The amount of information a.s.sociated thus with a black hole depends directly upon the area A of the hole: the larger the hole, the greater the amount of missing information.

When information enters into a black hole, it is no longer recoverable from outside. But the information which enters the black hole carries with it the energy by which the black hole becomes larger and increases its area. Viewed from outside, the information lost in the black hole now appears as entropy a.s.sociated with the area of the hole. The first to suspect something similar was the Israeli physicist Jacob Bekenstein.

But the situation is anything but clear because, as we have seen in the last chapter, black holes emit thermal radiation and very slowly evaporate, becoming smaller and smaller until they probably disappear, subsumed in that ocean of microscopic black holes which const.i.tutes s.p.a.ce at the Planck scale. Where does the information that has fallen into the black hole as the black hole shrinks end up? Theoretical physicists are debating the question, and no one has a completely clear answer.

All of this, I believe, indicates that in order to grasp the basic grammar of the world, we need to merge three basic ingredients, not just two: not just general relativity and quantum mechanics, but also the theory of heat, that is, statistical mechanics and thermodynamics, which we can also describe as information theory. But the thermodynamics of general relativity, that is to say, the statistical mechanics of quanta of s.p.a.ce, is as yet only in its first infancy. Everything is still confused, and there is a very great deal which remains to be understood.

All of this brings us to the last idea I describe in this book: thermal time.

Thermal time

The problem at the root of the idea of thermal time is simple. In Chapter 7, I showed that it is not necessary to use the notion of time to describe physics. It is better to forget time altogether. Time plays no role at the fundamental level of physics. Once we have understood this, it is easier to write the equations of quantum gravity.

There are many everyday notions which no longer have any role in the fundamental equations of the universe; for example, the notions of 'up' and 'down', or 'hot' and 'cold', so it is not particularly strange that shared quotidian notions disappear from fundamental physics. However, once we have accepted this idea, we obviously open up a second problem. How can we recover the notions of our everyday experience? How do they emerge, in our specific context?

For example, the notions of 'up' and 'down' don't enter into Newton's equations, but we know what they mean in a schema without absolute up and down. 'Up' and 'down' are meaningful near a large ma.s.s, like a planet. 'Down' indicates the direction towards which the large, near ma.s.s exerts gravitational pull; 'up' indicates the opposite direction. The same goes for 'hot' and 'cold': there are no 'hot' or 'cold' things at a microscopic level but, when we put together a large number of microscopic const.i.tuents and describe them in terms of averages, then the notion of 'heat' appears: a hot body is a body where the average speed of single const.i.tuents is raised. We are able to understand the meaning of 'up' or 'hot' in certain situations: the presence of a nearby ma.s.s, or the fact that we are dealing only with average values of many molecules, and so on.

Something similar must apply to 'time'. If the notion of time has no role to play at an elementary level, it certainly plays a significant role in our lives, just as 'up' and 'hot' do. What does 'the pa.s.sage of time' mean, if time plays no part in the fundamental description of the world?

The answer is simple. The origin of time may be similar to that of heat: it comes from averages of many microscopic variables. Let's look at this in detail.

That there is a link between time and temperature is an ancient and recurrent idea. If you think about it, all phenomena where we detect the pa.s.sage of time are co-involved with temperature. The salient characteristic of time is that it moves forwards and not backwards, that is to say, there are irreversible phenomena. Mechanical phenomena ones that don't involve heat are reversible. If we film them and then run the film backwards, we see something realistic. If we film a swinging pendulum, or a stone thrown upwards then falling, and then watch the film in reverse, we still see a plausible pendulum swinging, or a stone rising and dropping to the ground.

When the stone reaches the ground, it stops, you might object: if you watch the film reversed, you see a stone leaping up from the ground by itself, and this is implausible. But when the stone reaches the ground and stops, where does its energy go? It heats the ground! At the precise moment when heat is produced, the process is irreversible: the past differs from the future. It is always heat and only heat that distinguishes the past from the future.

This is universal. A burning candle is transformed into smoke the smoke cannot transform into a candle and a candle produces heat. A boiling-hot cup of tea cools down and does not heat up: it diffuses heat. We live and get old: producing heat. Our old bicycle wears out with time: producing heat through friction. Think of the solar system. At first approximation, it continues to turn like an immense mechanism always equal to itself. It doesn't produce heat and, in fact, if you watched it in reverse you wouldn't notice anything strange about it. But looked at more closely, there are also irreversible phenomena: the Sun is using up its combustible hydrogen and will eventually exhaust it and extinguish: the Sun, too, is getting older and, in fact, produces heat. The Moon also appears to orbit the Earth unchangingly and be always equal to itself, whereas in reality it is slowly moving away. This is because it raises tides, and the tides heat the sea a little, thus exchanging energy with the Moon. Whenever you consider a phenomenon certifying the pa.s.sage of time, it is through the production of heat that it does so. There is no preferred direction of time without heat.

But heat is our way to name averages over many variables.

The idea of thermal time reverses this observation. That is to say, instead of enquiring how time produces dissipation in heat, it asks how heat produces time.

Thanks to Boltzmann, we know that the notion of heat comes from the fact that we interact with averages. The idea of thermal time is that the notion of time, too, comes from the fact that we interact only with averages of many variables.fn53 As long as we have a complete description of a system, all the variables of the system are on the same footing; none of them acts as a time variable. That is to say: none is correlated to irreversible phenomena. But as soon as we describe the system by means of averages of many variables, we have a preferred variable that functions like common time. A time along which heat is dissipated. The time of our everyday experience.

Hence time is not a fundamental const.i.tuent of the world, but it appears because the world is immense, and we are small systems within the world, interacting only with macroscopic variables that average among innumerable small, microscopic variables. We, in our everyday lives, never see a single elementary particle, or a single quantum of s.p.a.ce. We see stones, mountains, the faces of our friends and each of these things we see is formed by myriads of elementary components. We are always correlated with averages. Averages behave like averages: they disperse heat and, intrinsically, generate time.

The difficulty of grasping this idea comes from the fact that it is hard for us to think of a world without time, and of time emerging in an approximate manner. We are too used to thinking of reality as existing in time. We are beings who live in time: we dwell in time, and are nourished by it. We are an effect of this temporality, produced by average values of microscopic variables. But the limitations of our intuitions should not mislead us. Understanding the world better often entails going against intuition. If this were not the case, understanding would be easy.

Time is an effect of our overlooking of the physical microstates of things. Time is information we don't have.

Time is our ignorance.

Reality and information

Why does information play such a central role as this? Perhaps because we must not confuse what we know about a system with the absolute state of the same system. What we know is something concerning the relation between the system and ourselves. Knowledge is intrinsically relational; it depends just as much on its object as upon its subject. The notion of the 'state' of a system refers, explicitly or implicitly, to another system. Cla.s.sical mechanics misled us into thinking that we could do without taking account of this simple truth, and that we could access, at least in theory, a vision of reality entirely independent of the observer. But the development of physics has shown that, at the end of the day, this is impossible.

Careful: when I say that we 'have information' about the temperature of cup of tea, or we 'don't have information' about the velocity of every single molecule, I am not saying something about mental states, or abstract ideas. I am only saying that the laws of physics determine a correlation between ourselves and the temperature (for instance, I've looked at a thermometer), but not between ourselves and the velocity of the individual molecules. It is the same notion of information as the one I started from in this chapter: the white ball in your hand 'has information' about the fact that the ball in my hand is black. We're dealing with physical facts, not mental notions. A ball has information, in this sense, even if the ball does not have mental states, just as a USB storage device contains information (the number of gigabytes printed on the device tells us how much information it can contain), even if a USB storage device does not think. Information in this sense correlation between states of systems is ubiquitous throughout the universe.

I believe that in order to understand reality we have to keep in mind that reality is this network of relations, of reciprocal information, which weaves the world. We slice up the reality surrounding us into objects. But reality is not made up of discrete objects. It is a variable flux. Think of an ocean wave. Where does a wave finish? Where does it begin? Think of mountains. Where does a mountain start? Where does it end? How far does it continue beneath the Earth's surface? These are questions without much sense, because a wave and a mountain are not objects in themselves; they are ways which we have of slicing up the world to apprehend it, to speak about it more easily. These limits are arbitrary, conventional, comfortable: they depend on us (as physical systems) more than on the waves or the mountains. They are ways of organizing the information which we have or, better, forms of information which we have.

It's the same for every object, properly considered, including living organisms. This is why it makes little sense to ask whether a half-cut fingernail is still 'me' or has become 'not-me'; or if the hairs left on my sofa by the cat are still part of the cat, or not; or precisely when a child's life begins. A child begins to live on the day when a person dreams of her for the first time, long before her conception, or when she forms her first self-image, or when she breathes for the first time, or when she recognizes her name, or when we apply any number of other conventions: they are all useful, but arbitrary. They are ways to think, and to orientate ourselves within the complexity of reality.

A living organism is a system which continually re-forms itself in order to remain itself, interacting ceaselessly with the external world. Of such organisms, only those continue to exist which are more efficient at doing so and, therefore, living organisms manifest properties which have suited them for survival. For this reason, they are interpretable, and we interpret them, in terms of intentionality, of purpose. The finalistic aspects of the biological world (this is Darwin's momentous discovery) are therefore the result of the selection of complex forms effective in persisting. But the effective way of continuing to exist in a changing environment is to manage correlations with the external world better, that is to say, information; to collect, store, transmit and elaborate information. For this reason, DNA exists, together with immune systems, sense organs, nervous systems, complex brains, languages, books, the library of Alexandria, computers and Wikipedia: they maximize the efficiency of information management the management of correlations favouring survival.

The statue that Aristotle sees in a block of marble is more than the block of marble: but it is not an abstract form that resides just in the statue. It is something residing in the correlations between the mind of Aristotle, or ours, and the marble; something that pertains to the information which the marble provides regarding something that is significant for Aristotle, or for us. It is something regarding a discus thrower, Phidias, Aristotle and the marble, and resides in the correlated dispositions of the atoms of the statue, and the correlations between these and a thousand others, in our minds or in Aristotle's. These speak of a discus thrower, just as the white ball in your hand tells you that the ball in mine is black. We are structured to manage precisely this information and remain in existence thanks to this.

Even from this brief overview it should be clear that the notion of information plays a central role in our attempts to understand the world. From communication to the basis of genetics, from thermodynamics to quantum mechanics and up to quantum gravity, the notion of information is gaining ground as a tool for understanding. The world should not be understood as an amorphous ensemble of atoms but rather as a game of mirrors, founded on the correlations between the structures formed by combinations of these atoms.

As Democritus said, it is not just a question of these atoms but also of the order in which they are arranged. Atoms are like the letters in an alphabet: an extraordinary alphabet, so rich as to be able to read, reflect and even think about itself. We are not atoms; we are orders in which atoms are arranged, capable of mirroring other atoms and mirroring ourselves.

Democritus gave a strange definition of 'man': 'Man is what we all know.'2 At first sight, this seems rather silly and empty, but it is not so.

Salomon Luria, the major scholar of Democritus, observes that it is not a ba.n.a.lity that Democritus is giving us. The nature of a man is not his internal structure but the network of personal, familial and social interactions within which he exists. It is these which 'make' us, these which guard us. As humans, we are that which others know of us, that which we know of ourselves, and that which others know about our knowledge. We are complex nodes in a rich web of reciprocal information.

All of this is not yet a theory. These are tracks we are following, I believe, in seeking to understand the world around us better. There still remains a great deal to understand. I'll speak of this in the final chapter.

13. Mystery

The truth is in the depths.

Democritus1 I've described what I think is the nature of things in the light of what we have learned to date. I've summarized the development of some key ideas of fundamental physics, and I have ill.u.s.trated the great discoveries made by physics in the twentieth century and the image of the world emerging from the research into a quantum theory of gravity.

Am I sure about all this? I am not.

One of the very first and most beautiful pages in the history of science is the pa.s.sage in Plato's Phaedo in which Socrates explains the shape of the Earth.

Socrates says he 'believes' the Earth is a sphere, with great valleys where men live. He's basically right, if a bit confused. He adds, 'I'm not sure.' This page is worth much more than all of the nonsense on the immortality of the soul which fills the rest of the dialogue. It is not just the oldest text to come down to us which speaks explicitly of the fact that the Earth must be spherical. More importantly, it shines with the crystalline clarity with which Plato acknowledges the limits of the knowledge of his time. 'I'm not sure,' says Socrates.

This acute awareness of our ignorance is the heart of scientific thinking. It is thanks to this awareness of the limits of our knowledge that we have learned so much. We are not certain of all which we suspect, just as Socrates was not sure of the spherical nature of the Earth. We are exploring at the borders of our knowledge.

Awareness of the limits of our knowledge is also awareness of the fact that what we know may turn out to be wrong, or inexact. Only by keeping in mind that our beliefs may turn out to be wrong is it possible to free ourselves from wrong ideas, and to learn. To learn something, it is necessary to have the courage to accept that what we think we know, including our most rooted convictions, may be wrong, or at least nave: shadows on the walls of Plato's cave.

Science is born from this act of humility: not trusting blindly in our past knowledge and our intuition. Not believing what everyone says. Not having faith in the acc.u.mulated knowledge of our fathers and grandfathers. We learn nothing if we think that we already know the essentials, if we a.s.sume that they were written in a book or known by the elders of the tribe. The centuries in which people had faith in what they believed were the centuries in which little new was learned. Had they trusted the knowledge of their fathers, Einstein, Newton and Copernicus would never have called things into question and would have never been able to move our knowledge forwards. If no one had raised doubts, we would be still worshipping pharaohs and thinking that the Earth is supported on the back of a giant turtle. Even our most efficacious knowledge, such as that found by Newton, may eventually turn out, as Einstein showed, to be simplistic.

Science is sometimes criticized for pretending to explain everything, for thinking that it has an answer to every question. It's a curious accusation. As every researcher working in every laboratory throughout the world knows, doing science means coming up hard against the limits of your ignorance on a daily basis the innumerable things which you don't know, and can't do. This is quite different from claiming to know everything. We don't know which particles we might see next year at CERN, or what our next telescopes will reveal, or which equations truly describe the world; we don't know how to solve the equations we have, and sometimes we don't understand what they signify; we don't know if the beautiful theory on which we are working is right. We don't know what there is beyond the Big Bang; we don't know how a storm works, or a bacterium, or an eye or the cells in our own bodies, or our thought processes. A scientist is someone who lives immersed in the awareness of our deep ignorance, in direct contact with our own innumerable limits, with the limits of our understanding.

But if we are certain of nothing, how can we possibly rely on what science tells us? The answer is simple. Science is not reliable because it provides certainty. It is reliable because it provides us with the best answers we have at present. Science is the most we know so far about the problems confronting us. It is precisely its openness, the fact that it constantly calls current knowledge into question, which guarantees that the answers it offers are the best so far available: if you find better answers, these new answers become science. When Einstein found better answers than Newton, he didn't question the capacity of science to give the best possible answers on the contrary, he confirmed it.

The answers given by science, then, are not reliable because they are definitive. They are reliable because they are not definitive. They are reliable because they are the best available today. And they are the best we have because we don't consider them to be definitive, but see them as open to improvement. It's the awareness of our ignorance that gives science its reliability.

And it is reliability that we need, not certainty. We don't have absolute certainty, and never will have it unless we accept blind belief. The most credible answers are the ones given by science, because science is the search for the most credible answers available, not for answers pretending to certainty.

Though rooted in previous knowledge, science is an adventure based on continuous change. The story I have told reaches back over millennia, tracing a narrative of science that has treasured good ideas but hasn't hesitated to throw ideas away when something which works better was found. The nature of scientific thinking is critical, rebellious and dissatisfied with a priori conceptions, with reverence and sacred or untouchable truth. The search for knowledge is not nourished by certainty: it is nourished by a radical distrust in certainty.

This means not giving credence to those who say they are in possession of the truth. For this reason, science and religion frequently find themselves on a collision course. Not because science pretends to know ultimate answers but precisely for the opposite reason: because the scientific spirit distrusts whoever claims to be the one having ultimate answers, or privileged access to Truth. This distrust is found to be disturbing in some religious quarters. It is not science which is disturbed by religion: there are certain religions that are disturbed by scientific thinking.

To accept the substantial uncertainty of our knowledge is to accept living immersed in ignorance and, therefore, in mystery, to accept living with questions to which we do not know the answers. Perhaps we don't know them yet or who knows? we never will.

To live with uncertainty may be difficult. There are those who prefer any certainty, even if unfounded, to the uncertainty which comes from recognizing our own limits. There are some who prefer to believe in a story just because it was believed by the tribe's ancestors rather than bravely to accept uncertainty.

Ignorance can be scary. Out of fear, we can tell ourselves calming stories: up there beyond the stars, there is an enchanted garden, with a gentle father who will welcome us into his arms. It doesn't matter if this is true, it is rea.s.suring.

There is always, in this world, someone who pretends to tell us the ultimate answers. The world is full of people who say that they have The Truth. Because they have got it from the fathers; they have read it in a Great Book; they have received it directly from a G.o.d; they have found it in the depths of themselves. There is always someone who has the presumption to be the depository of Truth, neglecting to notice that the world is full of other depositories of Truth, each one with his own real Truth, different from that of the others. There is always some prophet dressed in white, uttering the words, 'Follow me, I am the true way.'

I don't criticize those who prefer to believe in this: we are all free to believe in whatever we want. Maybe, after all, there is a grain of truth in the joke reported by St Augustine: What was G.o.d doing before creating the world? He was preparing h.e.l.l for those who seek to scrutinize deep mysteries.2 But these deep mysteries are precisely the 'depths' in which Democritus, in the quote that opens this chapter, invites us to seek the truth.

For my part, I prefer to look our ignorance in the face, accept it and seek to look just a bit further: to try to understand that which we are able to understand. Not just because accepting this ignorance is the way to avoid being entangled in superst.i.tions and prejudices but because to accept our ignorance in the first place seems to me to be the truest, the most beautiful and, above all, the most honest way.

To seek to look further, to go further, seems to me to be one of the splendid things which gives sense to life. Like loving, or looking at the sky. The curiosity to learn, to discover, to look over the next hill, the desire to taste the apple: these are the things which make us human. As Dante's Ulysses reminds his companions, we are not made 'to live like brutes, but to seek virtue and knowledge'.

The world is more extraordinary and profound than any of the fables told by our forefathers. I want to go and see it. To accept uncertainty doesn't detract from our sense of mystery. On the contrary: we are immersed in the mystery and the beauty of the world. The world revealed by quantum gravity is a new and strange one still full of mystery, but coherent with its simple and clear beauty.

It is a world which does not exist in s.p.a.ce and does not develop in time. A world made up solely of interacting quantum fields the swarming of which generates through a dense network of reciprocal interactions s.p.a.ce, time, particles, waves and light (figure 13.1) It continues, it continues, teeming life, and death Tender and hostile, clear and unknowable.

And the poet goes on: So much the eye can see, from this watching tower.3 A world without infinity, where the infinitely small does not exist, because there is a minimum scale to this teeming, beneath which there is nothing. Quanta of s.p.a.ce mingle with the foam of s.p.a.cetime, and the structure of things is born from reciprocal information which weaves the correlations between the regions of the world. A world which we know how to describe with a set of equations. Perhaps, to be corrected.

Figure 13.1 An intuitive representation of quantum gravity.

It's a vast world, with much still to clarify and explore. It's my fondest dream that someone one of the younger readers of this book, I hope will be able to voyage across it and illuminate it better. Beyond the next hill there are worlds still more vast, still to be discovered.

Notes.

1. Grains

1 On Anaximander and the Milesians, see Carlo Rovelli, The First Scientist: Anaximander and His Legacy, trans. Marion Lignana Rosenberg (Yardley, Westholme, 2007).

2 The Milesian origin of Leucippus is given, for instance, by Simplicius (see M. Andolfo, Atomisti antichi. Frammenti e testimonianze (Ancient Atomism. Fragments and Testimonies), (Milan, Rusconi, 1999), p. 103. But it is not certain. The reference to Miletus and to Elea is significant in relation to his cultural roots; the debt Leucippus owed to Zeno of Elea is discussed in the following pages.

3 Seneca, Naturales questiones, VII, 3, 2d.

4 Cicero, Academica priora, II, 23, 73.

5 s.e.xtus Empiricus, Adversus mathematicos, VIII, 135 (trans. R. G. Bury Against the Professors), (Loeb Cla.s.sical Library, 1989).

6 See Aristotle, On Generation and Corruption, A1, 315b 6, in The Complete Works of Aristotle, Vol. I, ed. Jonathan Barnes (Princeton, Princeton University Press, 1984).

7 A collection of ancient fragments and testimonies which speak of atomism is given in M. Andolfo's Ancient Atomists. A complete anthology of fragments and testimonies concerning Democritus has been published by Solomon Luria. (See entry 'Democritus' in the bibliography for an English alternative.) 8 For a brief and interesting recent work on the thought of Democritus, placing it in the context of humanism, see S. Martini, Democrito: filosofo della natura o filosofo dell'uomo? (Democritus: Philosopher of Nature or Philosopher of Man?), (Rome, Armando, 2002.) 9 Plato, Phaedo, ed. David Gallup (Oxford, Oxford University Press, 2009), XLVI.

10 Richard Feynman, The Feynman Lectures on Physics, Vol. 1, eds. Robert Leighton and Matthew Sands (London, Basic Books, 2011).

11 See Aristotle, On Generation and Decay, A2, 316a, in The Complete Works of Aristotle, Vol. I, ed. Jonathan Barnes (Princeton, Princeton University Press, 1984).

12 A good recent account of Zeno's paradoxes, and of their philosophical and mathematical relevance, is given by Vincenzo Fano in I paradossi di Zenone (Zeno's Paradoxes), (Rome, Carocci, 2012).

13 Amores (Love Poems), I, 15, 234.

14 Lucretius, On the Nature of the Universe, trans. E. A. Latham, (Harmondsworth, Penguin, 1951), p. 173 15 Ibid., p. 89.

16 Ibid., p. 27 17 Ibid., p. 60.

18 Guido Cavalcanti, Rime., trans. Leonard Cottrell.

19 For an account of the rediscovery of Lucretius's text and its impact upon European culture, see Stephen Greenblatt, The Swerve: How the World Became Modern (New York, Norton, 2011).

20 See M. Camarota, 'Galileo, Lucretius and Atomism', in F. Citti and M. Beretta (eds.), Lucrezio, la natura e la scienza (Lucretius, Nature and Science), (Florence, Leo S. Olshki, 2008), pp. 14175.

21 See R. Kargon, Atomism in England from Hariot to Newton (Oxford, Oxford University Press, 1966).

22 William Shakespeare, Romeo and Juliet, 1.4, 5560, Complete Works, eds. Jonathan Bate and Eric Ra.s.smussen (London, Macmillan/The Royal Shakespeare Company, 2007), p. 1690.

23 On The Nature of the Universe, pp. 634.

24 Piergiorgio Odifreddi has published a fine translation of and commentary on Lucretius's text, designed for use in schools. (Come stanno le cose. Il mio Lucrezio, la mia Venere (The Nature of Things. My Lucretius, My Venus), (Milan, Rizzoli, 2013). It would be wonderful if schools adopted this book and if this extraordinary text was more widely known. A reading of the text, and of its author, diametrically opposed to Odifreddi's is given by V. E. Alfieri in Lucrezio (Lucretius), (Florence, Le Monnier, 1929) and emphasizes the poignant, poetic qualities of the work, deriving from them a n.o.ble but bitter interpretation of the character of Lucretius.