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Laws, Natural Properties and Algorithmic Compression
Billy Wheeler1
Keywords: laws of nature; best system account; natural properties, package deal account; better best system account; algorithmic compression; Kolmogorov complexity
1. Introduction
The best system account (BSA) is often considered the most defensible Humean theory of laws. It is certainly the most widely adopted by contemporary Humeans. 2 One reason for its success is its promise to provide some degree of objectivity to lawhood. According to the BSA laws of nature are the axioms (and theorems) in a deductive systematisation of all true empirical statements achieving the best balance of strength (information content) and simplicity. Since statements about our world can satisfy these requirements regardless of whether anybody has thought of them or not, it makes lawhood more or less independent of human activity. Yet this supposed strength of the BSA was cast into serious doubt shortly after Lewis (1973) gave us its most canonical formulation. In his defence of realism concerning natural necessity, David Armstrong (1983) produces an argument which shows that the strength and simplicity of a system depends crucially on the language in which it is written. The upshot is that without a pre-agreed upon set of basic predicate terms, inter-system comparisons of strength and simplicity are impossible. But who is to decide what terms we should use? Since we are at liberty to choose the language of our system, so too are we at liberty to choose the axioms. Hence Armstrong’s argument shows that laws on the BSA are not, contrary to what was hoped, independent of human activity.
Lewis (1983) was quick to address ‘Armstrong’s problem’ for the BSA. His solution was to advocate realism about natural properties. These are the properties whose predicate terms can be used to describe the world without gaps or redundancy. If we place on the BSA a requirement of all candidate systems that they be formulated in the language of natural properties, then we can avoid Armstrong’s problem. In this paper I will argue that it is possible to solve Armstrong’s problem without having to appeal to natural properties. This should be desirable from a Humean point of view
1 King’s College, University of Cambridge, King’s Parade, CB2 1ST. Contact: [email protected] 2 One version or another of this view has been held by Jeremy Butterfield (1985), David Papineau (1988), John
Earman (1994), Barry Loewer (1996), John Roberts (1999), John Halpin (2003) and Jonathan Cohen and Craig Callender (2009).
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which is sceptical of any metaphysical primitives that make no observable difference to our world. This will be achieved by outlining an alternative Humean approach to laws which, whilst retaining the core intuition of the BSA that laws are abridged
descriptions of nature, is nonetheless immune from Armstrong’s problem.
The structure of the remainder of the paper will be as follows. In section 2 I will begin by outlining Armstrong’s problem in detail and discussing Lewis’ solution to it. Lewis gives a number of reasons to support his adoption of natural properties and whilst I will argue that these arguments do not fail on their own terms, I will suggest that they are not ones a consistent Humean should be willing to accept. Armstrong’s problem has not gone unnoticed in the literature and this is not the first attempt to solve it without appealing to natural properties. In section 3 I will discuss two recent attempts given by Barry Loewer (2007) and Jonathan Cohen and Craig Callender (2009) at reformulating the BSA in a hope to avoid natural properties. Both of these attempts fail to resolve the problem however, because they both share the same fatal assumption as Lewis. All these formulations of the BSA make a conflation of laws
with statements of law and this is what drives the dependency of lawhood on choice of vocabulary. Fortunately, this can be avoided if we think of laws, not as general statements, but as algorithms or rules. These rules allow us to compress raw empirical data sets in order to achieve abridged description of nature. I will call this approach the ‘algorithmic compression theory’ of laws (ACT) because laws are algorithms which aid in simplified scientific representation. In section 4 I will provide the background to this view and explain how a result from algorithmic information theory known as the ‘invariance theorem’ can be used to demonstrate that the ability of an algorithm to provide compression is largely independent of the language we choose to describe the world. I will finish in section 5 by pointing out the main similarities and differences to Lewis’ version of the BSA and highlight future avenues in which the ACT can be developed.
2. Armstrong’s Problem
2.1. Outline of the Problem
The objection was first given by David Armstrong in 1983 and can be summarised as a thought-experiment. Suppose we live in a world where scientists have discovered that the best system can be given by a theory T1, where T1 includes only the following generalisations as axioms:
(T1) (i) (x) (Fx Gx) (ii) (x) (Hx Jx) (iii) (x) (Kx Lx)
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According to Armstrong, scientists in this world would now be at liberty to simplify T1 further, should they wish. They could define two new predicates ‘M’ and ‘N’ such that:
Mx d.f. Fx Hx Kx Nx d.f. Gx Jx Lx
With these new predicates at hand, T1 can be simplified to T2, which contains just one single law:
(T2) (i) (x) (Mx Nx)
One can quickly see how this process could be generalised from any starting system of axioms. In fact, in the limit, we could define a property ‘P’ such that (x)(Px) and make this our only law in the universe (Lewis 1983, p. 215). Cleary this trivialises the concept of laws as a mind-independent features of the world waiting to be discovered and so Armstrong provides a reductio ad absurdum of the BSA.
It might be reasoned that systems T1 and T2 are not quite as equivalent as Armstrong supposes. For although by all measures T2 is simpler than T1, can it be said that it is as strong? Using the T1 system, if I know that x is F I can infer that it is G; but on the T2 system all I can know about x is that it is G or J or L. Surely this shows that T1 is a stronger system than T2 and therefore objectively preferable? Armstrong considers this line of thought and provides the following response:
Can it be said that, where the M-N ‘law’ has greater simplicity, it lacks the strength of the original generalizations? On the new system, given an F we can deduce it is an N. But on the old system, we can deduce that it is a G, and G is a further specialization of N. However, this seems to beg the question. Suppose that the old system could have been improved, so that a theoretical identification of properties could have been made: G = J = L. To say that an F was a G would then have been no more informative than saying that it is a G or a J or an L. But the new predicate N is being supposed to pick out a single property which the predicates ‘G’, ‘J’ and ‘L’ artificially break up. From the standpoint of the ‘property’ N, therefore, G is J, and J is L (1983, p. 68).
The reason why the response above does not save the BSA from Armstrong’s criticism is that it assumes the truth of a certain metaphysical partitioning of the world. In this response we are assuming that the strength of our system should be measured in relation to properties G, J and L. The advocate of T2, however, would not see things this way: he or she would measure the strength of the system against properties M and N, and, as far as they are concerned, this makes T2 as strong as T1. We cannot argue that T1 is stronger than T2, therefore, without begging the question against our opponent.
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What this suggests is that in order to compare the strength and simplicity of rival systems we need to have an agreed set of predicates—this would prevent systems like T2 or the more radical system ‘(x)(Px)’ from being considered legitimate alternatives. But who is to say what the right predicates are? Armstrong thinks it is nature which decides. On his preferred theory of properties, the universe contains an objective class of natural properties—the universals—which correspond to the right predicates.
2.2. Lewis’ Solution: Realism about Natural Properties
As is well known Armstrong and Lewis differ radically in their metaphysical interpretation of laws and properties. Armstrong is committed to the view that the only properties that exist are universals and where there is a law this is because two or more universals are connected by a second-order relation of natural necessity. Lewis on the other hand holds a form of class-nominalism that opens up class membership to both actual and possible individuals. Despite these differences Armstrong and Lewis agree on how they think the problem above ought to be resolved:
The remedy, of course, is not to tolerate such a perverse choice of primitive vocabulary. We should ask how candidate systems compare in simplicity when each is formulated in the simplest eligible way; or, if we count different formulations as different systems, we should dismiss the ineligible ones from candidacy. An appropriate standard of eligibility is not far to seek: let the primitive vocabulary that appears in the axioms refer only to perfectly natural properties (1983, p. 216).
Lewis remains largely agnostic concerning how natural properties ought to be understood metaphysically, i.e. whether they should be interpreted as universals, tropes, classes or something else entirely (1983, p. 194). What is important is that a distinction is made between the types of properties that exist: between natural properties on the one hand and abundant properties on the other. For Lewis the fundamental difference between these two is their ‘sparseness’: there are just enough natural properties to describe the world fully without redundancy. Presumably he also accepts certain metaphors about natural properties ‘carving nature at its joints’ but he thinks abundant properties can do this also it is just that they carve nature ‘everywhere else as well’ (1983, p.192).
If Armstrong’s problem can be resolved by admitting natural properties then should we not just supplement the BSA with them and consider the case closed? The difficulty with this is that natural properties sit uncomfortably with the outlook of most contemporary Humeans. For what difference would the ‘naturalness’ of a property have in our immediate experience of the world? Common everyday predicates such as ‘green’ or ‘flower’ are just as good and no better at describing the events that obtain than any other gerrymandered set of predicates we may choose. In the words
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of John Earman, natural properties fail the ‘empiricist loyalty test’ (1984, p. 195). Suppose we have two worlds W1 and W2 that agree in all respects except for the fact that in W1 some properties instantiate a second-order property of ‘naturalness’ whereas in W2 no properties have this feature. Intuitively W1 and W2 would not differ as regards the occurrent events that obtain, and as such, ‘naturalness’ in properties fails to pass the test.
This alone therefore sums up the prima facie case for why Humeans should be sceptical about the existence of natural properties. To be fair to Lewis, he does not justify accepting natural properties on the basis of observing a second-order property of ‘naturalness’: his reasons are largely pragmatic. Before turning to Lewis’s positive arguments I want to consider what might be termed the ‘common sense’ argument for realism about natural properties. Even the most isolated philosopher could not have failed to notice that people do, by and large, describe the world using the same set of basic predicates. Nearly everyone describes emeralds as ‘green’ and not ‘grue’ or ‘bleen’. One could argue: isn’t the best explanation for mass common assent the fact that terms like ‘green’, ‘sweet’ and ‘heavy’ refer to real properties in nature whereas terms like ‘grue’, ‘bleen’ or ‘emerose’ do not?
This line of argument faces too many difficulties to provide any grounds for accepting natural properties. Firstly, it is not even likely to be true that people do agree (entirely) on the predicates used to describe the world. We know there exist communities cut-off from the rest of civilisation that use different terms to classify and group objects in the natural world – classifications that would no doubt be rejected by others. But even closer to home people haven’t always agreed on basic kind terms: consider the predicate ‘...is a planet’ which once divided opinion concerning whether its extension should include the sun, earth or moon. Secondly, even if everybody did agree on predicates, it is unlikely that these would be thought to refer to natural properties. Scientific investigation has shown that terms such as ‘green’, ‘goat’ or ‘loud’ fail to refer to fundamental properties of nature and as such are likely to be one of Lewis’ abundant properties rather than a sparse property. Lastly there are no rational grounds for thinking there exists a relationship between common usage and the naturalness of a property. To see that this is so we need only answer the question: why is it that most people tend to agree on the basic properties? One plausible explanation is that our choice of predicates was determined early on by evolutionary pressures, so that the distinctions we made which helped us survive where more likely to be used and passed on. Yet as van Fraassen so eloquently put it ‘there is no such close connection between the jungle and the blackboard’ (1989, p. 52). Just because the conceptual distinctions we happened to make in the past helped us identify and escape danger, does not mean they correspond to any real divisions in nature as abundant properties could do this job just as well.
As mentioned already Lewis’s argument for realism about natural properties is largely pragmatic. In fact, according to Barry Taylor (1993), Lewis gives two distinct
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arguments: (1) natural properties are needed to explain ‘Moorean’ facts about sameness of type, and (2) natural properties could provide a unified understanding of many disparate metaphysical phenomena. My reason for disagreeing with Lewis here is not because I believe natural properties cannot fulfil the roles he outlines for them, but because, as a general strategy for realism, this would validate the existence of ‘natural necessities’, which undermines taking a Humean approach to laws in the first place.
Let us take the first argument. Like G. E. Moore (1925), Lewis accepts that there are certain commonsense facts about the world which, although not certain, are more certain than sceptical arguments to the contrary. The one which is relevant to natural properties is ‘sameness of type’ (1983, p.352). We routinely describe the world in terms of objects that are of the same type, i.e. the Sears tower and the Eiffel tower, although distinct entities are both instances of the same type of object: ‘tower’. By saying that the building in Canada and the building in France both possess the same natural property, we have a ready explanation for types.
I don’t believe, however, that this is an argument Humeans should be willing to make. According to Lewis, ‘A Moorean fact [is] one of those things that we know better than we know the premises of any philosophical argument to the contrary’ (1999, p. 549). If this holds for the relation ‘sameness of type’ then surely it also holds for the ‘cause-effect’ relation. What could be less open to doubt than the observed fact that some events are causally connected whilst others are not? But just like before we can ask ‘what does the causal relation consist in?’ and, following Lewis’s logic, we could say realism about natural necessities. Since natural necessity is anathema to all Humean accounts, I cannot see how Lewis can consistently use this argument to justify natural properties whilst also denying natural necessities.
What about Lewis’s second argument? This fails for a similar reason. To paraphrase Lewis, he says natural properties can ‘earn their living’ by doing ‘much needed work’ in philosophy (1983, p. 188). In particular by providing a systematic and unified treatment of topics as diverse as ‘duplication, supervenience, and divergent worlds; a minimal form of materialism; laws and causation; and the content of language and thought’ adding ‘perhaps this list could be extended’ (1983, p 189). It is the meta-theoretical virtue of unification that Lewis is here appealing to to warrant accepting natural properties. But once again, permitting this as a general formula for realism allows too much. Interpreting the causal relation as one of natural necessity would allow us to provide a uniform treatment of the following topics: laws of nature, freewill, inductive reasoning, moral responsibility, action and the foundations of quantum mechanics. Again this list could likely be extended. Hence I cannot see how this strategy for realism can be consistently accepted by Humeans.
Whether the arguments against Lewis are conclusive or not, it should be evident that a response to Armstrong’s problem that avoids realism about natural properties is
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going to be more appealing to the Humean. I now turn to examine two recent approaches that attempt to do just that.
3. Recent Solutions to Armstrong’s Problem
3.1. Barry Lower’s ‘Package Deal Account’
Barry Loewer (2007) agrees with David Lewis that the BSA needs supplementing with a class of elite properties in order to overcome Armstrong’s Problem. However, unlike Lewis, he does not believe this involves accepting natural properties as a metaphysical primitive. Instead he says it should be physics rather than metaphysics
which tell us what the scientifically significant properties are. He develops what he calls the ‘Package Deal Account’ where the final theory of physics gives us a ‘2-for-1 deal’ by supplying both the laws and the scientifically acceptable predicates.
The motivation for Loewer’s theory stems from a consideration first brought to light by van Fraassen (1989, p.53). We can imagine a system, call it ‘A’, which is accepted by the scientific community as the ‘final theory’ but which has had no requirement to be formulated using only natural property terms. Now it seems theoretically possible, according to van Fraassen and Loewer, that in the attempt to meet Lewis’ requirement by rewriting A in the language of natural properties, we take it further—not closer—to the ideals of simplicity and informativeness. Suppose scientists are given a choice of a theory: choose A, which best maximises the virtues of strength and simplicity but which is not written in natural predicate terms, or choose B, which is so written, but which does not maximise these virtues. In this scenario it seems highly likely that the scientific community would choose A over B despite the former not referring to natural properties.
From this thought experiment alone Loewer concludes that the search for natural properties is not itself a guiding principle or internal ambition of scientific practice and therefore not conceptually necessary for there to be a ‘Best System’. We can still ask what are the elite properties—that is, what are the properties that can do all the work
Lewis wants them to do in metaphysics? But then there is an easy answer to this: they are just the properties referred to by the predicate terms in the final theory actually accepted by the scientific community:
This is the ‘Package Deal Account’ (PDA) of laws since it identifies the laws and the nomological properties together. The PDA is not constrained by the requirement that its primitive predicates refer to Lewisian natural properties...I propose that the PDA is the right response to Van Fraassen’s problems and is a better version of the BSA. On a Humean account propositions count as laws in virtue of their connection to the best scientific summary of the world. If the best scientific summary of the world is best when formulated in FL [the language of the final theory] than any system formulated in NL [the language of natural properties] then it should be the law giver. (2007, 324)
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Loewer’s PDA has a number of features that make it an appealing theory of laws. Firstly, if successful in avoiding Armstrong’s problem, it provides a theory of laws that does not depend on realism about natural properties which—as we have seen above—is difficult to justify consistently on Humean grounds. Secondly, by embedding the Best System firmly in actual scientific practice in the form of the ‘final theory’ accepted by physicists, there is no danger of an uncoupling of scientific practice with the metaphysics of science: Loewer has produced a thoroughly naturalistic theory that ought to appeal to both scientists and philosophers alike. Lastly, the PDA achieves good theoretical economy by accounting for two concepts in a single analysis. This is especially advantageous here for it is commonly thought that laws and natural properties share a deep connection. By showing how they arise simultaneously in the attempt to meet the aims of science, Loewer has provided an explanation for why they are metaphysically connected.
Unfortunately, many 2-for-1 deals sound too good to be true and Loewer’s PDA is no exception to this. Let us take a closer look at how the PDA is supposed to avoid Armstrong’s problem. In the problem’s most extreme form, as given by Lewis, it is possible that the Best System could contain just one law: ‘(x)Fx’, where ‘Fx’ is defined in our language as a complex property that holds for all objects in the universe. Thus stated, the system is maximally informative and maximally simple. However, nobody would consider ‘(x)Fx’ a serious contender for a law of nature. Loewer thinks this is right and offers the following explanation:
From the perspective of the aims of science the obvious trouble with ‘(x)Fx’ is not that ‘Fx’ doesn’t refer to a perfectly natural property but that ‘(x)Fx’ is not a credible scientific theory. We don’t know what it says and it is completely lacking in explanatory value. While it might be maximally informative given Lewis’ characterization of information as excluding alternatives this merely shows that Lewis’ proposal for evaluating informativeness is not relevant to the way scientists evaluate informativeness. The information needs to be extractable in a way that connects with the problems that are of scientific interest. (2007, p. 325)
This seems a fair point to make here and it won’t do to say—as Armstrong does above—that ‘(x)Fx’ is maximally informative ‘by its own lights’; for the PDA states that all measures of information and simplicity must be made with respect to the predicates of the final theory.
The trouble with this explanation, however, is that we aren’t currently in possession of the final theory. This means that we only have actual theory by which to judge the credibility and ‘explanatory value’ of new terms. This poses a significant challenge in accounting for theory change and the growth of science in general. Who is to say that ‘(x)Fx’ is not likely to be the final best theory? Just because the predicate ‘Fx’ does not add explanatory value in the light of our current theories does not mean that it ought to be rejected outright. No doubt Special Relativity Theory and Quantum
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Mechanics would fail to be explanatory in the eyes of seventeenth century physicists; yet it is true that these theories are an improvement over Newtonian Mechanics. Likewise, it is possible that ‘(x)Fx’ could be an improvement over current theories even if we aren’t able to appreciate the fact yet. This shows that, contra Loewer, it is not obvious in light of current science what terms might be explanatory in some future theory.
Loewer is aware of this difficulty in his solution and says that new predicate terms ought to be evaluated with respect to current theories and with regards to how well they promote the ‘rational development’ of the sciences:
A candidate for a final theory is evaluated with respect to, among other virtues, the extent to which it is formulated in SL [the language of current science] or any language SL+ that may succeed SL in the rational development of the sciences. By ‘rational development’ I mean developments that are considered within the scientific community to increase the simplicity, coherence, informativeness, explanatoriness, and other scientific virtues of a theory. (2007, p. 325)
If ‘rational development’ is defined—even partly—in terms of informativeness and simplicity, I do not see how Loewer’s explanation can avoid falling into circularity here. Recall that the PDA is supposed to place a restriction on which predicates can be used in genuine scientific theories so that when we make our comparisons of simplicity, artificial systems such as ‘(x)(Fx)’ don’t come out as best. But without foresight into the final theory what grounds do we have for accepting or banning a predicate? According to Loewer it depends on whether that predicate helps promote rational development which includes, among other things, the simplicity of a theory. Yet this is the very virtue the restriction on predicates was to supposed to clarify! Either we define the ‘acceptable simplification’ of a theory in terms of the predicates or we define the ‘acceptable predicates’ in terms of the simplification of a theory—we cannot have it both ways.
Perhaps there might be a way to improve the PDA so that it does not involve this circularity. I suspect the crux of the worry with Armstrong’s problem is that artificial predicates make the virtue of simplicity trivial—at least as a deciding factor in accepting one theory over another. Could we just not place a restriction on theory choice so that no theory or system is considered ‘better’ than any other solely on the basis of simplicity? All else being equal the simplest theory is no better than the more complex one. We could add the proviso that simplicity must be conjoined with some other virtue—such as explanatoriness or comprehensiveness—in order to be a deciding factor. This would rule out system T2 being chosen over T1 in Armstrong’s example above since it’s only virtue over T1 is that it contains fewer generalisations. Yet even this amended version of the PDA faces problems. The first question one is likely to ask is: ‘what grounds do we have for ruling out simplicity as a singular
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deciding factor?’ Without a reasonable answer there will always be the suspicion this restriction was tailored just to meet Armstrong’s problem and so will be considered
ad hoc. There is also the worry that this restriction is too heavy-handed. Do we really want to reject simplicity as a guiding factor in theory choice when all else is equal? This maxim has been one of the great guiding principles of scientific method, and even if it has played a greater role in the popular imagination of science, rather than science as it is practiced day-to-day in the laboratory, there must be legitimate grounds for why it became so iconic in the first place. Finally, there is the issue of which other virtues simplicity should ‘piggy-back’ on when used as a deciding factor. In addition to informativeness, Loewer mentions ‘explanatoriness’ and ‘comprehensiveness’—but surely these notions are too closely related to simplicity to guarantee we have avoided the problem of circularity in its entirety.
3.2. Cohen and Callender’s ‘Relativized MRL’
In their insightful paper ‘A Better Best System Account of Lawhood’ (2009) Jonathan Cohen and Craig Callender analyse Armstrong’s problem for the BSA (what they call the ‘MRL’ view after its main progenitors Mill, Ramsey and Lewis) as trying to find a
transcendent measure of strength and simplicity where, they believe, none exists. Instead they argue we must make do with an immanent measure. We cannot compare the strength and simplicity of various competing systems with respect to a universally adopted set of basic kind terms, but instead must carry it out relative to the basic kind terms of a given particular systematisation. Potentially there are an infinite number of systematisations with their own basic kind vocabulary and each, according to Cohen and Callender, are as valid as each other.3 They anticipate that for many philosophers such law-relativism will be unappealing, for as we have seen already, one of the supposed benefits of the BSA over prior Humean accounts is its ability to deliver objective laws:
Prima facie, the realization that simplicity, strength, and balance are immanent rather than transcendent—what we’ll call the problem of immanent
comparisons—is a devastating blow to the MRL view. For what counts as a
law according to that view depends on what is a Best System; but the immanence of simplicity and strength undercut the possibility of intersystem comparisons, and therefore the very idea of something’s being a Best System (2009, p. 6).
Whether a given systematisation of the facts is or is not a Best System has to be decided relative to a set of basic kind terms used to describe the world. For each such set there will be one Best System whose axioms are laws; but according to
3 Cohen and Callender claim to accept the position known as ‘explosive realism’: ‘the world permits possibly
infinitely many distinct carvings up into kinds, each equally good from the perspective of nature itself, but differently congenial and significant to us given the kinds of creatures we are’ (2009, p. 22). Explosive realism therefore rejects a carving of properties into ‘natural’ and ‘artificial’; for this reason I see it as opposed to Lewis’ distinction between sparse and abundant properties and therefore anti-realist concerning natural properties as a
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Cohen and Callender there are no laws simpliciter only laws with respect to a given vocabulary. Suppose we have a basic kind structure K whose predicates are Pk then it is possible that ‘All Fs are Gs’ is a K-law and not a J-law where J is a competing basic kind structure that does not recognise the predicates F or G. Cohen and Callender call this version of the best system account relativized MRL.
What decides the basic kind terms for a given scientific community? It appears to come down to the scientists’ interests. For example if scientists are interested in
observable properties, then this class will form the basis for inter-system
comparisons. On the other hand if they prefer macroscopic properties then these will underlie all evaluations. The worry is of course is that this makes lawhood arbitrary and interest-relative. Cohen and Callender believe this worry is, however, overblown and provide an argument which aims to mitigate the arbitrariness of relativized laws. According to them any proponent of a fixed kind system can treat this ‘as a pro tanto, a posteriori and defeasible assumption that is not insulated from empirical inquiry’
(2009, p. 20). Because the choice of kinds is not itself an arbitrary choice but one subject to rational revision, then scientific advance can be understood as a rational process, even though there is no ‘privileged outsider position’ from which all systems could be compared.
Cohen and Callender fail to provide details to explain just how the basic kind background assumptions are revisable in light of empirical evidence or shared methodological principles. My concern is that if lawhood is a concept that is relative to a set of basic kinds then so too will be many other aspects of scientific practice. It is already well known that what constitutes evidence for a given hypothesis radically depends on a chosen kind vocabulary (Hempel, 1965) as well too is the conclusion one reaches from generalising inductively (Goodman, 1979). If comparisons of systems lack a transcendent basis by which to do business, it is difficult to see how this would not affect other areas of scientific enterprise and hence undermine the possibility of a shared basis for ‘rational revision’.
Another suggestion Cohen and Callender give in the attempt to mitigate the arbitrariness of relativized MRL is to appeal to the possibility of overlapping predicates during periods of scientific change: ‘scientists A and B disagree, but they might agree that getting (say) the observables, suitably characterized, is important. They could then formulate the Best System relative to the observables of interest, which would be a third preferred [set of kinds]’ (2009, p. 31). If this is the case then it would be possible to rationally reconstruct the shift, say from the Ptolemaic world view to the Copernican system, on the basis that both theories shared a common concern with explaining particular features of the world and that the ‘Best System’ was the one evaluated with respect to the vocabulary that picked out the common features.
It seems highly likely that scientists who advocate different theories do work with some common interests and a world where they have no basic kinds in common
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seems remote. But even if this is true, what would it show? There are two possibilities: (1) the overlapping agreement is purely accidental and represents a historical coincidence (although not one that is random) or (2) there is some sharing of methodological principles by all parties that explains their agreement in interests and predicates. Neither of these can be of any help to the relativized MRL though. Take the first, if it is purely accidental, then why should we appeal to those kind terms they have in common by which to judge the Best System? Just because they have them in common does not make them any less arbitrary, after all, at another time in history the intersecting kind terms could have been completely different. What then of the second suggestion? If there is some shared set of common empirical or methodological assumptions that favour one basic kind vocabulary over another then there is no longer any need for a relativized MRL. In the words of Cohen and Callender, we could produce a ‘stipulated MRL’ with respect to these shared predicates, and make that our transcendent basis by which to compare the strength and simplicity of all rival systems.
Cohen and Callender’s relativized MRL therefore faces a dilemma: either accept the laws are relative to a kind structure which is itself founded on arbitrary interests, or, favour one kind structure on rational grounds and stipulate these as the fundamental kind terms by which to compare all systems. Since Cohen and Callender are clear they wish to avoid Lewisian natural properties, it seems they would have no choice but to accept a thoroughly relativistic science for their version of the MRL to work.
4. The Algorithmic Compression Theory of Laws (ACT)
4.2. How Algorithms Compress Empirical Data
The BSA is not the only approach that places simplified representation at the centre of its analysis of lawhood. A much older expression of the view was given by Ernst Mach. Mach likens science to a kind of business ‘consisting of the completest possible presentment of the facts with the least possible expenditure of thought’ (1894, p. 586). Laws play a crucial role in this ‘economic description of the world’ by reducing the number of individual data points that need to be recorded or memorised:
The communication of scientific knowledge always involves description, that is, a mimetic reproduction of facts in thought, the object of which is to replace and save the trouble of new experience. Again, to save the labor [sic] of instruction and of acquisition, concise, abridged description is sought. This is really all the natural laws are. Knowing the value of the acceleration of gravity, and Galileo’s laws of descent, we possess simple and compendious directions for reproducing in thought all possible motions of falling bodies (1894, p. 193).
Another way of putting this is to say that laws help simplify our description of the world by compressing empirical data into shorter, more manageable, forms. As an
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illustration imagine we were to note the relative positions of the sun, earth and moon each day for one thousand years. We could then construct an astronomical table containing this data. No doubt the table would be very long and, if printed, consume volumes of paper and ink. However there is a much easier way of representing the same information: we merely need to record the positions of each on one particular day as well as Kepler’s laws of planetary motion. From this we could infer all the other positions by mechanically applying Kepler’s laws to the initial data. In a sense then, an initial recording plus the laws provides a compression of the possible observations one could make, thus saving valuable resources such as time, energy and recording space.
The concept of compression is especially important in the computer sciences where, since the dawn of the digital age, it has been subject to rigorous mathematical study. In the language of computation, a compression of data is usually described as consisting of two parts: (i) an unstructured string of symbols called the ‘input data’, and (ii) a list of instructions for interpreting the input data known as a ‘program’. The essential idea being that any simple machine which can read and write symbols in accordance with rules can produce (iii) a structured string of symbols called the ‘output data’ when given (i) and (ii). If the combined length of the program and the input data (measured as number of symbols in some particular computing language) is shorter than the length of the output data, then compression has been achieved. In most instances compression is possible because the data to be compressed is not the shortest it can be: usually it contains regularities or patterns that can be summarised by a rule. This is a familiar compression technique and is the principle behind Morse code and the popular JPEG image compressor. These types of compressors are examples of ‘algorithmic compression’ because an algorithmic procedure can be given which encodes an original structured string into a shorter unstructured string of symbols.
A number of scientists and philosophers have used the concept of compression to develop an explanatory model of scientific practice itself. The prominent mathematician and physicist Murray Gell-Mann writes:
The best way to compress an account of large numbers of facts in nature is to find a correct scientific theory, which we may regard as a way of writing down in a concise form a rule that describes all the cases of a phenomena that can be observed in nature...A scientific theory thus compresses a huge number of relationships among data into some very short statement. (1987, p.4)
The astronomer and physicist Paul Davies is even more explicit defining laws as algorithms which compress empirical data:
The existence of regularities may be expressed by saying that the world is
algorithmically compressible. Given some data set, the job of the scientist is to find a suitable compression, which expresses the causal linkages involved.
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For example, the positions of the planets in the solar system over some interval constitute a compressible data set, because Newton’s laws may be used to link these positions at all times to the positions (and velocities) at some initial time. In this case Newton’s laws supply the necessary algorithm to achieve the compression (1995, p. 249).
According to this picture of scientific activity the universe is seen as a vast information source. It provides scientists with a raw set of empirical data points. As there are regularities in nature, the data which is received contains repetitions and patterns. The job of the theoretician is then to discover algorithms which can best compress this raw data into shorter strings. Should the original data ever be required, one need only apply the algorithm to a subset of the original data points to retrieve the missing values, as was illustrated with Kepler’s laws above.
In addition, philosophers have also been attracted to the algorithmic picture of laws, citing explanatory advantage over other metaphysical accounts. Mauro Dorato (2005a & 2005b) advances a broadly algorithmic theory saying it is the best explanation for the ‘unreasonable effectiveness of mathematics’ in physics. According to him, the language of mathematics is the best one for capturing ‘nature’s programs’ which he also calls the ‘software of the universe’. Taking a slightly different approach, David Braddon-Mitchell (2001) believes an algorithmic theory is an improvement over Lewis’ version of the BSA since it is able to account for why there are generalisations in science that are considered ‘lawlike’ but which are known to have exceptions. His explanation proceeds by making a distinction (commonplace in computer science) between lossless and lossy compression. Just as in other areas of data representation, sometimes errors in the output data are justified if the gains in compression are high enough. This trade-off between accuracy in prediction and simplicity in representation is assumed to be an integral feature of scientific practice.
We don’t want to define lawhood too closely to scientific practice, however, or else we risk losing the concept of a law of nature as a mind-independent object that science aims to discover but might get wrong. Let us define the algorithmic compression theory of laws (ACT) in the following way: imagine the universe is a vast information source that provides us with a string of data (U). Then the laws of nature correspond to the algorithms in the best possible compression of U. Or in other words, suppose we had a computer with finite resources and wanted to produce U as output. Then the laws are the algorithms run by the computer on the shortest input data that returns U as output.
So far so good; however there is a residual issue that needs to be addressed. All theories that ground lawhood in regularity over natural necessity face the challenge of distinguishing genuine laws from so-called ‘accidental regularities’—and the ACT is no exception to this. Suppose I want to record all the values of the coins in my wallet, which contains a mixture of currencies. I remove one coin at a time; note the
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currency and then the value of the coin. Once the wallet is empty I see that all coins which are of British Stirling are a 50 pence piece. There is therefore a pattern in the data—‘all Stirling coins in my wallet are a 50 pence piece’—but this is clearly not a law of nature. How are we to explain this?
Such small-scale regularities as these are easily explained by the ACT. It is possible for a string of data to be algorithmically describable but not algorithmically compressible if the length of the algorithm plus the input is not less than the output. Structure in the original data does not always guarantee compression and this is obviously so when the original data set contained a small number of values, as happened in the case above. Nonetheless, it is clear medium-scale regularities could still pose a problem. This time imagine my stock-take of coins takes place in a bank that yields thousands if not millions of values. Surely if the same regularity came up (which would be highly unusual!) it would nevertheless be algorithmically compressible? How should medium-sized regularities that are not laws be accounted for?
I think in these cases advocates of the ACT would have to accept that lawhood comes in degrees and that algorithms such as those in the bank example above are
laws, although extremely ‘weak’ or ‘feeble’ ones. But why don’t we recognise them as laws if this is the case? This is explained by the fact that the algorithms scientists are concerned with are ones that achieve vast factors of compression, such that examples like the one above don’t seem lawlike at all by comparison. The lawhood of an algorithm depends on the degree to which it achieves compression: the greater the compression, the more lawlike the algorithm.
Yet event this suggestion might face a problem if it was shown that large-scale regularities that are non-laws exist. In his recent book, John Roberts has argued that such cosmic regularities do indeed exist and that this shows theories such as the BSA must be defective. He gives the following examples of widespread regularities that are non-laws (2008, p. 23):
(a) The stability and harmony of the solar system (b) The flatness and homogeneity of the universe (c) The constant and universal increase in entropy
(d) The availability of large amounts of condensed matter for the formation of galaxies, planets, and heavy elements
These pose a problem for the BSA, according to Roberts, because they are surely essential in any account of what our universe is like, but are not themselves consequences of the laws of nature. That means they would have to be axioms in the best system, making them fundamental laws, which, Roberts believes, they are not.
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I won’t enter a dispute with Roberts here concerning whether examples (a)-(d) constitute real laws of nature; however I do believe the ACT could in principle support a view which suggests they are not. The ACT says that to get the total amount of information produced by the universe (U), one needs two things: (i) an unstructured set of input data, and (ii) algorithms for interpreting the input data. For the facts (a)-(d) to form the ground for algorithms they need to present a pattern or regularity that can be exploited for the sake of compression. Yet it is unclear whether they actually do so. Instead it seems more plausible that (a)-(d) state singular facts about the universe as a whole rather than generalisations about the entities within it. This is indicated by the fact that all the examples above start with the preposition ‘the’. Therefore, if such information was essential for an output of U, it looks as if it is better included in the unstructured input data, rather than written into the algorithm itself. If, on the other hand, we had access to more than one universe and (a)-(d) where shown to be true throughout all the universes, then it would be cost-beneficial to include them in an algorithm.
Central to the ACT is the assumption that the universe (or more precisely data received from the universe) is compressible. But at least one philosopher4 has recently argued that raw empirical data sets are maximally complex and therefore already constitute their shortest possible description. If true this would leave no role for laws in the compression of empirical data and provide a decisive blow against the ACT. It is far from clear, however, whether this has been demonstrated and the debate is still ongoing.5 Whilst noting that this is an important issue and the ACT owes us some additional reason to think the universe is indeed compressible, I wish to put this aside for now in order to return to the main problem.
4.2 Solving Armstrong’s Problem without Natural Properties
It is now time to make good on the earlier promise and show how the ACT can solve Armstrong’s problem without the need to postulate natural properties. In order to do this it will be useful to have a more precise concept of an ‘algorithm’ at hand. This can be achieved through the idea of a Turing Machine. Let us define a Turing Machine as any machine which has the following properties: (i) an infinite amount of input tape on which symbols can be written, (ii) an infinite amount of blank output tape on which symbols can be written, (iii) a sensor that can read and write symbols, and, (iv) a program instructing the sensor to write new symbols in response to symbols it reads. From this simple model of a computing device it is clear that the program is the physical embodiment of an algorithm—albeit in some fixed computing language.
The program in the Turing Machine will only be able to respond to the right input data, that is, input data in the operating language of the computer. Insert data written in a different language from the Turing Machine and it will not be able to fulfil its task.
4
James McAllister (2003)
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Let us represent a Turing Machine in the following way, where ‘S’L1’ is the input data written in language L1, ‘
TM’ is the Turing Machine operating in L1 and ‘S’’L1’ is the output data written in L1:(1) S’L1
TM S’’L1A significant advance in twentieth-century computing came with the introduction of the concept of a Universal Turing Machine—a Turing Machine that could simulate any other fixed-program Turing Machine. It does this because a Universal Turing Machine can not only read unstructured strings on the input tape, but also instructions or programs on how to interpret those strings. In other words, Universal Turing Machines are programmable in much the same way personal computers are today. Let us represent a Universal Turing Machine in this way:
(2) S’L1 + PL1
UTM S’’L1Armstrong’s Problem, you will recall from section 2, aims to show that lawhood is language sensitive; so that we cannot say of a generalisation whether or not it is a genuine law unless we have first agreed upon the ‘right’ language in which the law is to be written. In the spirit of the ACT, this would mean that whether or not an algorithm achieves significant compression for some string of symbols S depends in a crucial sense on the language in which S is written. But it can be shown that this is not generally the case.
The argument depends on a result from algorithmic information theory known as the ‘Invariance Theorem’. This result was proven independently by Ray Solomonoff (1964a & 1964b) and Andrey Kolmogorov (1965) and can be summarised by the following equation:
(3) (S) K(S)L1 — K(S)L2 c
For any given string S, let us define its ‘complexity’ (K) as the length of the shortest input on a Universal Turing Machine that will output S and halt. The equation above states that for any string S the difference between its complexity when written in two distinct languages (L1 and L2) is no greater than some constant value c. What determines the value for c? Suppose I decided that I wanted a string of symbols S translated from one language to another. I could write a program for a Turing Machine that would translate each symbol it receives in L1 and print the corresponding symbol in L2:
(4) SL1
TM SL2We already know that a Universal Turing Machine can emulate any Turing Machine provided we insert a program for that algorithm as input. Let us define a program
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‘PL1L2’ which, when followed, performs the same function as (4). Then we can tell a
UTM to output the string in any language we like, provided we supply a translation program:
(5) SL1 PL1L2
UTM SL2From this it should be apparent that the size of c—the difference in complexity between a string in two separate languages—is equal to the length of the program PL1L2:
(6) c = PL1L2
What this shows is that language change will affect the complexity of a string, but it will be bounded to a constant value dependent on the size of the computational resources required to translate the two strings. According to the ACT, an algorithm is a law if it achieves very high compressions of empirical data obtained from the natural world. We can now check to see if changing our description language of the world will have any effect on the lawhood of a program P. Suppose I start by describing the world in L1. What I want to know is the complexity of U (the total information obtainable from the universe) when written in L1. Suppose I discover that this is given by input S and program P:
(7) SL1 PL1
UTM UL1According to Armstrong, if I now change my operating language to L2, PL1 should no longer be a law of nature. But all we need to add to the input above is the translation program to ensure that U arrives in L2 as output.
(8) SL2 PL1 PL1L2
UTM UL2Therefore in order to output UL2 we need only add the fixed-length translation program which is typically a very small addition when one considers the size of compression gained by using P to describe U. The fact the translation program is constant shows that language change does have an effect when the length of the output data is small, but its significance overall will diminish as the length of the output increases. Since the output for a scientific description of the universe U is going to be very large the addition of a further translation program will make little difference on the compression rates achieved. Hence the language one chooses to describe U has negligible effect on the compression achieved by an algorithm and consequently cannot affect its status as a law.
5. The ACT vs. the BSA: Parallels and Contrasts
At their core there is much that the ACT and BSA agree upon. Perhaps their most significant point of overlap is in their explanation of the existence of laws. Both
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subscribe to a functional explanation of laws and both give a similar view on what that job is: the job of laws is to provide an abridged description of nature. The world is a messy place and confronts us with a diverse array of particular matters of fact. But there is order in this complexity in the form of regularities and these regularities can be exploited to provide condensed or compressed representations of the world. Nothing is needed beyond matters of fact and regularities in both accounts to explain the existence of laws. Neither does each view attach lawhood to the actual
representations of scientists since we can idealize somewhat and imagine the ‘best’ abridgement of all the empirical facts (both discovered and undiscovered).
Where the BSA and ACT differ is in the theoretical framework used to articulate how laws achieve this functional role. The BSA appeals to the concepts of mathematical logic whereas the ACT appeals to computational information theory. By so doing we end up with quite different ontological views about what laws of nature are. For Lewis and the BSA laws are general statements that are true. They achieve the functional role described to them by being axioms or theorems in the best deductive system covering as many matters of fact as possible in the simplest possible way. By contrast, according to the ACT, laws are not statements; they are algorithms or rules that allow us to compress empirical data so as to provide the most accurate copy of ‘nature’s information’ using as fewer computational resources as possible.
Some might at this point object to the ACT on the grounds that by making laws rules we are ignoring one of the most common and intuitive beliefs about laws: that they
are truths to be discovered. Most contemporary empiricist philosophers with
sympathies towards Humeanism accept laws are some kind of true statement without much consideration (the exception to this seems to be Marc Lange 2000). Yet I wonder why this needs to be the case. Even on the most realistic sounding governing conception of laws, laws are not truths. For example, according to Brian Ellis’ (2001) theory of laws they are ‘essential natures’ and according to David Armstrong’s (1983) account they are ‘second-order relations of necessity’. It is no more logically appropriate to label these objects ‘true’ or ‘false’ anymore than it is to call a chair or table ‘true’ or ‘false’. Of course when people say laws are truths what they usually mean is that statements of law are truths, such as ‘The gravitational
force exerted by an object is proportional to its mass’. It is because realist approaches to laws can make this distinction between statements of law and their ontological ground that they can avoid Armstrong’s problem. Contrastingly, it is because Lewis’ BSA assimilates statements of law with their ontological ground that it suffers from Armstrong’s problem. By making laws themselves statements the BSA is confronted with the difficulty that a law’s ability to fulfil its function will depend radically on the vocabulary of predicate terms it is willing to recognise.
Like traditional realist accounts the ACT separates the ontological grounding of a law from a statement expressing it. The law itself is not a statement but an algorithm or program which allows us to concisely represent empirical data. Unlike realist accounts of law we have not needed to appeal to irreducible natural necessities to
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explain the existence of laws. Given any data set, wherever there is sufficient regularity and order, there is an algorithm for compression – we don’t need to go over and above the occurrent facts.
One might further object to the ACT here by citing a feeling of déjà vu: by identifying laws with rules are we not resurrecting the so-called inference-ticket-view as advocated by Schlick (1931), Ryle (1949) and Toulmin (1953)? This approach to laws asserts that laws of nature do not take on the function of statements in scientific arguments (for the purposes of explanation and prediction) but instead take on the role of extra-logical inference rules. Laws such as Snell’s law rarely feature as major premises in scientific arguments, according to Toulmin, but instead are used to license inferences from true premises about angles of incidence to true premises about angles of refraction. Ernest Nagel (1956) calls these rules ‘material rules of inference’, because unlike logical rules such as modus ponens they only preserve truth when the premises refer to particular individual objects.
The inference-ticket-view is widely believed to suffer from a number of fatal flaws. Perhaps the most challenging is expressed in an example given by Gavin Alexander (1958, p. 321). According to Alexander any general empirical claim, such as ‘All professors dress formally’ can be used as a material inference rule provided we are explicit about its subject matter and scope. But clearly this leads to a conflation of the law/accident distinction as such examples are not genuine laws of nature. It seems that the inference-ticket-rule lacks grounds for preferring one set of inference rules over any others. They could say that the laws are the rules which are actually used
by the scientific community, but this would render the view unappealingly relativist. Ironically the inference-ticket-view shares the same problem as the BSA: by relying exclusively on the framework of mathematical logic, it lacks the conceptual apparatus to provide an inegalitarian theory of rules. However the ACT is not so constrained. Algorithmic complexity theory gives us a definite meaning to the term ‘length of an algorithm’ as well as a means to measure the amount of information contained within it. Because of this the ACT can say that not all rules are equal: the best are those that can provide the shortest possible compression of empirical data whilst also containing the most amount of information. For the ACT, the objection from conflating the law/accident distinction does not arise.
The ACT provides real progress over the BSA and should be welcomed by Humeans sceptical about the existence of natural properties. No doubt the account sketched here needs supplementing with further details in order to provide a fully-fledged alternative theory of laws. For example, I have said very little about what information is, a concept crucial for comparing the merits of different compression algorithms. Much work has been done on the metaphysics of information and a number of competing theories exist.6 This suggests that there could be more than one viable
6
See the discussions of ‘theories of information’ in Adriaans & van Benthem (2008) and Luciano Floridi (2004 & 2013).
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version of the ACT depending on how one understands the attending notion of information. Which one is the best? That is a question that will have to await further investigation.
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