Should Maxwell’s equations be considered laws of our actual world? Consider the following remarks made by Einstein in his seminal 1905 paper on the light quantum:
The wave theory of light, which operates with continuous spatial functions, has proved itself splendidly in describing purely optical phenomena and will probably never be replaced by another theory. One should keep in mind, however, that optical observations apply to time averages and not to momentary values, and it is conceivable that despite the complete confirmation of the theories of diffraction, reflection, refraction, dispersion, etc., by experiment, the theory of light, which operates with continuous spatial functions, may lead to contradictions with experience when it is applied to the phenomena of production and transformation of light. (Einstein 1905, pp. 132-133.)
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Einstein takes Maxwell’s as a theory which is true only up to an approximation, the ap-proximation being a result of operating with time averages instead of momentary values of physical quantities. Even though he denies its exactness Einstein still uses a strong language to endorse Maxwell’s theory as it “has proved itself splendidly,” got “completely confirmed”
in several domains of phenomena and “will probably never be replaced by another theory.” It may be reasonable to interpret these statements as regarding the Maxwell equations as laws despite of their approximate truth; to the extent this sentiment is shared by other physicists it may point towards an acceptance of merely approximately true generalizations as laws.
To regard generalizations which are only true-to-an-approximation as laws likely does not mesh well with Governing accounts of laws. Approximate truth on the other hand does seem to be consistent with the spirit of the Best System account of laws. To ease discussion let us first highlight the difference between the ‘Best True System’ and the ‘Best Approximately True System’ accounts:
(BTS) Laws are propositions of the true deductive systems which best balance simplicity and informativeness.
(BATS) Laws are propositions of the deductive systems which best balance simplicity, approximation to truth, and informativeness.
BTS is a view that balances the virtues of informativeness and of simplicity assuming truth; BATS is a view that balances the virtues of informativeness, of simplicity, and of truth. According to BTS truth is a must; according to BATS, lots of simple informativeness may outweigh a small loss in exact truth. BATS does embrace truth but allows for the introduction of a certain coarse graining in the true description of the world. As BATS proposes a balance between all virtues without singling out one as absolute it seems to follow better the spirit of identifying laws as means to achieve effective organization of information about the mosaic of events than does BTS.
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Difference between BTS and BATS may be highlighted by an example. Which deductive system carries more the merit of lawhood: one which implies the exact value of the electro-magnetic field in one spacetime point but is completely silent about its values everywhere else, or another which implies the values of the electromagnetic field everywhere but only up to an approximation which lies beyond our measurement capabilities to detect? BTS would force us to choose the first option while BATS also allows for the second.
We do not aim to argue here for the validity of BATS. We merely submit that BATS is an account which should be amenable to defenders of a Best System account of laws. As we have already pointed out the Relativized Best System account implicitly already embraces BATS. Any other account that allows the special sciences to have laws is also likely to implicitly embrace BATS. If the same mosaic of events may be subject to laws of theories located on different levels of the proverbial layer cake then truth with approximation should be sufficient for laws as theories on a higher level typically operate with more coarse-grained descriptions than theories on the lower level. Allowing for approximate truth does come with a major disadvantage, though: it adds approximation to the laundry list of terms such as informativeness and simplicity which we need to make sense of35.
35Without denying the need for clarification I resort to making only one comment. Approximation is best accounted for by appealing to a notion of distance between different initial data, different solutions, and so on: a claim about a physical quantity is approximately true if the true value lies within a certain distance from the claimed value. But this notion of distance is logically independent of the notion of
‘distance’ referred to in debates about ‘closeness’ of possible worlds. An example for this latter would be the evaluation of counterfactuals a la Lewis: a counterfactual claim is true if in all sufficiently ‘close’ worlds in which the antecedent is true the consequent is also true. This ‘closeness’ is a notion which needs metaphysical grounding; approximation, on the other hand, is primarily tied to epistemic interests, and it relates to issues of measurement precision. A typical physical theory already comes equipped with a notion of distance which reflects the connection of the mathematical apparatus to measurement, and hence this distance notion might be less problematic than the hypothetical closeness-of-possible-worlds relation of Lewis. One may of course connect these two interests, but they do not necessarily need to coincide, and hence we need to keep them separate.
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