Intervening for the Normativity of Scientific Laws — Epoche B2
Beyond Explanation: Intervening for the Normativity of Scientific Laws Scientific laws are not merely reported [1] , they are relied on: they tell us what to expect and, more strongly, what to do. The question is where that authority comes from. The standard answer is that a law earns it by explaining — the more phenomena it subsumes, the more we ought to believe it and act on it. This essay argues that the order is the wrong way round. A law's authority comes from its licensing of interventions : from its telling us, correctly, what would happen if we changed something. The argument is not that explanation is worthless but that explanatory success is a symptom of interventional reliability rather than its source, and the way to see this is to look at cases where the two come apart and can be measured against each other. The view being opposed Two distinct positions are often run together and should be separated. David Hume's is a thesis about what a law is [2] : nothing over and above a regularity, so that "all $F$s are $G$s" states a constant conjunction and adds no necessity to it. The explanation-centric account of a law's authority is a later and separate thesis, given its sharpest form by Carl Hempel and Paul Oppenheim in 1948 [3] . On their deductive-nomological model, to explain an event is to derive a statement of it from laws together with statements of the initial conditions: $$ \{L_1,\dots,L_m\} \cup \{C_1,\dots,C_n\} \;\vdash\; E, $$ where each $L_i$ is a general law, each $C_j$ a particular fact about the case, $E$ the statement describing what happened, and $\vdash$ denotes valid deduction. The model requires the laws to be true, because a valid derivation from false premises establishes nothing. Explanatory power on this account is a matter of how much can be put on the right-hand side, and the normative force of a law follows: believe what explains most. The difficulty is that the requirement of truth is not met by the laws that do the most explaining. This is Nancy Cartwright's argument in How the Laws of Physics Lie (1983 [4] ): the fundamental laws achieve their generality by describing situations that do not occur — point masses, no friction, no other forces — and are, read as descriptions of actual systems, false. If the premises of the deduction are false, the deduction explains nothing, and the account collapses at exactly the laws it was meant to underwrite. A law with a measurable error Rather than leave "false" as a rhetorical flourish, take a law and measure its falsity. The textbook law of the simple pendulum is $$ T_0 = 2\pi\sqrt{\frac{L}{g}}, $$ with $T_0$ the period in seconds, $L$ the length in metres and $g = 9.81\ \mathrm{m\,s^{-2}}$. It contains no reference to the amplitude, and this is presented as a virtue: the period is independent of how far the bob swings. That is not true. Solving the pendulum equation without the small-angle approximation gives a period depending on the initial angular amplitude $\theta_0$: $$ T = T_0\left(1 + \frac{\theta_0^{2}}{16} + \frac{11\,\theta_0^{4}}{3072} + \cdots\right). $$ At an unremarkable amplitude of $30^\circ = 0.524\,$rad the correction is $1.74\%$. A pendulum clock built on the textbook law and swung that far would run slow by $0.0174\times86400 \approx 1500$ seconds a day, or twenty-five minutes. The law is not approximately right in any sense that would satisfy a clockmaker; it is wrong by an amount anyone would notice within an afternoon. Now notice what the same equation does when read as an instruction rather than a description. Demand a clock accurate to one second a day, a fractional error of $1/86400 = 1.16\times10^{-5}$. Setting $\theta_0^{2}/16$ equal to that and solving, $$ \theta_0 \le \sqrt{16\times1.16\times10^{-5}} = 0.0136\ \mathrm{rad} = 0.78^{\circ}. $$ Hold the swing below about three quarters of a degree and the false law becomes true to the required tolerance. This is not a philosopher's reconstruction of what clockmakers do; it is what an escapement is for. The law's authority here plainly does not rest on its being a true description of pendulums, because it is not one. It rests on its telling us exactly which quantity to control, and by how much, in order to make the world resemble it. What intervention adds that observation does not The interventionist account of causation, developed in detail by James Woodward in Making Things Happen (2003 [5] ) and given its formal machinery by Judea Pearl [6] , takes this as the definition rather than a happy side effect. $X$ counts as a cause of $Y$ relative to a set of background variables if some intervention on $X$ — a change made to $X$ alone, cutting it off from whatever normally determines it — changes the distribution of $Y$. The crucial notational move is that intervening is written differently from observing: $$ P\bigl(Y \mid do(X=x)\bigr) \ne P\bigl(Y \mid X=x\bigr) \quad\text{in general}, $$ where the left-hand side is the distribution of $Y$ when $X$ is set to $x$, and the right-hand side its distribution among cases where $X$ is found to equal $x$. That these differ is not a subtlety; it is the ordinary situation, and it can be made exact. Consider three variables related by $$ Z = \eta_Z, \qquad X = \alpha Z + \eta_X, \qquad Y = \beta X + \gamma Z + \eta_Y, $$ with $\eta_Z,\eta_X,\eta_Y$ independent noise terms of unit variance and coefficients $\alpha=1$, $\beta=2$, $\gamma=3$. Here $Z$ is a common cause of $X$ and $Y$ — a confounder. From these, $\mathrm{Var}(X) = \alpha^2+1 = 2$ and $\mathrm{Cov}(Z,X) = \alpha = 1$, so the slope obtained by regressing $Y$ on observed $X$ is $$ b_{\text{obs}} = \frac{\mathrm{Cov}(X,Y)}{\mathrm{Var}(X)} = \frac{\beta\,\mathrm{Var}(X) + \gamma\,\mathrm{Cov}(Z,X)}{\mathrm{Var}(X)} = \frac{2\cdot 2 + 3\cdot 1}{2} = 3.5, $$ whereas an intervention that sets $X$ replaces the second equation entirely, leaving $Y = \beta x + \gamma Z + \eta_Y$, so that the effect of setting $X$ is $b_{\text{int}} = \beta =