The origins of this book date to a conversation between the authors a short time before both were due to (formally) retire. Sadly little had been achieved before the more experienced author died. As a consequence any shortcomings of the book must be attributed to the surviving author.

Exact solutions of any system of partial differential equations attract attention. This must be particularly true of the Navier–Stokes equations which, for the best part of 200 years, have been the foundation for the significant and worldwide study of the behaviour of fluids in motion. The subject burgeoned in the twentieth century from stimuli as diverse as international conflict, and a desire to create a better understanding of the environment. In the nineteenth century theoretical advance was slow, and until the approximate or, as we would rather view them, asymptotic theories of Stokes and Prandtl for small and large values of the Reynolds number were devised, only exact solutions, and few at that, were available. In spite of the advances in asymptotic methods during the first half of the twentieth century, and the increasing use of computational methods in its later decades, exact solutions of the Navier–Stokes equations have been pursued. At best these provide an insight into the behaviour of fluids in motion; they may also provide a vehicle for novel mathematical methods or a useful check for a computer code. Some, it must be admitted, provide little of value in either of these senses.

This monograph elaborates a fundamental topic of the theory of fluid dynamics which is introduced in most textbooks on the theory of flow of a viscous fluid. A knowledge of this introductory background, for which reference may be made to Batchelor (1967), will be assumed here. However, it will be helpful to summarise a little of the background wherever we need it. In particular, we begin by introducing the scope of the book by loosely defining the terms of the title.

The Navier–Stokes equations are the system of non-linear partial differential equations governing the motion of a Newtonian fluid, which may be liquid or gas. In essence, they represent the balance between the rate of change of momentum of an element of fluid and the forces on it, as does Newton's second law of motion for a particle, where the stress is linearly related to the rate of strain of the fluid. Newton himself did not understand well the nature of the forces between elemental particles in a continuum, but he did (Newton 1687, Vol. II, Section IX, Hypothesis, Proposition LI) initiate the theory of the dynamics of a uniform viscous fluid in an intuitive and imaginative way. It was many years later that the Navier–Stokes equations, as we now know them, were deduced from various physical hypotheses, and in various forms, by Navier (1827), Poisson (1831), Saint-Venant (1843) and Stokes (1845).

Science and natural philosophy largely abandoned ideas about parallel worlds of mind and matter in the years following Descartes and his dualistic philosophy. By the twentieth century, most of science exhibited an unhesitant materialistic metaphysics. The present investigation occasions an opportunity to reexamine ideas about materialism.

What is materialism?

The standard conception of materialism is the thesis that all events in the world consist of ordinary physical matter, energy, and other physical properties, denying the existence or causal influence of other things. It does not deny the possibility of using nonphysical properties to characterize physical things; civilization's use of numbers to quantify physical dimensions would suffer greatly were this so. But it does deny that these nonphysical characterizations play any physical role.

One should note that materialism exhibits an open-ended character. When philosophers first bruited materialism, it referred to everything being the tangible, visible stuff of the world. Eventually this conception required enlargement to include the invisible, intangible stuff—energy, electromagnetic fields, spin, neutrinos—that later physics developed as physical entities or properties, even though some of these are far removed from the direct experience characteristic of the original conceptions of physical materials.

Mechanics has enjoyed some four centuries of sustained development without producing results in psychology or economics. The mental sciences have enjoyed a couple centuries of sustained development without requiring mechanical intervention. To use the standard economic argument, if there was a connection worth pursuing, would not one have already been made?

In fact, people have made numerous attempts at connecting mechanics and mind. Although those attempts at establishing such connections have failed, there are identifiable changes in scientific circumstances that explain why a mechanical approach to psychology and economics should prove more fruitful now.

To see the reasons for the lack of successful connections in the past, this chapter examines some of the difficulties prevailing at earlier times and how they have undercut historical attempts at connecting physics and psychology. Readers wishing to proceed to mechanics proper can skip ahead to Chapter 4 or Chapter 5 without loss of understanding.

Impediments to understanding

Why have the mental sciences lagged the physical so markedly? The answer could involve social factors, such as the stimulus to physical discovery made by war and trade, but one might expect that discoveries about the mind might benefit these activities to some extent as well, as was assumed by Joseph Göbbels and is known by advertising agencies today.

As was noted earlier, the traditional conception of what we call mechanical computation or computation by machine relies on a purely kinematical conception of mechanics. It entirely omits any notion of force and focuses attention only on abstract states and motion between them. In this it follows a trend in mechanical formalism that moved away from considering forces and spatial motions to considering mainly Hamiltonian motion through abstract spaces, with no mention of either the central notion of force or the key notion of mass (cf. Hermann 1990, Sussman & Wisdom 2001).

This disconnect between mechanical computation and mechanics comes closest to being bridged in the related field of information theory, in which some authors have viewed information content as a type of mass measure (Manthey & Moret 1983) and have produced formal relations between information content and thermostatic theories of entropy (Chaitin 1975). These ropes tossed across the gap lack tether to the notion of force and still leave the crossing perilous.

Let us now reconsider the notion of computation from the mechanical point of view, to treat “mechanizability”—viewed in terms of machines—as mechanizability—viewed in terms of mechanics. We seek to understand the notion of effectiveness as involving not just abstract kinematics but also those fundamental concepts that distinguish mechanics from geometry, especially the concepts of rate of motion limited by limits on force and bounds on the rate of work.

Discussions in previous chapters have touched on these ideas already.

The preceding treatment of reasoning indicates how we can interpret psychological rationality in terms of mechanical processes. Let us now look at the ways in which mechanical concepts enter into characterizing forms of economic rationality.

Limits on rationality

The difficulty and slowness with which real agents change their mental state constitutes one of the most evident limitations on rationality. As noted earlier, we can see reflections of the mechanical connection between momentum and force in “the more you need to change, the more you have to force yourself,” “the more you know, the harder it is to change your mind,” and other truisms of popular psychology. We can read the first of these truisms as stating a monotonicity relation between the size of changes and the size of the required forces and work done, and the second as stating a monotonicity relation between the mass and the force required for given changes. Notions of monotonicity and proportionality among the numerical magnitudes of momentum and force are familiar in traditional mechanics, but how do these apply in the discrete mechanical setting?

A mechanical interpretation of thinking also naturally relates slowness of change to inertia. From the same perspective, the unreality of ideal rationality appears because when we determine actions by finding the maxima of an expected utility function generated by instantaneous beliefs and desires, large changes can come from small impulses.

The preceding chapters presented the beginnings of a mathematical and mechanical theory of mind.

We began by examining the curious divorce between mechanical understandings of mind and nature that occurred when natural philosophy developed mathematical techniques useful in characterizing physical mechanics but inapplicable to mental mechanics. The mathematical study of mental materials developed separately, but with the key mathematical theories of logical and economic rationality lacking any connection to mechanics. The mechanical reconciliation of mind and nature began to take shape only when the development of artificial computers enabled construction of artificial minds precise and concrete enough to relate to a new rational mechanics broad enough to encompass mental as well as physical materials. The reconciliation promises not only to open traditional philosophical questions to new forms of technical analysis, but also to provide a new formal vocabulary for describing agents of limited rationality and for engineering computational and social systems based on such agents.

We then examined two sides of the reconciliation of physical and mental mechanics. On the physical side, we recast the axioms of modern rational mechanics so as to cover discrete mechanical systems and their hybrids with physical mechanical systems.