The Beginning of Infinity

by David Deutsch

  Because explanation plays this central role in science, and because testability is of little use in the case of bad explanations, I myself prefer to call myths, superstitions and similar theories unscientific even when they make testable predictions. But it does not matter what terminology you use, so long as it does not lead you to conclude that there is something worthwhile about the Persephone myth, or the prophet’s apocalyptic theory or the gambler’s delusion, just because is it testable. Nor is a person capable of making progress merely by virtue of being willing to drop a theory when it is refuted: one must also be seeking a better explanation of the relevant phenomena. That is the scientific frame of mind.

  As the physicist Richard Feynman said, ‘Science is what we have learned about how to keep from fooling ourselves.’ By adopting easily variable explanations, the gambler and prophet are ensuring that they will be able to continue fooling themselves no matter what happens. Just as thoroughly as if they had adopted untestable theories, they are insulating themselves from facing evidence that they are mistaken about what is really there in the physical world.

  The quest for good explanations is, I believe, the basic regulating principle not only of science, but of the Enlightenment generally. It is the feature that distinguishes those approaches to knowledge from all others, and it implies all those other conditions for scientific progress I have discussed: It trivially implies that prediction alone is insufficient. Somewhat less trivially, it leads to the rejection of authority, because if we adopt a theory on authority, that means that we would also have accepted a range of different theories on authority. And hence it also implies the need for a tradition of criticism. It also implies a methodological rule – a criterion for reality – namely that we should conclude that a particular thing is real if and only if it figures in our best explanation of something.

  Although the pioneers of the Enlightenment and of the scientific revolution did not put it this way, seeking good explanations was (and remains) the spirit of the age. This is how they began to think. It is what they began to do, systematically for the first time. It is what made that momentous difference to the rate of progress of all kinds.

  Long before the Enlightenment, there were individuals who sought good explanations. Indeed, my discussion here suggests that all progress then, as now, was due to such people. But in most ages they lacked contact with a tradition of criticism in which others could carry on their ideas, and so created little that left any trace for us to detect. We do know of sporadic traditions of good-explanation-seeking in narrowly defined fields, such as geometry, and even short-lived traditions of criticism – mini-enlightenments – which were tragically snuffed out, as I shall describe in Chapter 9. But the sea change in the values and patterns of thinking of a whole community of thinkers, which brought about a sustained and accelerating creation of knowledge, happened only once in history, with the Enlightenment and its scientific revolution. An entire political, moral, economic and intellectual culture – roughly what is now called ‘the West’ – grew around the values entailed by the quest for good explanations, such as tolerance of dissent, openness to change, distrust of dogmatism and authority, and the aspiration to progress both by individuals and for the culture as a whole. And the progress made by that multifaceted culture, in turn, promoted those values – though, as I shall explain in Chapter 15, they are nowhere close to being fully implemented.

  Now consider the true explanation of seasons. It is that the Earth’s axis of rotation is tilted relative to the plane of its orbit around the sun. Hence for half of each year the northern hemisphere is tilted towards the sun while the southern hemisphere is tilted away, and for the other half it is the other way around. Whenever the sun’s rays are falling vertically in one hemisphere (thus providing more heat per unit area of the surface) they are falling obliquely in the other (thus providing less).

  The true explanation of seasons (not to scale!)

  That is a good explanation – hard to vary, because all its details play a functional role. For instance, we know – and can test independently of our experience of seasons – that surfaces tilted away from radiant heat are heated less than when they are facing it, and that a spinning sphere in space points in a constant direction. And we can explain why, in terms of theories of geometry, heat and mechanics. Also, the same tilt appears in our explanation of where the sun appears relative to the horizon at different times of year. In the Persephone myth, in contrast, the coldness of the world is caused by Demeter’s sadness – but people do not generally cool their surroundings when they are sad, and we have no way of knowing that Demeter is sad, or that she ever cools the world, other than the onset of winter itself. One could not substitute the moon for the sun in the axis-tilt story, because the position of the moon in the sky does not repeat itself once a year, and because the sun’s rays heating the Earth are integral to the explanation. Nor could one easily incorporate any stories about how the sun god feels about all this, because if the true explanation of winter is in the geometry of the Earth–sun motion, then how anyone feels about it is irrelevant, and if there were some flaw in that explanation, then no story about how anyone felt would put it right.

  The axis-tilt theory also predicts that the seasons will be out of phase in the two hemispheres. So if they had been found to be in phase, the theory would have been refuted, just as, in the event, the Persephone and Freyr myths were refuted by the opposite observation. But the difference is, if the axis-tilt theory had been refuted, its defenders would have had nowhere to go. No easily implemented change could make tilted axes cause the same seasons all over the planet. Fundamentally new ideas would have been needed. That is what makes good explanations essential to science: it is only when a theory is a good explanation – hard to vary – that it even matters whether it is testable. Bad explanations are equally useless whether they are testable or not.

  Most accounts of the differences between myth and science make too much of the issue of testability – as if the ancient Greeks’ great mistake was that they did not send expeditions to the southern hemisphere to observe the seasons. But in fact they could never have guessed that such an expedition might provide evidence about seasons unless they had already guessed that seasons would be out of phase in the two hemispheres – and if that guess was hard to vary, which it could have been only if it had been part of a good explanation. If their guess was easy to vary, they might just as well have saved themselves the boat fare, stayed at home, and tested the easily testable theory that winter can be staved off by yodelling.

  So long as they had no better explanation than the Persephone myth, there should have been no need for testing. Had they been seeking good explanations, they would immediately have tried to improve upon the myth, without testing it. That is what we do today. We do not test every testable theory, but only the few that we find are good explanations. Science would be impossible if it were not for the fact that the overwhelming majority of false theories can be rejected out of hand without any experiment, simply for being bad explanations.

  Good explanations are often strikingly simple or elegant – as I shall discuss in Chapter 14. Also, a common way in which an explanation can be bad is by containing superfluous features or arbitrariness, and sometimes removing those yields a good explanation. This has given rise to a misconception known as ‘Occam’s razor’ (named after the fourteenth-century philosopher William of Occam, but dating back to antiquity), namely that one should always seek the ‘simplest explanation’. One statement of it is ‘Do not multiply assumptions beyond necessity.’ However, there are plenty of very simple explanations that are nevertheless easily variable (such as ‘Demeter did it’). And, while assumptions ‘beyond necessity’ make a theory bad by definition, there have been many mistaken ideas of what is ‘necessary’ in a theory. Instrumentalism, for instance, considers explanation itself unnecessary, and so do many other bad philosophies of science, as I shall discuss in Chapter 12.

  When a formerly good
explanation has been falsified by new observations, it is no longer a good explanation, because the problem has expanded to include those observations. Thus the standard scientific methodology of dropping theories when refuted by experiment is implied by the requirement for good explanations. The best explanations are the ones that are most constrained by existing knowledge – including other good explanations as well as other knowledge of the phenomena to be explained. That is why testable explanations that have passed stringent tests become extremely good explanations, which is in turn why the maxim of testability promotes the growth of knowledge in science.

  Conjectures are the products of creative imagination. But the problem with imagination is that it can create fiction much more easily than truth. As I have suggested, historically, virtually all human attempts to explain experience in terms of a wider reality have indeed been fiction, in the form of myths, dogma and mistaken common sense – and the rule of testability is an insufficient check on such mistakes. But the quest for good explanations does the job: inventing falsehoods is easy, and therefore they are easy to vary once found; discovering good explanations is hard, but the harder they are to find, the harder they are to vary once found. The ideal that explanatory science strives for is nicely described by the quotation from Wheeler with which I began this chapter: ‘Behind it all is surely an idea so simple, so beautiful, that when we grasp it – in a decade, a century, or a millennium – we will all say to each other, how could it have been otherwise? [my italics].’ Now we shall see how this explanation-based conception of science answers the question that I asked above: how do we know so much about unfamiliar aspects of reality?

  Put yourself in the place of an ancient astronomer thinking about the axis-tilt explanation of seasons. For the sake of simplicity, let us assume that you have also adopted the heliocentric theory. So you might be, say, Aristarchus of Samos, who gave the earliest known arguments for the heliocentric theory in the third century BCE.

  Although you know that the Earth is a sphere, you possess no evidence about any location on Earth south of Ethiopia or north of the Shetland Islands. You do not know that there is an Atlantic or a Pacific ocean; to you, the known world consists of Europe, North Africa and parts of Asia, and the coastal waters nearby. Nevertheless, from the axis-tilt theory of seasons, you can make predictions about the weather in the literally unheard-of places beyond your known world. Some of these predictions are mundane and could be mistaken for induction: you predict that due east or west, however far you travel, you will experience seasons at about the same time of year (though the timings of sunrise and sunset will gradually shift with longitude). But you will also make some counter-intuitive predictions: if you travel only a little further north than the Shetlands, you will reach a frozen region where each day and each night last six months; if you travel further south than Ethiopia, you will first reach a place where there are no seasons, and then, still further south, you will reach a place where there are seasons, but they are perfectly out of phase with those everywhere in your known world. You have never travelled more than a few hundred kilometres from your home island in the Mediterranean. You have never experienced any seasons other than Mediterranean ones. You have never read, nor heard tell, of seasons that were out of phase with the ones you have experienced. But you know about them.

  What if you’d rather not know? You may not like these predictions. Your friends and colleagues may ridicule them. You may try to modify the explanation so that it will not make them, without spoiling its agreement with observations and with other ideas for which you have no good alternatives. You will fail. That is what a good explanation will do for you: it makes it harder for you to fool yourself.

  For instance, it may occur to you to modify your theory as follows: ‘In the known world, the seasons happen at the times of year predicted by the axis-tilt theory; everywhere else on Earth, they also happen at those times of year.’ This theory correctly predicts all evidence known to you. And it is just as testable as your real theory. But now, in order to deny what the axis-tilt theory predicts in the faraway places, you have had to deny what it says about reality, everywhere. The modified theory is no longer an explanation of seasons, just a (purported) rule of thumb. So denying that the original explanation describes the true cause of seasons in the places about which you have no evidence has forced you to deny that it describes the true cause even on your home island.

  Suppose for the sake of argument that you thought of the axis-tilt theory yourself. It is your conjecture, your own original creation. Yet because it is a good explanation – hard to vary – it is not yours to modify. It has an autonomous meaning and an autonomous domain of applicability. You cannot confine its predictions to a region of your choosing. Whether you like it or not, it makes predictions about places both known to you and unknown to you, predictions that you have thought of and ones that you have not thought of. Tilted planets in similar orbits in other solar systems must have seasonal heating and cooling – planets in the most distant galaxies, and planets that we shall never see because they were destroyed aeons ago, and also planets that have yet to form. The theory reaches out, as it were, from its finite origins inside one brain that has been affected only by scraps of patchy evidence from a small part of one hemisphere of one planet – to infinity. This reach of explanations is another meaning of ‘the beginning of infinity’. It is the ability of some of them to solve problems beyond those that they were created to solve.

  The axis-tilt theory is an example: it was originally proposed to explain the changes in the sun’s angle of elevation during each year. Combined with a little knowledge of heat and spinning bodies, it then explained seasons. And, without any further modification, it also explained why seasons are out of phase in the two hemispheres, and why tropical regions do not have them, and why the summer sun shines at midnight in polar regions – three phenomena of which its creators may well have been unaware.

  The reach of an explanation is not a ‘principle of induction’; it is not something that the creator of the explanation can use to obtain or justify it. It is not part of the creative process at all. We find out about it only after we have the explanation – sometimes long after. So it has nothing to do with ‘extrapolation’, or ‘induction’, or with ‘deriving’ a theory in any other alleged way. It is exactly the other way round: the reason that the explanation of seasons reaches far outside the experience of its creators is precisely that it does not have to be extrapolated. By its nature as an explanation, when its creators first thought of it, it already applied in our planet’s other hemisphere, and throughout the solar system, and in other solar systems, and at other times.

  Thus the reach of an explanation is neither an additional assumption nor a detachable one. It is determined by the content of the explanation itself. The better an explanation is, the more rigidly its reach is determined – because the harder it is to vary an explanation, the harder it is in particular to construct a variant with a different reach, whether larger or smaller, that is still an explanation. We expect the law of gravity to be the same on Mars as on Earth because only one viable explanation of gravity is known – Einstein’s general theory of relativity – and that is a universal theory; but we do not expect the map of Mars to resemble the map of Earth, because our theories about how Earth looks, despite being excellent explanations, have no reach to the appearance of any other astronomical object. Always, it is explanatory theories that tell us which (usually few) aspects of one situation can be ‘extrapolated’ to others.

  It also makes sense to speak of the reach of non-explanatory forms of knowledge – rules of thumb, and also knowledge that is implicit in the genes for biological adaptations. So, as I said, my rule of thumb about cups-and-balls tricks has reach to a certain class of tricks; but I could not know what that class is without the explanation for why the rule works.

  Old ways of thought, which did not seek good explanations, permitted no process such as science for correcting errors a
nd misconceptions. Improvements happened so rarely that most people never experienced one. Ideas were static for long periods. Being bad explanations, even the best of them typically had little reach and were therefore brittle and unreliable beyond, and often within, their traditional applications. When ideas did change, it was seldom for the better, and when it did happen to be for the better, that seldom increased their reach. The emergence of science, and more broadly what I am calling the Enlightenment, was the beginning of the end of such static, parochial systems of ideas. It initiated the present era in human history, unique for its sustained, rapid creation of knowledge with ever-increasing reach. Many have wondered how long this can continue. Is it inherently bounded? Or is this the beginning of infinity – that is to say, do these methods have unlimited potential to create further knowledge? It may seem paradoxical to claim anything so grand (even if only potentially) on behalf of a project that has swept away all the ancient myths that used to assign human beings a special significance in the scheme of things. For if the power of the human faculties of reason and creativity, which have driven the Enlightenment, were indeed unlimited, would humans not have just such a significance?

  And yet, as I mentioned at the beginning of this chapter, gold can be created only by stars and by intelligent beings. If you find a nugget of gold anywhere in the universe, you can be sure that in its history there was either a supernova or an intelligent being with an explanation. And if you find an explanation anywhere in the universe, you know that there must have been an intelligent being. A supernova alone would not suffice.

  But – so what? Gold is important to us, but in the cosmic scheme of things it has little significance. Explanations are important to us: we need them to survive. But is there anything significant, in the cosmic scheme of things, about explanation, that apparently puny physical process that happens inside brains? I shall address that question in Chapter 3, after some reflections about appearance and reality.