Synthese manuscript No.(will be inserted by the editor)

The Generality of Scientific Models

A Measure Theoretic Approach

Cory Travers Lewis · Christopher

Belanger

Received: date / Accepted: date

Abstract Scientific models are often said to be more or less general depend-ing on how many’ cases they cover. In this paper we argue that the cardinalityof cases is insufficient as a metric of generality, and we present a novel accountbased on measure theory. This account overcomes several problems with thecardinality approach, and additionally provides some insight into the natureof assessments of generality. Specifically, measure theory affords a natural andquantitative way of describing local spaces of possibility. The generality ofmodels can be understood as the measure of possibilities to which the modelsapply. In order to illustrate our view, we consider the example of StructuralGenomics, the ongoing project of building three-dimensional models of bio-logical macromolecules like proteins. Using measure theory, we interpret thepractice of homology modeling, where such models are understood to applywidely but imperfectly to the space of possible proteins.

Keywords philosophy of science · modelling · generality · measure theory ·

structural genomics

Acknowledgements We would like to thank audiences at Duke University, the CanadianSociety for the History and Philosophy of Science 2014 annual meeting, and at the IHPSTToronto, for their invaluable feedback on earlier versions of this paper. We would also liketo thank Denis Walsh, Philippe Huneman, Burkhard Rost, and our reviewers, for helpfulcomments.

C.T. LewisIHPST, University of Toronto, Room 316, Victoria College, 91 Charles Street West, Toronto,Ontario M5S 1K7, CanadaTel.: 416-978-5397Fax: 416-978-3003E-mail: cory.lewis@utoronto.ca

C. BelangerIHPST, University of Toronto, Canada

Title Page with all author information

Synthese manuscript No.(will be inserted by the editor)

The Generality of Scientific Models

A Measure Theoretic Approach

Received: date / Accepted: date

Abstract Scientific models are often said to be more or less general dependingon how many cases they cover. In this paper we argue that the cardinality ofcases is insufficient as a metric of generality, and we present a novel account basedon measure theory. This account overcomes several problems with the cardinalityapproach, and additionally provides some insight into the nature of assessments ofgenerality. Specifically, measure theory affords a natural and quantitative way ofdescribing local spaces of possibility. The generality of models can be understoodas the measure of possibilities to which the models apply. In order to illustrateour view, we consider the example of Structural Genomics, the ongoing projectof building three-dimensional models of biological macromolecules like proteins.Using measure theory, we interpret the practice of homology modeling, where suchmodels are understood to apply widely but imperfectly to the space of possibleproteins.

Keywords philosophy of science · modelling · generality · measure theory ·structural genomics

1 Introduction

Generality—which we might gloss intuitively as the property of applying widely—plays a pivotal role in some of the central debates in the philosophy of science.Consider, for example, its role in discussions of the laws of nature. Early treatmentsof laws of nature treated the general and the universal interchangeably, as a markof lawhood (Hempel, 1942). But this has proven naıve, since the strictures ofuniversality can be satisfied by hyper-specific conditions which apply only narrowly(Sober, 1997). And further, those constructs which have traditionally been called‘laws of nature’ typically admit of exceptions, or hold only “ceteris paribus,” andtherefore may not be truly universal (Cartwright, 1999; Lange, 2002). Universalityand generality have therefore been teased apart. Some have taken the failure ofuniversality to mean that we should give up on the idea of laws of nature altogether

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(Giere, 1999), while others have embraced the idea that applying widely, or being“stable” across counterfactual situations, could rehabilitate laws (Mitchell, 1997,2000; Lange and Lundberg, 2005).

Generality and its cognates also feature prominently in recent attempts tocharacterize causal relations. For instance, Woodward (2003, 2006, 2010) arguesthat “invariant” relations between variables, which is to say relations holding withsome degree of generality, are a sine qua non of causality. And in the tradition ofFriedman (1974) and Kitcher (1989), Strevens (2004, 2008) argues that applyingwidely is one of the features that makes scientific knowledge properly explanatory.

But despite its importance, philosophers of science have not spent much energyon the project of characterizing generality. A few proposals exist in the literature,but they are either suggestive though incomplete sketches, or more formal treat-ments which apply only in limited cases. In the latter group we could place vanFraassen’s brief remarks in Laws and Symmetry, where he writes, “To be trulygeneral, a statement must be covariant, it must have this logical status: it is eithertrue in all frames of reference or true in none” (van Fraassen, 1989, 287). Thissuggestion may well be appropriate for models in physics, but it is not at all clearhow it could apply in other domains. What would the analogous principle be forpopulation genetics, where inertial frames of reference do not appear in the formal-ism at all? It seems that this proposal can only be taken metaphorically in mostcontexts. In the former group of suggestive sketches, we could include the commonsuggestions that to be more general is to cover “more cases” or a “wider range” ofcases. These proposals are only helpful to the extent that we know exactly what“more” or “wider” mean.1 As we will argue, this is by no means always clear.

This paper attempts to provide an account of the minimum formal machineryrequired to make precise, quantitative assessments of generality across a varietyof contexts. That is, we attempt to provide a formalism which describes gener-ality, generally. Specifically, we tackle the issue of generality in scientific models,which has also attracted considerable attention (Levins, 1966; Orzack and Sober,1993; Orzack, 2005; Odenbaugh, 2002; Weisberg, 2006; Matthewson and Weis-berg, 2009). Models are a natural object to consider in this context, since theyare usually understood as being intermediate entities, less general than laws, yetmore than mere descriptions of particular states of affairs (Morgan and Morrison,1999).

In speaking informally about generality, philosophers have a tendency to as-sume that models are general to the degree that they apply to a large number ofcases. We will call this the ‘cardinality account,’ though it is unclear that anyonehas ever defended it seriously. We articulate it here as a foil for our own view,because it is an intuitive gloss which is commonly employed. Thinking about gen-erality in terms of the number of cases covered has the virtue of being quantitative,and so reasonably precise where it is applicable. But it obscures the quite complexjudgements that need to be in place in order to individuate cases, and assign themnumerical values. Simply counting cases gives the mistaken impression that therewill be one canonical way of determining the generality of a given phenomenon.We argue that a more serious formal treatment shows that this will not be thecase. Rather, determining the right way to individuate cases in a given scientificcontext is a research question in its own right.

1 See for example Hitchcock and Woodward (2003) and Weslake (2010).

The Generality of Scientific Models 3

So in place of cardinality, we propose a measure theoretic account of the gen-erality of scientific models, which we show to be a more powerful and illuminatingformalization.2 Our measure theoretic account only applies in cases where it ispossible to be quantitative about what “more” and “wider” mean, but if a subjectis irredeemably qualitative then no amount of philosophical precision will changethat fact. It is our belief that our account is robust and flexible enough to applyto any context in which it is possible to be quantitative.

If measure theory is the right way of thinking about generality, a few thingsfollow. First, we are presented with a very natural way of thinking about generalityin much more local and empirically embedded ways than standard modal talkaffords. As we will try to show through our case study, measure theory lendsitself nicely to describing local spaces of possibility, like the space of possibleprotein types. Modal logic and the possible-worlds semantics which go with itare very precise tools for talking about the modal extremes: truth in all possibleworlds, and truth in at least one possible world. But science occupies itself witha network of much more specific and local spaces of possibility. We worry aboutthe possibilities for our climate, or the spectrum of possible reactions to a drug.State-spaces may be understood as a type of modal space: a space of the possiblevalues of the variables in some equations. State-spaces are dynamical, but not alllocal spaces of possibility describe changes over time. We will present a case study,homology modeling, which deals with the space of possible protein types - a non-dynamical space of possibilities. A measure theoretic treatment of generality givesus a modal language well suited to dealing with this diversity of actual judgementsof generality.

Adopting measure theory as a tool for analyzing generality also brings out thedegree to which assessments of generality are conventional, or at least interest-relative. A great many different measures can be defined over any space of pos-sibilities, and the relevance of various measures will depends on the investigativeproject at hand. Whether one thinks there is either a single correct measure, orobjectively better or worse measures, or merely useful and useless measures, willdepend on one’s broader philosophical commitments. But measure theory at leastbrings out the process by which assessments of generality are constructed. Ourcase study is meant to illustrate this process by example, showing how one mea-sure can be preferred over another, given particular investigative interests. Thisis the sense in which measure theory provides a way of thinking about generalityassessments in a way that is emprically embedded.

The structure of the paper is as follows. We begin in section 2 by outlining thecardinality account of generality, and present some problem cases to motivate ourown account. In Section 3 we introduce the minimal amount of measure theoryneeded to ground our account, and in Section 4 we explain how to interpret thisformalism as an account of generality. Section 5 considers what this means for ourability to make assessments of generality. Specifically, we highlight the fact thatmaking sense of our assessments of the generality of ordinary scientific modelsrequires the use of relatively local spaces of possibility. We show that treating allassessments of generality as relative to the space of logical possibility, for example,

2 Measure theory is identified by Michael Strevens as the appropriate formalism for capturing‘abstractness,’ a property related to generality (Strevens, 2004, 170). Our proposal is largelyinspired by this suggestion.

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has some deeply unappealing consequences. In section 6 we consider as a casestudy one such local space of possibilities: the space of possible proteins. This casestudy highlights the way in which assessments of generality must be motivatedby specific research questions and interests, as well as the exquisitely fine-grainedmanner in which measure theory allows us to tune our measure of generality inorder to adequately track those research interests. We close in Section 7.

2 The cardinality account of generality

The essence of the cardinality account of generality is that to be more general isto cover “more” target phenomena, in the sense of applying to a larger number.This is an intuitively appealing treatment, and is widely used, though it is unclearthat anyone has defended it seriously. Michael Weisberg makes a number of helpfulclarifications of this apparently commonsense position, which we will adopt. First,he distinguishes between “a-generality” and “p-generality,” which correspond tocovering more actual or possible cases respectively (Weisberg, 2004, 1076). Fol-lowing Potochnik (2007), Strevens (2004), Strevens (2008), Weslake (2010), andHitchcock and Woodward (2003), we take p-generality to be our primary target.We note that Weisberg’s formulation of p-generality corresponds almost exactlywith the graded modality described by Fine (1972), where a fact is partially neces-sary to the degree that it is true in a large number of possible worlds. P-generalityis in this sense a graded measure of modal robustness, whereas a-generality is amore accidental feature of the world.

Matthewson and Weisberg (2009) also clarify the status of the crucial phrase“applies to.” They point out that there is no fixed degree of accuracy requiredfor a model to apply to a given phenomenon. The ‘applies to’ relation could, insome contexts, demand high accuracy. In others, it could require none at all. Doesthe standard model of a frictionless pendulum apply to actual pendulums? Theredoes not seem to be a single, principled answer that can be given. Matthewsonand Weisberg (2009) distinguish several senses of ‘applies to’ which could be rel-evant. These include isomophism, partial isomorphism, homomorphism, “metricrelationships, such as distance in a state space between model and target,” aswell as less specified notions of similarity (Matthewson and Weisberg, 2009, 180).Like Matthewson and Weisberg, we prefer to remain neutral on this question. Itis worth noting that more strict readings of ‘applies to,’ requiring a more precisematch between model and target, will typically reduce the generality of the model.

Similarly, there will be a tradeoff between precision and generality. Precision,here meant in the sense of Levins (1966), is some function of the range of val-ues taken by the parameters in the model description. Precision is being used as atechnical term here, and is to be sharply distinguished from accuracy, which tracksthe degree of fit between the model and the world. For a description of a model tobe more precise is, in a rough sense, for its parameters to have a smaller range.3

Following Giere (1988), Weisberg (2004) makes a distinction between model de-scriptions, models, and model instantiations. Model descriptions can take variousforms, including causal graphs, pictures, or even prose, but here we will focus on

3 For a more precise treatment of precision, and a discussion of the potential tradeoff betweenprecision and generality, see Weisberg (2004).

The Generality of Scientific Models 5

mathematical models. In that restricted case, a model description is the set ofequations. Model instantiations are ways of filling out the equations, giving spe-cific values to the variables. A model is an abstract structure which is picked outby an instantiated model description. A model description (what is colloquiallycalled ‘the model’) typically picks out a group of such models, which are called amodel family.

Precision is then a feature of model-descriptions, unlike accuracy which is arelational property between models and the world (Weisberg, 2006). The moreprecise our model description is (the more finely specified its parameters) the lessgenerally we should expect it to apply, for a given value of the ‘applies to’ relation(Weisberg, 2004).

As the literature following Levins (1966) demonstrates, generality is impor-tantly, perhaps constitutively related to nearby concepts like accuracy, simplicityand precision. The relevance of a given assessment of generality will depend onwhat is going on with these other theoretical virtues in a context. That a modelis extremely general may not mean much, if it achieves this at the cost of beingwildly inaccurate, for example. The exact relationship between precision, accuracyand generality is quite complex, and beyond the scope of this paper. But gettingclear on the basic meaning of these terms will be important in what follows, aswe now turn to describe a key limitation of the cardinality account of generality.That we wish to remain ecumenical about what ‘applies to’ means, or that we donot provide an account of accuracy, may be seen as limitations of our view. Ourgoal is modest, however: to provide an incremental improvement to our picture ofjust one of these theoretical virtues.

Such an improvement is warranted, because it is clear that defining generalityas a function of the number of target phenomena a model applies to is not aworkable approach for many commonly used types of models. This is because,as Strevens (2004) notes, models with real-valued variables will range over anuncountably infinite number of target phenomena. All such models therefore coverthe same number of cases, at least from a cadinality point of view. We take suchdifficulties in dealing with real-valued variables to be one of the main motivatorsfor going beyond the intuitive cardinality account.

But the cardinality account is not straightforwardly inapplicable to modelswith real-valued variables. Weisberg (2004) points out that the generality of suchmodels can be compared in cases where the set of model instantiations of onemodel is a proper sub-set of the model instantiations of another. Consider hisexample of the “ball and spring” model of covalent bonds, which treats atoms aspoint masses separated by ideal springs (Weisberg, 2004, 1074). The energy of acovalent bond between two particles permits of the following model description:

E = k(r − r0)2 (1)

Where k is a constant, r is the separation of the particles, and r0 is the “natural”equilibrium distance between the particles.

The value of k determines the precise models described by Equation 1. Sincek is unspecified, it does not pick out any specific models. We could easily choosea precise value for k, which would then give us a set of model instantiations.However, Weisberg notes, our model descriptions are often not completely precise,and their parameters may take on a range of values. He considers the following

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two equations (Weisberg, 2004, 1077):

E1 = 2.0± 0.1(r − r0)2 (2)

E2 = 2.01± 0.01(r − r0)2 (3)

These descriptions, Weisberg argues, demonstrate an important relationship be-tween precision and generality: more precise models will tend to be less general,since they pick out fewer possibilities. Because the interval (2.0, 2.02) is inside theinterval (1.9, 2.1), the set of model instantiations picked out by the second equationforms a proper subset of the model instantiations picked out by the first, and so thesecond is more precise. Furthermore, Weisberg notes, we can make claims aboutthe relative p-generalities of these model families. Generality is a relation betweenmodels and target systems, and in this example the target is logical possibilities.Although we cannot strictly compare the number of logical possibilities the modelsof E1 and E2 apply to, we can tell that the model family picked out by E1 is moregeneral than the one picked out by E2 because the latter’s instantiations form aproper subset of the former’s. In other words, the model family picked out by E1

covers all the models picked out by E2 and then some, so the models of E1 willapply to more logical possibilities, and are therefore more p-general than E2.

This works well as long as the model families picked out are proper subsetsof each other, but there are cases where model instantiations seem to be clearlymore or less general than each other despite neither one being a proper subset ofthe other. Consider the following model description:

E3 = 2.1± 0.01(r − r0)2 (4)

E3 has exactly the same precision as E2, since their parameters have the sameerror terms. So it would seem natural to say that E1 is likewise more general thanE3. Unfortunately this is not a consequence of the set-theoretic account, becauseE3’s model instantiations are not a proper subset of E1’s. We cannot say whichof these two model families is larger. This would hold even if we sharpened theparameter in E4 arbitrarily, and the model description:

E4 = 2.1± 0.00000001(r − r0)2 (5)

cannot be said to be less general than E1, even though it is orders of magnitudemore precise.

As a final observation, consider the following model description:

E5 = 3.0± 0.1(r − r0)2 (6)

E5 has the exact same precision in its parameter as E1, but on the set-theoreticaccount they cannot be said to be equally general.

So this approach works well in specific cases, but it does not apply outsideof the narrowly delimited situation where model families are proper subsets ofeach other. It intuitively seems like we should be able to compare non-overlappingmodel families ranging over real-valued variables, but so far it seems we cannot.Fortunately, there exists a quite standard mathematical formalism for dealingwith exactly this type of question. In the next section, we turn to this formalism,measure theory.

The Generality of Scientific Models 7

3 Measure theory introduced

Measure theory is a set of well-worn mathematical tools, which (intuitively speak-ing) lets us measure the ‘size’ of a set. Using it we can, for example, define ‘length’along the real-number line, or ‘volume’ in a real-valued space. It is probably mostfamiliar to philosophers of science as the foundation of probability theory, wherethe relative ‘size’ of a region of some outcome space is interpreted as its probabil-ity. Here we present some basic elements of measure theory, which will receive aphilosophical interpretation in the next section.

We begin with a set X. If our measure ranges over the real number line, forexample, X will be the set of all real numbers. If we are using measure theory inthe context of defining p-generality, then X will be some set of possibilities. Inprobability theory, X is the set of possible outcomes, for example.

We then construct a set of subsets of X, denoted ΣX , which is called a σ-algebra. The σ-algebra can be thought of as a way of carving up the space ofpossibilities. It defines the regions over which quantitative values will be assigned.The σ-algebra must have the following properties:

1. The null set is a member of ΣX ;2. If a set E is in ΣX , then E’s set-theoretic complement X\E is in ΣX ;3. For every countable sequence of sets En in ΣX , the union of this sequence of

set is also in ΣX .

To take the two simplest examples, ∅, X and the power set of X—that is, theset of all subsets of X—are both σ-algebras of X.

We then define a measure function µ which takes elements of ΣX to the positivereals. This function µ is what assigns quantitative values to sub-sets of X, givingthem a precise ‘size’ or ‘volume.’ But those metaphors are slightly misleading,because ordinarily we think of things as having a definite ‘size’ or ‘volume’. Incontrast, we have an enormous amount of flexibility in specifying µ, and differentfunctions need not preserve even the ordinal ranking of different measures. µ can beabout anything we like, within certain formal constraints. Specifically, µ : ΣX →[0,∞] such that:

1. µ(∅) = 0;2. If the sets En, n ∈ N, are disjoint members of ΣX , then the measure of their

union is the sum of their measures: µ(⋃

n∈NEn) =∑∞

n=0 µ(En).

An ordered triple (X,ΣX , µ) is called a measure space. In this paper we refer to µas a “measure function,” and reserve the term “measure” for the value µ assignsto a given set. Note that while µ(∅) = 0, a non-empty set can have measure zero.

It can be helpful to think of the measure of a region of X as its ‘size,’ but notethat many different measure functions can be defined over the same ΣX . Differentfunctions can assign different measures to the same sets, and so there is no oneunique measure of a given set. By way of example, consider two such functions.Let X be the real numbers R, and ΣX be the Lebesgue-measurable subsets of R.First we introduce µL, the Lebesgue measure function. In the one-dimensional realcase, the Lebesgue measure of an interval corresponds to its length in the ordinarysense. If we let the sets A = [0, 1] and B = [0,∞], then we have µL(A) = 1 andµL(B) = ∞. Now consider µG, the Gaussian measure function. µG is a measure-theoretic analogue of the Gaussian probability distribution, which here we take to

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have standard deviation of 1 and to be centred around 0. For the same sets A andB, we have µG(A) ≈ 0.341 and µG(B) = 0.5. These two measure functions assigndifferent values to the same sets.

This is just to say that different µ functions will measure the space differently.There will not be a single, obviously canonical way to measure a given spaceX, or even a given ΣX . Further, since µ is a function of ΣX , which is a set ofsubsets of X, in general measure functions over different spaces X and Y will beincommensurable. As we will discuss in more detail below, the plasticity of theformalism, and the ease with which measures can be incommensurable with eachother, are some of the philosophically interesting consequence of taking measuretheory to be the appropriate way of thinking about generality.

4 Measure as generality

If for a model to be more general is for it to cover more cases, we can formalize thisusing measure theory as follows. First, we let X be our possibility space, the set ofall the possibilities our model M could cover. Our discussion of the ball-and-springexample was framed in terms of logical possibility, but measure theory imposesno formal constraint on what species of possibility generality needs to range over.The view that all possibility is logical possibility can be accommodated; however,in Section 5 we will argue that logical possibility is very unlikely to be the kind ofpossibility relevant to judgements of generality in scientific contexts. Alternately,X could be defined over the set of metaphysical, physical, chemical or biologicalpossibilities, among others. The formalism can handle spaces of possibility whichare arbitrarily local. We take this flexibility to be one of the virtues of usingmeasure theory in this context.

Next, we define the σ-algebra ΣX . ΣX should include all relevant subsets ofX, and so sensible choices will typically be the power set of X or the Lebesgue-measurable subsets of X. The power set is attractive because it is guaranteed tocapture all of the relevant subsets, as it is the set of all subsets. The Lebesgue-measurable subsets of X present an appealing σ-algebra for real-valued systemsbecause the Lebesgue measure corresponds reasonably well with an intuitive notionof length or volume of uncountably infinite sets of things. The Lebesgue measure ofthe uncountably infinite set of real numbers between 0 and 1 is 1, and the measurebetween 0 and 10 is 10, for example.

Finally, we need a measure function µ. The appropriate µ will be determinedby the model and the domain of inquiry. The measure function µ is especiallyimportant, since it actually assigns values of generality. This formalism gives usa great deal of freedom in how we choose µ. Nothing compels us to choose onefunction over another, so the choice of measure function is conventional. Beingconventional, however, is not the same as being arbitrary, and if we want ourformalism to give a plausible measure of generality then our options will be limited.Just as our domain of inquiry will suggest limits on X, the sensible choice for µwill depend on the situation at hand. As we hope to demonstrate in our case studybelow, determining which measure is appropriate for a given modeling context ispart of the work of practicing scientists. Anti-realists about scientific knowledgewill tend to understand such assessments as practical. That is, scientists will lookfor the most instrumentally useful way of measuring generality. Realists will be

The Generality of Scientific Models 9

more bullish, holding that scientists are to some degree able to determine whichmeasure of generality is the right one, given specific research interests. We clearlyare not in a position to resolve that debate here, but merely flag it as one interestingway in which broader issues in philosophy of science intersect with this discussion.

With all that in place, we can now give a formal statement of our view. Takea model M , that applies to some possible systems. If we have chosen X properlythen all of these systems will be members of X. If we let capital M denote themodel and lower-case m denote the possibilities it applies to, we can associate Mwith a subset m of X. If we’ve set our apparatus up properly, then m should bea member of ΣX .

Once we have our measure space (X,ΣX , µ) and a model M associated with amember m of ΣX , we can define the generality of a model as follows: the generalityof a model is the measure assigned to the set of possibilities it applies to. Formally,the generality of a model M is µ(m).

5 Consequences of the formalism

The main appeal of treating generality in this way is that using it, we can representboth commensurability between non-overlapping sets of cases, and the incommen-surability of more deeply non-similar spaces of possibility. As we showed in theprevious section, measure theory lets us compare things which intuitively shouldbe comparable, like non-overlapping regions of the same state-space. But the ap-paratus of measure theory also gives us a nice way of formally representing theincommensurability of radically different spaces of possibility. Is the Lorenz modelin fluid dynamics more or less general than the Volterra-Lotka equations in ecol-ogy? There is no obvious way of making such comparisons, and measure theorygives us a nice formal explanation for why this should be so: those models rangeover different spaces of possibility, that is, different Xs. Any measures defined overdifferent Xs will be strictly incommensurable. So if someone wants to maintainthat there is a clear ordering of the generality of those models, the way to demon-strate this would be to produce the state-space of which those models both pickout regions, and somehow motivate a particular measure over it. However, we willargue in this section that this is not an easy thing to do.

To see why, consider the the question of which variety of possibility, or varietiesof possibility, are appropriate for thinking about the generality of scientific models.The cardinality account of generality appears to have no bearing on this question,but our measure theoretic account does, because letting X be the set of all logicalpossibilities and defining µ uniformly across X has catastrophic consequences forassessing the generality of scientific models. Since it is not logically contradictoryto suppose that the physical constants, or even the laws of nature themselves,could have been different, the set of physical possibilities is infinitesimally smallcompared to the set of logical possibilities. This means that if we make X theset of logical possibilities and define µ in a way that does not privilege the actualphysics we observe ad hoc, the measure of what is physically possible goes tozero. By extension, the generality of our current physical models goes to zero.In fact, even a perfect model of the laws of nature would be minimally generalon this account. Those extremely committed to logical possibility as the relevant

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possibility space might accept this result, but we doubt it will be attractive tothose who think of generality as a virtue relevant to theory selection.

If we take X to be the set of all physical possibilities then presumably wehave at least some physical models with non-zero generality. But now, absent adhoc tinkering with µ, many ‘non-fundamental’ models, for example those from thebiological sciences, will turn out not to be general. For instance, the generalization“Baboon dominance heierarchies are transitive” is useful and appears general inthe context of primatology. It is robust across a number of contexts, and canbe used to model a variety of counterfactuals of interest to primatologists. Butsince baboons form an infinitesimally small proportion of the physically possibleconfigurations of matter countenanced by fundamental physics, this generalizationranges over only an infinitesimal set of possibilities, and so has zero generality.In other words, if we let X be the set of physical possibilities and let µ assignmeasure uniformly across the set, only fundamental physical theories will have anygenerality at all. A radical physicalist might accept that chemical and biologicalmodels—to say nothing of psychological or sociological models—are minimallygeneral. Still, the intuition that some non-fundamental models have some degreeof generality will motivate others to reject the imposition of X as the set of allphysical possibilities.

Measure theory provides an easy and natural way of clarifying disagreementsbetween the reductionist and the non-reductionist about generality. The physi-cal reductionist embeds all models in the space of all physical possibilities andevaluates generality with the function µP , while the baboon primatologist embedsmodels in the relevant space of baboon possibilities and evaluates generality withrespect to µB . Because these measures of generality are produced by differentfunctions over different spaces they are incommensurable. Baboon models are cer-tainly not general in the space of physical possibility, but once we recognize thatprimatological models are evaluated with respect to a more local X we lose thetemptation to judge them by the standards of physics. Our definition of general-ity does not force this move, but it is an affordance of the formalism. It is also,the physical reductionist should note, the same move necessary to preserve thegenerality of physical theories in the face of the full space of logical possibility.

So while adopting measure theory to describe generality clearly does not re-solve the reductionism/anti-reductionism debate, it affords a slightly adjusted, andhopefully clarified, perspective on it. Assuming generality is a key virtue of scien-tific knowledge, the reductionist needs to hold that all models are somehow em-beddable in one overarching space of possibilities. The anti-reductionist can denythat this is possible. Some canonical anti-reductionist arguments appeal to thegreater generality of high-level descriptions (??). On a measure-theoretic accountof generality, this thought can be cashed out in terms of the generality of a modelwith respect to some specific space of possibilities, rather than with respect toall of what is physically or logically possible. This seems intuitively plausible. Forexample, our assessment of the generality of the Volterra-Lotka equations wouldbe unaffected if we discovered a new principle in fundamental physics, becausethat is not the space of possibilties we are evaluating it with respect to. Rather,it is general with respect to the space of possible ecological scenarios.

So far we have only considered the consequences of selecting one or many X’s.There is also the important question of how to define ΣX and µ. In the nextsection we will look at an example of model-based science which illustrates what

The Generality of Scientific Models 11

sort of empirical motivation there can be for specifying ΣX and µ in a particularway, in a given context. With that example described, we can then further discussthe consequences of our proposal.

6 Example: Structural Genomics and Homology Modeling

In this section, we will look at homology modeling, a practice within structuralgenomics. The point of this section is to provide an example to fill out the quiteabstract description provided above. First we’ll describe the overall research pro-gram of structural genomics, and then focus on the strategy of homology modeling.The point of this example is largely to illustrate the kind of case-specific reasonsscientists may have for picking on particular measure of generality over another.But it also exemplifies a more general point about why scientists would care aboutthe generality of their models in the first place.

Structural genomics is the sub-field of molecular biology that builds models ofthe three-dimensional structure of biological molecules, like proteins. Determiningthe linear sequence of amino acid types that make up a particular protein hasbecome a relatively cheap and easy process over the last few decades (Mardis,2011). DNA can now be mass-sequenced, and sequences that code for proteinscan be identified. Once this is accomplished, it is possible to infer the sequence ofamino acids that are coded for by that particular region of DNA. Millions of suchsequences are stored in huge databases, like UniProt (Consortium, 2011).

The problem is that sequence data does not tell us much about a particularprotein’s function. In order to make inferences about how a protein functions, weneed to know more than just the sequence of amino acids that make it up. We needto at least know how a given sequence folds in three-dimensional space. That is, weneed a three-dimensional model of the protein. Structural genomics is the projectof building such models. With knowledge of the three-dimensional structure ofa protein in hand, we can then begin to predict and explain what the proteinwill do in the context of an organism. So this is a classic case of models-basedscience, where relatively local models are built on the basis of empirical data, andwhich we hope in the long run will form the basis for prediction, explanation, andunderstanding.

Unfortunately, creating such models is much costlier and more time-consumingthan obtaining sequence data. The standard empirical methods, X-Ray crystallog-raphy and Nuclear Magnetic Resonance imaging, can only handle one protein ata time (Dessailly et al, 2009). With millions of protein sequences being added toUniProt every year, this has resulted in a substantial ‘sequence/structure gap.’Which is to say, our knowledge of protein sequences has far out-paced our knowl-edge of their structure. This is something of a worry, since again, it is structuralknowledge which for the most part facilitates prediction and explanation of inter-esting biological phenomena.

Thus, some researchers have adopted a strategy called Homology Modeling(Orry and Abagyan, 2012; Hillisch et al, 2004; Cavasotto and Phatak, 2009). Thisis the practice of taking one painstakingly constructed model of the structure ofa protein, and using it as a probe to understand other, similar proteins. Again,this is a prototypical strategy in model-based science, where we sacrifice some ofthe accuracy of our model in order to obtain greater generality. Researchers have

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found that if two proteins share more than about 50% of their sequence, inferencescan reliably be made about the structure of one by looking at the structure of theother4. This is a heuristic strategy, of course, but when dealing with the manymillions of proteins involved in living things, it has proven to be fruitful.

Let us consider an example of how this approach has indeed been empiricallyfruitful. Liu et al (2007) define what they call ‘Novel Leverage,’ which is a quantita-tive measure of the generality of protein models.5 Novel leverage is just how muchof the set of protein types a model applies to (we could say, leverages) which noprevious model has already covered. They use this measure to show that Homol-ogy Modeling has provided more new knowledge about the structure of proteinsthan traditional laboratory methods, on a per-dollar basis. With millions of dollarsinvested in Structural Genomics, there is unsurprisingly interest in assessing therelative effectiveness of different methodologies. Liu et al. try to show that Ho-mology Modeling yields the best investment, by measuring the generality of themodels produced by it, versus the standard empirical methods. Thus, generalityis being treated as a methodological success signal. That method which producesthe most general models is taken to be the one to spend resources on.

But as we emphasized above, in general there will not be a single, obviousmeasure for a given space of possibility. In this case, we are dealing with the so-called protein universe, the space of all possible proteins (Ladunga, 1992). Buthow should we assess the generality of a given model, or family of models? Thesimplest approach to the problem would be to simply count the number of proteinsequences modeled. Call this the naıve cardinality account, because we expectthat sophisticated defenders of the cardinality approach would rightly reject it.Certainly those biologists working on the problem reject it as not tracking whatis important in this context. Chadonia and Brenner write, “Although every newexperimental structure adds to our repository of structural data, most structuralbiologists would agree that novel structures [...] are especially valuable” (Chando-nia and Brenner, 2006, 347). The emphasis is not just on cranking out as manymodels of as many proteins as possible, but rather on targeting research effortstowards finding models which illuminate “novel” neighborhoods of the protein uni-verse. Some models apparently count for more than others, in terms of how muchof the protein universe they apply to.

Sensitive to this thought, Liu et al. define two ways of assessing the novelleverage of a model. We will describe both presently, but focus on the slightlysimpler notion of ‘leverage’ for ease of exposition. Leverage is just the degree ofapplicability of a model across the space of protein types, where novel leverage isthe degree to which the leverage of a model covers previously unmodeled proteintypes.

The first step in calculating the leverage of a model is to enter its amino acidsequence into PSI-BLAST,6 an algorithm which searches the massive sequence

4 In some contexts 50% sequence similarity is much more than is necessary for proteins toexhibit structural similarities. Under the influence of natural selection, evolved proteins oftendiffer by 75% or more of their sequence and retain structural similarities (Rost, 1999).

5 For more details on how novel leverage is calculated, see theSupplementary Information to Liu et al (2007), available online:http://www.nature.com/nbt/journal/v25/n8/extref/nbt0807-849-S1.pdf

6 http://blast.ncbi.nlm.nih.gov/Blast.cgi

The Generality of Scientific Models 13

library UniProt7 for similar sequences to the originally modeled protein. Thatis, it searches for statistically improbable string matches, rejecting matches thatare plausibly attributed to chance. This search returns a set of ‘significant’ High-Scoring Segment Pairs (HSPs).

This set of HSPs is then used to calculate the leverage of the model. Liu et al.provide two distinct ways of doing this: leverage at the protein level, and leverageat the residue (amino acid) level. At the protein level, leverage is defined as thenumber of proteins that have at least 50 residues or 50% of their total residuescovered by HSPs. So models apply to any protein which they cover either halfof, or a goodly chunk of. This is relatively coarse grained, and could still be readas a slightly more elaborate form of counting. At the residue level, on the otherhand, the leverage of a model is defined as the number of residues contained insignificant HSPs. So higher leverage values are assigned to larger HSPs, meaning amodel can apply to a protein in a more fine-grained way. Larger and more complexproteins will count for more leverage as well. This is identified by Liu et al (2007)as one of the reasons to prefer the residue to the protein definition of leverage.Characterizing a larger or more complex protein is a more significant empiricalsuccess, so our metrics of the productivity of a research program should reflectthat fact. Merely counting the number of proteins that a model applies to wouldmiss this difference.

This second version of leverage is therefore not merely tracking ‘how many’proteins a model applies to. The residue level definition of leverage does countsomething, but not the number of cases each model ranges over. Rather, it is thenumber of residues contained in HSPs common to both the modeled protein andthe library of protein sequences in UniProt.

To make the connection between leverage and generality as a measure theoreticconcept more explicit, it may help to formulate the residue-level version of leveragein a measure. First let our possibility space X be the protein universe, and let oursigma-algebra σX be the power set (that is, the set of all subsets) of U . Our measurefunction µ assigns leverage to a set of modelled proteins P ∈ σX as follows. First,we define a similarity function s(p, r) which takes two protein sequences p andr as inputs and returns the number of residues contained in HSPs in commonbetween p and r. For each individual protein p ∈ P , we then sum the similarity ofp with all sequences in the UniProt library. Finally, the leverage of the initial setof sequences P is the sum of the leverages of its constituents p. In mathematicalterms:

µ(P ) =∑p∈P

∑r∈X

s(p, r) (7)

Where P is a set of protein sequences, p is a single protein sequence, X is theprotein universe as represented by UniProt, and s(p, r) is a similarity functionbetween the protein sequence p and some other sequence s from the UniProtlibrary, as determined by PSI-BLAST at some specific search parameters. Thefunction s(p, r) returns the number of residues contained in HSPs common betweenthe modeled protein p and a sequence from UniProt s. This is just to say that themeasure of p is equal to the sum, for all members of X, of the similarity of p toeach member; and the measure of a set of sequences P is equal to the sum of themeasures of each individual sequence.

7 http://www.uniprot.org/

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To return to the point of this case study, notice how many finely specifiableparameters go into defining leverage. Some threshold must be chosen for howsimilar proteins must be before a model of one can apply to another, and anygiven threshold must be motivated empirically. Similarly, leverage is not definedindifferently over the entire space of possible proteins. Rather, UniProt standsin for the protein universe here, so leverage is focused on those protein typeswhich are most likely to have biological significance (having proven their worthby actually appearing in an organism). We can imagine even more sophisticatedways of structuring the space of possibility, according to structural or functionalfamilies, for example.

All of this is meant to show, first, that scientists are actually concerned aboutthe generality of their models. They want to know how broadly their models apply,and as in the case of homology modeling, these models often do not fall into neatsub-set relations. But further, it exemplifies the way in which the specific researchquestions we ask will directly shape the kind of measures that ought to be appliedin assessing generality. Scientific realists may be inclined to think that given a clearand specific research question, there will be a ‘correct’ measure, or a restricted setof measures which best capture reality. Anti-realists will of course tend to say theopposite, that our choice of measure is conventional and pragmatic. Like in thecase of choosing X and the reductionism v. anti-reductionism, we make no claim tohave resolved this deep and ongoing debate. However, we do hope to have offereda somewhat novel way of framing the disagreement.

It is also interesting to note that what motivated Liu et al’s interest in leveragewas the need to make a comparison between methodologies. There is a commonintuition that generality is a comparative concept, and this case supports that in-tuition. Technically, measure theory affords absolute (non-comparative) judgmentsabout the generality of a model. But it is hard to imagine why anyone would careabout such non-comparative measures. In the above case study, the need to com-pare the generality of two or more models provides guidance about what measureis the relevant one. Although we leave it open as a conceptual possibility thatsomeone could be interested in non-comparative measures of generality, a desireto compare models appears to be necessary to motivate specific measures.

Finally, it is worth wondering whether these structural models are actuallyexplanatory. Above we mentioned the debate over scientific explanation as oneof the points where generality is important, the thought being that one of thefeatures that makes an explanation good is that it is ‘deep,’ in the sense of beinggeneral Strevens (2004, 2008).

In fact, according to that way of seeing, we should not expect structural mod-els of proteins to be particularly explanatory, since they cover so little of theirlocal space of possibilities. This seems right: they are mostly descriptive. But nowimagine a future in which we have adequately described the protein universe, andhave moved on to creating models which don’t just cover it piecemeal, but applyto it generally. If we were in posession of models which, for example, facilitated thedesign of novel proteins with desired functional properties, it would be much moretempting to call those models explanatory. We should not suppose that simplybeing general exhausts what it is to be explanatory, since accuracy and simplicityare clearly also important. But we do wish to suggest that explanatory depth, inthe sense of generality, may be captured by our formalism in a reasonably preciseand explicit way.

The Generality of Scientific Models 15

7 Conclusion

Above we have argued that measure theory provides the minimum formal machin-ery required to characterize generality across contexts. Its most obvious advantageover what we have dubbed the cardinality view is that it allows us to deal withmodels which are defined over real-valued variables. But, as we saw in the casestudy of homology modeling, it also lets us describe the way in which models mayapply widely but imperfectly across cases.

Perhaps more significantly, measure theory has some important affordancesand limitations which, if we accept it as the right way of thinking about gener-ality, have consequences for key debates in the philosophy of science. It affords apluralism about the relevant set of possibilities over which generality should bedefined, which is congenial to anti-reductionism. And it is limited, in the sensethat only measures which at least share a space of possibilities can be meaning-fully compared. Similarly, measure theory affords an almost unlimited flexibilityin terms of the function which assigns measures to regions of a given space of pos-sibilities. This conventionality seems congenial to the anti-realist with respect toassessments of the modal depth of models. However, the possibility remains openfor a scientific realist to argue that some measures capture reality better thanothers. It may be the case that a specific empirical situation will impose enoughconstraints on our choice of measure that it can meaningfully be said to reflect“the” generality of a model.

There may be no algorithm for calculating generality in all contexts, but wehope our framework is robust and flexible enough—and, indeed, general enough—to capture the scientific usage of the term.

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