Two Non-senses of “Probability”

This chapter engages in some ground-clearing. Two concepts have been proposed to play the role of objective probability. One is associated with the idea that probability involves mere counting of possibilities (often wrongly attributed to Laplace). The other is frequentism, the idea that probability can be defined as long-run relative frequency in some actual or hypothetical sequence of events. Associated with the idea that probability is merely a matter of counting of possibilities is a temptation to believe that there is a principle, called the Principle of Indifference, which can generate probabilities out of ignorance. In this chapter the reasons that neither of these approaches can achieve its goal are rehearsed, with reference to historical discussions in the eighteenth and nineteenth centuries. It includes some of the prehistory of discussions of what has come to be known, misleadingly, as Bertrand’s paradox.

Epilogue

This chapter reflects on the contents of earlier chapters and ties them in with other currents in the philosophy of science. These involve concepts that go beyond the familiar dichotomy of subjective and objective. A complete description of the world need not invoke concepts of this sort. They are, however, significant for agents such as ourselves, situated in a world to which we have limited epistemic access.

The Puzzle of Predictability

This chapter begins with a puzzle: how is it that reliable prediction is ever possible, in physics? The reason that this is puzzling is that, even if the systems we are making predictions about are governed by deterministic laws that are known to us, the information available to us is a minuscule fraction of what might in principle be required to make a prediction. The answer to the puzzle lies in the phenomenon of statistical regularity, first identified in the social sciences. In a sufficiently large population, reliable predictions can be made about the total number of events that, taken individually, are unpredictable. Aggregate order arises out of individual disorder. This means that, as James Clerk Maxwell perceived already in the nineteenth century, all observed regularities are statistical regularities. To understand these requires the use of probabilistic concepts. This means that probabilistic reasoning is required even in our most certain predictions. Probability permeates physics, and we are going to have to make sense of it.

Probabilities in Quantum Mechanics

This chapter examines the role played by probabilities on each of the major approaches to understanding quantum mechanics. It is argued that the sorts of considerations brought up in previous chapters, having to do with limitations on precise knowledge of physical states, and the result of applying dynamical evolution to agents’ degrees of belief about those states, have a part to play on each of those approaches. The chapter includes an introduction to the basic formalism of quantum mechanics.

What Could a “Natural Measure” Be?

The invocation of probabilistic considerations in physics often involves, implicitly or explicitly, some notion of relative sizes, or measures, of sets of possibilities. In equilibrium statistical mechanics, certain standard measures are introduced explicitly. It is often said that these measures are “natural,” in some sense. This chapter explores what that could mean. It does so by means of a toy example, a fictitious machine that I call the parabola gadget. The dynamics of the parabola gadget pick out a measure on the space of states of the gadget that other measures converge towards. In this sense, that measure is a natural one to use for systems that have been evolving freely long enough for the requisite washing-out of disagreements among input distributions to have taken place. We have good reason to think that the standard measures evoked in equilibrium statistical mechanics are of this sort One upshot of this is that this notion of standard measure is of no use for making judgments about probability or improbability of conditions in the early universe.

Epistemic Chances, or “Almost-Objective” Probabilities

In this chapter is introduced the key concept of this book, a concept that I call epistemic chance. The definition of an epistemic chance invokes two kinds of considerations. One is limitations of knowledge on the part of an agent assigning probabilities to events. The other is a physical consideration, involving the dynamics of the system under consideration. Epistemic chances, therefore, do not fall neatly into the familiar dichotomy of chance and credence. They are hybrid probabilities.

Thermodynamics: The Science of Heat and Work

This chapter provides an introduction to thermodynamics, with an emphasis on the distinction that defines the subject, the distinction between energy transfer as heat and as work. The Zeroth, First, and Second Laws of Thermodynamics are introduced, as well as the concept of entropy. The difficulties associated with the reduction of thermodynamics to mechanics are introduced, with reference to the history of their introduction. It is argued that thermodynamics must be reconceived in light of these difficulties.

Two Senses of “Probability”

The word “probability” has long been used in (at least) two distinct senses. One sense has to do with a rational agent’s degree of belief, commonly called credence in the philosophical literature. The other sort of probability is thought to be characteristic of a physical system, such as a roulette wheel; these are “in the world” rather than in our heads. This concept is called chance. It would be a mistake to think of these as rivals for the title of the single correct interpretation of probability. Rather, they are both useful concepts, with different roles to play. This chapter is an introduction to these concepts and their relations. It includes a discussion of the proper formulation and justification of a principle that links the two concepts, the Principal Principle. It is argued that neither of these concepts is dispensable. This raises the question of whether there is a notion of probability that can play the role of objective chance and is compatible with deterministic laws of physics.

Statistical Mechanics: The Basics

This chapter introduces the reader to the basics of statistical mechanics. Gibbsian and neo-Boltzmannian approaches are outlined. It includes a statistical-mechanical analogue of the second law of thermodynamics, and a proof of the Poincaré recurrence theorem. It is argued that the differences between Gibbsian and neo-Boltzmannian approaches have been exaggerated.

Probabilities in Statistical Mechanics

This chapter deals with the question of how we should think of the probabilities introduced in statistical mechanics. It is usually said that these probabilities are introduced on the basis of our ignorance of the precise state of the system, or our incapacity to treat analytically the equations of motion in order to deliver a detailed account of the evolution of the system, or both. This suggests an epistemic reading. However, we make predictions on the basis of them, that are verified by experiment. This suggests that they be thought of as objective chances. It looks as if we have to employ both the epistemic and the objective concept, in an inconsistent way. Moreover, the standard equilibrium probabilities, applied to a system that has relaxed to equilibrium while isolated from its environment, cannot be either credences or chances. It is argued that the hybrid concept, epistemic chance, resolves these puzzles and is suited to play the role required of probability in statistical mechanics.