28. The combination of events that are at spacelike distance from here.
29. Among the first to realize this was Kurt Gödel; see “An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation,” Reviews of Modern Physics 21 (1949): 447–50. In his own words: “The notion of ‘now’ is nothing more than a certain relation between a certain observer and the rest of the universe.”
30. Transitive.
31. Even the existence of a relation of partial order might be too strong with regard to reality, if closed temporal curves exist. On this subject see, for example, Marc Lachièze-Rey, Voyager dans le temp: La Physique moderne et la temporalité, (Paris: Editions du Seuil, 2013).
32. The fact that there is nothing logically impossible about travels to the past is demonstrated clearly in an engaging article by one of the great philosophers of the last century: David Lewis, “The Paradoxes of Time Travel,” American Philosophical Quarterly 13 (1976): 145–52, reprinted in The Philosophy of Time, eds. R. Le Poidevin and M. MacBeath (Oxford: Oxford University Press, 1993).
33. This is the representation of the causal structure of a black hole metric in Eddington-Finkelstein coordinates.
34. Among the dissenting voices, there are those of two great scientists for whom I have a particular friendship, affection, and admiration: Lee Smolin (Time Reborn [Boston: Houghton Mifflin Harcourt, 2013]) and George Ellis (“On the Flow of Time,” Fqxi Essay, 2008, https://arxiv.org/abs/0812.0240; “The Evolving Block Universe and the Meshing Together of Times,” Annals of the New York Academy of Sciences 1326 [2014]: 26–41; How Can Physics Underlie the Mind? [Berlin: Springer, 2016]). Both insist that there must exist a privileged time and a real present, even if these are not captured by current physics. Science is like affection: those who are dearest to us are those with whom we have the liveliest disagreements. An articulate defense of the fundamental aspect of the reality of time can be found in Roberto M. Unger and Lee Smolin, The Singular Universe and the Reality of Time (Cambridge, UK: Cambridge University Press, 2015). The point of view of Smolin and Ellis is defensible. But is it fruitful? The choice is between forcing the description of the world so that it adapts to our intuition, or learning instead to adapt our intuition to what we have discovered about the world. I have few doubts that the second strategy is the most fruitful one.
4. LOSS OF INDEPENDENCE
35. On the effects of drugs on time perception, see R. A. Sewell et al., “Acute Effects of THC on Time Perception in Frequent and Infrequent Cannabis Users,” Psychopharmacology 226 (2013): 401–13; the direct experience is astonishing.
36. Valtteri Arstila, “Time Slows Down during Accidents,” Frontiers in Psychology 3 (2012): 196.
37. In our cultures. There are others with a profoundly different notion of time: D. L. Everett, Don’t Sleep, There are Snakes, (New York: Pantheon, 2008).
38. Matthew 20:1–16.
39. Peter Galison, Einstein’s Clocks, Poincaré’s Maps (New York: Norton, 2003), p. 126.
40. An excellent panoramic history of the way in which technology has progressively modified our concept of time can be found in Adam Frank, About Time: Cosmology and Culture at the Twilight of the Big Bang (New York: Free Press, 2001).
41. D. A. Golombek, I. L. Bussi, and P. V. Agostino, “Minutes, Days and Years: Molecular Interactions among Different Scales of Biological Timing,” Philosophical Transactions of the Royal Society. Series B: Biological Sciences 369 (2014).
42. Time is: “number of change, with regard to before and after” (Aristotle, Physics IV.219b2; see also 232b22‒3).
43. Aristotle, Physics, trans. Robin Waterfield, with an introduction and notes by David Bostock (Oxford: Oxford University Press, 1999), p. 105.
44. Isaac Newton, Philosophiae Naturalis Principia Mathematica, Book I, def. VIII, scholium.
45. Ibid.
46. An introduction to the philosophy of space and of time can be found in B. C. van Fraassen, An Introduction to the Philosophy of Time and Space (New York: Random House, 1970).
47. Newton’s fundamental equation is F = m d2x/dt2. Note that time t is squared: this reflects the fact that the equation does not distinguish t from ‒t, that is to say, it is the same backward or forward in time, as I explain in chapter 2.
48. Curiously, many contemporary manuals of the history of science present the discussion between Leibniz and the Newtonians as if Leibniz were the heterodox figure with audacious and innovative relationist ideas. In reality, the opposite was the case: Leibniz defended (with a new wealth of arguments) the dominant traditional understanding of space, which from Aristotle to Descartes had always been relationist.
49. Aristotle’s definition is more precise: the place of a thing is the inner boundary of that which surrounds the thing, an elegant and rigorous definition.
5. QUANTA OF TIME
50. I speak of this in more depth in Reality Is Not What It Seems, trans. Simon Carnell and Erica Segre (New York: Riverhead Books, 2017).
51. It is not possible to locate a degree of liberty in a region of its phase space within a volume smaller than the Planck constant.
52. The speed of light, the Newton constant and the Planck constant.
53. Maimonides, The Guide for the Perplexed I.73.106a.
54. We can try to infer the thought of Democritus from the discussions of Aristotle (for example, in Physics IV.213), but the evidence seems insufficient to me. See Democrito. Raccolta dei frammenti, interpretazione e commentario di Salomon Luria (Milan: Bompiani, 2007).
55. Unless the de Broglie-Bohm theory is true, in which case it has it—but hides it from us. Which is perhaps not so different in the end.
56. Carlo Rovelli, “Relational Quantum Mechanics,” International Journal of Theoretical Physics 35 (1996): 1637, http://arxiv.org/abs/quant-ph/9609002. See also: “The Sky Is Blue and Birds Fly Through It,” http://arxiv.org/abs/1712.02894.
57. Grateful Dead, “Walk in the Sunshine.”
6. THE WORLD IS MADE OF EVENTS, NOT THINGS
58. Nelson Goodman, The Structure of Appearance (Cambridge, MA: Harvard University Press, 1951).
7. THE INADEQUACY OF GRAMMAR
59. For opposing views, see note 34.
60. In the terminology of a celebrated article by John McTaggart (“The Unreality of Time,” Mind, N.S. 17 (1908): 457–74; reprinted in The Philosophy of Time, Le Poidevin and MacBeath), this is equivalent to denying the reality of the A-series (the organization of time into “past-present-future”). The meaning of temporal determinations would then be reduced to only the B-series (the organization of time into “before-it, after-it”). For McTaggart, this implies denying the reality of time. To my mind, McTaggart is too inflexible: the fact that my car works differently from how I’d imagined it and how I’d originally defined it in my head does not mean that my car is not real.