Darwin's theory of evolution is an unsettling idea. It provides explanations for the existence of many features of the biological world. Not least, it provides explanations for the existence of human traits and behaviours which we typically take for granted. These sorts of explanations can threaten our ordinary self-understanding.

Normative concepts, like rationality, reason, right, and wrong do not play any essential role in the explanations offered to us by evolutionary theory. Moreover, it appears as though the very existence of our concepts of rationality, reason, right, and wrong might be susceptible to being explained – in large part – by evolutionary theory. The question then arises: are the explanations of these concepts that we can obtain from evolutionary theory capable of vindicating our ordinary practice in deploying those concepts? Having understood evolutionary theory, should we be content to employ normative concepts in much the same way that we did before? Or does an evolutionary account debunk the normative realm? Does it show that, implicit in our normative practices, there is something misguided, erroneous, or otherwise incorrect?

In trying to develop a theory of chance, I have so far drawn largely on a metaphysical picture inspired by classical physics, and have used statistical mechanics as my central example of chance arising in physical theories. It has proven difficult to develop an adequate theory of chance. But perhaps things will look different if we turn to quantum mechanics. The theory of quantum mechanics is extremely well confirmed. It is doubtful whether it is strictly true in its current form, due to the well-known conflict between quantum mechanics and the general theory of relativity, but it is an outstandingly good example of a successful physical theory, and it represents a stark break from earlier models of the physical world. Perhaps by tapping into these resources, we can develop an adequate metaphysics of chance.

The quantum mechanical world

It is widely believed that quantum mechanics is starkly opposed to classical physics, because quantum mechanics claims that the world is governed by fundamentally indeterministic laws. As it happens, this common belief oversimplifies somewhat. Quantum mechanics is a theory that is formulated in relatively mathematical terms, quite removed from concepts of directly observable physical entities. Consequently, there is a great deal of room for interpretation of the meaning of the mathematics. Indeed, there are at least three interpretations of quantum mechanics which are serious candidates for giving an adequate account of how the mathematics relates to reality.

When I began writing this book, I believed that I had identified a realist theory of chance which – though not entirely novel – had not been defended as well as it might have been. My book was to have been the definitive presentation and defence of a realist account.

Roughly six years later, I have come to appreciate much better the enormous difficulties facing not only that theory, but all realist accounts of chance, and I find myself in the mildly embarrassing position of writing the preface to a book in which I defend a modest form of anti-realism. In some sense, I now believe, Hume was correct to say that chance has no ‘real being’ in nature (Hume 1902 [1777]: §8, part I).

During this gradual conversion, becoming better acquainted with the literature, I frequently found the going rather difficult. Much of the literature is very technical, to the point of being inaccessible to many readers, including myself. This is unfortunate. Our best physical theories strongly suggest that chances are a fundamental part of reality. If we are to understand and evaluate these claims, we need to understand philosophical and scientific debates about chance. In consequence, I have written this book, not merely as a vehicle for my own ideas, but also to introduce the philosophy of chance to the broadest possible audience. While I don't pretend that the material is always easy, I expect it should at least be accessible to any tertiary-level reader.

Obviously, our knowledge of the world is quite unlike our hypothetical deity's. We don't know exactly how many particles there are. We don't know their exact properties. We don't know their exact positions.

One important way to distinguish our state of knowledge from the deity's is that for us, there are lots of ways the world might be. If we were shown the deity's memento, even if we had time to examine it, we still could not say with confidence that it is an accurate representation of the world.

This suggests a link between knowledge and possibility. Our state of knowledge is one that is compatible with a number of different ways the world might be. And this is a key difference between our state of knowledge and the state of a being who knows everything. Such a being is in a state that is compatible with only one way the world is. In this chapter, I will explore this link further, and consider how we can represent certain sorts of knowledge in terms of ways the world might be.

A multitude of lists

The deity's memento is a perfect record of the world at a particular time. It consists simply of a table that lists every particle and the physically important properties of each particle: mass, charge, position, velocity, and any others that might feature in the laws.

For a long time physicists regarded the world as conforming very closely to what I will call ‘the classical picture’. The classical picture is most importantly based upon Newtonian mechanics, a beautiful and very powerful theory which is still used to make very accurate predictions about a wide variety of phenomena. But the classical picture is not restricted solely to Newtonian theories: by this term I mean a whole family of theories which were developed in the period before Einstein, and which were in broad agreement with Newton about fundamental matters. For example, what is known as classical electrodynamics is a theory that goes well beyond Newtonian mechanics in the sort of phenomena it describes, but it is still recognisably part of the classical picture.

In the following sections, I will present a heavily simplified version of the classical picture. Many of the ideas will be familiar to many readers, and it might seem unnecessary to rehearse them. It is often unappreciated, however, that these features interact to generate an overall conception of the world. To make the classical picture vivid, I will ask you to imagine a fictitious being, much like a deity, attempting to record in perfect detail exactly how the world is at a particular time.

In the preceding chapters, we studied integrable systems and their perturbations. We noted that integrability is rare among dynamical systems, and that, while the perturbative approach is quite successful in any finite order, the perturbation series cannot be counted on to converge in the generic case. As we shall soon see, the perturbative convergence problem can be overcome if the perturbation is small enough and certain other hypotheses are satisfied, thanks to the famous theorem of Kolmogorov, Arnol'd, and Moser (KAM) [26, 27, 28]. There are several approaches (none of them easy!) to the statement and proof of this theorem. In this chapter we will rely mainly on that of [28]. A helpful discussion of the theorem, without detailed proofs, can be found in [29].

Perhaps the main message of the KAM theorem is that if we label the invariant n-tori of the unperturbed integrable model by the n oscillation frequencies ω1, …, ωn, and if the perturbation is weak enough, then a fraction, arbitrarily close to unity, of the tori will be preserved. This is the main result concerning “order” in Hamiltonian systems. No comparably strong statement exists concerning what replaces those tori which break up under the perturbation. Here we rely mainly on numerical investigations in a variety of models. These suggest certain universal features, principally island chains and deterministic chaos.

In the present chapter we will introduce the KAM theorem in the context of nonlinear stability of equilibrium states.