Abstract
Virtually all philosophers of science have construed fundamental theories as descriptions of entities, properties, and/or structures. Call this the “descriptive-ontological” view. I argue that this view is incorrect, at least insofar as physical theories are concerned. I propose a novel construal of theories that I call the “prescriptive-dynamical” view. The central tenet of this view, roughly put, is that the essential content of fundamental physical theories is a prescription for interfacing with natural systems and translating local data into compact theoretical language. The descriptive-ontological aspects of theories, if any, are taken as inessential content on this view: they do not contribute to the predictive success of the theory. Rather than describing what is there, the essence of a physical theory is to tell us what to do when interfacing with a physical system.
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Notes
The notion of essential content is akin to Kitcher’s working posits (1993, 149 ff.) and Psillos’s “indispensable” or “essential contributors” (1999, 108 ff.), which are characterized as those parts of the theory that actually contribute to its empirical success, as opposed to any idle baggage that might come with the theory. See Sest. 6.1 below for further discussion.
Some constructs that may be classified as “theories”, such as special relativity and statistical mechanics, are not dynamical theories and thus do not fall under the two-tier analysis. However, this is not a problem for my account, which concerns predictive theories. Statistical mechanics is a mathematical tool for bridging micro- and macro-theories, and any prediction that may be derived from it must originate in the dynamics of the micro-theory, rather than the posits of statistical mechanics itself. The case of special relativity is more complicated, but the sketch of a response can be provided as follows. Special relativity may be considered as a set of geometrical coordination rules, i.e. Lorentz transformations (which can still be taken as prescriptive though not open-ended) for aligning measuring rods and clocks in a world in which the maximum possible speed is some constant c. This theory makes no predictions of its own unless c is specified. Now, the statement that c is finite and equals the speed of light may be seen as external to the coordination rules of special relativity, either as a directly-verifiable empirical fact or as derived from Maxwell’s dynamical equations. And as said, the light postulate is the sole source of predictive power in special relativity, to the extent that such power exists. Therefore, statistical mechanics and special relativity are not autonomous theories and yield no predictions of their own unless supplemented with dynamical theories. As such, they do not undermine my account.
Note also that sometimes a theory has both a dynamical formulation (with states evolving into other similar states) and a non-dynamical formulation. For example, General Relativity is sometimes presented as a yes/no function that takes a specified spacetime and matter distribution and tells us whether that combination is a physically possible universe. That is not a dynamical theory. But GR can also be formulated as a constrained Hamiltonian system, which takes states of the form \((g^{ij}, \pi ^{ij})\) to other states of the same form, where \(g^{ij}\) is the 3-metric and \(\pi ^{ij}\) its conjugate momentum (Lagrangian mechanics with its variational and dynamical formulations is another example).
Cartwright has argued for the malleability of phenomenological laws but she, too, considers fundamental laws to be fixed. My claim is precisely that the latter are partially open-ended. More on this below.
I will use “auxiliaries” to refer to background facts and theories of instrument, and “correspondence rules” to broadly mean any set of principles that connect theoretical terms to observable and/or measurable quantities (Halvorson 2016, 6).
Hempel proposes the following schema for such predictive inferences:
\(C_1\), \(C_2\), ..., \(C_k\)
\(L_1\), \(L_2\), ..., \(L_r\)
——————
E
Here, \(C_1\), \(C_2\), \(C_k\) are statements of particular occurrences (e.g., of the position and momenta of certain celestial bodies at a specified time), and \(L_1, \ldots , L_r\) are general laws (e.g., those of Newtonian mechanics); finally, E is a sentence stating whatever is being explained, predicted, or postdicted. (Hempel 1958, 37–38)
Both assumptions are clear in this passage: the theory is i) a set of “general laws” that ii) leads to empirical predictions in conjunction with “particular occurrences” such as initial conditions.
That is, unless one has a way of calculating the secondary formula from microscopic equations that govern the particles in the spring, but that is more wishful thinking than anything related to the actual practice of physics.
This is not to deny that such accounts can be useful in discussing theoretical change over time.
Halvorson does criticize both syntactic and semantic views for taking theories to be “flat” rather than “structured”. However, what Halvorson means by “structure” here is inferential hierarchy, which involves commitments to what is a logical consequence of what (including what is analytic and what is synthetic), and so on (Halvorson 2012, 2016). But Halvorson’s “structured” account still takes theories to be fixed.
While it is true that Giere talks about theories as “rules” for constructing concrete representational models, and that might sounds quite similar to “prescriptions”, this talk for Giere is not specific to the fixed part of the theory: he considers the entire conjunction of the fixed and movable part as a set of rules for creating fully specified models of particular systems.
The rules instruct one to locate the relevant masses and forces, and then to equate the product of the mass and acceleration of each body with the force impressed upon it. With luck one can solve the resulting equations of motion... . (Giere 1999, 94-95)
Giere’s “rules” clearly encompass Newton’s laws of motion plus a specific force function. My “prescriptions”, by contrast, explain precisely how the latter is constructed from the combination of the former and local data in the first place.
Halvorson makes this more precise in what he calls “pointwise isomorphism of models”: “\(\mathcal {M}\) is the same theory as \(\mathcal {M}'\), just in case there is a bijection \(F: \mathcal {M} \rightarrow \mathcal {M}'\) such that each model \(m \in \mathcal {M}\) is isomorphic to its paired model \(F(m) \in \mathcal {M}'\). (Halvorson 2012, 190).
There have been important recent developments in the modeling literature that add complexity to the traditional semantic accounts regarding the relationship between theory, model, and experiment (see: Morgan and Morrison 1999). However, all such nuanced views still fail to note the open-ended character of theories in the sense defended here. See, e.g., ibid, 3-4, for an endorsement of theoretically fixed models.
This is of course assuming that one has not previously constructed a suitable LEMP for the same system and the same context of interaction such that one already expects \(f(x, \dot{x}, t, \ldots )\) to be a certain way).
Since LEMPs are abundant, the only reason to abandon a theory in this framework is if the state assignments of the theory become inapplicable to the system at hand. A thorough discussion of this condition is beyond the scope of this paper, but an illustration can be provided using the case of quantum mechanics.
One might begin by asking: if LEMPs are abundant, could we not find a modified force formula that would allow \(F=ma\) to fit quantum phenomena? The answer is of course yes, we can. That is precisely what Bohmian quantum mechanics accomplishes. In Bohmian mechanics, one adds a “quantum potential” Q to the classical potential energy function V, and calculates the force as the gradient of this augmented potential function. The addition of Q accounts for all strange quantum behavior, evidenced by the fact that Bohmians can draw continuous trajectories for particles in a double-slit experiment.
So, then we ask: why do physicist consider quantum phenomena a reason to abandon \(F=ma\) rather than modify the force formula? The prescriptivist answer would be that \(F=ma\) has lost operational relevance: the state assignments of classical physics (namely x and \(\dot{x}\)) are still theoretically possible but no longer operationally assignable to quantum systems. This is because the correlations baked into the quantum potential Q make it impossible to prepare or detect the system in an arbitrary \((x, \dot{x})\) pair: attempting to prepare or detect the system in a particular x causes one to lose control over \(\dot{x}\) and vice versa. So, while \(F=ma\) is not technically “refuted” by quantum phenomena, it has lost its experimental relevance. This is why many practicing physicists state that Bohmian mechanics has no empirical relevance—a statement that makes perfect sense from a prescriptivist standpoint but is difficult to understand descriptively, given that Bohmian mechanics captures all of the same empirical data as its standard alternative.
In short, physicists switch to a new theory when the old state assignment rules become operationally inapplicable to the system. This can happen despite the fact that a modified LEMP can still be found for the theory to “save the phenomena”.
For a derivation, see Brown [manuscript].
Mathematically-minded readers would disagree that Lagrangian and Hamiltonian mechanics assign the same states, for the state-spaces of the two theories have different geometrical structure (one is a tangent bundle, the other a symplectic manifold). For extensive discussion of this issue see: North (2009) and Curiel (2014). However, from a prescriptive standpoint, the two theories yield the same equations of motion whenever they are both applicable. The two state assignments are thus dynamically equivalent.
The other requirements are similarly discarded in favor of predictive power. The requirement that forces be compliant with Newton’s third law (action and interaction) would exclude constraint forces (Wilson [manuscript], 5) and arguably electromagnetic forces (assuming a vectorial form of the third law); demanding an identifiable source entity for all forces disqualifies entropy and viscous forces (Wilson 2018, 26–27); the requirement that forces be conservative leaves out dissipative forces; and so on.
ABUNDANCE is in direct contrast to Giere’s views (see above), for according to ABUNDANCE there is a set of necessary and sufficient conditions for what secondary formulae are allowed: namely that they be fitted to local data. Moreover, ABUNDANCE implies that new models should not be required a priori to have any sort of “family resemblance” to existing ones. In other words, the descriptive attributes of LEMPs are not treated by the scientist as global constraints, but as local facts of experience. Hence the adjective “local” in “local empirical mediating principles”.
Broadly speaking, there have been two species of local philosophy of science: methodological and semantic/ontic. The methodological variety—i.e. the notion that different areas of science require different heuristics and principles of reasoning—includes Cartwright’s “dappled world” (Cartwright 1999), Norton’s material theory of induction (Norton 2016) and Wimsatt’s “piecewise approximations” and “heuristic strategies” (Wimsatt 2007). The semantic/ontic version of locality—i.e. the idea that theoretical descriptions of the world aquire different meaning or ontological character in different local contexts—includes Norton’s arguments against causal fundamentalism (Norton 2003, 2008), Wilson’s arguments that concepts of classical mechanics often have “wandering significance” which allows them to bridge different “patches” of applied mathematics (Wilson 2008, 2018), Ruetsche’s “locavore” ontology of quantum field theories (Ruetsche 2015, 2016), as well as Lloyd’s “nesting models” account of evolutionary biology (Lloyd 1988[1988], Lloyd 2013). Of interest to my project is the semantic/ontic variety, which can be easily accommodated in a prescriptive-dynamical framework.
To be fair, Ruetsche comes closest, for instance in her occasional reference to the cannonical commutation relations as “recipes” (Ruetsche 2015, 3437 ff.). Nevertheless, she remains committed to descriptivism, evidenced by her commitment to Fine’s Natural Ontological Attitude, which still takes theories as guides to ontology, notwithstanding the emphasis on the multiplicity of local ontologies (see ibid, 3426). I should recognize the sole exception at this point: in The Quantum Revolution in Philosophy (Healey 2017), Richard Healey explicitly argues against descriptive readings of quantum mechanics and uses prescriptive language to interpret quantum mechanical state assignments. Healey’s account is the closest to mine, with two differences. Firstly, Healey’s are prescriptions of credences. To be sure, the credences are derived from the state assignments, which are prescriptions in my view. Nevertheless, Healey is highlighting a second sense in which quantum mechanics is prescriptive, one compatible with but slightly different from mine. Secondly, Healey’s prescriptiveness is unique to quantum mechanics, whereas mine is more general. (Prof. Healey has expressed his agreement with the foregoing assessment of the relationship between his view and mine in personal communication.) Nevertheless, Healey provides the essential roadmap for dispelling oneself of (quantum) descriptivism. More on this in Sest. 6.1.
I do not mean to claim that the inventory of the fundamental building blocks of nature itself is inexhaustible (I doubt that we can have such knowledge one way or another). Rather, my claim is that our theories do not tell us when this inventory has been exhausted.
I do not mean to imply that the fixed formulae are “a priori” in Friedman’s neo-Kantian sense (Friedman 2001). The master formulae are not conditions for the possibility of empirical knowledge, although they are instructions for constructing LEMPs.
The same goes for non-classical theories. In Quantum Field Theory, for instance, one holds the canonical commutation relations of fermionic and bosonic fields fixed, and experimentally reads off interaction Lagrangian terms of the form \(\mathcal {L}_\text {int} = -i e \bar{\psi } \gamma _\mu A^\mu \psi\) for various species of particles and the interactions they participate in. Thus, there is no theoretical prediction for the number, form, or strength of terms in the interaction Lagrangian: the number of generations of fermions and the existence of right-handed neutrinos, for instance, are open-ended in the Standard Model. Each interaction term corresponds to a particular set of experimental findings. This means that if new transition amplitudes—e.g. corresponding to some of the many versions of the inflaton field—turn up in future experiments, we must simply accept them into the family. However, the Standard Model does present a potential trouble case for my view, insofar as the Weinberg-Salam electroweak Lagrangian (a LEMP) predicted the existence of the Higgs field. A satisfactory treatment of this case is beyond this paper, but my short response is as follows: sometimes the state assignments plus some empirical data imply certain coherence conditions which essentially state that either the transition rules must obey certain constraints or else the state assignments lead to contradiction. It is therefore not precisely theoretical considerations that fixed the electroweak Lagrangian, but logical and empirical ones, namely the fact that the previous LEMPs required gauge invariance for the theory to be coherent, and guage invariance required the Higgs field.
What if the “new system” can be modeled as composed of several component systems with known LEMPs? Could one deduce the LEMP of the total system from those of the components? I do not deny the possibility of educated guesses about the LEMP of the total system; but these guesses inevitably hinge on certain substantive assumptions about compositionality/additivity of the components’ LEMPs. However, compositionality and additivity are ultimately empirical assumptions, i.e. it is ultimately up to experience to decide whether LEMPs are holistic or not. As such, it would not be a violation of Newtonian mechanics if the force formula for a three-body gravitational system were different from the sum of pair-wise gravitational forces.
Thanks to an anonymous referee for discussion of this objection.
Note: I use “transition” in a more general sense than change in time. The “transitions” of Poisson’s equation (\(\nabla ^2 \phi = f(x, y, z)\)), for example, are purely spatial. Thus, the transition operator may well be \(D_{\delta x}\), etc.
This argument is independent of where one falls on the debate about the (in)equivalence of Newtonian, Lagrangian, and Hamiltonian mechanics, for the point can be made about a specific system or group of systems to which more than one formulation of classical mechanics applies.
How about the existence of masses or various families of particles? For instance, doesn’t the Standard Model of particle physics contain the claim that there are massive vector bosons such as the Z and W? To see if this is the case, first of all note that the Standard Model is a LEMP in my terminology for which the canonical commutation relations of quantum field theory operate as the fixed master formulae. Quantum field theory in itself does not imply the existence of anything; it merely lays out a number of ways one could assign quantum field operators to systems. Thus, even if the Standard Model implies the existence of various kinds of fields/particles, that is consistent with the prescriptive-dynamical framework, insofar as the latter locates descriptive content in the local LEMPs.
But regardless, the appearance that each kind of quantum field represents a specific kind of entity is misleading anyway. A massive vector boson, for example, is replaced through the Higgs mechanism with two massless bosons (one vectorial and one scalar), without affecting the resulting equation of motion. If descriptive claims that such and such particles of such and such masses exist were essential to the Standard Model or quantum field theory, either the same equation of motion could not be derived without them, or else one would face a vexing underdetermination problem.
Once again, this is unproblematic from a dynamical point of view, for the latter construes each type of field operator (scalar, vector, spinor, etc.) as instructions for dynamical state assignments, which instructions can be scaffolded with a number of different ontologies. It is not surprising that a dynamical state with three degrees of freedom (massive vector) can be swapped with one state with two degrees of freedom (massless vector) plus one with one degree of freedom (real massless scalar). The ontologies differ radically, but the state assignments are equivalent, and underdetermination is avoided. (A detailed treatment of how quantum operators can be treated as prescriptions for state assignment is explored in Healey 2017.)
To be clear, the fact that a certain state assignment is the best prescription for a given context implies a descriptive fact about how the world works, namely something to the effect of “the (hidden) underlying dynamics of the system is such that these state assignments track the true transitions of the system.” But this is hardly much of a description at all, and certainly not comparable to ordinary descriptive statements such as “there are particles and forces and they are related in such and such a way,” which the prescriptivist considers as either embellishment or scaffolding for the state space.
One can extend the argument to an account with movable pieces (such as Giere’s) by asking: should one include all possible family resemblances or only known ones? Since family resemblance is not a local or empirical matter, Giere’s view faces a similar dilemma as the traditional, fully-fixed accounts.
Besides, what circumstances are the \(C_i\)s meant to range over? If they are meant to cover all possible circumstances, then \((C_1 \vee C_2 \vee \ldots )\) is vacuously true, and the massive conjunction simply boils down to L itself, which once again is not the entire content of the theory. If, on the other hand, the \(C_i\) are only meant to range over known physical circumstances in which the theory has been tested, then all of the conditionals become vacuously true in all untested circumstances, which means that the theory is empirically impotent outside of known circumstances. But again, INEXHAUSTIBILITY means that new LEMPs are always possible, and ABUNDANCE implies that the theory can make predictions about new systems and contexts as long as its master formula is still applicable. Including only known LEMPs would once again deprive the theory of future success in any amenable contexts that have not been encountered before.
This is a manifestation of Hempel’s Theoretican’s Dilemma. Thus, the prescriptive-dynamical view provides a new solution to the Theoretician’s Dilemma, one that unlike Psillos’ (1999, 22 ff.) does not require questioning the observational-theoretical distinction itself. But that is beyond this paper.
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Acknowledgements
I would like to thank John D. Norton, John Earman, Jim Woodward, Richard Healey, Gal Ben Porath, as well as the audiences of Pitt HPS Work-In-Progress talks, the 43rd Annual Midsouth Philosophy Conference, and the anonymous reviewers of Philosophy of Science, the British Journal for the Philosophy of Science, Synthese, and Erkenntnis for invaluable comments and illuminating discussions.
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Kaveh, S. Physical Theories are Prescriptions, not Descriptions. Erkenn 88, 1825–1853 (2023). https://doi.org/10.1007/s10670-021-00429-2
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DOI: https://doi.org/10.1007/s10670-021-00429-2