# Models and the dynamics of theory-building in physics. Part II—Case studies

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Studies in History and Philosophy of

Modern Physics 38 (2007) 683723

1. Introduction

Denition 1.1.1. H- modeling aims to explore the connections to be established between thecoreespecially the physical postulatesand its physical semantic relevance: in particular

ARTICLE IN PRESS

www.elsevier.com/locate/shpsb1355-2198/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.shpsb.2006.10.005

E-mail address: gge@math.u.edu.1Permanent address: Department of Mathematics, University of Florida, Gainesville FL 32611-8105, USA.1.1. Summary of Part I

Considering the view that scientic theories are developing as the nodes of Quines webof knowledge, I proposed that these nodes have a structure which I compared to that of acell, composed as they are of a core surrounded by experiments and models. I focused onthe role of modeling strategies as agents of change. I claimed they provide much of thedynamics governing the cells evolution, especially before it becomes rmly embedded inQuines web as a closed theory. For this process to happen, I specied two mainmodeling strategies from which the models proceed. I advanced the following denitions.Models and the dynamics of theory-building inphysics. Part IICase studies

Gerard G. Emch1

All Souls College, University of Oxford OX1 4AL, UK

Received 4 April 2005; received in revised form 5 October 2006; accepted 10 October 2006

Abstract

In Part I, it was argued that models are best explained by considering the strategies from which they

issue. A distinction was proposed between two classes of modeling that contribute to theory-building: H-

modeling and L-modeling. Case studies are presented in this Part II to illustrate the characteristic features

of these modeling strategies; examples are drawn from classical statistical mechanics and quantum physics.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Epistemology; Models; Theory construction

the observability of its concepts and their adequacy to describe the world as apprehendedby laboratory experiments.

Denition 1.1.2. L- modeling is designed to test the correctness and economy of the syntax,the logical consistency and independence of its axioms, the formal value of its assertions

ARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723684and theorems.

Denition 1.1.3. The products of these two modeling activities are called respectivelyH-models and L-models.

Denition 1.1.4. By default, a mere model designates an H-model or an L-model separatedpurposefully or notfrom the modeling strategy that was followed to construct it.

1.2. Methodology for Part II

In this second part of the paper, I present case studies that show how the different typesof modeling strategies effectively contribute to the evolution of a cell. In choosing whichmodels to include, I kept in mind three constraints; namely, the illustrations ought to be:(a) familiar and accessible; (b) detailed enough for their roles in H- and L-modelings to beconvincing; (c) varied enough to indicate how extended my project is intended to be.I collect my illustrations in four sequences belonging each to a different eld of physics.

More models will be briey listed in the Appendix according to the strategy from whichthey proceed.

2. From equilibrium thermodynamics to statistical mechanics

This section might also be entitled: roles that modeling played in the beginnings of thereduction of thermodynamics to statistical mechanics.

2.1. Empirical presentation of the ideal gas

I designate by ideal gas what is also called the perfect gas, to emphasize that it is anidealized description of a gas in thermodynamical equilibrium, namely

pV NmoleRT , (2.1.1)where p; V ; T label pressure, volume and temperature; Nmole measures the quantity of thegas,2 and R is an universal constant.3 In particular, note the proportionality of increments

a along isotherms i:e: at constant temperature : Dp aDr with r 1V

b along isochores i:e: at constant volume : Dp aDT

9=;.(2.1.2)

2The modern denition is: a mole is the amount of a substance that contains as many elementary particles

(atoms, molecules, ions, etc.) as the number of carbon atoms in 12 g of C12 (carbon-12). The number of carbon

atoms contained in 12 g of C12 is approximately 6:02 1023; this is called Avogadros number. A mole, then, is anamount of any substance that weighs, in grams, as much as the numerically equivalent atomic weight of that

substance.

3The modern value of R is 8:3143 107 erg deg1 mol1.

Empirical background: Boyle (1660, 1662) and Mariotte (1679) performed crucialexperiments on air, the results of which they described by writing that underotherwise similar circumstances the pressure and volume of a gas vary only in such a

wh

2.2

The rst facet is its purpose as an explanatory activity: explaining the observable in

ARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723 685terms of the unobservable, the observable phenomena being described by relations (2.1.1)or (2.1.2), while the unobservable is the putative existence of the particles constitutingthe gas. The other facet is the exploration of the unobserved itself, through itsconsequences on the observed. As these specic models unfold, I will point outthe contributions each makes to the development of a core, the microscopic viewof matter.

4This was valid until signicant deviations were systematically observed by Andrews (1869); see Section 2.5.5This arbitrariness later found a justication in the extensive observations of Gay-Lussac (1802, 1807) and the

theoretical considerations of Fourier (1822) and Thomson (1848). Mentioning these here would mask thegashisis idealization.

. Microscopic modelings for the ideal gas

The following three modelings explore possible microscopic structures of the idealin terms of the mechanics of material points. They exhibit two facets of H-modeling.prothontons thermometer is linear in T . This completes the description of the relation 2.1.1ich gives the empirical, macroscopic description of the ideal gas.The purposes of this section are: (i) to present the microscopic modeling strategiesmpted by this empirical description; and (ii) to indicate how real gases may depart fromconAmway that the product pV remains constant.4 To specify the operational meaning of whatBoyle and Mariotte perceived as similar circumstances requires one to postulate thatthese could be characterized by a parameter having to do with what one calls todaytemperature. The delineation of this notion of temperature involves several issues in thehistory and philosophy of science (e.g. Chang, 2004). Here, I focus on the followingdevelopments.

1. The construction of thermometers, measuring the expansion of diverse substances,gaseous air with Amontons (1702) and liquid mercury with Fahrenheit (1724); and thecalibration of these thermometers, resulting from Fahrenheits remark that Amontonshad discovered that water boils at a xed degree of heat (Magie, 1969, p. 131) andthat the same holds for other liquids as well, each with its characteristic boilingtemperature.

2. The discoveryor was it an invention?of Newtons (1701) cooling equation_T / T Tu, the universality of which is predicated on the assumption that thetemperature is dened up to a change of scale T ! aT Tob where a; b and To areuniversal constants.

3. The empirical fact that Nmole can be accounted for in units adapted to the gasconsidered in such a way that R becomes a universal constant.

Once one accepts these observations/idealizations, one can arbitrarily choose5 thevenience of a temperature scale in which a 1; and To 0 with b 1, so that thetorical context of discovery for the modeling strategies to be described in the next subsection.

2.2.1.

ThPrinc

does

ARTICLE IN PRESSG.G. Emch / Studies in History and Philosophy of Modern Physics 38 (2007) 683723686that the density of air is either exactly or at least very nearly as the compressing force. (p.696) This is indeed (2.1.2(a)). As a microscopic model for this empirical relation, Newtonoffers an assembly of particles that are repelled from one another by forces that areinversely proportional to the distances between their center. (p. 697) He then purports toshow that these particles constitute a uid whose density is proportional to thecompression. (p. 697)

Remarks 2.2.1. (1) As detailed in Guicciardini (2003), Newton argues in a heuristicmanner, characteristic of Euclid: Newton states rst what he wants to prove; then hemakes the assumptions that would allow him to prove it; and nally he derives what hesaid he would prove.(2) Note the semantic distinctionquite remarkable for his timethat Newton makes

between the proposition he has just demonstrated and the world out there: he commentsthat he is satised with his mathematical argument, and that . . . whether elastic uidsconsist of particles that repel one another is, however, a question for physics. (Newton,1726, p. 699) Hence, Newton knows when he does what I call H-modeling, and he warns usof the limits such an exploration necessarily involves.(3) This model marks a rst systematic step in what became later the core of a

programme: reducing thermodynamics to some form of mechanics.(4) Yet, Newton models a gas constituted of particles at rest, i.e. in a state of static

equilibrium. This strategic option did not survive subsequent modelings of the phenomena.(5) Moreover, Newtons hypothesis on the space-dependence of the interparticle forces

did not survive either. Sometimes a models value lies less in its immediate explanatorypower than

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