Most philosophers believe that, other things being equal, simplertheories are better. But what exactly does theoretical simplicityamount to? Syntactic simplicity, or elegance, measures the number andconciseness of the theory’s basic principles. Ontological simplicity,or parsimony, measures the number of kinds of entities postulated bythe theory. One issue concerns how these two forms of simplicityrelate to one another. There is also an issue concerning thejustification of principles, such as Occam’s Razor, which favor simpletheories. The history of philosophy has seen many approaches todefending Occam’s Razor, from the theological justifications of theEarly Modern period, to contemporary justifications employing resultsfrom probability theory and statistics.

• 1. Introduction
• 2. Ontological Parsimony
• 3.A Priori Justifications of Simplicity
• 4. Naturalistic Justifications of Simplicity
• 5. Probabilistic/Statistical Justifications of Simplicity
• 6. Other Issues Concerning Simplicity
  • 6.1 Quantitative Parsimony
  • 6.2 Principles of Plenitude
  • 6.3 Simplicity and Induction
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

There is a widespread philosophical presumption that simplicity is atheoretical virtue. This presumption that simpler theories arepreferable appears in many guises. Often it remains implicit;sometimes it is invoked as a primitive, self-evident proposition;other times it is elevated to the status of a ‘Principle’and labeled as such (for example, the ‘Principle ofParsimony’). However, it is perhaps best known by the name‘Occam’s (or Ockham’s) Razor.’ Simplicity principles havebeen proposed in various forms by theologians, philosophers, andscientists, from ancient through medieval to modern times. ThusAristotle writes in hisPosterior Analytics,

We may assume the superiorityceteris paribus of thedemonstration which derives from fewer postulates orhypotheses. (Posterior Analytics, p. 150)

Moving to the medieval period, Aquinas writes:

If a thing can be done adequately by means of one, it is superfluousto do it by means of several; for we observe that nature does notemploy two instruments where one suffices (Aquinas, [BW], p. 129).

Kant—in theCritique of Pure Reason—supports themaxim that “rudiments or principles must not be unnecessarilymultiplied (entia praeter necessitatem non essemultiplicanda)” and argues that this is a regulative ideaof pure reason which underlies scientists’ theorizing about nature(Kant, 1781/1787, pp. 538–9). Both Galileo and Newton acceptedversions of Occam’s Razor. Indeed Newton includes a principle ofparsimony as one of his three ‘Rules of Reasoning inPhilosophy’ at the beginning of Book III ofPrincipiaMathematica (1687):

Rule I: We are to admit no more causes of natural things than such asare both true and sufficient to explain their appearances.

Newton goes on to remark that “Nature is pleased withsimplicity, and affects not the pomp of superfluous causes”(Newton 1687, p. 398). Galileo, in the course of making a detailedcomparison of the Ptolemaic and Copernican models of the solar system,maintains that “Nature does not multiply things unnecessarily;that she makes use of the easiest and simplest means for producing hereffects; that she does nothing in vain, and the like” (Galileo1632, p. 397). Nor are scientific advocates of simplicity principlesrestricted to the ranks of physicists and astronomers. Here is thechemist Lavoisier writing in the late 18^th Century

If all of chemistry can be explained in a satisfactory manner withoutthe help of phlogiston, that is enough to render it infinitely likelythat the principle does not exist, that it is a hypotheticalsubstance, a gratuitous supposition. It is, after all, a principle oflogic not to multiply entities unnecessarily (Lavoisier 1862, pp.623–4).

Compare this to the following passage from Einstein, writing 150 yearslater.

[T]he grand aim of all science…is to cover the greatestpossible number of empirical facts by logical deductions from thesmallest possible number of hypotheses or axioms (Einstein, quoted inNash 1963, p. 173).

Editors of a volume on simplicity sent out surveys to 25 recentNobel laureates in economics. Almost all replied that simplicityplayed a role in their research, and that simplicity is a desirablefeature of economic theories (Zellner et al. 2001, p.2). Riesch (2010)interviewed 40 scientists and found a range of attitudes towards thenature and role of simplicity principles in science.

Within philosophy, Occam’s Razor (OR) is often wielded againstmetaphysical theories which involve allegedly superfluous ontologicalapparatus. Thus materialists about the mind may use OR againstdualism, on the grounds that dualism postulates an extra ontologicalcategory for mental phenomena. Similarly, nominalists about abstractobjects may use OR against their platonist opponents, taking them totask for committing to an uncountably vast realm of abstractmathematical entities. The aim of appeals to simplicity in suchcontexts seem to be more about shifting the burden of proof, and lessabout refuting the less simple theory outright.

The philosophical issues surrounding the notion of simplicity arenumerous and somewhat tangled. The topic has been studied in piecemealfashion by scientists, philosophers, and statisticians (though for aninvaluable book-length philosophical treatment see Sober 2015). Theapparent familiarity of the notion of simplicity means that it isoften left unanalyzed, while its vagueness and multiplicity ofmeanings contributes to the challenge of pinning the notion down precisely.^[1] A distinction is often made between two fundamentally distinct sensesof simplicity: syntactic simplicity (roughly, the number andcomplexity of hypotheses), and ontological simplicity (roughly, thenumber and complexity of things postulated).^[2] These two facets of simplicity are often referred to aselegance andparsimony respectively. For thepurposes of the present overview we shall follow this usage andreserve ‘parsimony’ specifically for simplicity in theontological sense. It should be noted, however, that the terms‘parsimony’ and ‘simplicity’ are usedvirtually interchangeably in much of the philosophical literature.

Philosophical interest in these two notions of simplicity may beorganized around answers to three basic questions;

(i)

How is simplicity to be defined?[Definition]

(ii)

What is the role of simplicity principles in different areas ofinquiry?[Usage]

(iii)

Is there a rational justification for such simplicity principles?[Justification]

As we shall see, answering the definitional question, (i), is morestraightforward for parsimony than for elegance. Conversely, moreprogress on the issue, (iii), of rational justification has been madefor elegance than for parsimony. It should also be noted that theabove questions can be raised for simplicity principles both withinphilosophy itself and in application to other areas of theorizing,especially empirical science.

With respect to question (ii), there is an important distinction to bemade between two sorts of simplicity principle. Occam’s Razor may beformulated as anepistemic principle: if theory \(T\) issimpler than theory \(T^*\), then it is rational (other thingsbeing equal) to believe \(T\) rather than \(T^*\). Or it maybe formulated as amethodological principle: if \(T\) issimpler than \(T^*\) then it is rational to adopt \(T\) asone’s working theory for scientific purposes. These two conceptions ofOccam’s Razor require different sorts of justification in answer toquestion (iii).

In analyzing simplicity, it can be difficult to keep its twofacets—elegance and parsimony—apart. Principles such asOccam’s Razor are frequently stated in a way which is ambiguousbetween the two notions, for example, “Don’t multiplypostulations beyond necessity.” Here it is unclear whether‘postulation’ refers to the entities being postulated, orthe hypotheses which are doing the postulating, or both. The firstreading corresponds to parsimony, the second to elegance. Examples ofboth sorts of simplicity principle can be found in the quotationsgiven earlier in this section.

While these two facets of simplicity are frequently conflated, it isimportant to treat them as distinct. One reason for doing so is thatconsiderations of parsimony and of elegance typically pull indifferent directions. Postulating extra entities may allow a theory tobe formulated more simply, while reducing the ontology of a theory mayonly be possible at the price of making it syntactically more complex.For example the postulation of Neptune, at the time not directlyobservable, allowed the perturbations in the orbits of other observedplanets to be explained without complicating the laws of celestialmechanics. There is typically a trade-off between ontology andideology—to use the terminology favored by Quine—in whichcontraction in one domain requires expansion in the other. This pointsto another way of characterizing the elegance/parsimony distinction,in terms of simplicity of theory versus simplicity of world respectively.^[3] Sober (2001) argues that both these facets of simplicity can beinterpreted in terms of minimization. In the (atypical) case oftheoretically idle entities, both forms of minimization pull in thesame direction; postulating the existence of such entities makes bothour theories (of the world) and the world (as represented by ourtheories) less simple than they might be.

2. Ontological Parsimony

Perhaps the most common formulation of the ontological form of Occam’sRazor is the following:

(OR)

Entities are not to be multiplied beyond necessity.

It should be noted that modern formulations of Occam’s Razor areconnected only very tenuously to the 14^th-century figureWilliam of Ockham. We are not here interested in the exegeticalquestion of how Ockham intended his ‘Razor’ to function,nor in the uses to which it was put in the context of medieval metaphysics.^[4] Contemporary philosophers have tended to reinterpret OR as aprinciple of theory choice: OR implies that—other things beingequal—it is rational to prefer theories which commit us tosmaller ontologies. This suggests the following paraphrase of OR:

(OR₁)

Other things being equal, if \(T_1\) is moreontologically parsimonious than \(T_2\) then it isrational to prefer \(T_1\) to \(T_2\).

What does it mean to say that one theory is more ontologicallyparsimonious than another? The basic notion of ontological parsimonyis quite straightforward, and is standardly cashed out in terms ofQuine’s concept ofontological commitment. A theory,\(T\), is ontologically committed to \(F\)s if and only if\(T\) entails that \(F\)’s exist (Quine 1981, pp.144–4). If two theories, \(T_1\) and\(T_2\), have the same ontological commitments exceptthat \(T_2\) is ontologically committed to \(F\)sand \(T_1\) is not, then \(T_1\) is moreparsimonious than \(T_2\). More generally, a sufficientcondition for \(T_1\) being more parsimonious than\(T_2\) is for the ontological commitments of\(T_1\) to be a proper subset of those of\(T_2\). Note that OR₁ is considerablyweaker than the informal version of Occam’s Razor, OR, with which westarted. OR stipulates only that entities should not be multipliedbeyond necessity. OR₁, by contrast, states thatentities should not be multipliedother things being equal,and this is compatible with parsimony being a comparatively weaktheoretical virtue.

One ‘easy’ case where OR₁ can bestraightforwardly applied is when a theory, \(T\), postulatesentities which are explanatorily idle. Excising these entities from\(T\) produces a second theory, \(T^*\), which has the sametheoretical virtues as \(T\) but a smaller set of ontologicalcommitments. Hence, according to OR₁, it is rational topick \(T^*\) over \(T\). (As previously noted, terminologysuch as ‘pick’ and ‘prefer’ is cruciallyambiguous between epistemic and methodological versions of Occam’sRazor. For the purposes of defining ontological parsimony, it is notnecessary to resolve this ambiguity.) However, such cases arepresumably rare, and this points to a more general worry concerningthe narrowness of application of OR₁. First, how often doesit actually happen that we have two (or more) competing theories forwhich ‘other things are equal’? As biologist KentHolsinger remarks,

Since Occam’s Razor ought to be invoked only when several hypothesesexplain the same set of facts equally well, in practice its domainwill be very limited…[C]ases where competing hypotheses explaina phenomenon equally well are comparatively rare (Holsinger 1980, pp.144–5).

Second, how often are one candidate theory’s ontological commitments aproper subset of another’s? Much more common are situations whereontologies of competing theories overlap, but each theory haspostulates which are not made by the other. Straightforwardcomparisons of ontological parsimony are not possible in suchcases.

Before setting aside the definitional question for ontologicalparsimony, one further distinction should be mentioned. Thisdistinction is betweenqualitative parsimony (roughly, thenumber of types (or kinds) of thing postulated) andquantitative parsimony (roughly, the number of individualthings postulated).^[5] The default reading of Occam’s Razor in the bulk of the philosophicalliterature is as a principle of qualitative parsimony. Thus Cartesiandualism, for example, is less qualitatively parsimonious thanmaterialism because it is committed to two broad kinds of entity(mental and physical) rather than one. Section 6.1 contains a briefdiscussion of quantitative parsimony; apart from this the focus willbe on the qualitative notion. It should be noted that interpretingOccam’s Razor in terms ofkinds of entity brings with it someextra philosophical baggage of its own. In particular, judgments ofparsimony become dependent on how the world is sliced up into kinds.Nor is guidance from extra-philosophical usage—and in particularfrom science—always clearcut. For example, is a previouslyundiscovered subatomic particle made up of a novel rearrangement ofalready discovered sub-particles a new ‘kind’? What abouta biological species, which presumably does not contain any novelbasic constituents? Also, ought more weight to be given to broad andseemingly fundamental divisions of kind—for example between themental and physical—than between more parochial divisions?Intuitively, the postulation of a new kind of matter would seem torequire much more extensive and solid justification than thepostulation of a new sub-species of spider.^[6]

The third and final question from Section 1 concerns potentialjustifications for principles of ontological parsimony such as Occam’sRazor. The demand for justification of such principles can beunderstood in two importantly distinct ways, corresponding to thedistinction between epistemic principles and methodological principlesmade at the end of Section 1. Justifying an epistemic principlerequires answering an epistemic question: why are parsimonioustheories more likely to be true? Justifying a methodological principlerequires answering a pragmatic question: why does it make practicalsense for theorists to adopt parsimonious theories?^[7] Most attention in the literature has centered on the first, epistemicquestion. It is easy to see how syntactic elegance in a theory canbring with it pragmatic advantages such as being more perspicuous,being easier to use and manipulate, and so on. But the case is moredifficult to make for ontological parsimony.^[8] It is unclear what particular pragmatic disadvantages accrue totheories which postulate extra kinds of entities; indeed—as wasmentioned in the previous section—such postulations can oftenbring with them striking syntactic simplification.

Before looking at approaches to answering the epistemic justificationquestion, mention should be made of two positions in the literaturewhich do not fall squarely into either the pragmatic or epistemiccamp. The first position, associated primarily with Quine, argues thatparsimony carries with it pragmatic advantages and that pragmaticconsiderations themselves provide rational grounds for discriminatingbetween competing theories (Quine 1966, Walsh 1979). The Quineanposition bases an answer to the second question on the answer to thefirst, thus blurring the boundary between pragmatic and epistemicjustification. The second position, due to Sober, rejects the implicitassumption in both the above questions that some global justificationof parsimony can be found (Sober 1988, 1994). Instead Sober arguesthat appeals to parsimony always depend on local backgroundassumptions for their rational justification. Thus Sober writes:

The legitimacy of parsimony stands or falls, in a particular researchcontext, on subject matter specific (anda posteriori)considerations. […] What makes parsimony reasonable in onecontext may have nothing in common with why it matters in another(Sober 1994).

Philosophers who reject these arguments of Quine and Sober, and thustake the demand for a global, epistemic justification seriously, havedeveloped a variety of approaches to justifying parsimony. Most ofthese approaches can be collected under two broad headings:

(A)

A priori philosophical, metaphysical, or theologicaljustifications.

(B)

Naturalistic justifications, based on appeal to scientificpractice.

As we shall see, the contrast between these two sorts of approachmirrors a broader divide between the rival traditions of rationalismand empiricism in philosophy as a whole.

As well as parsimony, the question of rational justification can alsobe raised for principles based on elegance, the second facet ofsimplicity distinguished in Section 1. Approaches to justifyingelegance along the lines of (A) and (B) are possible, but much of therecent work falls under a third category;

(C)

Justifications based on results from probability theory and/orstatistics.

The next three sections examine these three modes of justification ofsimplicity principles. Thea priori justifications incategory (A) concern simplicity in both its parsimony and eleganceforms. The justifications falling under category (B) pertain mostly toparsimony, while those falling under category (C) pertain mostly toelegance.

3.A Priori Justifications of Simplicity

The role of simplicity as a theoretical virtue seems so widespread,fundamental, and implicit that many philosophers, scientists, andtheologians have sought a justification for principles such as Occam’sRazor on similarly broad and basic grounds. This rationalist approachis connected to the view that makinga priori simplicityassumptions is the only way to get around the underdetermination oftheory by data. Until the second half of the 20^th Centurythis was probably the predominant approach to the issue of simplicity.More recently, the rise of empiricism within analytic philosophy ledmany philosophers to argue disparagingly thata priorijustifications keep simplicity in the realm of metaphysics (seeZellner et al. 2001, p.1). Despite its changing fortunes, therationalist approach to simplicity still has its adherents. Forexample, Richard Swinburne writes:

I seek…to show that—other things being equal—thesimplest hypothesis proposed as an explanation of phenomena is morelikely to be the true one than is any other available hypothesis, thatits predictions are more likely to be true than those of any otheravailable hypothesis, and that it is an ultimatea prioriepistemic principle that simplicity is evidence for truth (Swinburne1997, p. 1).

3.1 Theological Justifications

The post-medieval period coincided with a gradual transition fromtheology to science as the predominant means of revealing the workingsof nature. In many cases, espoused principles of parsimony continuedto wear their theological origins on their sleeves, as with Leibniz’sthesis that God has created the best and most complete of all possibleworlds, and his linking of this thesis to simplifying principles suchas light always taking the (time-wise) shortest path. A similarattitude—and rhetoric—is shared by scientists through theearly modern and modern period, including Kepler, Newton, andMaxwell.

Some of this rhetoric has survived to the present day, especiallyamong theoretical physicists and cosmologists such as Einstein and Hawking.^[9] Yet there are clear dangers with relying on a theologicaljustification of simplicity principles. Firstly, many—probablymost—contemporary scientists are reluctant to linkmethodological principles to religious belief in this way. Secondly,even those scientists who do talk of ‘God’ often turn outto be using the term metaphorically, and not necessarily as referringto the personal and intentional Being of monotheistic religions.Thirdly, even if there is a tendency to justify simplicity principlesvia some literal belief in the existence of God, such justification isonly rational to the extent that rational arguments can be given forthe existence of God.^[10]

For these reasons, few philosophers today are content to rest with atheological justification of simplicity principles. Yet there is nodoubting the influence such justifications have had on past andpresent attitudes to simplicity. As Smart (1994) writes:

There is a tendency…for us to take simplicity…as a guideto metaphysical truth. Perhaps this tendency derives from earliertheological notions: we expect God to have created a beautifuluniverse (Smart 1984, p. 121).

3.2 Metaphysical Justifications

One approach to justifying simplicity principles is to embed suchprinciples in some more general metaphysical framework. Perhaps theclearest historical example of systematic metaphysics of this sort isthe work of Leibniz. The leading contemporary example of thisapproach—and in one sense a direct descendent of Leibniz’smethodology—is the possible worlds framework of David Lewis. Inone of his earlier works, Lewis writes,

I subscribe to the general view that qualitative parsimony is good ina philosophical or empirical hypothesis (Lewis 1973, p. 87).

Lewis has been attacked for not saying more about what exactly hetakes simplicity to be (see Woodward 2003). However, what is clear isthat simplicity plays a key role in underpinning his metaphysicalframework, and is also taken to be aprima facie theoreticalvirtue.

Though Occam’s Razor has arguably been a longstanding and importanttool in the rise of analytic metaphysics, it has only beencomparatively recently that there has been much debate amongmetaphysicians concerning the principle itself. Cameron (2010),Schaffer (2010), and Sider (2013) each argue for a version of Occam’sRazor that focuses specifically onfundamental entities.Schaffer (2015, p. 647) dubs this version "The Laser" and formulatesit as an injunction not to multiply fundamental entities beyondnecessity, together with the implicit understanding that there is nosuch injunction against multiplying derivative entities. Baron andTallant (2018) attack ‘razor-revisers’ such as Schaffer, arguing thatprinciples such as The Laser fail to mesh with actual patterns oftheory-choice in science and are also not vindicated by some of thelines of justification for Occam’s Razor.

3.3 ‘Intrinsic Value’ Justifications

Some philosophers have approached the issue of justifying simplicityprinciples by arguing that simplicity has intrinsic value as atheoretical goal. Sober, for example, writes:

Just as the question ‘why be rational?’ may have nonon-circular answer, the same may be true of the question ‘whyshould simplicity be considered in evaluating the plausibility ofhypotheses?’ (Sober 2001, p. 19).

Such intrinsic value may be ‘primitive’ in some sense, orit may be analyzable as one aspect of some broader value. For thosewho favor the second approach, a popular candidate for this broadervalue is aesthetic. Derkse (1992) is a book-length development of thisidea, and echoes can be found in Quine’s remarks—in connectionwith his defense of Occam’s Razor—concerning his taste for“clear skies” and “desert landscapes.” Ingeneral, forging a connection between aesthetic virtue and simplicityprinciples seems better suited to defending methodological rather thanepistemic principles.

3.4 Justifications via Principles of Rationality

Another approach is to try to show how simplicity principles followfrom other better established or better understood principles of rationality.^[11] For example, some philosophers just stipulate that they will take‘simplicity’ as shorthand for whatever package oftheoretical virtues is (or ought to be) characteristic of rationalinquiry. A more substantive alternative is to link simplicity to someparticular theoretical goal, for example unification (see Friedman1983). While this approach might work for elegance, it is less clearhow it can be maintained for ontological parsimony. Conversely, a lineof argument which seems better suited to defending parsimony than todefending elegance is to appeal to a principle of epistemologicalconservatism. Parsimony in a theory can be viewed as minimizing thenumber of ‘new’ kinds of entities and mechanisms which arepostulated. This preference for old mechanisms may in turn bejustified by a more general epistemological caution, or conservatism,which is characteristic of rational inquiry.

Note that the above style of approach can be given both a rationalistand an empiricist gloss. If unification, or epistemologicalconservatism, are themselvesa priori rational principles,then simplicity principles stand to inherit this feature if thisapproach can be carried out successfully. However, philosophers withempiricist sympathies may also pursue analysis of this sort, and thenjustify the base principles either inductively from past success ornaturalistically from the fact that such principles are in fact usedin science.

To summarize, the main problem witha priori justificationsof simplicity principles is that it can be difficult to distinguishbetween ana priori defense andno defense(!).Sometimes the theoretical virtue of simplicity is invoked as aprimitive, self-evident proposition that cannot be further justifiedor elaborated upon. (One example is the beginning of Goodman andQuine’s 1947 paper, where they state that their refusal to admitabstract objects into their ontology is “based on aphilosophical intuition that cannot be justified by appeal to anythingmore ultimate.”) (Goodman & Quine 1947, p. 174). It isunclear where leverage for persuading skeptics of the validity of suchprinciples can come from, especially if the grounds provided are notthemselves to beg further questions. Misgivings of this sort have ledto a shift away from justifications rooted in ‘firstphilosophy’ towards approaches which engage to a greater degreewith the details of actual practice, both scientific and statistical.These other approaches will be discussed in the next two sections.

4. Naturalistic Justifications of Simplicity

The rise of naturalized epistemology as a movement within analyticphilosophy in the second half of the 20^th Century haslargely sidelined the rationalist style of approach. From thenaturalistic perspective, philosophy is conceived of as continuouswith science, and not as having some independently privileged status.The perspective of the naturalistic philosopher may be broader, buther concerns and methods are not fundamentally different from those ofthe scientist. The conclusion is that science neither needs—norcan legitimately be given—external philosophical justification.It is against this broadly naturalistic background that somephilosophers have sought to provide an epistemic justification ofsimplicity principles, and in particular principles of ontologicalparsimony such as Occam’s Razor.

The main empirical evidence bearing on this issue consists of thepatterns of acceptance and rejection of competing theories by workingscientists. Einstein’s development of Special Relativity—and itsimpact on the hypothesis of the existence of the electromagneticether—is one of the episodes most often cited (by bothphilosophers and scientists) as an example of Occam’s Razor in action(see Sober 1981, p. 153). The ether is by hypothesis a fixed mediumand reference frame for the propagation of light (and otherelectromagnetic waves). The Special Theory of Relativity includes theradical postulate that the speed of a light ray through a vacuum isconstant relative to an observer no matter what the state of motion ofthe observer. Given this assumption, the notion of a universalreference frame is incoherent. Hence Special Relativity implies thatthe ether does not exist.

This episode can be viewed as the replacement of an empiricallyadequate theory (the Lorentz-Poincaré theory) by a moreontologically parsimonious alternative (Special Relativity). Hence itis often taken to be an example of Occam’s Razor in action. Theproblem with using this example as evidence for Occam’s Razor is thatSpecial Relativity (SR) has several other theoretical advantages overthe Lorentz-Poincaré (LP) theory in addition to being moreontologically parsimonious. Firstly, SR is a simpler and more unifiedtheory than LP, since in order to ‘save the phenomena’ anumber ofad hoc and physically unmotivated patches had beenadded to LP. Secondly, LP raises doubts about the physical meaning ofdistance measurements. According to LP, a rod moving with velocity,\(v\), contracts by a factor of \((1 - v^2 /c^2)^{1/2}\). Thusonly distance measurements that are made in a frame at rest relativeto the ether are valid without modification by a correction factor.However, LP also implies that motion relative to the ether is inprinciple undetectable. So how is distance to be measured? In otherwords, the issue here is complicated by the fact that—accordingto LP—the ether is not just an extra piece of ontology but anundetectable extra piece. Given these advantages of SR overLP, it seems clear that the ether example is not merely a case ofontological parsimony making up for an otherwise inferior theory.

A genuine test-case for Occam’s Razor must involve an ontologicallyparsimonious theory which is not clearly superior to its rivals inother respects. An instructive example is the following historicalepisode from biogeography, a scientific subdiscipline which originatedtowards the end of the 18^th Century, and whose centralpurpose was to explain the geographical distribution of plant andanimal species.^[12] In 1761, the French naturalist Buffon proposed the following law;

(BL)

Areas separated by natural barriers have distinct species.

There were also known exceptions to Buffon’s Law, for example remoteislands which share (so-called) ‘cosmopolitan’ specieswith continental regions a large distance away.

Two rival theories were developed to explain Buffon’s Law and itsoccasional exceptions. According to the first theory, due to Darwinand Wallace, both facts can be explained by the combined effects oftwo causal mechanisms—dispersal, and evolution by naturalselection. The explanation for Buffon’s Law is as follows. Speciesgradually migrate into new areas, a process which Darwin calls“dispersal.” As natural selection acts over time on thecontingent initial distribution of species in different areas,completely distinct species eventually evolve. The existence ofcosmopolitan species is explained by “improbabledispersal,” Darwin’s term for dispersal across seeminglyimpenetrable barriers by “occasional means of transport”such as ocean currents, winds, and floating ice. Cosmopolitan speciesare explained as the result of improbable dispersal in the relativelyrecent past.

In the 1950’s, Croizat proposed an alternative to the Darwin-Wallacetheory which rejects their presupposition of geographical stability.Croizat argues that tectonic change, not dispersal, is the principalcausal mechanism which underlies Buffon’s Law. Forces such ascontinental drift, the submerging of ocean floors, and the formationof mountain ranges have acted within the time frame of evolutionaryhistory to create natural barriers between species where at previoustimes there were none. Croizat’s theory was the sophisticatedculmination of a theoretical tradition which stretched back to thelate 17^th Century. Followers of this so-called“extensionist” tradition had postulated the existence ofancient land bridges to account for anomalies in the geographicaldistribution of plants and animals.^[13]

Extensionist theories are clearly less ontologically parsimonious thanDispersal Theories, since the former are committed to extra entitiessuch as land bridges or movable tectonic plates. Moreover,Extensionist theories were (given the evidence then available) notmanifestly superior in other respects. Darwin was an early critic ofExtensionist theories, arguing that they went beyond the“legitimate deductions of science.” Another critic ofExtensionist theories pointed to their “dependence on ad hochypotheses, such as land bridges and continental extensions of vastextent, to meet each new distributional anomaly” (Fichman 1977,p. 62) The debate over the more parsimonious Dispersal theoriescentered on whether the mechanism of dispersal is sufficient on itsown to explain the known facts about species distribution, withoutpostulating any extra geographical or tectonic entities.

The criticisms leveled at the Extensionist and Dispersal theoriesfollow a pattern that is characteristic of situations in which onetheory is more ontologically parsimonious than its rivals. In suchsituations the debate is typically over whether the extra ontology isreally necessary in order to explain the observed phenomena. The lessparsimonious theories are condemned for profligacy, and lack of directevidential support. The more parsimonious theories are condemned fortheir inadequacy to explain the observed facts. This illustrates arecurring theme in discussions of simplicity—both inside andoutside philosophy—namely, how the correct balance betweensimplicity and goodness of fit ought to be struck. This theme takescenter stage in the statistical approaches to simplicity discussed inSection 5.

Less work has been done on describing episodes in science whereelegance—as opposed to parsimony—has been (or may havebeen) the crucial factor. This may just reflect the fact thatconsiderations linked to elegance are so pervasive in scientifictheory choice as to be unremarkable as a topic for special study. Anotable exception to this general neglect is the area of celestialmechanics, where the transition from Ptolemy to Copernicus to Keplerto Newton is an oft-cited example of simplicity considerations inaction, and a case study which makes much more sense when seen throughthe lens of elegance rather than of parsimony.^[14]

Naturalism depends on a number of presuppositions which are open todebate. But even if these presuppositions are granted, thenaturalistic project of looking to science for methodological guidancewithin philosophy faces a major difficulty, namely how to ‘readoff’ from actual scientific practice what the underlyingmethodological principles are supposed to be. Burgess, for example,argues that what the patterns of scientific behavior show is not aconcern with multiplying entitiesper se, but a concern morespecifically with multiplying ‘causal mechanisms’ (Burgess1998). And Sober considers the debate in psychology over psychologicalegoism versus motivational pluralism, arguing that the former theorypostulates fewer types of ultimate desire but a larger number ofcausal beliefs, and hence that comparing the parsimony of these twotheories depends on what is counted and how (Sober 2001, pp.14–5). Some of the concerns raised in Sections 1 and 2 alsoreappear in this context; for example, how the world is sliced up intokinds effects the extent to which a given theory‘multiplies’ kinds of entity. Justifying a particular wayof slicing becomes more difficult once the epistemological naturalistleaves behind thea priori, metaphysical presuppositions ofthe rationalist approach.

One philosophical debate where these worries over naturalism becomeparticularly acute is the issue of the application of parsimonyprinciples toabstract objects. The scientific datais—in an important sense—ambiguous. Applications ofOccam’s Razor in science are always to concrete, causally efficaciousentities, whether land-bridges, unicorns, or the luminiferous ether.Perhaps scientists apply an unrestricted version of Occam’s Razor tothat portion of reality in which they are interested, namely theconcrete, causal, spatiotemporal world. Or perhaps scientists apply a‘concretized’ version of Occam’s Razor unrestrictedly.Which is the case? The answer determines which general philosophicalprinciple we end up with: ought we to avoid the multiplication ofobjects of whatever kind, or merely the multiplication of concreteobjects? The distinction here is crucial for a number of centralphilosophical debates. Unrestricted Occam’s Razor favors monism overdualism, and nominalism over platonism. By contrast,‘concretized’ Occam’s Razor has no bearing on thesedebates, since the extra entities in each case are not concrete.

5. Probabilistic/Statistical Justifications of Simplicity

The two approaches discussed in Sections 3 and 4—apriori rationalism and naturalized empiricism—are both insome sense extreme. Simplicity principles are taken either to have noempirical grounding, or to have solely empirical grounding. Perhaps asa result, both these approaches yield vague answers to certain keyquestions about simplicity. In particular, neither seems equipped toanswer how exactly simplicity ought to be balanced against empiricaladequacy. Simple but wildly inaccurate theories are not hard to comeup with. Nor are accurate theories which are highly complex. But howmuch accuracy should be sacrificed for a gain in simplicity? Theblack-and-white boundaries of the rationalism/empiricism divide maynot provide appropriate tools for analyzing this question. Inresponse, philosophers have recently turned to the mathematicalframework of probability theory and statistics, hoping in the processto combine sensitivity to actual practice with the‘trans-empirical’ strength of mathematics.

Philosophically influential early work in this direction was done byJeffreys and by Popper, both of whom tried to analyze simplicity inprobabilistic terms. Jeffreys argued that “the simpler laws havethe greater prior probability,” and went on to provide anoperational measure of simplicity, according to which the priorprobability of a law is \(2^{-k}\), where\(k =\) order + degree + absolute values of the coefficients,when the law is expressed as a differential equation (Jeffreys 1961,p. 47). A generalization of Jeffreys’ approach is to look not atspecific equations, but atfamilies of equations. Forexample, one might compare the family, LIN, of linear equations (ofthe form \(y = a + bx)\) with the family, PAR,of parabolic equations (of the form \(y = a + bx + cx^2)\). Since PAR is of higher degreethan LIN, Jeffreys’ proposal assigns higher probability to LIN. Lawsof this form are intuitively simpler (in the sense of being moreelegant).

Popper (1959) points out that Jeffreys’ proposal, as it stands,contradicts the axioms of probability. Every member of LIN is also amember of PAR, where the coefficient, \(c\), is set to 0. Hence‘Law, \(L\), is a member of LIN’ entails ‘Law,\(L\), is a member of PAR.’ Jeffreys’ approach assignshigher probability to the former than the latter. But it follows fromthe axioms of probability that when \(A\) entails \(B\), theprobability of \(B\) is greater than or equal to the probabilityof \(A\). Popper argues, in contrast to Jeffreys, that LIN haslower prior probability than PAR. Hence LIN is—inPopper’s sense—more falsifiable, and hence should be preferredas the default hypothesis. One response to Popper’s objection is toamend Jeffrey’s proposal and restrict members of PAR to equationswhere \(c \ne 0\).

More recent work on the issue of simplicity has borrowed tools fromstatistics as well as from probability theory. It should be noted thatthe literature on this topic tends to use the terms‘simplicity’ and ‘parsimony’ more-or-lessinterchangeably (see Sober 2003). But, whichever term is preferred,there is general agreement among those working in this area thatsimplicity is to be cashed out in terms of the number of free (or‘adjustable’) parameters of competing hypotheses. Thus thefocus here is totally at the level of theory. Philosophers who havemade important contributions to this approach include Forster andSober (1994) and Lange (1995).

The standard case in the statistical literature on parsimony concernscurve-fitting.^[15] We imagine a situation in which we have a set of discrete data pointsand are looking for the curve (i.e. function) which has generatedthem. The issue of what family of curves the answer belongs in (e.g.in LIN or in PAR) is often referred to asmodel-selection.The basic idea is that there are two competing criteria for modelselection—parsimony and goodness of fit. The possibility ofmeasurement error and ‘noise’ in the data means that thecorrect curve may not go through every data point. Indeed, if goodnessof fit were the only criterion then there would be a danger of‘overfitting’ the model to accidental discrepanciesunrepresentative of the broader regularity. Parsimony acts as acounterbalance to such overfitting, since a curve passing throughevery data point is likely to be very convoluted and hence have manyadjusted parameters.

If proponents of the statistical approach are in general agreementthat simplicity should be cashed out in terms of number of parameters,there is less unanimity over what the goal of simplicity principlesought to be. This is partly because the goal is often not madeexplicit. (An analogous issue arises in the case of Occam’s Razor.‘Entities are not to be multiplied beyond necessity.’ Butnecessity forwhat, exactly?) Forster distinguishes twopotential goals of model selection, namely probable truth andpredictive accuracy, and claims that these are importantly distinct(Forster 2001, p. 95). Forster argues that predictive accuracy tendsto be what scientists care about most. They care less about theprobability of an hypothesis being exactly right than they do about ithaving a high degree of accuracy.

One reason for investigating statistical approaches to simplicity is adissatisfaction with the vagaries of thea priori andnaturalistic approaches. Statisticians have come up with a variety ofnumerically specific proposals for the trade-off between simplicityand goodness of fit. However, these alternative proposals disagreeabout the ‘cost’ associated with more complex hypotheses.Two leading contenders in the recent literature on model selection arethe Akaike Information Criterion [AIC] and the Bayesian InformationCriterion [BIC]. AIC directs theorists to choose the model with thehighest value of \(\{\log L(\Theta_k )/n\} - k/n,\) where \(\Theta_k\)is the best-fitting member of the class of curves of polynomial degree\(k\), \(\log L\) is log-likelihood, and \(n\) is the sample size. Bycontrast, BIC maximizes the value of \(\{\log L(\Theta_k )/n\} -k\log[n]/2n.\) In effect, BIC gives an extra positive weighting tosimplicity by a factor of \(\log[n]/2\) (where \(n\) is the size ofthe sample).^[16]

Extreme answers to the trade-off problem seem to be obviouslyinadequate. Always picking the model with the best fit to the data,regardless of its complexity, faces the prospect (mentioned earlier)of ‘overfitting’ error and noise in the data. Alwayspicking the simplest model, regardless of its fit to the data, cutsthe model free from any link to observation or experiment. Forsterassociates the ‘Always Complex’ and the ‘AlwaysSimple’ rule with empiricism and rationalism respectively (ibid.). All the candidate rules that are seriously discussed by statisticiansfall in between these two extremes. Yet they differ in their answersover how much weight to give simplicity in its trade-off againstgoodness of fit. In addition to AIC and BIC, other rules includeNeyman-Pearson hypothesis testing, and the minimum description length(MDL) criterion.

There are at least three possible responses to the varying answers tothe trade-off problem provided by different criteria. One response,favored by Forster and by Sober, is to argue that there is no genuineconflict here because the different criteria have different aims. ThusAIC and BIC might both be optimal criteria, if AIC is aiming tomaximize predictive accuracy whereas BIC is aiming to maximizeprobable truth. Another difference that may influence the choice ofcriterion is whether the goal of the model is to extrapolate beyondgiven data or interpolate between known data points. A secondresponse, typically favored by statisticians, is to argue that theconflict is genuine but that it has the potential to be resolved byanalyzing (using both mathematical and empirical methods) whichcriterion performs best over the widest class of possible situations.A third, more pessimistic, response is to argue that the conflict isgenuine but is unresolvable. Kuhn (1977) takes this line, claimingthat how much weight individual scientists give a particulartheoretical virtue, such as simplicity, is solely a matter of taste,and is not open to rational resolution. McAllister (2007) drawsontological morals from a similar conclusion, arguing that sets ofdata typically exhibit multiple patterns, and that different patternsmay be highlighted by different quantitative techniques.

Aside from this issue of conflicting criteria, there are otherproblems with the statistical approach to simplicity. One problem,which afflicts any approach emphasizing the elegance aspect ofsimplicity, is language relativity. Crudely put, hypotheses which aresyntactically very complex in one language may be syntactically verysimple in another. The traditional philosophical illustration of thisproblem is Goodman’s ‘grue’ challenge to induction. Arestatistical approaches to the measurement of simplicity similarlylanguage relative, and—if so—what justifies choosing onelanguage over another? It turns out that the statistical approach hasthe resources to at least partially deflect the charge of languagerelativity. Borrowing techniques from information theory, it can beshown that certain syntactic measures of simplicity are asymptoticallyindependent of choice of measurement language.^[17]

A second problem for the statistical approach is whether it canaccount not only for our preference for small numbers over largenumbers (when it comes to picking values for coefficients or exponentsin model equations), but also our preference for whole numbers andsimple fractions over other values. In Gregor Mendel’s originalexperiments on the hybridization of garden peas, he crossed peavarieties with different specific traits, such as tall versus short orgreen seeds versus yellow seeds, and then self-pollinated the hybridsfor one or more generations.^[18] In each case one trait was present in all the first-generationhybrids, but both traits were present in subsequent generations.Across his experiments with seven different such traits, the ratio ofdominant trait to recessive trait averaged 2.98 : 1. On thisbasis, Mendel hypothesized that the true ratio is 3 : 1.This ‘rounding’ was made prior to the formulation of anyexplanatory model, hence it cannot have been driven by anytheory-specific consideration. This raises two related questions.First, in what sense is the 3 : 1 ratio hypothesissimpler than the 2.98 : 1 ratio hypothesis? Second,can this choice be justified within the framework of the statisticalapproach to simplicity? The more general worry lying behind thesequestions is whether the statistical approach, in defining simplicityin terms of number of adjustable parameters, is replacing the broadissue of simplicity with a more narrowly—and perhapsarbitrarily—defined set of issues.

A third problem with the statistical approach concerns whether it canshed any light on the specific issue of ontological parsimony. Atfirst glance, one might think that the postulation of extra entitiescan be attacked on probabilistic grounds. For example, quantummechanics together with the postulation ‘There existunicorns’ is less probable than quantum mechanics alone, sincethe former logically entails the latter. However, as Sober has pointedout, it is important here to distinguish betweenagnosticOccam’s Razor andatheistic Occam’s Razor. Atheistic ORdirects theorists to claim that unicorns donot exist, in theabsence of any compelling evidence in their favor. And there is norelation of logical entailment between \(\{QM +\) there existunicorns\(\}\) and \(\{QM +\) there do not exist unicorns\(\}\). This alsolinks back to the terminological issue. Models involving circularorbits are more parsimonious—in the statisticians’ sense of‘parsimonious’—than models involving ellipticalorbits, but the latter models do not postulate the existence of anymore things in the world.

6. Other Issues Concerning Simplicity

This section addresses three distinct issues concerning simplicity andits relation to other methodological issues. These issues concernquantitative parsimony, plenitude, and induction.

6.1 Quantitative Parsimony

Theorists tend to be frugal in their postulation of new entities. Whena trace is observed in a cloud-chamber, physicists may seek to explainit in terms of the influence of a hitherto unobserved particle. But,if possible, they will postulate one such unobserved particle, nottwo, or twenty, or 207 of them. This desire to minimize the number ofindividual new entities postulated is often referred to asquantitative parsimony. (For arguments that the qualitativeparsimony / quantitative parsimony distinction is not so clear, seeSendlak 2018.) David Lewis articulates the attitude of manyphilosophers when he writes:

I subscribe to the general view that qualitative parsimony is good ina philosophical or empirical hypothesis; but I recognize nopresumption whatever in favour of quantitative parsimony (Lewis 1973,p. 87).

Is the initial assumption that one particle is acting to cause theobserved trace more rational than the assumption that 207 particlesare so acting? Or is it merely the product of wishful thinking,aesthetic bias, or some other non-rational influence?

Nolan (1997) examines these questions in the context of the discoveryof the neutrino.^[19] Physicists in the 1930’s were puzzled by certain anomalies arisingfrom experiments in which radioactive atoms emit electrons duringso-called Beta decay. In these experiments the total spin of theparticles in the system before decay exceeds by½ the total spin of the (observed) emittedparticles. Physicists’ response was to posit a ‘new’fundamental particle, the neutrino, with spin½ and to hypothesize that exactly one neutrinois emitted by each electron during Beta decay.

Note that there is a wide range of very similar neutrino theorieswhich can also account for the missing spin.

(H₁)

1 neutrino with a spin of ½ is emitted ineach case of Beta decay.

(H₂)

2 neutrinos, each with a spin of ¼ areemitted in each case of Beta decay.

and, more generally, for any positive integer \(n\),

(H_\(n\))

\(n\) neutrinos, each with a spin of½ are emitted in each case of Betadecay.

Each of these hypotheses adequately explains the observation of amissing ½-spin following Beta decay. Yet themost quantitatively parsimonious hypothesis, H₁, is theobvious default choice.^[20]

One promising approach is to focus on the relative explanatory powerof the alternative hypotheses, H₁, H₂, …H_n. When neutrinos were first postulated in the 1930’s,numerous experimental set-ups were being devised to explore theproducts of various kinds of particle decay. In none of theseexperiments had cases of ‘missing’⅓-spin, or ¼-spin, or\(^1 /_{100}\)-spin been found. The absence of thesesmaller fractional spins was a phenomenon which competing neutrinohypotheses might potentially help to explain.

Consider the following two competing neutrino hypotheses:

(H₁)

1 neutrino with a spin of ½ is emitted ineach case of Beta decay.

(H₁₀)

10 neutrinos, each with a spin of \(^1 /_{20}\), areemitted in each case of Beta decay.

Why has no experimental set-up yielded a ‘missing’spin-value of \(^1 /_{20}\)? H₁ allows abetter answer to this question than H₁₀ does, forH₁ is consistent with a simple and parsimoniousexplanation, namely that there exist no particles with spin\(^1 /_{20}\) (or less). In the case of H₁₀,this potential explanation is ruled out because H₁₀explicitly postulates particles with spin \(^1 /_{20}\).Of course, H₁₀ is consistent withother hypotheseswhich explain the non-occurrence of missing\(^1 /_{20}\)-spin. For example, one might conjoin toH₁₀) the law that neutrinos are always emitted in groups often. However, this would make the overall explanation lesssyntactically simple, and hence less virtuous in other respects. Inthis case, quantitative parsimony brings greater explanatory power.Less quantitatively parsimonious hypotheses can match this power onlyby adding auxiliary claims which decrease their syntactic simplicity.Thus the preference for quantitatively parsimonious hypotheses emergesas one facet of a more general preference for hypotheses with greaterexplanatory power.

One distinctive feature of the neutrino example is that it is‘additive.’ It involves postulating the existence of acollection of qualitatively identical objects which collectivelyexplain the observed phenomenon. The explanation is additive in thesense that the overall phenomenon is explained by summing theindividual positive contributions of each object.^[21] Whether the above approach can be extended to non-additive casesinvolving quantitative parsimony is an interesting question. Janssonand Tallant (2017) argue that it can, and they offer a probabilisticanalysis that aims to bring together a variety of different caseswhere quantitative parsimony plays a role in hypothesis selection.Consider a case in which the aberrations of a planet’s orbit can beexplained by postulating a single unobserved planet, or it can beexplained by postulating two or more unobserved planets. In order forthe latter situation to be actual, the multiple planets must orbit incertain restricted ways so as to match the effects of a single planet.Prima facie this is unlikely, and this counts against theless quantitatively parsimonious hypothesis.

6.2 Principles of Plenitude

Ranged against the principles of parsimony discussed in previoussections is an equally firmly rooted (though less well-known)tradition of what might be termed “principles of explanatory sufficiency.”^[22] These principles have their origins in the same medievalcontroversies that spawned Occam’s Razor. Ockham’s contemporary,Walter of Chatton, proposed the following counter-principle to Occam’sRazor:

[I]f three things are not enough to verify an affirmative propositionabout things, a fourth must be added, and so on (quoted in Maurer1984, p. 464).

A related counter-principle was later defended by Kant:

The variety of entities should not be rashly diminished (Kant1781/1787, p. 541).
Entium varietates non temere esse minuendas.

There is no inconsistency in the coexistence of these two families ofprinciples, for they are not in direct conflict with each other.Considerations of parsimony and of explanatory sufficiency function asmutual counter-balances, penalizing theories which stray intoexplanatory inadequacy or ontological excess.^[23] What we see here is an historical echo of the contemporary debateamong statisticians concerning the proper trade-off between simplicityand goodness of fit.

There is, however, a second family of principles which do appeardirectly to conflict with Occam’s Razor. These are so-called‘principles of plenitude.’ Perhaps the best-known versionis associated with Leibniz, according to whom God created the best ofall possible worlds with the greatest number of possible entities.More generally, a principle of plenitude claims that if it ispossible for an object to exist then that objectactually exists. Principles of plenitude conflict withOccam’s Razor over the existence of physically possible butexplanatorily idle objects. Our best current theories presumably donot rule out the existence of unicorns, but nor do they provide anysupport for their existence. According to Occam’s Razor we oughtnot to postulate the existence of unicorns. According to aprinciple of plenitude weought to postulate theirexistence.

The rise of particle physics and quantum mechanics in the20^th Century led to various principles of plenitude beingappealed to by scientists as an integral part of their theoreticalframework. A particularly clear-cut example of such an appeal is thecase of magnetic monopoles.^[24] The 19^th-century theory of electromagnetism postulatednumerous analogies between electric charge and magnetic charge. Onetheoretical difference is that magnetic charges must always come inoppositely-charged pairs, called “dipoles” (as in theNorth and South poles of a bar magnet), whereas single electriccharges, or “monopoles,” can exist in isolation. However,no actual magnetic monopole had ever been observed. Physicists beganto wonder whether there was some theoretical reason why monopolescould not exist. It was initially thought that the newly developedtheory of quantum mechanics ruled out the possibility of magneticmonopoles, and this is why none had ever been detected. However, in1931 the physicist Paul Dirac showed that the existence of monopolesis consistent with quantum mechanics, although it is not required byit. Dirac went on to assert the existence of monopoles, arguing thattheir existence is not ruled out by theory and that “under thesecircumstances one would be surprised if Nature had made no use ofit” (Dirac 1930, p. 71, note 5). This appeal to plenitude waswidely—though not universally—accepted by otherphysicists.

One of the elementary rules of nature is that, in the absence of lawsprohibiting an event or phenomenon it is bound to occur with somedegree of probability. To put it simply and crudely: anything thatcan happendoes happen. Hence physicists must assumethat the magnetic monopole exists unless they can find a law barringits existence (Ford 1963, p. 122).

Others have been less impressed by Dirac’s line of argument:

Dirac’s…line of reasoning, when conjecturing the existence ofmagnetic monopoles, does not differ from 18^th-centuryarguments in favour of mermaids…[A]s the notion of mermaids wasneither intrinsically contradictory nor colliding with currentbiological laws, these creatures were assumed to exist.^[25]

It is difficult to know how to interpret these principles ofplenitude. Quantum mechanics diverges from classical physics byreplacing of a deterministic model of the universe with a model basedon objective probabilities. According to this probabilistic model,there are numerous ways the universe could have evolved from itsinitial state, each with a certain probability of occurring that isfixed by the laws of nature. Consider some kind of object, sayunicorns, whose existence is not ruled out by the initial conditionsplus the laws of nature. Then one can distinguish between a weak and astrong version of the principle of plenitude. According to the weakprinciple, if there is a small finite probability of unicorns existingthen given enough time and space unicorns will exist. According to thestrong principle, it follows from the theory of quantum mechanics thatif it is possible for unicorns to exist then they do exist. One way inwhich this latter principle may be cashed out is in the‘many-worlds’ interpretation of quantum mechanics,according to which reality has a branching structure in which everypossible outcome is realized.

6.3 Simplicity and Induction

The problem of induction is closely linked to the issue of simplicity.One obvious link is between the curve-fitting problem and theinductive problem of predicting future outcomes from observed data.Less obviously, Schulte (1999) argues for a connection betweeninduction and ontological parsimony. Schulte frames the problem ofinduction in information-theoretic terms: given a data-stream ofobservations of non-unicorns (for example), what general conclusionshould be drawn? He argues for two constraints on potential rules.First, the rule should converge on the truth in the long run (so if nounicorns exist then it should yield this conclusion). Second, the ruleshould minimize the maximum number of changes of hypothesis, givendifferent possible future observations. Schulte argues that the‘Occam Rule’—conjecture that \(\Omega\) does not existuntil it has been detected in an experiment—is optimal relativeto these constraints. An alternative rule—for example,conjecturing that \(\Omega\) exists until 1 million negative results havebeen obtained—may result in two changes of hypothesis if, say,\(\Omega\)’s are not detected until the 2 millionth experiment. The OccamRule leads to at most one change of hypothesis (when an \(\Omega\) isfirst detected). (See also Kelly 2004, 2007.) Schulte (2008) appliesthis approach to the problem of discovering conservation laws inparticle physics. The analysis has been criticized by Fitzpatrick(2013), who raises doubts about why long-run convergence to the truthshould matter when it comes to predicting the outcome of the very nextexperiment, and the debate is continued in Kelly et al. (2016).

With respect to the justification question, arguments have been madein both directions. Scientists are often inclined to justifysimplicity principles on broadly inductive grounds. According to thisargument, scientists select new hypotheses based partly on criteriathat have been generated inductively from previous cases of theorychoice. Choosing the most parsimonious of the acceptable alternativehypotheses has tended to work in the past. Hence scientists continueto use this as a rule of thumb, and are justified in so doing oninductive grounds. One might try to bolster this point of view byconsidering a counterfactual world in which all the fundamentalconstituents of the universe exist in pairs. In such a‘pairwise’ world, scientists might well prefer pairwisehypotheses in general to their more parsimonious rivals. This line ofargument has a couple of significant weaknesses. Firstly, one mightlegitimately wonder just how successful the choice of parsimonioushypotheses has been; examples from chemistry spring to mind, such asoxygen molecules containing two atoms rather than one. Secondly, andmore importantly, there remains the issue of explainingwhythe preference for parsimonious hypotheses in science has been assuccessful as it has been.

Making the justificatory argument in the reverse direction, fromsimplicity to induction, has a strong historical precedent inphilosophical approaches to the problem of induction, from Humeonwards. Justifying the ‘straight rule’ of induction byappeal to some general Principle of Uniformity is an initiallyappealing response to the skeptical challenge. However, in the absenceof a defense of the underlying Principle itself (and one which doesnot, on pain of circularity, depend inductively on past success), itis unclear how much progress this represents. There have also beenattempts (see e.g. Steel 2009) to use simplicity considerations torespond to Nelson Goodman’s ‘new riddle of induction.’

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computability and complexity |induction: problem of |learning theory, formal |logic: inductive |naturalism |Ockham [Occam], William