Or, if it has two or more neighbours, ‘Because none of its neighbours ever fall.’
To be sure, this is correct as far as it goes, but it doesn’t go very far. It just passes the buck to a different domino.
In fact one could keep passing the buck from domino to domino, to provide ever more detailed answers that were ‘silly, but correct as far as they go’. Eventually, after one had passed the buck billions of times (many more times than there are dominoes, because the program ‘loops’), one would arrive at that first domino – the ‘on switch’.
At that point, the reductive (to high-level physics) explanation would be, in summary, ‘That domino did not fall because none of the patterns of motion initiated by knocking over the “on switch” ever include it.’ But we knew that already. We can reach that conclusion – as we just have – without going through that laborious process. And it is undeniably true. But it is not the explanation we were looking for because it is addressing a different question – predictive rather than explanatory – namely, if the first domino falls, will the output domino ever fall? And it is asking at the wrong level of emergence. What we asked was: why does it not fall? To answer that, Hofstadter then adopts a different mode of explanation, at the right level of emergence:
The second type of answer would be, ‘Because 641 is prime.’ Now this answer, while just as correct (indeed, in some sense it is far more on the mark), has the curious property of not talking about anything physical at all. Not only has the focus moved upwards to collective properties . . . these properties somehow transcend the physical and have to do with pure abstractions, such as primality.
Hofstadter concludes, ‘The point of this example is that 641’s primality is the best explanation, perhaps even the only explanation, for why certain dominoes did fall and certain others did not fall.’
Just to correct that slightly: the physics-based explanation is true as well, and the physics of the dominoes is also essential to explaining why prime numbers are relevant to that particular arrangement of them. But Hofstadter’s argument does show that primality must be part of any full explanation of why the dominos did or did not fall. Hence it is a refutation of reductionism in regard to abstractions. For the theory of prime numbers is not part of physics. It refers not to physical objects, but to abstract entities – such as numbers, of which there is an infinite set.
Unfortunately, Hofstadter goes on to disown his own argument and to embrace reductionism. Why?
His book is primarily about one particular emergent phenomenon, the mind – or, as he puts it, the ‘I’. He asks whether the mind can consistently be thought of as affecting the body – causing it to do one thing rather than another, given the all-embracing nature of the laws of physics. This is known as the mind–body problem. For instance, we often explain our actions in terms of choosing one action rather than another, but our bodies, including our brains, are completely controlled by the laws of physics, leaving no physical variable free for an ‘I’ to affect in order to make such a choice. Following the philosopher Daniel Dennett, Hofstadter eventually concludes that the ‘I’ is an illusion. Minds, he concludes, can’t ‘push material stuff around’, because ‘physical law alone would suffice to determine [its] behaviour’. Hence his reductionism.
But, first of all, physical laws can’t push anything either. They only explain and predict. And they are not our only explanations. The theory that the domino stands ‘because 641 is a prime (and because the domino network instantiates a primality-testing algorithm)’ is an exceedingly good explanation. What is wrong with it? It does not contradict the laws of physics. It explains more than any explanation purely in terms of those laws. And no known variant of it can do the same job.
Second, that reductionist argument would equally deny that an atom can ‘push’ (in the sense of ‘cause to move’) another atom, since the initial state of the universe, together with the laws of motion, has already determined the state at every other time.
Third, the very idea of a cause is emergent and abstract. It is mentioned nowhere in the laws of motion of elementary particles, and, as the philosopher David Hume pointed out, we cannot perceive causation, only a succession of events. Also, the laws of motion are ‘conservative’ – that is to say, they do not lose information. That means that, just as they determine the final state of any motion given the initial state, they also determine the initial state given the final state, and the state at any time from the state at any other time. So, at that level of explanation, cause and effect are interchangeable – and are not what we mean when we say that a program causes a computer to win at chess, or that a domino remained standing because 641 is a prime.
There is no inconsistency in having multiple explanations of the same phenomenon, at different levels of emergence. Regarding microphysical explanations as more fundamental than emergent ones is arbitrary and fallacious. There is no escape from Hofstadter’s 641 argument, and no reason to want one. The world may or may not be as we wish it to be, and to reject good explanations on that account is to imprison oneself in parochial error.
So the answer ‘Because 641 is a prime’ does explain the immunity of that domino. The theory of prime numbers on which that answer depends is not a law of physics, nor an approximation to one. It is about abstractions, and infinite sets of them at that (such as the set of ‘natural numbers’ 1, 2, 3, . . ., where the ellipsis ‘ . . . ’ denotes continuation ad infinitum). It is no mystery how we can have knowledge of infinitely large things, like the set of all natural numbers. That is just a matter of reach. Versions of number theory that confined themselves to ‘small natural numbers’ would have to be so full of arbitrary qualifiers, workarounds and unanswered questions that they would be very bad explanations until they were generalized to the case that makes sense without such ad-hoc restrictions: the infinite case. I shall discuss various sorts of infinity in Chapter 8.
When we use theories about emergent physical quantities to explain the behaviour of water in a kettle, we are using an abstraction – an ‘idealized’ model of the kettle that ignores most of its details – as an approximation to a real physical system. But when we use a computer to investigate prime numbers, we are doing the reverse: we are using the physical computer as an approximation to an abstract one which perfectly models prime numbers. Unlike any real computer, the latter never goes wrong, requires no maintenance, and has unlimited memory and unlimited time to run its program.
Our own brains are, likewise, computers which we can use to learn about things beyond the physical world, including pure mathematical abstractions. This ability to understand abstractions is an emergent property of people which greatly puzzled the ancient Athenian philosopher Plato. He noticed that the theorems of geometry – such as Pythagoras’ theorem – are about entities that are never experienced: perfectly straight lines with no thickness, intersecting each other on a perfect plane to make a perfect triangle. These are not possible objects of any observation. And yet people knew about them – and not just superficially: at the time, such knowledge was the deepest knowledge, of anything, that human beings had ever had. Where did it come from? Plato concluded that it – and all human knowledge – must come from the supernatural.
He was right that it could not have come from observation. But then it could not have even if people had been able to observe perfect triangles (as arguably they could today, using virtual reality). As I explained in Chapter 1, empiricism has multiple fatal flaws. But it is no mystery where our knowledge of abstractions comes from: it comes from conjecture, like all our knowledge, and through criticism and seeking good explanations. It is only empiricism that made it seem plausible that knowledge outside science is inaccessible; and it is only the justified-true-belief misconception that makes such knowledge seem less ‘justified’ than scientific theories.
As I explained in Chapter 1, even in science, almost all rejected theories are rejected for being bad explanations, without ever being tested. Experimental testing
is only one of many methods of criticism used in science, and the Enlightenment has made progress by bringing those other methods to bear in non-scientific fields too. The basic reason that such progress is possible is that good explanations about philosophical issues are as hard to find as in science – and criticism is correspondingly effective.
Moreover, experience does play a role in philosophy – only not the role of experimental testing that it plays in science. Primarily, it provides philosophical problems. There would have been no philosophy of science if the issue of how we can acquire knowledge of the physical world had been unproblematic. There would be no such thing as political philosophy if there had not first been a problem of how to run societies. (To avoid misunderstanding, let me stress that experience provides problems only by bringing already-existing ideas into conflict. It does not, of course, provide theories.)
In the case of moral philosophy, the empiricist and justificationist misconceptions are often expressed in the maxim that ‘you can’t derive an ought from an is’ (a paraphrase of a remark by the Enlightenment philosopher David Hume). It means that moral theories cannot be deduced from factual knowledge. This has become conventional wisdom, and has resulted in a kind of dogmatic despair about morality: ‘you can’t derive an ought from an is, therefore morality cannot be justified by reason’. That leaves only two options: either to embrace unreason or to try living without ever making a moral judgement. Both are liable to lead to morally wrong choices, just as embracing unreason or never attempting to explain the physical world leads to factually false theories (and not just ignorance).
Certainly you can’t derive an ought from an is, but you can’t derive a factual theory from an is either. That is not what science does. The growth of knowledge does not consist of finding ways to justify one’s beliefs. It consists of finding good explanations. And, although factual evidence and moral maxims are logically independent, factual and moral explanations are not. Thus factual knowledge can be useful in criticizing moral explanations.
For example, in the nineteenth century, if an American slave had written a bestselling book, that event would not logically have ruled out the proposition ‘Negroes are intended by Providence to be slaves.’ No experience could, because that is a philosophical theory. But it might have ruined the explanation through which many people understood that proposition. And if, as a result, such people had found themselves unable to explain to their own satisfaction why it would be Providential if that author were to be forced back into slavery, then they might have questioned the account that they had formerly accepted of what a black person really is, and what a person in general is – and then a good person, a good society, and so on.
Conversely, advocates of highly immoral doctrines almost invariably believe associated factual falsehoods as well. For instance, ever since the attack on the United States on 11 September 2001, millions of people worldwide have believed it was carried out by the US government, or the Israeli secret service. Those are purely factual misconceptions, yet they bear the imprint of moral wrongness just as clearly as a fossil – made of purely inorganic material – bears the imprint of ancient life. And the link, in both cases, is explanation. To concoct a moral explanation for why Westerners deserve to be killed indiscriminately, one needs to explain factually that the West is not what it pretends to be – and that requires uncritical acceptance of conspiracy theories, denials of history, and so on.
Quite generally, in order to understand the moral landscape in terms of a given set of values, one needs to understand some facts as being a certain way too. And the converse is also true: for example, 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.
We should not be surprised at these connections. The truth has structural unity as well as logical consistency, and I guess that no true explanation is entirely disconnected from any other. Since the universe is explicable, it must be that morally right values are connected in this way with true factual theories, and morally wrong values with false theories.
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.
This also illustrates the emptiness of reductionism in philosophy. For if I ask you for advice about what objectives to pursue in life, it is no good telling me to do what the laws of physics mandate. I shall do that in any case. Nor is it any good telling me to do what I prefer, because I don’t know what I prefer to do until I have decided what sort of life I want to lead or how I should want the world to be. Since our preferences are shaped in this way, at least in part, by our moral explanations, it does not make sense to define right and wrong entirely in terms of their utility in meeting people’s preferences. Trying to do so is the project of the influential moral philosophy known as utilitarianism, which played much the same role as empiricism did in the philosophy of science: it acted as a liberating focus for the rebellion against traditional dogmas, while its own positive content contained little truth.
So there is no avoiding what-to-do-next problems, and, since the distinction between right and wrong appears in our best explanations that address such problems, we must regard that distinction as real. In other words, there is an objective difference between right and wrong: those are real attributes of objectives and behaviours. In Chapter 14 I shall argue that the same is true in the field of aesthetics: there is such a thing as objective beauty.
Beauty, right and wrong, primality, infinite sets – they all exist objectively. But not physically. What does that mean? Certainly they can affect you – as examples like Hofstadter’s show – but apparently not in the same sense that physical objects do. You cannot trip over one of them in the street. However, there is less to that distinction than our empiricism-biased common sense assumes. First of all, being affected by a physical object means that something about the physical object has caused a change, via the laws of physics (or, equivalently, that the laws of physics have caused a change via that object). But causation and the laws of physics are not themselves physical objects. They are abstractions, and our knowledge of them comes – just as for all other abstractions – from the fact that our best explanations invoke them. Progress depends on explanation, and therefore trying to conceive of the world as merely a sequence of events with unexplained regularities would entail giving up on progress.
This argument that abstractions really exist does not tell us what they exist as – for instance, which of them are purely emergent aspects of others, and which exist independently of the others. Would the laws of morality still be the same if the laws of physics were different? If they were such that knowledge could best be obtained by blind obedience to authority, then scientists would have to avoid what we think of as the values of scientific inquiry in order to make progress. My guess is that morality is more autonomous than that, and so it makes sense to say that such laws of physics would be immoral, and (as I remarked in Chapter 4) to imagine laws of physics that would be more moral than the real ones.
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. A theory
can have infinite reach even if the person who originated it is unaware that it does. However, a person is an abstraction too. And there is a kind of infinite reach that is unique to people: the reach of the ability to understand explanations. And this ability is itself an instance of the wider phenomenon of universality – to which I turn next.
TERMINOLOGY
Levels of emergence Sets of phenomena that can be explained well in terms of each other without analysing them into their constituent entities such as atoms.
Natural numbers The whole numbers 1, 2, 3 and so on.
Reductionism The misconception that science must or should always explain things by analysing them into components (and hence that higher-level explanations cannot be fundamental).
Holism The misconception that all significant explanations are of components in terms of wholes rather than vice versa.
Moral philosophy Addresses the problem of what sort of life to want.
MEANINGS OF ‘THE BEGINNING OF INFINITY’ ENCOUNTERED IN THIS CHAPTER
– The existence of emergent phenomena, and the fact that they can encode knowledge about other emergent phenomena.
– The existence of levels of approximation to true explanations.
– The ability to understand explanations.
– The ability of explanation to escape from parochialism by ‘letting our theories die in our place’.
SUMMARY
Reductionism and holism are both mistakes. In reality, explanations do not form a hierarchy with the lowest level being the most fundamental. Rather, explanations at any level of emergence can be fundamental. Abstract entities are real, and can play a role in causing physical phenomena. Causation is itself such an abstraction.
6
The Jump to Universality
The Beginning of Infinity Page 15