclical part is inference. We found no passage in which Newton teaches that history repeats, and his prophecies run toward a Kingdom of God: a manuscript of about 1704 works out, counting 1,260 years from the year 800, that a prophesied span could end no earlier than the year 2060, and adds that he mentions it only to stop rash guessing. What he did hold was a pattern of corruption and restoration, and a belief that the world’s machinery runs down. In the last Query of his Opticks he wrote that motion “is much more apt to be lost than got, and is always upon the Decay,” and that irregularities among the planets would grow “till this System wants a Reformation.”

This course draws a paradox from the clock. Every clock counts cycles: the turning of the earth, the swing of a pendulum, the vibration of a quartz crystal or a cesium atom. Yet the society that lives by clocks experiences time as a line that does not return and does not wait: the schedule, the deadline, “time is money” (Franklin’s phrase of 1748, from Lesson 2). By then the line also had a philosophy. In 1750 the young Turgot, later a finance minister of Louis XVI, set out a “philosophical review” of the successive advances of the human mind, often called the first complete statement of the idea of progress, and Condorcet’s Sketch for a Historical Picture of the Progress of the Human Mind, written in hiding during the Revolution, appeared in 1795, after his death. (The five stages of history later taught in the Soviet Union are a late version of the same line, though not Marx’s own: his 1859 preface names four modes of production, and the five-step scheme was codified in Stalin’s 1938 pamphlet.)

The day of two noons

On the front of the Exchange in Bristol, a clock first installed in 1822 has two minute hands: a red one for Bristol time and a pink one for Greenwich time. Bristol lies 2°35′ west of Greenwich, so when the sun is highest there it is already ten minutes past noon in London. Until the railways, every English town kept its own sun time, and the difference rarely mattered, because nothing moved fast enough to show it. In November 1840 the Great Western Railway began running on London time. In 1847 the companies’ common office, the Railway Clearing House, told its members to adopt Greenwich time, and most big lines switched on 1 December. Parliament made it the law of Great Britain on 2 August 1880.

The United States had the same problem on a larger stage. Before 1883 every city kept its own sun time, more than 300 of them, and the railroads ran on about 50 different railway times; a junction served by several lines might have a clock for each. William F. Allen, secretary of the railroads’ General Time Convention, proposed in 1881 to replace the 50 with a few zones, and in October 1883 the railroad managers met at the Grand Pacific Hotel in Chicago and agreed. The change came at noon on Sunday, 18 November 1883, later nicknamed “the day of two noons”: as standard-time noon reached each zone, station clocks were reset. In a town east of its zone’s meridian the clocks were set back, and noon came twice. Four zones were adopted: Eastern, Central, Mountain, and Pacific. Federal law followed only thirty-five years later, with the Standard Time Act of 19 March 1918, which also began daylight saving time (repealed in 1919).

Meanwhile the world needed a zero. In October 1884 delegates from more than twenty nations met in Washington at the request of President Chester A. Arthur and voted 22 to 1, with two abstentions (San Domingo against; France and Brazil abstaining), to make the meridian through the transit instrument at Greenwich the zero of longitude. The conference chose a meridian and a universal day; the zones themselves spread by railroad timetable and then by national law.

China took the opposite road. The country spans about 61 degrees of longitude, from roughly 73°E to 135°E, about four hours of sun time, and its clocks run on one zone: Beijing time, UTC+8 (北京时间, Běijīng shíjiān). The Republic’s Central Observatory proposed five zones in 1918 and the government adopted them in 1939, though they were not fully observed; after 1949–50 most regions switched to Beijing time. In Xinjiang many people also use an unofficial “Xinjiang time,” two hours behind, which keeps the clock nearer the sun but can confuse an arrangement between people using different times. In Kashgar, 44 degrees of longitude west of the 120°E reference meridian, the sun is highest at about 14:56 on a Beijing clock.

This course adds an interpretation. A time set by a distant office and the same across a whole zone is a time stripped of its circumstances. In this reading, the clock keeps the number and loses the 时机 (shíjī), the sense of the right moment: if you are late, the question used to be what happened on the road, and now it is only by how many minutes. Whether or not you accept that, two things are plain. The sun was demoted: noon stopped meaning when the sun is highest. And a late train made time everybody’s business, because a delay in one place was a delay along the whole line.

Time as a coordinate

The Greeks measured time by the sky and tied it to change. In Plato’s Timaeus the maker of the world made time as “a moving image of eternity,” born together with the heavens, and the sun, moon, and planets were made “to distinguish and preserve the numbers of time” (Jowett’s translation). Aristotle’s definition, in the Physics (IV.11), is more austere: time is “number of motion in respect of ‘before’ and ‘after’,” and “neither does time exist without change” (Hardie and Gaye’s translation). For Aristotle, time measures motion; it is not a stage on which motion happens.

Galileo changed the question. He wanted motion as a relation between two measured quantities, distance and time, and for that he needed a clock. In Two New Sciences (1638) he describes rolling a bronze ball down a polished groove in a plank about twelve cubits long (roughly 6 meters, or 20 feet) and timing each run with a water clock, a vessel whose thin jet was caught in a glass and weighed on a balance. The weights gave the times. Time had become something you could weigh, and the numbers showed that the distance fallen grows as the square of the time. In this course’s reading, that is the turning point: Aristotle’s “motion” meant every kind of change, from a seed sprouting to a fire dying, and Galileo kept only change of place, so that motion became a function of place and time. Once time is a variable t, it is natural to draw it as an axis, a ruler laid beside the road, with every moment a point.

The next step was theological as much as mathematical. Isaac Barrow, the first Lucasian Professor at Cambridge (from 1663) and Newton’s teacher, argued that time does not depend on motion at all: “whether we sleep or wake,” he wrote, time “flows perpetually with an equal Tenor” (1735 English translation of his Geometrical Lectures). It is sometimes said that Barrow called time God; we did not find that in the passages we could check, where time is only the possibility of continuing to exist. The divine step is Newton’s. In 1687 his Scholium said that absolute time “flows equably” without relation to anything external (Course 19 tells this, and Newton’s own doubt that any real clock keeps it), and in the General Scholium, added in 1713, he wrote that God, “by existing always and every where,” constitutes “Duration and Space” (Motte’s 1729 translation).

Here, this course argues, is the reversal that made modern life run on schedules. For Aristotle, time followed motion. For Newton, motion happened in time, and time was there first, flowing whether or not anyone watched. A thing that is there first and flows by itself is something you can fall behind. Before, time served life; now life serves time: the calendar, the shift, the deadline. Newton himself drew no such moral, but it is the world his absolute time helped build.

A block with no now

Course 19 told how Einstein in 1905 asked what “at the same time” means for two distant events, so that simultaneity depends on the observer. What came next is the subject here. On 21 September 1908, at a meeting of German scientists in Cologne, Einstein’s former teacher Hermann Minkowski gave the theory its geometry and a famous opening: “space by itself, and time by itself, are doomed to fade away into mere shadows,” and only a kind of union of the two would keep an independent reality (in the English translation of 1923 by Perrett and Jeffery). In that union, called spacetime, every event has a place and a date, and a thing’s whole life is a line through it, its worldline. Draw the whole picture and nothing in it flows. It is a block: past, present, and future laid out together, the “block universe.” In this reading, Einstein’s physics in effect flattens time into a fourth dimension of an iron block.

Einstein himself said something like this, in sorrow. His lifelong friend Michele Besso died on 15 March 1955. On 21 March Einstein wrote to Besso’s family that Michele had left this strange world a little ahead of him, and that it meant nothing: for believing physicists, the distinction between past, present, and future is only “a stubbornly persistent illusion” (the usual English wording of the German letter). Einstein died less than four weeks later, on 18 April.

Yet the Now troubled him. The philosopher Rudolf Carnap recalled that Einstein once said the problem of the Now worried him seriously: the experience of the Now means something special to us, essentially different from past and future, but this difference does not and cannot occur within physics. And in 1949 the logician Kurt Gödel published a solution of Einstein’s own equations, a rotating universe, in which a traveler could in principle follow a path that leads back to his own past. Einstein replied, in a volume of essays in his honor that year, that such solutions raised the question whether they should be excluded on physical grounds. (A story that he smiled and said nothing is told; we could not confirm it.) No one has found evidence that our universe rotates that way.

The sun was demoted a second time in the definition of the second. For centuries the second was 1/86,400 of a day (24 × 60 × 60). In 1956 it was redefined as a fraction of a year, and the international conference on weights and measures adopted that in 1960. Meanwhile Louis Essen and Jack Parry had built the first practical cesium atomic clock in England in 1955. Essen, with William Markowitz of the U.S. Naval Observatory, then measured the atom’s frequency in the astronomers’ seconds, and in 1967 the conference defined the second as 9,192,631,770 periods of the radiation from a transition in the cesium-133 atom. Now the earth is what gets checked against the clock: 27 leap seconds were added from 1972 to 2016, and in 2022 the conference decided to phase them out by 2035. In the voice of this course: in physics, time is a parameter, a label on a point, and nothing in the equations marks which instant is the present. What we feel as the flow of time, in this reading, is exactly what the clock and the block leave out.

A cup that only cools

Put a hot cup of coffee on a desk. It cools until it matches the room, and it never heats itself from the room’s air, though energy would balance either way. Now look at the laws. For forces that depend only on position, Newton’s second law has time in it only through a second derivative, so reversing the direction of time (replacing t by −t) turns one solution into another solution, and Einstein’s equations behave the same way. A film of two billiard balls colliding, run backward, shows a motion that obeys the same laws. Yet nobody has seen a cup reheat.

Course 5 tells the second law and entropy; the time story is this. Sadi Carnot’s 1824 essay on the motive power of fire started the subject, and in 1865 Rudolf Clausius coined “entropy,” choosing a word as similar as possible to “energy.” In 1852 William Thomson, later Lord Kelvin, published “On a Universal Tendency in Nature to the Dissipation of Mechanical Energy,” and by 1862 he wrote of “universal rest and death” if the universe were finite. Helmholtz and Rankine helped popularize the idea as the “heat death” (热寂, rèjì). It is said to have alarmed the late nineteenth century; we found plenty of discussion but no good evidence of alarm.

Newton would not have been surprised that the world runs down. In that same Query he thought the world’s motion was always decreasing, so that there is “a necessity of conserving and recruiting it by active Principles,” gravity among them. His critics noticed. In a letter of November 1715 that began the famous exchange with Newton’s spokesman Samuel Clarke, Leibniz mocked a God who must wind up his watch from time to time. Then came Laplace, whose Celestial Mechanics (published 1799 to 1825) explained the planets’ wobbles by gravity alone. The story, told by W. W. Rouse Ball, is that when Napoleon remarked that the book never mentioned God, Laplace answered that he had had no need of that hypothesis. The anecdote is second-hand, the date is unknown, and the astronomer Hervé Faye wrote in 1884 that Laplace meant only that Newton was wrong to call on God to adjust the machine now and then (Course 4 tells Laplace’s demon).

The deeper puzzle came from the atoms. If a gas is only particles obeying Newton’s laws, then for every motion in which a gas spreads there is a reversed motion in which it gathers. In 1876 Josef Loschmidt put exactly this to Ludwig Boltzmann, whose statistical account of entropy is told in Course 5. Boltzmann first answered that the reversed motions exist but are extremely improbable.

A second objection followed. Henri Poincaré had shown in 1890 that a system like this must, given enough time, come back close to its starting state, and in 1896 Ernst Zermelo turned the theorem against Boltzmann. The recurrence time for anything macroscopic is staggeringly long, far longer than the age of the universe, but the argument stung. It is often said that Boltzmann killed himself because he could not solve the puzzle. He did die by suicide, on 5 September 1906 at Duino near Trieste, during a holiday with his wife and daughter. But his mental and physical health had been failing, and in May he had resigned his chair because of a worsening mental condition. The Stanford Encyclopedia of Philosophy concludes that the suicide seems to have been due to his personal life, not to academic matters.

Arthur Eddington gave the idea its famous name, the arrow of time, in 1927, and in his book of 1928 he wrote that the law of increasing entropy “holds, I think, the supreme position among the laws of Nature.” The modern answer to Loschmidt keeps his point: the laws are symmetric, so the arrow must come from the starting condition. If the early universe began in a state of very low entropy, entropy has risen ever since, in every cup and every star. Philosophers call this the Past Hypothesis, defended by David Albert (2000) and in different forms by Richard Feynman and Roger Penrose; critics such as John Earman doubt that “the entropy of the universe” is even well defined in general relativity. Course 7 tells how Lemaître and Hubble found the galaxies running away from each other.

Life seems to run the other way. Course 15 explains how open systems can build order without breaking the second law. Ilya Prigogine, who won the 1977 Nobel Prize in Chemistry “for his contributions to non-equilibrium thermodynamics, particularly the theory of dissipative structures,” went further and argued that irreversibility is built into nature, not just into its starting conditions; that stronger claim is a minority view. This course counts four kinds of time: none in Newton, Einstein, and quantum theory; decay, in thermodynamics; expansion, in cosmology; and the growth of order, in evolution. It is a tally, not a theorem, and the three arrows do not obviously point the same way: entropy says order decays, evolution says order grows, and expansion says space grows. The cosmic and thermodynamic arrows are plausibly linked through the Past Hypothesis. Whether the others are one arrow or several is open.