Scientific Explanation through Models — Epoche B2
Beyond Immutable Laws: Re-evaluating Scientific Explanation through Models The Traditional View of Natural Laws and Scientific Explanation For a long stretch of the history of science and philosophy, natural laws were understood as fundamental ontological entities [1] : basic, universal, exceptionless truths existing independently of human formulation, governing all phenomena. On that view laws are not descriptions of regularities but the underlying structures that dictate how the world operates [2] , and Newton's laws of motion were read as immutable principles explaining the movement of all bodies. This understanding of laws led to a particular view of scientific explanation, articulated by Carl Hempel and Paul Oppenheim in 'Studies in the Logic of Explanation' (1948 [3] ). In their deductive-nomological model, explaining a phenomenon means deducing it from one or more general laws together with a set of initial conditions: one shows that its occurrence was to be expected. The fall of an apple is explained by deducing it from the law of universal gravitation and the conditions of release. Laws, on this picture, are the bedrock of understanding, supplying the ultimate "why" for everything observed. A Worked Case: the Ideal Gas Law The argument that follows turns on how a fundamental law behaves when confronted with a real substance, so it is worth taking one law and measuring it. The ideal gas law states $$ PV = nRT, $$ with $P$ the pressure in pascals, $V$ the volume in cubic metres, $n$ the amount of substance in moles, $T$ the absolute temperature in kelvin, and $R = 8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}$ the gas constant. It is derived from a model in which molecules occupy no volume and exert no forces on one another except in perfectly elastic collisions. Neither assumption is true of any gas. The size of the discrepancy is quantified by the compressibility factor , $$ Z \equiv \frac{P V_m}{RT}, $$ where $V_m = V/n$ is the molar volume; $Z = 1$ exactly for an ideal gas. Measure carbon dioxide at $273.15\,\mathrm{K}$ and one atmosphere and its molar volume is $22.26\,\mathrm{L\,mol^{-1}}$ rather than the ideal $22.414$, so $Z = 0.9931$. The law is wrong by $0.69\%$, at ordinary laboratory conditions, for one of the commonest gases there is. It is not wrong because the measurement was poor; it is wrong because carbon dioxide molecules attract one another and the law says they do not. The standard repair is van der Waals' equation of state, $$ \left(P + \frac{a}{V_m^{2}}\right)\left(V_m - b\right) = RT, $$ in which $b$ (units $\mathrm{m^3\,mol^{-1}}$) is the volume excluded by the molecules themselves and $a$ (units $\mathrm{Pa\,m^6\,mol^{-2}}$) measures their mutual attraction, which lowers the pressure the walls actually feel. Expanding for a dilute gas gives the leading correction in terms of the second virial coefficient $B(T)$: $$ Z = 1 + \frac{B(T)}{V_m} + \cdots, \qquad B(T) = b - \frac{a}{RT}. $$ With the constants tabulated for carbon dioxide in Atkins and de Paula's Physical Chemistry [4] , $a = 0.364\,\mathrm{Pa\,m^6\,mol^{-2}}$ and $b = 4.27\times10^{-5}\,\mathrm{m^3\,mol^{-1}}$, this gives $B = -1.18\times10^{-4}\,\mathrm{m^3\,mol^{-1}}$ and $Z = 0.9948$ — against the measured $0.9931$, an error of $0.17\%$ rather than $0.69\%$. Two extra parameters have bought a fourfold improvement. Now notice what those parameters are. They are not derived from any deeper law: $a$ and $b$ differ for every substance and are fitted, usually to the measured critical point. They do not even reproduce the quantity they were meant to explain very well — the experimentally determined second virial coefficient of carbon dioxide at this temperature is about $-1.5\times10^{-4}\,\mathrm{m^3\,mol^{-1}}$, a quarter larger in magnitude than van der Waals predicts. Nothing here is an approximation to an exceptionless truth being gradually uncovered. It is a sequence of models, each fitted to its target, each with a domain and an error bar. The Emergence of Scientific Models as Explanatory Tools A closer examination of scientific practice reveals that scientists frequently rely for explanation on entities other than universal laws: on models. A scientific model is a representation of a target system that simplifies and idealises certain aspects while omitting others. Models are not meant to be perfect replicas of reality but tools for understanding, predicting and manipulating specific features of the world. A physicist might model a planetary system as point masses interacting under gravity, ignoring their internal structure and their atmospheres. Two strategies recur. Idealisation attributes to a system properties it does not possess, or removes properties it does — frictionless planes, ideal gases, perfectly rational agents. Abstraction focuses on some features and deliberately ignores others as irrelevant to the question at hand. Both mean that a model is a constructed representation tailored to a purpose, not an unvarnished description of what there is. Scientists also make heavy use of phenomenological models, which describe observed regularities and predict well without claiming to represent fundamental mechanisms. Fourier's law of heat conduction, $q = -k\,\nabla T$, relates heat flux to temperature gradient through a thermal conductivity $k$ that is measured rather than derived, and it governed the design of steam engines and buildings for a century before anyone could compute $k$ from atomic interactions. Van der Waals' $a$ and $b$ are of the same kind. Such models are highly successful at prediction and control while articulating no fundamental law at all. Challenging the Primacy of Laws: Cartwright and Giere The pervasive and effective use of models has led philosophers of science to question the traditional view of laws. Nancy Cartwright [5] , in How the Laws of Physics Lie (1983), argues that the fundamental laws of physics are not true descriptions of what happens. Her claim is deliberat