Abstract
Asymptotic models in which singular limits are taken are very common in physics. They are often used to investigate the general behaviour of systems undergoing rapid, discontinuous, changes. The singularities in the mathematics of these systems have no physical counterparts; these models operate by containing non-physically interpretable fictional elements. As such there is an intuition that states that asymptotics only offer descriptions of systems not explanations of them. By contrast, in different areas of science other models containing fictional elements which are physically interpretable are claimed to be explanatory. I argue that both types of fictional model possess modal content, and therefore it is unclear why models that contain unphysical fictions are merely descriptions, but models that contain physical fictions are explanations. One must either reject both as unexplanatory because they use false ontology to explain, or one should accept both types as explanatory.
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Notes
In this case the breakup would look the same if we, say, double viscosity, multiply the characteristic internal drop size by 4 and the time scale by 8.
A classic example is the thermodynamic limit: in this we have to set the number of particles in our model system to infinity, even though in the real system it is finite, to recover some thermodynamic quantities and to see discontinuous phase changes in them. To fully understand and explain water turning to ice, we need the thermodynamic limit: it is not a mere idealisation which can be removed to produce a more accurate model. In asymptotic reasoning in general, we do not have a smooth transition from one qualitative regime of behaviour into another. Rather, the change is discontinuous, like being told as a child that you can play football in the garden, but not in the house. There is a qualitative difference to the allowed behaviours depending upon where you are. It is all or nothing: you cannot play a little bit of football inside the house.
Should the explanatory role of asymptotics be decided by scientists? Why evaluate it from within one theory of explanation, this opens us up to a relative account where a different theory of explanation will come to a different conclusion. Of course in a sense this is correct. Scientists will use asymptotics and describe such models as explanatory, and we should aim to recover this pragmatic aspect of how scientists view asymptotics. But this is very different from taking the science at face value. For instance when a scientist uses asymptotics to explain something are they explaining epistemically or ontically? Such a distinction is pragmatically irrelevant, since most explanations contain elements of both, but to the philosophical debate such a distinction is potentially very important. B&F’s claim is that asymptotics can never be explanatory on principle in an ontic sense, they make no such claim about an epistemic sense of explanation. I don’t think the question of whether asymptotic models are explanatory in an epistemic or ontic sense is decided by simply looking at the science alone (or how scientists self-describe, since they are largely uninterested in such philosophical distinctions).
Modal information is produced whenever we experiment to determine what depends upon what or theorise to the same end in a manner that is consistent with observation. How each case does this will be specific to it, but in general terms we change input variables to either observe in experiment/computer simulation or deduce from theory/model what the resultant change on an output variable is. Once we can systematically connect input and output variables for a system then we can say we have mapped (part of) the modal structure of that system: we can say how things would (or wouldn’t) be different under changes in those variables, therefore we answer w-questions when we explain. (We may do others things as well, such as unify systems, or discover mechanisms, but we must always provide modal content. For instance it is difficult to imagine giving a mechanistic explanation in which the mechanism doesn’t allow one to say how circumstances could have resulted in a different outcome.)
I am grateful to a referee for pointing out that the notion of constrain is ambiguous here. In one sense the surplus structure does constrain the model by providing us with rules of mathematics, but for B&F what is at stake is the form of that constraint. The surplus structure constrains mathematics, the form of that mathematics has its own rules and those impose boundaries on the model but only at the representational level. For B&F anti-realists about maths, mathematics is only a representational device and nothing more. So the laws of mathematics are only laws about certain ways of representing the world, not about the physical world itself. One can think of this in analogy with language. To say something in one language rather than another imposes constraints on what is being said, but for B&F those constraints are merely syntactical, not semantical. The meaning of a sentence in any language can be translated even where the precise wording cannot (this is not necessarily my position but again I wish to fight B&F on their own ground as much as possible to keep the debate focused and not beg the question in relation to the assumptions I’m making about mathematics. B&F’s description of mathematics does rely on a set of assumptions that can be challenged, but my aim is to challenge them within those assumptions.)
I am grateful to Bueno (private communication) for raising a potential ambiguity here. An objection might be raised that inferences are not meant to be explanatory. After all inferences are just true or false, whereas it is models and theories are explanatory. I believe this is quite true as far as it goes, but part of why being able to make an inference from a model to a target system matters is that by doing so we can extract modal information from the model and apply it to the system. We believe, because of our ability to draw inferences in the model that relate to the target system, that the model is giving us justified modal information. The inferences are a mechanism for the transmission of modal information: “if system X is like system Y in respect of relationship between K & J then the modal architecture of X is the same as Y in relation to K& J”. We can use inferences we make to help us to answer w-questions or to justify the answers to w-questions that our model gives us. In the asymptotic cases the fact we can account for them in the inferentialist framework does not mean that they are explanatory or not explanatory, inferences (in general) are neutral when it comes to the question of explanation, so the fact we cannot make direct inferences from certain parts of a model does not rule that model out as explanatory.
That said the source of the confusion here may be Bokulich’s own analysis. Bokulich does describe “manipulating” wave-functions and classical orbits directly as well as “measuring” them. I believe this is a confusion in Bokulich’s presentation, the classical orbits do not exist hence they can neither be manipulated nor measured! As regards manipulating classical orbits of course in Woodwardian terms this makes sense for classical systems, but what is at issue is describing a purely quantum system, the quantum billiard, by representing it using classical orbits, none of which are taken to exist (in that quantum system at least!). We cannot manipulate the fictional classical descriptive element for the quantum system since we can never causally manipulate the purely imaginary.
I am grateful to S. French for pointing this objection out (French 2013 private communication)
In the extension of manipulationism developed by Pexton (2013) sources of asymmetry are not a problem as we have an analogue of manipulations that can apply in non-causal cases.
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Acknowledments
I would like to thank Otávio Bueno, Stephen French and Juha Saatsi for helpful comments on an earlier draft as well as the extremely useful comments of the referees.
Funding
This work was supported by the Arts and Humanities Research Council at the University of Leeds and by the Templeton foundation at the University of Durham
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Pexton, M. Can asymptotic models be explanatory?. Euro Jnl Phil Sci 4, 233–252 (2014). https://doi.org/10.1007/s13194-014-0084-7
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DOI: https://doi.org/10.1007/s13194-014-0084-7