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Tractability and laws

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According to the Best System Account (BSA) of lawhood, laws of nature are theorems of the deductive systems that best balance simplicity and strength. In this paper, I advocate a different account of lawhood which is related, in spirit, to the BSA: according to my account, laws are theorems of deductive systems that best balance simplicity, strength, and also calculational tractability. I discuss two problems that the BSA faces, and I show that my account solves them. I also use my account to illuminate the nomological character of special science laws.

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  1. For brevity, I will not discuss probabilistic laws here. For an account that does, see (Lewis 1994).

  2. I say ‘more-or-less’ because strictly speaking, \(W^{\prime }\) is marginally more calculationally tractable than W. The greater tractability of \(W^{\prime }\) comes from the fact that its additional axiom occasionally allows for slightly shorter computations. But the small increase in tractability allotted to systems which include that additional, redundant axiom is outweighed by the simplicity allotted to systems which leave it out.

  3. The additional axiom in \(W^{\prime }\) is redundant because it can be derived from Newton’s equations.

  4. The ‘\(\uparrow \)’, which is Knuth’s up-arrow notation, represents a generalization of exponentiation. For example, \(2\uparrow 4=2^{4}=16\), \(2\uparrow \uparrow 4=2\uparrow \big (2\uparrow (2\uparrow 2)\big )=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65536\), \(2\uparrow \uparrow \uparrow 4=2\uparrow \uparrow \big (2\uparrow \uparrow (2\uparrow \uparrow 2)\big )=2\uparrow \uparrow (2\uparrow \uparrow 4)=2\uparrow \uparrow 65536=2\uparrow (2\uparrow \ldots )\), where the ‘\(\ldots \)’ contains a total of 65534 instances of ‘2’. So a number like \((7\uparrow \uparrow \uparrow \uparrow \uparrow 9)-(3\uparrow \uparrow \uparrow \uparrow \uparrow 8)\) is unimaginably massive. For more discussion of this notation, see (Knuth 1976).

  5. The encoding consists of a series of metasemantic rules which map (i) digits in various places of the binary expansion of \((7\uparrow \uparrow \uparrow \uparrow \uparrow 9)-(3\uparrow \uparrow \uparrow \uparrow \uparrow 8)\), to (ii) the positions of particles at various times. These metasemantic rules may be highly gerrymandered and complex, of course. But the complexity of these metasemantic rules does not increase the complexity of the corresponding deductive system, just as the complexity of the interpretation function used to construct a set-theoretic model of a theory in first-order logic does not increase the complexity of that theory.

  6. And so understood, the BUSA goes some way towards addressing the concerns raised by Woodward about the definition—or really, lack of a definition—of simplicity (2014). For the BUSA contains a more precise characterization of simplicity than the BSA does.

  7. Thanks to two anonymous reviewers for pressing me on which, of these many different procedures, is best for the purposes of formulating the BUSA.

  8. It would be interesting to explore, for instance, whether these different ranking procedures tend to produce more-or-less the same rankings of deductive systems—and therefore, tend to agree on which statements count as laws. I suspect that they might, since I see no reason to think otherwise; and they all seem to imply that W is a better deductive system than both \(W^{\prime }\) and Y.

  9. According to another version of the BUSA, calculational tractability is ‘transcendent’: like the accounts of simplicity and strength offered by Lewis (1973) and Loewer (2004), the calculational tractability of a system is evaluated independently—or at least, somewhat independently—of that system’s language. I take theoretical virtues to be transcendent, because immanent approaches tend to imply an overabundance of (relativized) laws: they often imply that for every regularity r, there is a choice of basic kinds and predicates such that relative to those kinds and predicates, r is nomological. But the choice between immanence and transcendence is not forced by the BUSA. Fans of immanent accounts of theoretical virtues can still subscribe to the BUSA, since the BUSA is neutral with respect to whether theoretical virtues are immanent or transcendent.

  10. There are also similarities between the BUSA and the approach to lawhood discussed by Urbaniak and Leuridan (2018). I discuss those similarities—and some differences as well—in Sect. 3.1.

  11. Similarly, the BUSA allows deductive systems to be calculationally tractable in more ways than Braddon-Mitchell’s account (2001) allows.

  12. A related but distinct worry: exactly which calculations are relevant for determining the degree to which a deductive system is tractable? Calculations of observables, or calculations of all quantities posited by a particular theoretical system, or calculations of something else entirely? The answer: for each collection of relevant calculations, there is a corresponding version of the BUSA. My preferred version of the BUSA takes all calculations—of all quantities, properties, constants of nature, and so on—to be relevant. But it is beyond the scope of this paper to investigate whether this version of the BUSA is better than those others.

  13. The ‘worst case running time’ of a Turing machine T is the maximum number of steps (that is, the time) required for T to halt on an input string of length n. So if T never requires more than \(n^{2}\) steps to halt on an input of length n, but always requires a number of steps asymptotically close to \(n^{2}\), then the worst case running time of T is \(n^2\). A ‘maximally efficient Turing machine’ is a machine whose worst case running time exhibits the slowest asymptotic growth in n (for a given problem). For further discussion, see (Cormen et al. 2009, pp. 27–29).

  14. It is possible to drop this assumption by doing everything in terms of relative positions in space and time. But then the example becomes needlessly complicated.

  15. A also includes an “and that’s all” clause—that is, a clause specifying that there are no positions or times other than those described by the triples.

  16. Putting a 0 to the right of a string is equivalent, in binary notation, to multiplying that string by two.

  17. The situation is somewhat more complicated than I have suggested. To see why, let \(B^{*}\) be the deductive system which results from supplementing B with a specific triple from A: whichever triple describes the position of the sphere at the initial time. Clearly, \(B^{*}\) and A are equally strong, but \(B^{*}\) is far simpler. In addition, \(B^{*}\) is much stronger than—but only slightly less simple than—B. And if computational utility is only determined by worst case running time—that is, if computational utility is measured as in the Worst Case Account only—then (i) B and \(B^{*}\) are equally calculationally tractable, and (ii) both are much more calculationally tractable than A. So on balance, if computational utility is only determined by worst case running time, then \(B^{*}\)—rather than B—seems like the best deductive system. To avoid this somewhat unintuitive result, I suggest measuring computational utility using more than just the notions in the Worst Case Account. In particular, the notions invoked in an account of computational utility described in Sect. 3.2—called the ‘Variational Account’—should be used to measure the computational utilities of the systems A, B, and \(B^{*}\). Then it follows that B and \(B^{*}\) are not equally calculationally tractable: B is far more calculationally tractable than \(B^{*}\). And so plausibly, B is indeed the best deductive system (thanks to an anonymous reviewer for discussion).

  18. For this reason, when I formulated the Worst Case Account using a biconditional, I was making a simplifying assumption. To keep the example concise, I assumed that no other features of deductive systems A and B contribute, in a substantial way, to those systems’ computational utility. As explained in footnote 17, however, those systems’ computational utilities are indeed affected by other features, such as the features described by the Variational Account in Sect. 3.2.

  19. Following Lewis, in this formulation of the problem, I take propositions to be classes of worlds, and I take implication to be the subclass relation between propositions (1983, p. 367). So the proposition expressed by one sentence implies the proposition expressed by another just in case the first proposition is a subclass of the second.

  20. For more criticisms of Lewis’s use of naturalness in the BSA, see Massimi (2017).

  21. For this reason, I prefer the BUSA’s solution to the trivialization problem over Urbaniak and Leuridan’s solution. There is much to like about the measure of strength which invokes possible worlds. And the BUSA is attractive, insofar as it is compatible with measuring strength in that way.

  22. For the sorts of reasons mentioned in footnote 18, in order to keep the following discussion simple, I formulate the Variational Account using a biconditional. Strictly speaking, free variation among parameters only partially contributes to the overall computational utility of any given deductive system.

  23. This version of the Variational Account is inspired by the ideas in (Hicks, 2018).

  24. Of course, Malthus’s law is only accurate in circumscribed domains. It only provides a reasonably accurate description of population growth over limited periods of time, in sufficiently isolated ecosystems, and so on. So strictly speaking, there is another theoretical virtue which contributes to the nomological character of Malthus’s law: accuracy. Following Braddon-Mitchell (2001, pp. 266–267), accuracy can be analyzed in terms of data compression. Altogether, then, Malthus’s law really is a law because it helps deductive systems strike a good balance between four theoretical virtues—strength, tractability, simplicity, and also accuracy—rather three. So to accommodate the fact that accuracy often contributes to the nomological character of special science laws, amend the formulation of the BUSA so that it balances those four virtues rather than the three virtues from earlier: in other words, the best system is the one which strikes the best balance between strength, tractability, simplicity, and accuracy.

  25. Tractability even contributes to the nomological character of certain generalizations about chaotic dynamics. That might be surprising: for in chaotic dynamical systems, many analytic generalizations—about subtle variations in the orbital parameters of the planets, say—are utterly intractable. Nevertheless, in chaotic systems, there are tractable generalizations which seem nomological. For examples of generalizations like those, and how they are used in astronomy and in the design of spacecraft trajectories, see (Wilhelm 2019).

  26. For accessible summaries of those procedures, along with explanations of why quantum field theories can be regarded as special sciences, see (Fraser 2020; Ruetsche 2011; Williams 2019).

  27. This complements other accounts of special science laws. For instance, it complements the view that special science laws derive from the Mentaculus, a fundamental physical theory of the world that assigns a probability to every physically possible event (Loewer 2008, pp. 159–160). According to the BUSA, the postulates of the Mentaculus—like the Past Hypothesis or the Statistical Postulate (Albert 2000, p. 96)—are laws: they increase the calculational tractability of the Mentaculus by implying that special science generalizations like Malthus’s law, which are extremely calculationally tractable, are highly probable.


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Thanks to Craig Callender, Ashley Chay, Eddy Chen, Jonathan Cohen, Chris Dorst, Véronica Gómez, Eugene Ho, Barry Loewer, Michael Townsen Hicks, Jill North, Eric Winsberg, Qiantong Wu, audiences at the 2016 Western Canadian Philosophical Association and at the 2017 Pacific APA, two anonymous referees, and especially Jonathan Schaffer, for much helpful feedback and discussion.

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Wilhelm, I. Tractability and laws. Synthese 200, 318 (2022).

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  • Laws
  • Best system account
  • Calculational tractability