The Order of Time

NOTES

PERHAPS TIME IS THE GREATEST MYSTERY

1. Aristotle, Metaphysics I.2.982.

2. The layering of the notion of time is discussed in depth, for example, in J. T. Fraser, Of Time, Passion, and Knowledge (New York: Braziller, 1975).

3. The philosopher Mauro Dorato has insisted on the necessity to render the elementary conceptual framework of physics coherent with our experience; see Mauro Dorato, Che cos’è il tempo? (Rome: Carocci, 2013).

1. LOSS OF UNITY

4. This is the essence of the theory of general relativity. Albert Einstein, “Die Grundlage der algemeinen Relativitätstheorie,” Annalen der Physik 49 (1916): 769–822.

5. In the approximation of a weak field, the metrics can be written ds2 = (1 + 2ϕ(x)) dt² – dx², where ϕ(x) is the potential of Newton. Newtonian gravity follows from the sole modification of the temporal component of the metrics goo, that is, from the local slowing down of time. The geodesics of these metrics describe the fall of bodies: they bend toward the lowest potentiality, where time slows. (These and similar notes are for those who have some familiarity with theoretical physics.)

6. Carlo Rovelli, Che cos’è la scienza: La rivoluzione di Anassimandro (Milan: Mondadori, 2011); English translation, The First Scientist: Anaximander and His Legacy (Chicago: Westholme, 2011).

7. For example: (ttable − tground) = gh/c2tground where c is the speed of light, g = 9.8m/s2 is the acceleration of Galileo, and h is the height of the table.

8. They can also be written with a single variable, t, the “temporal coordinate,” but this does not indicate the time measured by a clock (determined by ds, not by dt) and may be changed arbitrarily without changing the world described. This t does not represent a physical quantity. What clocks measure is the proper time along a line of the universe γ, given by tγ = ∫γ√ gab (x)dxadxb. The physical relation between this quantity and gab(x) is discussed further on.

2. LOSS OF DIRECTION

9. Rainer Maria Rilke, Duineser Elegien, in Sämtliche Werke, vol. 1, I, vv. 83–85 (Frankfurt: Insel, 1955).

10. The French Revolution was an extraordinary moment of scientific vitality in which the bases of chemistry, biology, analytic mechanics, and much else were founded. The social revolution went hand in hand with the scientific one. The first revolutionary mayor of Paris was an astronomer; Lazare Carnot was a mathematician; Marat considered himself to be, above all else, a physicist. Antoine Lavoisier was active in politics. The mathematician Joseph-Louis Lagrange was honored by the different governments that succeeded each other in that tormented and magnificent moment in the history of humanity. See Steve Jones, Revolutionary Science: Transformation and Turmoil in the Age of the Guillotine (New York: Pegasus, 2017).

11. Changing what is opportune: for instance, the sign of the magnetic field in the equations of Maxwell, charge and parity of elementary particles, etc. It is the invariance under CPT (Charge, Parity, and Time reversal symmetry) that is relevant.

12. The equations of Newton determine how things accelerate, and the acceleration does not change if I project a film backward. The acceleration of a stone thrown upward is the same as that of a falling stone. If I imagine years running backward, the moon turns around the Earth in the opposite direction but appears equally attracted to the Earth.

13. The conclusion does not change by adding quantum gravity. On the efforts to fund the origin of the direction of time, see, for example, H. D. Zeh, Die Physik der Zeitrichtung (Berlin: Springer, 1984).

14. Rudolf Clausius, “Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie,” Annalen der Physik 125 (1865): 353–400; 390.

15. In particular as a quantity of heat that escapes from a body divided by temperature. When the heat escapes from a hot body and enters a cold one, the total entropy increases because the difference in temperature makes it so that the entropy due to heat that escapes is less than that owed to the heat that enters. When all the bodies reach the same temperature, the entropy has reached its maximum: equilibrium has been reached.

16. Arnold Sommerfeld.

17. Wilhelm Ostwald.

18. The definition of entropy requires a coarse graining, that is to say, the distinction between microstates and macrostates. The entropy of a macrostate is determined by the number of corresponding microstates. In classic thermodynamics, the coarse graining is defined the moment it is decided to treat some variables of the system as “manipulable” or “measurable” from outside (the volume or pressure of a gas, for instance). A macrostate is determined by fixing these macroscopic variables.

19. That is to say, in a deterministic manner if you overlook quantum mechanics, and in a probabilistic manner if you take account of quantum mechanics instead. In both cases, in the same way for the future as for the past.

20. S = k log W. S is the entropy, W is the number of microscopic states, or the corresponding volume of phase space, and k is just a constant, today called Boltzmann’s constant, that adjusts the (arbitrary) dimensions.

3. THE END OF THE PRESENT

21. General relativity (A. Einstein, “Die Grundlage der algemeinen Relativitätstheorie,” op. cit.).

22. Special relativity (Albert Einstein, “Zur Elektrodynamik bewegter Körper,” Annalen der Physik 17 [1905]: 891–921).

23. J. C. Hafele and Richard E. Keating, “Around-the-World Atomic Clocks: Observed Relativistic Time Gains,” Science 177 (1972): 168–70.

24. That depends as much on t as on your speed and position.

25. Poincaré. Lorentz had tried to give a physical interpretation to t, but in a quite convoluted way.

26. Einstein frequently maintained that the experiments of Michelson and Morley were of no importance in allowing him to arrive at special relativity. I believe this to be true, and that it illustrates an important factor in the philosophy of science. In order to make advances in our understanding of the world, it is not always necessary to have new data. Copernicus had no more observational data than Ptolemy: he was able to deduce heliocentrism from the data available to Ptolemy by interpreting it better—as Einstein did with regard to Maxwell.

27. If I see my sister through a telescope celebrating her twentieth birthday and send her a radio message that will arrive on her twenty-eighth birthday, I can say that now is her twenty-fourth birthday: halfway between when the light departed from there (20) and when it returned (28). It’s a nice idea (not mine: it’s Einstein’s definition of “simultaneity”). But this does not define a common time. If Proxima b is moving away, and my sister uses the same logic to calculate the moment simultaneous to her twenty-fourth birthday, she does not obtain the present moment here. In other words, in this way of defining simultaneity, if for me a moment A in her life is simultaneous with a moment B in mine, the contrary is not the case: for her, A and B are not simultaneous. Our different speeds define different surfaces of simultaneity. Not even in this way do we obtain a notion of a common “present.”

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.

61. Letter by Einstein to the son and sister of Michele Besso, March 1955, in Albert Einstein and Michele Besso, Correspondence, 1903–1955 (Paris: Hermann, 1972).

62. The classic argument for the block universe is given by the philosopher Hilary Putnam in a famous article published in 1967 (“Time and Physical Geometry,” Journal of Philosophy 64 (1967): 240–47). Putnam uses Einstein’s definition of simultaneity. As we have seen in note 27, if the Earth and Proxima b move with respect to one another, say they are approaching each other, an event A on Earth is simultaneous (for an earthling) to an event B on Proxima b, which in turn is simultaneous (for those on Proxima b) to an event C on Earth, that is in the future of A. Putnam assumes that “being simultaneous” implies “being real now,” and deduces that the event in the future (such as C) is real now. The error is to assume that Einstein’s definition of simultaneity has an ontological value, whereas it is only a definition of convenience. It serves to identify a relativistic notion that may be reduced to the nonrelativistic one through an approximation. But nonrelativistic simultaneity is a notion that is reflexive and transitive, whereas Einstein’s is not, hence it makes no sense to assume that the two have the same ontological meaning beyond the approximation.

63. That the discovery by physics of the impossibility of presentism implies that time is illusory is an argument put forward by Gödel in “A Remark about the Relationship between Relativity Theory and Idealistic Philosophy,” in Albert Einstein: Philosopher-Scientist, ed. P. A. Schlipp (Evanston, IL: Library of Living Philosophers, 1949). The error always lies in defining time as a single conceptual block that is either all there or not there at all. The point is discussed lucidly by Dorato, Che cos’è il tempo?, p. 77.

64. See, for instance, W. V. O. Quine, “On What There Is,” Review of Metaphysics 2 (1948): 21–38, and the fine discussion of the meaning of reality in J. L. Austin, Sense and Sensibilia (Oxford: Clarendon Press, 1962).

65. De Hebdomadibus of Boethius II.24, cited in C. H. Kahn, Anaximander and the Origins of Greek Cosmology (New York: Columbia University Press, 1960), pp. 84–85.

66. Some examples of important arguments where Einstein has strongly supported a thesis that he later changed his mind about: 1. The expansion of the universe (first ridiculed, then accepted); 2. The existence of gravitational waves (first taken as obvious, then rejected, then accepted again); 3. The equations of relativity do not admit solutions without matter (a long-defended thesis that was abandoned—rightly so); 4. Nothing exists beyond the horizon of Schwarzschild (wrong, though perhaps he never came to realize this); 5. The equations of the gravitational field cannot be general-covariant (asserted in the work with Grossmann in 1912; three years later, Einstein argued the opposite); 6. The importance of the cosmological constant (first affirmed, then denied—having been right the first time.) . . .

8. DYNAMICS AS RELATION

67. The general form of a mechanical theory that describes the development of a system in time is given by a phase space and a Hamiltonian H. Evolution is described by the orbits generated by H, parametrized by the time t. The general form of a mechanical theory that describes the evolutions of variable with respect to each other is instead given by a phase space and a constraint C. The relations between the variables are given by the orbits generated by C in the subspace C=0. The parametrization of these orbits has no physical meaning. A detailed technical discussion can be found in chapter 3 of Carlo Rovelli, Quantum Gravity (Cambridge, UK: Cambridge University Press, 2004). For a concise technical account, see Carlo Rovelli, “Forget Time,” Foundations of Physics 41 (2011): 1475–90, https://arxiv.org/abs/0903.3832.

68. An accessible account of loop quantum gravity can be found in Rovelli, Reality Is Not What It Seems (op. cit.).

69. Bryce S. DeWitt, “Quantum Theory of Gravity. I. The Canonical Theory,” Physical Review 160 (1967): 1112–48.

70. J. A. Wheeler, “Hermann Weyl and the Unity of Knowledge,” American Scientist 74 (1986): 366–75.

71. J. Butterfield and C. J. Isham, “On the Emergence of Time in Quantum Gravity,” in The Arguments of Time, ed. J. Butterfield (Oxford: Oxford University Press, 1999), pp. 111–68 (http://philsci-archive.pitt.edu/1914/1/EmergTimeQG=9901024.pdf); Zeh, Die Physik der Zeitrichtung; Craig Callender and Nick Huggett, eds., Physics Meets Philosophy at the Planck Scale (Cambridge, UK: Cambridge University Press, 2001); Sean Carroll, From Eternity to Here: The Quest for the Ultimate Theory of Time (New York: Dutton, 2010).

72. The general form of a quantum theory that describes the evolution of a system in time is given by a Hilbert space and the Hamiltonian operator H. The evolution is described by Schrödinger’s equation iħ∂tΨ = HΨ. The probability of measuring a pure state Ψ a time t after having measured a state Ψ′ is given by the transition amplitude 〈Ψ | exp[–iHt/ħ] | Ψ′〉. The general form of a quantum theory that describes the evolution of the variables with respect to one another is given by a Hilbert space and Wheeler-DeWitt equation CΨ = 0. The probability of measuring the state Ψ after having measured the state Ψ′ is determined by the amplitude 〈Ψ | ∫ dt exp[iCt/ħ]|Ψ′〉. A detailed technical discussion can be found in chapter 5 of Rovelli, Quantum Gravity (op. cit.). For a concise technical version, see Rovelli, “Forget Time” (op. cit.)

73. Bryce S. DeWitt, Sopra un raggio di luce (Rome: Di Renzo, 2005).

74. There are three: they define the Hilbert space of the theory where the elementary operators are defined, whose eigenstates describe the quanta of space and the probability of transitions between these.

75. Spin is the quantity that enumerates the representations of the group SO(3), the group of spatial symmetry. The mathematics that describes the spin networks has this feature in common with the mathematics of ordinary physical space.

76. These arguments are covered in detail in Rovelli, Reality Is Not What It Seems. (op. cit.).

9. TIME IS IGNORANCE

77. Ecclesiastes 3:2–4.

78. More precisely, the Hamiltonian H, that is, the energy as a function of position and speed.

79. dA/dt = {A, H}, where { , } are the Poisson brackets and A is any variable.

80. Ergodic.

81. The equations are more readable in the canonical formations of Boltzmann than in the microcanonical form to which I make reference in the text: the state ρ = exp[‒H/kT] is determined by the Hamiltonian H that generates evolution of time.

82. H = ‒klog[ρ] determines a Hamiltonian (up to a multiplicative constant), and thus a “thermal” time, starting from the state ρ.

83. Roger Penrose, The Emperor’s New Mind (Oxford: Oxford University Press, 1989); also, Penrose, The Road to Reality (London: Cape, 2004).

84. In the language of quantum mechanics manuals, it is conventionally referred to as “measure.” Once again, there is something misleading about this language, inasmuch as it speaks about physics laboratories rather than speaking about the world.

85. The theorem of Tomita-Takesaki shows that a state on a von Neumann algebra defines a flow (a one-parameter family of modular automorphisms). Connes has shown that the flows defined by different states are equivalent up to internal automorphisms, and therefore define an abstract flow determined only by the noncommutative structure of algebra.

86. The internal automorphisms of the algebra referred to in the above note.

87. In a von Neumann algebra, the thermal time of a state is exactly the same as Tomita’s flow! The state is KMS with respect to this flow.

88. See Carlo Rovelli, “Statistical Mechanics of Gravity and the Thermodynamical Origin of Time,” Classical and Quantum Gravity 10 (1993): 1549–66; Alain Connes and Carlo Rovelli, “Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories,” Classical and Quantum Gravity 11 (1994): 2899–918.

89. Alain Connes, Danye Chéreau, and Jacques Dixmier, Le Théâtre quantique (Paris: Odile Jacob, 2013).

10. PERSPECTIVE

90. There are many confused aspects to this question. An excellent, cogent critique can be found in John Earman, “The ‘Past Hypothesis’: Not Even False,” Studies in History and Philosophy of Modern Physics 37 (2006): 399–430. In the text, “low initial entropy” is intended in the more general sense that, as Earman argues in this article, is far from being well understood.

91. Friedrich Nietzsche, The Gay Science (Cambridge, UK: Cambridge University Press, 2001), v–354.

92. The technical details can be found in Carlo Rovelli, “Is Time’s Arrow Perspectival?” in The Philosophy of Cosmology, eds. K. Chamcham et al. (Cambridge, UK: Cambridge University Press, 2017), https://arxiv.org/abs/1505.01125.

93. In the classical formulation of thermodynamics, we describe a system by specifying in the first place some variables on which we assume we can act from the outside (moving a piston, for example), or which we assume we can measure (a relative concentration of components, for xample). These are the “thermodynamic variables.” Thermodynamics is not a true description of the system; it is a description of these variables of the system—those through which we assume we are able to interact with the system.

94. For example, the entropy of the air in this room has a value taken from air as a homogeneous gas, but it changes (diminishes) if I measure its chemical composition.

95. A contemporary philosopher who has shed light on these aspects of the perspectival nature of the world is Jenann T. Ismael, The Situated Self (New York: Oxford University Press, 2007). Ismael has also written an excellent book on free will: How Physics Makes Us Free (New York: Oxford University Press, 2016).

96. David Z. Albert, in Time and Chance (Cambridge, MA: Harvard University Press, 2000), proposes to elevate this fact to a natural law, and calls it “past hypothesis.”

11. WHAT EMERGES FROM A PARTICULARITY

97. This is another common source of confusion, because a condensed cloud seems more “ordered” than a dispersed one. It isn’t, because the speed of the molecules of a dispersed cloud are all small (in an ordered manner), while, when the cloud condenses, the speeds of the molecules increase and spread in phase space. The molecules concentrate in physical space but disperse in phase space, which is the relevant one.

98. See, in particular, Stuart A. Kauffman, Humanity in a Creative Universe (New York: Oxford University Press, 2016).

99. The importance of the existence of this ramified structure of interactions in the universe for the understanding of the growth of local entropy is discussed, for instance, by Hans Reichenbach, in The Direction of Time (Berkeley: University of California Press, 1956). Reichenbach’s text is fundamental for whoever has doubts about these arguments or is interested in pursuing them in more depth.

100. On the precise relation between traces and entropy, see Reichenbach, The Direction of Time, in particular the discussion on the relation among entropy, traces, and common cause, and Albert, Time and Chance. A recent approach can be found in D. H. Wolpert, “Memory Systems, Computation and the Second Law of Thermodynamics,” International Journal of Theoretical Physics 31 (1992): 743–85.

101. On the difficult question of what “cause” means to us, see Nancy Cartwright, Hunting Causes and Using Them: Approaches in Philosophy and Economics (Cambridge, MA: Cambridge University Press, 2007).

102. “Common cause,” in Reichenbach’s terminology.

103. Bertrand Russell, “On the Notion of Cause,” Proceedings of the Aristotelian Society, N. S. 13 (1912–1913): 1–26.

104. Cartwright, Hunting Causes and Using Them.

105. For a lucid discussion on the question of the direction of time, see Huw Price, Time’s Arrow and Archimedes’ Point (Oxford: Oxford University Press, 1996).

12. THE SCENT OF THE MADELEINE

106. Milinda Pañha (The Questions of King Milinda) II.1, in T. W. Rhys Davids, Sacred Books of the East, vol. XXXV (Oxford: Clarendon Press, 1890).

107. Carlo Rovelli, Meaning = Information + Evolution, 2016, https://arxiv.org/abs/1611.02420.

108. G. Tononi, O. Sporns, and G. M. Edelman, “A Measure for Brain Complexity: Relating Functional Segregation and Integration in the Nervous System,” Proceedings of the National Academy of Sciences USA 91 (1994): 5033–37.

109. Jakob Hohwy, The Predictive Mind (Oxford: Oxford University Press, 2013).

110. See, for example, V. Mante, D. Sussillo, K. V. Shenoy, and W. T. Newsome, “Context-dependent Computation by Recurrent Dynamics in Prefrontal Cortex,” Nature 503 (2013): 78–84, and the literature cited in this article.

111. Dean Buonomano, Your Brain Is a Time Machine: The Neuroscience and Physics of Time (New York: Norton, 2017).

112. La Condemnation parisienne de 1277, ed. D. Piché (Paris: Vrin, 1999).

113. Edmund Husserl, Vorlesungen zur Phänomenologie des inneren Zeitbewusstseins (Halle: Niemeyer, 1928).

114. In the cited text, Husserl insists that this does not constitute a “physical phenomenon.” To a naturalist, this sounds like a statement of principle: he does not want to see memory as a physical phenomenon because he has decided to use phenomenological experience as the starting point of his analysis. The study of the dynamics of neurons in our brain shows how the phenomenon manifests itself in physical terms: the present of the physical state of my brain “retains” its past state, and this is gradually more faded the farther away we are from that past. See, for example, M. Jazayeri and M. N. Shadlen, “A Neural Mechanism for Sensing and Reproducing a Time Interval,” Current Biology 25 (2015): 2599–609.

115. Martin Heidegger, “Einführung in die Metaphysik” (1935), in Gesamtausgabe, vol. XL, p. 90 (Frankfurt: Klostermann, 1983).

116. Martin Heidegger, Sein und Zeit (1927), in Gesamtausgabe, vol. II, passim (Frankfurt: Klostermann, 1977); trans. Being and Time.

117. Marcel Proust, Du côté du chez Swann, in À la Recherche du temps perdu, vol. I (Paris: Gallimard, 1987), pp. 3–9.

118. Ibid., p. 182.

119. G. B. Vicario, Il tempo. Saggio di psicologia sperimentale (Bologna: Il Mulino, 2005).

120. The observation, a quite common one, can be found, for example, in the introduction to J. M. E. McTaggart, The Nature of Existence, vol. I (Cambridge, UK: Cambridge University Press, 1921).

121. Lichtung, perhaps; in Martin Heidegger, Holzwege (1950), in Gesamtausgabe, vol. V, passim (Frankfurt: Klostermann, 1977).

122. For Durkheim, one of the founders of sociology, like the other great categories of thought, the concept of time has its origins in society—and in particular in the religious structure that constitutes its primary form; in Les Formes élémentaires de la vie religieuse (Paris: Alcan, 1912). If this can be true for complex aspects of the notion of time—for the “more external layers” of the notion of time—it seems to me difficult to extend it to include our direct experience of the passage of time: other mammals have brains roughly similar to ours, and consequently experience the passage of time like we do, without any need for a society or a religion.

123. On the foundational aspect of time for human psychology, see William James’s classic The Principles of Psychology (New York: Henry Holt, 1890).

124. Mahāvagga I.6.19, in Rhys Davids, Sacred Books of the East, vol. XIII (1881). For the concepts relating to Buddhism, I have drawn particularly on Hermann Oldenburg, Buddha (Milan: Dall’Oglio, 1956).

125. Hugo von Hofmannstahl, Der Rosenkavalier, act I.

13. THE SOURCE OF TIME

126. Ecclesiastes 3:2.

127. For a lighthearted, engaging but informed exposition of these aspects of time, see Craig Callender and Ralph Edney, Introducing Time (Cambridge, UK: Icon Books, 2001).

THE SISTER OF SLEEP

128. Mahābhārata III.297.

129. Ibid., I.119.

130. A. Balestrieri, “Il disturbo schizofrenico nell’evoluzione della mente umana. Pensiero astratto e perdita del senso naturale della realtà,” Comprendre 14 (2004): 55–60.

131. Roberto Calasso, L’ardore (Milan: Adelphi, 2010).

132. Ecclesiastes 12:6–7.