chapter six
The Mystery of the
Very Large Numbers

‘History is the science of things which are not repeated.’

SPOOKY NUMBERS

‘Although we talk so much about coincidence we do not really believe in it. In our heart of hearts we think better of the universe, we are secretly convinced that it is not a slipshod, haphazard affair, that everything in it has meaning.’

J.B. Priestley

The greatest mystery surrounding the values of the constants of Nature is without doubt the ubiquity of certain huge numbers that seem to appear in a variety of apparently quite unrelated considerations. Eddington's number is a notable example. The total number of protons that lie within the encompass of the observable Universe ² is close to the number

10⁸⁰

Then, if we ask for the ratio of the strengths of electromagnetic and gravitational forces between two protons, the answer does not depend on their separation,³ but is equal to approximately

10⁴⁰

This is slightly sinister. It is peculiar enough for pure numbers involving the constants of Nature to turn out very different from numbers within a factor of a hundred or so of 1, but 10⁴⁰ and its square, 10⁸⁰, is bizarre! Nor does it end there. If we were to follow Max Planck and compute an estimate for the ‘action’⁴ of the observable universe in units of the fundamental Planck units of action,⁵ then we get

10¹²⁰

We have already seen that Eddington was inclined to relate the number of particles in the observable universe to some quantity involving the cosmological constant. This quantity has had a fairly quiet history since that time, occasionally re-emerging when theoretical cosmologists need to find a way of accommodating awkward new observations. Recently this scenario has been played out again. New observations of unprecedented reach and accuracy, made possible by the Hubble space telescope working in co-operation with sensitive ground-based telescopes, has detected supernovae in far distant galaxies. Their characteristic brightening and fading pattern allows their distance to be deduced from their apparent brightnesses. Remarkably, they turn out to be receding from us far more quickly than anyone expected. The expansion of the universe has turned from a state of deceleration into one of acceleration. These observations imply the existence of a positive cosmological constant. If we express its numerical value as a pure dimensionless number by measuring in units of the square of the Planck length, then we get a number very close to

10^–120

No smaller number has ever been encountered in a real physical investigation.

What are we to make of all these large numbers? Is there something cosmically significant about 10⁴⁰ and its squares and cubes?

A BOLD HYPOTHESIS

‘Look what happens to people when they get married!’

George Gamow ⁶

The appearance of some of these large numbers had been a source of amazement ever since they were first noticed by Hermann Weyl in 1919. Eddington had tried to build a theory that made their appearance understandable. But he failed to convince a significant body of cosmologists that he was on the right track. Yet Eddington succeeded in persuading people that there was something that needed explaining. Completely unexpectedly, it was one of his famous neighbours in Cambridge who wrote the short letter to the journal Nature which succeeded in fanning interest in the problem with an idea that remains a viable possibility even to this day.

Paul Dirac was the Lucasian Professor of Mathematics at Cambridge for some of the time when Eddington was living at the Observatories. Stories of Dirac's simple and entirely logical approach to life and social interaction are legion and it is entirely in keeping with their peculiar tenor to find that his unexpected foray into the issue of Large Numbers was written whilst on his honeymoon, in February 1937.⁷

Unpersuaded by Eddington's numerological approach to the presence of ‘large numbers’ amongst the constants of Nature,⁸ Dirac argued that very large dimensionless numbers taking values like 10⁴⁰ and 10⁸⁰ are most unlikely to be independent and unrelated accidents: there must exist some undiscovered mathematical formula linking the quantities involved. They must be consequences rather than coincidences. This is Dirac's Large Numbers Hypothesis (LNH):

‘Any two of the very large dimensionless numbers occurring in Nature are connected by a simple mathematical relation, in which the coefficients are of the order of unity.’⁹

The large numbers that Dirac marshalled to motivate this daring new hypothesis drew on Eddington's work and were three in number:

Here, t is the present age of the Universe, me is the mass of an electron, m_pr is the proton mass, G the constant of gravitation, c the speed of light, and e the electron charge.

According to Dirac's hypothesis, the numbers N1, N2 and ?N were actually equal up to small numerical factors of order unity. By this, he meant that there must be laws of Nature that require formulae like 1= N2 or even N1 = 2N2. A number like 2, or 3, not terribly different from 1 is permitted because it is so much smaller than the large numbers involved in the formula; this is what he meant by the ‘coefficients … of the order of unity.’

This hypothesis of equality between Large Numbers is not in itself original to Dirac. Eddington and others had written down such relations before, but Eddington had not distinguished between the number of particles in the entire universe – which might be infinite – and the number of particles in the observable Universe, which is defined to be a sphere about us with radius equal to the speed of light times the present age of the Universe. The radical change precipitated by Dirac's LNH is that it requires us to believe that a collection of traditional constants of Nature, like N2 must be changing as the universe ages in time, t:

N₁≈N₂≈√N

Because Dirac had included two combinations which included the age of the Universe, t, in his catalogue of Large Numbers, the relation he proposes requires that a combination of three of the traditional constants of Nature is not constant at all but must increase steadily in value as the Universe ages, so

e²/Gm_pr αt(*)

Dirac chose to accommodate this requirement by abandoning the constancy of Newton's gravitation constant, G. He suggested that it was decreasing in direct proportion to the age of the Universe over cosmic time scales, as

G αl/t

Thus in the past G was bigger and in the future it will be smaller than it is measured to be today. One now sees that N₁≈N₂≈N ?t and the huge magnitude of the three Large Numbers is a consequence of the great age of the universe:¹⁰ they all get larger as time goes on.¹¹

Dirac's proposal provoked a stir amongst a group of vociferous scientists who filled the letters pages of the journal Nature with arguments for and against.¹² Meanwhile, Dirac maintained his customary low profile, but wrote about his belief in the importance of large numbers for our understanding of the universe in words that might easily have been written by Eddington, so closely do they mirror the philosophy of his ill-fated ‘Fundamental Theory’:

‘Might it not be that all present events correspond to properties of this large number [10⁴⁰], and, more generally, that the whole history of the universe corresponds to properties of the whole sequence of natural numbers…? There is thus a possibility that the ancient dream of philosophers to connect all Nature with the properties of whole numbers will some day be realised.’¹³

Dirac's approach has two significant elements. First, he seeks to show that what might previously have been regarded as coincidences are consequences of a deeper set of relationships that have been missed. Second, he sacrifices the constancy of the oldest known constant of Nature. Unfortunately, Dirac's hypothesis did not survive for long. The proposed change in the value of G was just too dramatic. In the past gravity would have been much stronger; the energy output of the Sun would be changed and the Earth would have been far hotter in the past than usually assumed.¹⁴ In fact, as the American physicist Edward Teller showed ¹⁵ in 1948, the oceans would have been boiling in the pre-Cambrian era, 200–300 million years ago, and life as we know it would not have survived, yet geological evidence available then showed that life had existed on Earth for at least 500 million years. Teller, an Hungarian émigré, was a high-profile physicist who played an important role in the development of the hydrogen bomb. He and Stan Ulam at Los Alamos were the two individuals who came up with the key idea (discovered independently by Andrei Sakharov in the Soviet Union) that showed how a nuclear bomb could be detonated. Later, Teller played a controversial role in the trial of Robert Oppenheimer and became an extreme hawk during the cold war period. He was the model for the character of Dr Strangelove so memorably portrayed by Peter Sellers in the film of that name. He is still an influential figure in weapons science and energy studies in the United States.

The exuberant George Gamow was a good friend of Teller's and responded to the boiling-ocean problem by suggesting that it could be ameliorated if it was assumed that Dirac's coincidences were created by a time variation in e, the electron charge, with e ² increasing with time as the equation (*) on p. 101 requires.¹⁶

This suggestion didn't survive for long either. Unfortunately, Gamow's proposal for varying e had all sorts of unacceptable consequences for life on Earth. It was soon realised that Gamow's theory would have resulted in the Sun exhausting all its nuclear fuel long ago. The Sun would not be shining today if e ² grows in proportion to the age of the universe. It would have been too small in the past to allow stars like the Sun to form.

Gamow had a number of discussions with Dirac about these variants to his hypothesis of varying G. Dirac made an interesting response to Gamow with regard to his idea that the charge on the electron, and hence the fine structure constant, might be varying. No doubt recalling Eddington's early belief that the fine structure constant was a rational number, he writes to Gamow in 1961 about the cosmological consequences of its variation as the logarithm of the age of the Universe, that

‘It is difficult to make any firm theories about the early stages of the universe because we do not know whether hc/e² is a constant or varies proportional to log(t). If hc/e²were an integer it would have to be a constant, but the experimenters now say it is not an integer, so it might very well be varying. If it does vary, the chemistry of the early stages would be quite different, and radio-activity would also be affected. When I started work on gravitation I hoped to find some connection between it and neutrinos, but this has failed.’¹⁷

Dirac was not readily going to subscribe to varying e as a solution of the Large Numbers conundrum. His most important scientific work had involved understanding the structure of atoms and the behaviour of the electron. All this was based upon the assumption, shared by everyone else, that e was a true constant, the same at all times and in all places in the Universe. Even Gamow soon gave up his theory about varying e and concluded that

‘The value of e stands as the Rock of Gibraltar for the last 6 × 10⁹ years!’¹⁸

Dirac's suggestion attracted wide interest from scientists of many unexpected quarters. Alan Turing, a pioneer in cryptography and the theory of computation, was fascinated by the idea of changing gravity and speculated about whether it would be possible to test the idea from fossil evidence, asking if

‘a paleontologist could tell from the footprint of an extinct animal, whether its weight was what it was supposed to be.’¹⁹

The great biologist J.B.S. Haldane ²⁰ became fascinated by the possible biological consequences of cosmological theories in which traditional ‘constants’ change with time or where gravitational processes unfolded with respect to a different cosmic clock than atomic processes. Such two-timing universes had been proposed by Milne and were the first suggestions that G might not be constant. Processes, like radioactive decay or molecular interaction rates, might be constant on one timescale but significantly variable with respect to the other. This gave rise to a scenario in which life-sustaining biochemistry only became possible after a particular cosmic epoch. Haldane suggests that

‘There was, in fact, a moment when life of any sort first became possible and the higher forms of life may only have become possible at a later date. Similarly a change in the properties of matter may account for some of the peculiarities of pre-Cambrian geology.’

This imaginative scenario is not dissimilar to that now known as ‘punctuated equilibrium’ in which evolution occurs in a staccato succession of accelerated bouts interspersed by long periods of slow change. However, Haldane provides an explanation for the changes.

What all these diverse responses to the ideas of Eddington and Dirac have in common is a growing appreciation that constants of Nature play a vital cosmological role: that there is a link between the structure of the Universe as a whole and the local conditions within it that are necessary for life to evolve and persist. If the traditional constants vary then astronomical theories have big consequences for biology, geology and life itself.

OF THINGS TO COME AT LARGE

‘The baby figure of the giant mass
Of things to come at large.’

William Shakespeare, Troilus and Cressida

The short-term legacy of the early interest in large numbers involving the constants of Nature was a focus of interest upon the possibility that some traditional constants of Nature might be varying very slowly over the billions of years of cosmic history. New theories of gravity were developed, extending Einstein's general theory of relativity to include varying gravity. Instead of being treated as a constant, Newton's G was like temperature, able to vary in strength from place to place and with the passage of time. Fortunately, this is not as hopelessly unconstrained as it might at first sound. In order that the changes in G respect the laws of cause and effect, not propagate changes at speeds faster than light, and don't violate the conservation of energy, there is a single type of theory which fits the bill. Many scientists found parts of this theory but the simplest and most complete representation of it was written down by the American physicist Robert Dicke and his research student, Carl Brans, in 1961.

Dicke was a rare physicist. He was equally at home as a mathematician, an experimental physicist, a distiller of complicated astronomical data, or the designer of sophisticated measuring instruments. He had the widest possible scientific interests. He realised that the idea of a varying ‘constant’ of gravity could be subjected to a plethora of observational tests, using the data of geology, palaeontology, astronomy and laboratory physics. Nor was he motivated simply by a desire to explain the Large Numbers. During the mid 1960s there was a further motivation for developing an extension of Einstein's theory of gravity that included varying G. For a while it appeared that Einstein's predictions about the wobble in the orbit of the planet Mercury did not agree with observations when the slightly non-spherical shape of the Sun was taken into account.

Dicke showed that if a variation in G with time was allowed then its rate of change could be chosen to have a value that would agree with the observations of Mercury's orbit. Sadly, years later this was all found to be something of a wild goose chase. The disagreement with Einstein's theory was being created by inaccuracies in our attempts to measure the diameter of the Sun which made it seem that the Sun was a different shape than it really was. The Sun's size is not so easy to measure at the levels of accuracy required because of the turbulent activity on the solar surface. When this problem was resolved in 1977 the need for a varying G to reconcile the observations with theory disappeared.²¹

In 1957, whilst beginning to develop theories with varying G, Dicke prepared a major review about the geophysical, palaeontological and astronomical evidences for possible variations of the traditional physical constants. He made the interesting remark that the issue of explaining the ‘large numbers’ of Eddington and Dirac must have some biological aspect:²²

‘The problem of the large size of these numbers now has a ready explanation … there is a single large dimensionless number which is statistical in origin. This is the number of particles in the Universe. The age of the Universe “now” is not random but is conditioned by biological factors … [because changes in the values of Large Numbers] would preclude the existence of man to consider the problem.’

Four years later, he elaborated this important insight in more detail, with special reference to Dirac's Large Number coincidences, in a short letter published in the journal Nature. Dicke argued that biochemical life-forms like ourselves owe their chemical basis to such elements as carbon, nitrogen, oxygen and phosphorus which are synthesised after billions of years of main-sequence stellar evolution. (The argument applies with equal force to any life-form based upon any atomic elements heavier than helium.) When stars die these ‘heavy’ biological elements are dispersed throughout space by supernovae from whence they are incorporated into grains, planetesimals, planets, self-replicating ‘smart’ molecules like DNA and, ultimately, into ourselves. Observers cannot arise until roughly the hydrogen-burning lifetime of a main-sequence star has elapsed and it is difficult for them to survive after the stars have burnt out. This time scale is controlled by fundamental constants of Nature to be

t(star) ≈ (Gmpr²/hc)^–1 h/mprc² ≈ 10⁴⁰ × 10^–23 seconds ≈ 10 billion years

We would not expect to be observing the Universe at times significantly in excess of t(star), since all stable stars would have expanded, cooled and died. Nor would we be able to see the Universe at times much less than t(star) because we could not exist! There would be neither stars nor heavy elements like carbon. We seem strait-jacketed by the facts of biological life to gaze at the Universe and develop cosmological theories after a time t(star) has elapsed since the Big Bang. Thus the value of Dirac's Large Number, N(t) is by no means random. It must have a value close to the value taken by N(t) when t is close in value to t(star).

If we look at the value of N at the time t(star) we find that it is precisely Dirac's Large Number coincidence. All Dirac's coincidence is saying is that we live at a time in cosmic history after the stars have formed and before they die. This is not surprising. Dicke is telling us that we could not fail to observe Dirac's coincidence: it is a prerequisite for life of our sort to exist. There is no need to give up Einstein's theory of gravitation by requiring G to vary, as Dirac implicitly required, nor do we need to deduce some numerological connection between the strength of gravity and the number of particles in the universe as Eddington had thought. The Large Number coincidence is no more surprising than the existence of life itself.

Dirac's response, his first written remarks about cosmology for more than twenty years, to this unusual perspective upon cosmological observations was rather bland:

‘On Dicke's assumption habitable planets could exist only for a limited period of time. With my assumption they could exist indefinitely in the future and life need never end. There is no decisive argument for deciding between these assumptions. I prefer the one that allows the possibility of endless life.’

Although he was willing to admit that life would be unlikely to exist before the stars had formed he was unwilling to concede that it could not continue long after they had burnt out. With Dirac's idea of varying G the coincidences would continue to be seen at all times but on Dicke's hypothesis they would only be seen near the present epoch. Dirac didn't think there was any problem with having habitable planets in the far future on his theory. However, if gravity is getting weaker it is not clear that stars and planets would be able to exist in the far future. At the very least, other constants would need to vary to maintain the balance between gravity and the other forces of Nature that make their existence possible.

It is very striking that other notable cosmologists like Milne had previously argued in the opposite way to Dicke. Milne regarded the appearance of Large Number coincidences in Eddington's theories as suspicious. He didn't believe that any ‘Fundamental Theory’ of Nature could possibly hope to explain coincidences between large numbers precisely because the large numbers involved the present age of the Universe. Since there was nothing special about the present time we were living at, no fundamental theory of physics could predict it or pick it out and so it could not explain the coincidences:

‘There is necessarily an empirically defined quantity, t [the present age of the universe] occurring in these expressions, for this simply measures the position of the instant at which we happen to be viewing the universe. This, of course, is incapable of prediction … The circumstance that Eddington's theory of the constants of nature appears to predict this … on a priori grounds seems to me an argument against Eddington's theory … for it appears to be equivalent to the feat of predicting the age of the universe at the moment we happen to be viewing it; which would be absurd.’²³

Dicke showed that, on the contrary, you certainly could predict something very particular about the age of the Universe if carbon-based beings are doing the predicting.

Dicke's point can be restated in an even more striking fashion. In order for a Big Bang universe to contain the basic building blocks ²⁴ necessary for the subsequent evolution of biochemical complexity it must have an age at least as long as the time it takes for the nuclear reactions in stars to produce these elements. This means that the observable Universe must be at least ten billion years old and so, since it is expanding, it must be at least ten billion light years in size. We could not exist in a universe that was significantly smaller.

Despite Dirac's dislike of Dicke's approach, we can find an unusual application of a similar idea being introduced by him to save his theory that G falls as the universe ages. After Edward Teller and others had discovered the problems that such a radical variation of gravity would create for the history of stars and life on Earth, there were attempts to keep the varying-G theory alive by assuming that stars like the Sun periodically passed through dense clouds from which they accreted material fast enough to offset the effects of decreasing G on the Sun's gravitational pull. Gamow thought that such an assumption

‘would be extremely unelegant, so that the total amount of elegance in the entire theory would have decreased quite considerably even though the elegant assumption [G ? t^–1] would be saved. So, we are thrown back to the hypothesis that 10⁴⁰ is simply the largest number the almighty God could write during the first day of creation.’

It is interesting to note Gamow's stress on the ‘inelegance’ of fudging the theory in this way, since Dirac always urged others to look for ‘beauty’ (which is not necessarily the same thing as simplicity, he liked to point out) in the equations describing a physical theory. Indeed, he once wrote to Heisenberg about one of his proposed theories that

‘My main objection to your work is that I do not think your basic … equation has sufficient mathematical beauty to be a fundamental equation of physics. The correct equation, when it is discovered, will probably involve some new kind of mathematics and will excite great interest among pure mathematicians.’²⁵

Yet Dirac was happy to defend the accretion idea, no matter how improbable it might appear, on the ground that it might be necessary for life to exist:

‘I do not see your objection to the accretion hypothesis. We may assume that the sun has passed through some dense clouds, sufficiently dense for it to pick up enough matter to keep the earth at a habitable temperature for 10⁹ years. You may say that it is improbable that the density should be just right for this purpose. I agree. It is improbable. But this kind of improbability does not matter. If we consider all the stars that have planets, only a very small fraction of them will have passed through clouds of the right density to maintain their planets at an equable temperature long enough for advanced life to develop. There will not be so many planets with men on them as we previously thought. However, provided there is one, it is sufficient to fit the facts. So there is no objection to assuming our sun has had a very unusual and improbable history.’²⁶

This is a remarkable about-face, six years after his initial opposition to Dicke's inclusion of human life as a factor in assessing the likelihood of unusual situations arising.

A beautifully simple argument regarding the inevitability of the large size of the Universe for us appears first in the text of the Bampton Lectures given by the Oxford theologian Eric Mascall. They were published in 1956 under the title Christian Theology and Natural Science and he attributes the basic idea to Gerald Whitrow. Stimulated by Whitrow's suggestions, he writes:

‘if we are inclined to be intimidated by the mere size of the Universe, it is well to remember that on certain cosmological theories there is a direct connection between the quantity of matter in the Universe and the conditions in any limited portion of it, so that in fact it may be necessary for the Universe to have the enormous size and complexity which modern astronomy has revealed, in order for the earth to be a possible habitation for living beings.’²⁷

This simple observation can be extended to provide us with a profound understanding of the subtle links that exist between superficially different aspects of the Universe we see around us and the properties that are needed if a Universe is to contain living beings of any sort.

BIG AND OLD, DARK AND COLD

‘It's a funny old world – a man's lucky if he gets out of it alive.’

W. C. Fields ²⁸

We have seen that the process of stellar alchemy takes time – billions of years of it. And because our Universe is expanding it needs to be billions of light years in size if it is to have enough time to produce the building blocks for living complexity. A universe that was only as big as our Milky Way galaxy, with its 100 billion stars, would be little more than a month old. Another consequence of an old expanding universe, besides its large size, is that it is cold, dark and lonely. When any ball of gas or radiation is expanded in volume, the temperature of its constituents falls off in proportion to the increase in its size. A universe that is big and old enough to contain the building blocks of complexity will be very cold and the levels of average radiant energy so low that space will everywhere appear dark.

It is sobering to reflect upon all the metaphysical and religious responses there have been, over the centuries, to the darkness of the night sky and the patterns of stars embroidered upon it; to the vastness of space and our incidental place within it, a mere dot in the grand scheme of things. Modern cosmology shows that these features are not random accidents. They are part and parcel of the whole interconnectedness of the universe. They are, in fact, necessary features of any universe that contains living observers. Remarkably, the metaphysical effect of this type of universe upon its inhabitants may well be another inescapable by-product for any sentient beings elsewhere as well. The Universe has the curious property of making living beings think that its unusual properties are unsympathetic to the existence of life when in fact they are essential for it.

If we were to smooth out all the material in the Universe into a uniform sea of atoms we would see just how little of anything there is. There would be little more than about 1 atom in every cubic metre of space. No laboratory on Earth could produce an artificial vacuum that was anywhere near as empty as that. The best vacuum achievable today contains approximately 1000 billion atoms in a cubic metre.

This way of looking at the Universe provides some important new insights into the properties it displays to us. Many of its most striking features – its vast size and huge age, the loneliness and darkness of space – are all necessary conditions for there to be intelligent observers like ourselves. We should not be surprised that extraterrestrial life, if it exists, is so rare and so far away. The low average density of matter in the Universe means that if we were to aggregate material into stars or galaxies, then we should expect huge distances to lie between these objects on average. In Figure 6.1, the density of material in the Universe is expressed in a variety of different ways which shows how far apart we should expect planets, stars and galaxies to be.

In Figure 6.2 we show the expanding trajectory of our Universe as time passes. Gradually, the environment within in the Universe cools off and allows atoms, molecules, galaxies, stars and planets to form. We are located in a particular niche of cosmic history between the birth and death of the stars.

Figure 6.1 The density of matter in our Universe expressed in a number of different units of volume that show how rare galaxies, stars, planets, and atoms actually are on the average. We should not be surprised to find that extraterrestrial life is very rare.

It appears that the existentialist philosopher Karl Jaspers was also provoked by Eddington's writings to consider the significance of our existence in a particular locale at a particular epoch of cosmic history. In his influential book,²⁹ written in 1949, soon after Eddington's death, he asks

‘Why do we live and accomplish our history in infinite

Figure 6.2The changing environment in an expanding universe like our own. As the universe cools and ages it is possible for atoms, molecules, galaxies, stars, planets, and living organisms to form. In the future the stars will extinguish their nuclear fuel and die. There is a niche of cosmic history in which our sort of biological evolution must first occur if it is ever to occur.

space at precisely this point, on a minute grain of dust in the universe, as though in an out-of-the-way corner? Why just now in infinite time? These are questions whose unanswerability makes us conscious of an enigma.

The fundamental fact of our existence is that we appear to be isolated in the cosmos. We are the only articulate rational beings in the silence of the universe. In the history of the solar system there has arisen on the earth, for a so far infinitesimally short period, a condition in which humans evolve and realise knowledge of themselves and of being … Within the boundless cosmos, on a tiny planet, for a tiny period of a few millennia, something has taken place as though this planet were the all-embracing, the authentic. This is the place, a mote in the immensity of the cosmos, at which being has awakened with man.'

There are some big assumptions here about the uniqueness of human life in the Universe. Yet, the question is raised, although not answered, as to why we are here at the time and place that we are. We have seen that modern cosmology can provide an illuminating response to this question.

THE BIGGEST NUMBER OF ALL

‘Al-Gore-rithm, n. a mathematical operation which is repeated many times until it converges to the desired result, especially in Florida.’

The Grapevine

Astronomers are used to huge numbers. They are challenged to explain to outsiders just what billions and billions of stars really means with some homespun analogy. It was only when the American national debt grew to astronomical levels that there were suddenly numbers in the financial pages of newspapers that were larger than the number of stars in the Milky Way or galaxies in the Universe.³⁰ Yet, curiously, if you want really big numbers, numbers that dwarf even the 10⁸⁰s of Eddington and Dirac, astronomy is not the place to look. The big numbers of astronomy are additive. They arise because we are counting stars, planets, atoms and photons in a huge volume. If you want really huge numbers you need to find a place where the possibilities multiply rather than add. For this you need complexity. And for complexity you need biology.

In the seventeenth century the English physicist Robert Hooke made a calculation ‘of the number of separate ideas the mind is capable of entertaining’.³¹ The answer he got was 3,155,760,000. Large as this number might appear to be (you would not live long enough to count up to it!) it would now be seen as a staggering underestimate. Our brains contain about ten billion neurons, each of which sends out feelers, or axons, to link it to about one thousand others. These connections play some role in creating our thoughts and memories. How this is done is still one of Nature's closely guarded secrets. Mike Holderness suggests that one way of estimating ³² the number of possible thoughts that a brain could conceive is to count all those connections. The brain can do many things at once so we could view it as some number, say a thousand, little groups of neurons. If each neuron makes a thousand different links to the ten million others in the same group then the number of different ways in which it could make connections in the same neuron group is 10⁷ × 10⁷ × 10⁷ × … one thousand times. This gives 10⁷⁰⁰⁰ possible patterns of connections. But this is just the number for one neuron group. The total number for 10⁷neurons is 10⁷⁰⁰⁰ multiplied together 10⁷ times. This is 10^{70,000,000,000}. If the 1000 or so groups of neurons can operate independently of each other then each of them contributes 10^{70,000,000,000} possible wirings, increasing the total to the Holderness number, 10^{70,000,000,000,000}.