The Beginning of Infinity: Explanations That Transform the World

by David Deutsch

Scientific theories are explanations: assertions about what is out there and how it behaves. Where do these theories come from? For most of the history of science, it was mistakenly believed that we ‘derive’ them from the evidence of our senses – a philosophical doctrine known as empiricism:

Experience is indeed essential to science, but its role is different from that supposed by empiricism. It is not the source from which theories are derived. Its main use is to choose between theories that have already been guessed. That is what ‘learning from experience’ is.

Fallibilists expect even their best and most fundamental explanations to contain misconceptions in addition to truth, and so they are predisposed to try to change them for the better.

There is nothing more deceptive than an obvious fact.’

But one thing that all conceptions of the Enlightenment agree on is that it was a rebellion, and specifically a rebellion against authority in regard to knowledge.

Rejecting authority in regard to knowledge was not just a matter of abstract analysis. It was a necessary condition for progress, because, before the Enlightenment, it was generally believed that everything important that was knowable had already been discovered, and was enshrined in authoritative sources such as ancient writings and traditional assumptions.

What was needed for the sustained, rapid growth of knowledge was a tradition of criticism. Before the Enlightenment, that was a very rare sort of tradition: usually the whole point of a tradition was to keep things the same. Thus the Enlightenment was a revolution in how people sought knowledge: by trying not to rely on authority. That is the context in which empiricism – purporting to rely solely on the senses for knowledge – played such a salutary historical role, despite being fundamentally false and even authoritative in its conception of how science works.

That is to say, the theory must make predictions which, if the theory were false, could be contradicted by the outcome of some possible observation. Thus, although scientific theories are not derived from experience, they can be tested by experience – by observation or experiment. For example, before the discovery of radioactivity, chemists had believed (and had verified in countless experiments) that transmutation is impossible. Rutherford and Soddy boldly conjectured that uranium spontaneously transmutes into other elements. Then, by demonstrating the creation of the element radium in a sealed container of uranium, they refuted the prevailing theory and science progressed. They were able to do that because that earlier theory was testable: it was possible to test for the presence of radium. In contrast, the ancient theory that all matter is composed of combinations of the elements earth, air, fire and water was untestable, because it did not include any way of testing for the presence of those components. So it could never be refuted by experiment. Hence it could never be – and never was – improved upon through experiment. The Enlightenment was at root a philosophical change.

Testability is now generally accepted as the defining characteristic of the scientific method. Popper called it the ‘criterion of demarcation’ between science and non-science.

The reason that testability is not enough is that prediction is not, and cannot be, the purpose of science.

the explanation of a conjuring trick were evident in its appearance, there would be no trick. If the explanations of physical phenomena were evident in their appearance, empiricism would be true and there would be no need for science as we know it.

Knowledge that is both familiar and uncontroversial is background knowledge. A predictive theory whose explanatory content consists only of background knowledge is a rule of thumb.

For example, if we are simply curious about something, it means that we believe that our existing ideas do not adequately capture or explain it. So, we have some criterion that our best existing explanation fails to meet. The criterion and the existing explanation are conflicting ideas. I shall call a situation in which we experience conflicting ideas a problem.

I think that there is only one way to science – or to philosophy, for that matter: to meet a problem, to see its beauty and fall in love with it; to get married to it and to live with it happily, till death do ye part – unless you should meet another and even more fascinating problem or unless, indeed, you should obtain a solution. But even if you do obtain a solution, you may then discover, to your delight, the existence of a whole family of enchanting, though perhaps difficult, problem children . . . Realism and the Aim of Science (1983)

Since theories can contradict each other, but there are no contradictions in reality, every problem signals that our knowledge must be flawed or inadequate. Our misconception could be about the reality we are observing, or about how our perceptions are related to it, or both.

This is another general fact about scientific explanation: if one has a misconception, observations that conflict with one’s expectations may (or may not) spur one into making further conjectures, but no amount of observing will correct the misconception until after one has thought of a better idea; in contrast, if one has the right idea one can explain the phenomenon even if there are large errors in the data.

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.

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.

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.

Conjectures are the products of creative imagination. But the problem with imagination is that it can create fiction much more easily than truth.

The inventor Thomas Edison once said, ‘None of my inventions came by accident. I see a worthwhile need to be met and I make trial after trial until it comes. What it boils down to is one per cent inspiration and ninety-nine per cent perspiration.’

Every additional layer of physical separation requires further levels of theory to relate the resulting perceptions to reality.

‘Principle of Mediocrity’: there is nothing significant about humans (in the cosmic scheme of things). As the physicist Stephen Hawking put it, humans are ‘just a chemical scum on the surface of a typical planet that’s in orbit round a typical star on the outskirts of a typical galaxy’.

The biosphere only ever achieves stability – and only temporarily at that – by continually neglecting, harming, disabling and killing individuals. Hence the metaphor of a spaceship or a life-support system, is quite perverse: when humans design a life-support system, they design it to provide the maximum possible comfort, safety and longevity for its users within the available resources; the biosphere has no such priorities. Nor is the biosphere a great preserver of species. In addition to being notoriously cruel to individuals, evolution involves continual extinctions of entire species. The average rate of extinction since the beginning of life on Earth has been about ten species per year (the number is known only very approximately), becoming much higher during the relatively brief periods that palaeontologists call ‘mass extinction events’. The rate at which species have come into existence has on balance only slightly exceeded the extinction rate, and the net effect is that the overwhelming majority of species that have ever existed on Earth (perhaps 99.9 per cent of them) are now extinct. Genetic evidence suggests that our own species narrowly escaped extinction on at least one occasion. Several species closely related to ours did become extinct. Significantly, the ‘life-support system’ itself wiped them out – by means such as natural disasters, evolutionary changes in other species, and climate change. Those cousins of ours had not invited extinction by changing thei...

This increasingly intimate connection between explaining the world and controlling it is no accident, but is part of the deep structure of the world. Consider the set of all conceivable transformations of physical objects. Some of those (like faster-than-light communication) never happen because they are forbidden by laws of nature; some (like the formation of stars out of primordial hydrogen) happen spontaneously; and some (such as converting air and water into trees, or converting raw materials into a radio telescope) are possible, but happen only when the requisite knowledge is present – for instance, embodied in genes or brains. But those are the only possibilities. That is to say, every putative physical transformation, to be performed in a given time with given resources or under any other conditions, is either – impossible because it is forbidden by the laws of nature; or – achievable, given the right knowledge. That momentous dichotomy exists because if there were transformations that technology could never achieve regardless of what knowledge was brought to bear, then this fact would itself be a testable regularity in nature. But all regularities in nature have explanations, so the explanation of that regularity would itself be a law of nature, or a consequence of one. And so, again, everything that is not forbidden by laws of nature is achievable, given the right knowledge.

In the unique case of humans, the difference between a hospitable environment and a deathtrap depends on what knowledge they have created.

The difference between humans and other species is in what kind of knowledge they can use (explanatory instead of rule-of-thumb) and in how they create it (conjecture and criticism of ideas, rather than the variation and selection of genes). It is precisely those two differences that explain why every other organism can function only in a certain range of environments that are hospitable to it, while humans transform inhospitable environments like the biosphere into support systems for themselves. And, while every other organism is a factory for converting resources of a fixed type into more such organisms, human bodies (including their brains) are factories for transforming anything into anything that the laws of nature allow. They are ‘universal constructors’.

Note that this conclusion does not depend on the assumption that humans, or anyone, will colonize the galaxy and take control of any supernovae: the assumption that they will not is equally a theory about the future behaviour of knowledge. Knowledge is a significant phenomenon in the universe, because to make almost any prediction about astrophysics one must take a position about what types of knowledge will or will not be present near the phenomena in question. So all explanations of what is out there in the physical world mention knowledge and people, if only implicitly.

‘proxy’: a physical variable which can be measured as a way of measuring another variable.

biological adaptations are both created by evolution in the broad sense: the variation of existing information, alternating with selection. In the case of human knowledge, the variation is by conjecture, and the selection is by criticism and experiment.

But the truth is always that knowledge must be first conjectured and then tested. That is what Darwin’s theory says: first, random mutations happen (they do not take account of what problem is being solved); then natural selection discards the variant genes that are less good at causing themselves to be present again in future generations.

Is it sheer luck, then, that most genes do usually confer some, albeit less than optimal, functional benefits on their species, and on their individual holders? No. Organisms are the slaves, or tools, that genes use to achieve their ‘purpose’ of spreading themselves through the population.

Ideas can be replicators too. For example, a good joke is a replicator: when lodged in a person’s mind, it has a tendency to cause that person to tell it to other people, thus copying it into their minds. Dawkins coined the term memes (rhymes with ‘dreams’) for ideas that are replicators.

Therefore the existence of an unsolved problem in physics is no more evidence for a supernatural explanation than the existence of an unsolved crime is evidence that a ghost committed it.

It will be, as Wheeler put it, ‘an idea so simple . . . that . . . we will all say to each other, how could it have been otherwise?’ In other words, the problem has been not that the world is so complex that we cannot understand why it looks as it does, but it is that it is so simple that we cannot yet understand it. But this will be noticeable only with hindsight.

Einstein remarked, ‘There could be no fairer destiny for any physical theory than that it should point the way to a more comprehensive theory in which it lives on as a limiting case.’

the philosopher David Hume pointed out, we cannot perceive causation, only a succession of events.

as the philosopher Jacob Bronowski pointed out, success at making factual, scientific discoveries entails a commitment to all sorts of values that are necessary for making progress. The individual scientist has to value truth, and good explanations, and be open to ideas and to change. The scientific community, and to some extent the civilization as a whole, has to value tolerance, integrity and openness of debate.

Moral philosophy is basically about the problem of what to do next – and, more generally, what sort of life to lead, and what sort of world to want. Some philosophers confine the term ‘moral’ to problems about how one should treat other people. But such problems are continuous with problems of individuals choosing what sort of life to lead, which is why I adopt the more inclusive definition. Terminology aside, if you were suddenly the last human on Earth, you would be wondering what sort of life to want. Deciding ‘I should do whatever pleases me most’ would give you very little clue, because what pleases you depends on your moral judgement of what constitutes a good life, not vice versa.

The reach of ideas into the world of abstractions is a property of the knowledge that they contain, not of the brain in which they may happen to be instantiated.

The jump to computational universality should have happened in the 1820s, when the mathematician Charles Babbage designed a device that he called the Difference Engine – a mechanical calculator which represented decimal digits by cogs, each of which could click into one of ten positions. His original purpose was parochial: to automate the production of tables of mathematical functions such as logarithms and cosines, which were heavily used in navigation and engineering. At the time, they were compiled by armies of clerks known as ‘computers’ (which is the origin of the word), and were notoriously error-prone. The Difference Engine would make fewer errors, because the rules of arithmetic would be built into its hardware.

Genes are replicators that can be interpreted as instructions in a genetic code. Genomes are groups of genes that are dependent on each other for replication. The process of copying a genome is called a living organism. Thus the genetic code is also a language for specifying organisms. At some point, the system switched to replicators made of DNA, which is more stable than RNA and therefore more suitable for storing large amounts of information.

At the present state of the field, a useful rule of thumb is: if it can already be programmed, it has nothing to do with intelligence in Turing’s sense. Conversely, I have settled on a simple test for judging claims, including Dennett’s, to have explained the nature of consciousness (or any other computational task): if you can’t program it, you haven’t understood it.

Becoming better at pretending to think is not the same as coming closer to being able to think.

Whenever we refer to infinity, we are making use of the infinite reach of some idea. For whenever an idea of infinity makes sense, that is because there is an explanation of why some finite set of rules for manipulating finite symbols refers to something infinite.

Only the laws of physics determine what is finite in nature. Failure to realize this has often caused confusion. The paradoxes of Zeno of Elea, such as that of Achilles and the tortoise, were early examples. Zeno managed to conclude that, in a race against a tortoise, Achilles will never overtake the tortoise if it has a head start – because, by the time Achilles reaches the point where the tortoise began, the tortoise will have moved on a little. By the time he reaches that new point, it will have moved a little further, and so on ad infinitum. Thus the ‘catching-up’ procedure requires Achilles to perform an infinite number of catching-up steps in a finite time, which as a finite being he presumably cannot do. Do you see what Zeno did there? He just presumed that the mathematical notion that happens to be called ‘infinity’ faithfully captures the distinction between finite and infinite that is relevant to that physical situation. That is simply false. If he is complaining that the mathematical notion of infinity does not make sense, then we can refer him to Cantor, who showed that it does. If he is complaining that the physical event of Achilles overtaking the tortoise does not make sense, then he is claiming that the laws of physics are inconsistent – but they are not. But if he is complaining that there is something inconsistent about motion because one could not experience each point along a continuous path, then he is simply confusing two different things that both ha...

Computations, as understood in Turing’s theory, are essentially the same thing as proofs: every valid proof can be converted to a computation that computes the conclusion from the premises, and every correctly executed computation is a proof that the output is the outcome of the given operations on the input. Now, a computation can also be thought of as computing a function that takes an arbitrary natural number as its input and delivers an output that depends in a particular way on that input. So, for instance, doubling a number is a function. Infinity Hotel typically tells guests to change rooms by specifying a function and telling them all to compute it with different inputs (their room numbers). One of Turing’s conclusions was that almost all mathematical functions that exist logically cannot be computed by any program. They are ‘non-computable’ for the same reason that most logically possible reallocations of rooms in Infinity Hotel cannot be effected by any instruction by the management: the set of all functions is uncountably infinite, while the set of all programs is merely countably infinite. (That is why it is meaningful to say that ‘almost all’ members of the infinite set of all functions have a particular property.) Hence also – as the mathematician Kurt Gödel had discovered using a different approach to Hilbert’s challenge – almost all mathematical truths have no proofs. They are unprovable truths.

Problems are conflicts between ideas.

Moreover, recall that finding proofs is not the purpose of mathematics: it is merely one of the methods of mathematics. The purpose is to understand, and the overall method, as in all fields, is to make conjectures and to criticize them according to how good they are as explanations. One does not understand a mathematical proposition merely by proving it true. This is why there are such things as mathematics lectures rather than just lists of proofs. And, conversely, the lack of a proof does not necessarily prevent a proposition from being understood.

But there is a crucial difference between the human condition and Russian roulette: the probability of winning at Russian roulette is unaffected by anything that the player may think or do. Within its rules, it is a game of pure chance. In contrast, the future of civilization depends entirely on what we think and do. If civilization falls, that will not be something that just happens to us: it will be the outcome of choices that people make. If civilization survives, that will be because people succeed in solving the problems of survival, and that too will not have happened by chance.