Robert W. Batterman

On the explanatory role of mathematics in empirical science

This paper examines contemporary attempts to explicate the explanatory role of mathematics in the physical sciences. Most such approaches involve developing so-called mapping accounts of the relationships between the physical world and mathematical structures. The paper argues that the use of idealizations in physical theorizing poses serious difficulties for such mapping accounts. A new approach to the applicability of mathematics is proposed. 1. Introduction2. Mathematical Explanations I: Entities3. Mathematical Explanations II: Operations4. Mapping Accounts: Strengths5. Mapping Accounts: Idealizations 5.1. Pincock and matching models5.2. Bueno, Colyvan, and the inferential conception6. Mapping Accounts: Limitations7. Suggestions for a New Approach8. Conclusion Introduction Mathematical Explanations I: Entities Mathematical Explanations II: Operations Mapping Accounts: Strengths Mapping Accounts: Idealizations 5.1. Pincock and matching models5.2. Bueno, Colyvan, and the inferential conception Pincock and matching models Bueno, Colyvan, and the inferential conception Mapping Accounts: Limitations Suggestions for a New Approach Conclusion

Minimal Model Explanations

This article discusses minimal model explanations, which we argue are distinct from various causal, mechanical, difference-making, and so on, strategies prominent in the philosophical literature. We contend that what accounts for the explanatory power of these models is not that they have certain features in common with real systems. Rather, the models are explanatory because of a story about why a class of systems will all display the same large-scale behavior because the details that distinguish them are irrelevant. This story explains patterns across extremely diverse systems and shows how minimal models can be used to understand real systems.

Idealization and modeling

This paper examines the role of mathematical idealization in describing and explaining various features of the world. It examines two cases: first, briefly, the modeling of shock formation using the idealization of the continuum. Second, and in more detail, the breaking of droplets from the points of view of both analytic fluid mechanics and molecular dynamical simulations at the nano-level. It argues that the continuum idealizations are explanatorily ineliminable and that a full understanding of certain physical phenomena cannot be obtained through completely detailed, nonidealized representations.

Asymptotics and the Role of Minimal Models

A traditional view of mathematical modeling holds, roughly, that the more details of the phenomenon being modeled that are represented in the model, the better the model is. This paper argues that often times this ‘details is better’ approach is misguided. One ought, in certain circumstances, to search for an exactly solvable minimal model—one which is, essentially, a caricature of the physics of the phenomenon in question.

Why Equilibrium Statistical Mechanics Works: Universality and the Renormalization Group

Discussions of the foundations of Classical Equilibrium Statistical Mechanics (SM) typically focus on the problem of justifying the use of a certain probability measure (the microcanonical measure) to compute average values of certain functions. One would like to be able to explain why the equilibrium behavior of a wide variety of distinct systems (different sorts of molecules interacting with different potentials) can be described by the same averaging procedure. A standard approach is to appeal to ergodic theory to justify this choice of measure. A different approach, eschewing ergodicity, was initiated by A. I. Khinchin. Both explanatory programs have been subjected to severe criticisms. This paper argues that the Khinchin type program deserves further attention in light of relatively recent results in understanding the physics of universal behavior.

Response to Belot’s “Whose Devil? Which Details?”*

I respond to Belot’s argument and defend the view that sometimes ‘fundamental theories’ are explanatorily inadequate and need to be supplemented with certain aspects of less fundamental ‘theories emeritus’.

Theories between theories : asymptotic limiting intertheoretic relations

This paper addresses a relatively common “scientific” (as opposed to philosophical) conception of intertheoretic reduction between physical theories. This is the sense of reduction in which one (typically newer and more refined) theory is said to reduce to another (typically older and “coarser”) theory in the limit as some small parameter tends to zero. Three examples of such reductions are discussed: First, the reduction of Special Relativity (SR) to Newtonian Mechanics (NM) as (v/c)2→0; second, the reduction of wave optics to geometrical optics as λ → 0; and third, the reduction of Quantum Mechanics (QM) to Classical Mechanics (CM) asħ→0. I argue for the following two claims. First, the case of SR reducing to NM is an instance of a genuine reductive relationship while the latter two cases are not. The reason for this concerns the nature of the limiting relationships between the theory pairs. In the SR/NM case, it is possible to consider SR as a regular perturbation of NM; whereas in the cases of wave and geometrical optics and QM/CM, the perturbation problem is singular. The second claim I wish to support is that as a result of the singular nature of the limits between these theory pairs, it is reasonable to maintain that third theories exist describing the asymptotic limiting domains. In the optics case, such a theory has been called “catastrophe optics”. In the QM/CM case, it is semiclassical mechanics. Aspects of both theories are discussed in some detail.

RANDOMNESS AND PROBABILITY IN DYNAMICAL THEORIES: ON THE PROPOSALS OF THE PRIGOGINE SCHOOL*

I discuss recent work in ergodic theory and statistical mechanics, regarding the compatibility and origin of random and chaotic behavior in deterministic dynamical systems. A detailed critique of some quite radical proposals of the Prigogine school is given. I argue that their conclusion regarding the conceptual bankruptcy of the classical conceptions of an exact microstate and unique phase space trajectory is not completely justified. The analogy they want to draw with quantum mechanics is not sufficiently close to support their most radical conclusion.

Chaos and algorithmic complexity

Our aim is to discover whether the notion of algorithmic orbit-complexity can serve to define “chaos” in a dynamical system. We begin with a mostly expository discussion of algorithmic complexity and certain results of Brudno, Pesin, and Ruelle (BRP theorems) which relate the degree of exponential instability of a dynamical system to the average algorithmic complexity of its orbits. When one speaks of predicting the behavior of a dynamical system, one usually has in mind one or more variables in the phase space that are of particular interest. To say that the system is unpredictable is, roughly, to say that one cannot feasibly determine future values of these variables from an approximation of the initial conditions of the system. We introduce the notions of restrictedexponential instability and conditionalorbit-complexity, and announce a new and rather general result, similar in spirit to the BRP theorems, establishing average conditional orbit-complexity as a lower bound for the degree of restricted exponential instability in a dynamical system. The BRP theorems require the phase space to be compact and metrizable. We construct a noncompact kicked rotor dynamical system of physical interest, and show that the relationship between orbit-complexity and exponential instability fails to hold for this system. We conclude that orbit-complexity cannot serve as a general definition of “chaos.”

Hydrodynamics versus Molecular Dynamics: Intertheory Relations in Condensed Matter Physics

This paper considers the relationship between continuum hydrodynamics and discrete molecular dynamics in the context of explaining the behavior of breaking droplets. It is argued that the idealization of a fluid as a continuum is actually essential for a full explanation of the drop breaking phenomenon and that, therefore, the less “fundamental,” emergent hydrodynamical theory plays an ineliminable role in our understanding.