## 1. Introduction



There is a widespread philosophical presumption that simplicity is a
theoretical virtue. This presumption that simpler theories are
preferable 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 of
Parsimony'). However, it is perhaps best known by the name
'Occam's (or Ockham's) Razor.' Simplicity principles have
been proposed in various forms by theologians, philosophers, and
scientists, from ancient through medieval to modern times. Thus
Aristotle writes in his *Posterior Analytics*,



> 
> We may assume the superiority *ceteris paribus* of the
> demonstration which derives from fewer postulates or
> hypotheses. (*Posterior Analytics*, p. 150)
>  



Moving to the medieval period, Aquinas writes:



> 
> If a thing can be done adequately by means of one, it is superfluous
> to do it by means of several; for we observe that nature does not
> employ two instruments where one suffices (Aquinas, [BW], p. 129).
> 



Kant--in the *Critique of Pure Reason*--supports the
maxim that "rudiments or principles must not be unnecessarily
multiplied (*entia praeter necessitatem non esse
multiplicanda*)" and argues that this is a regulative idea
of pure reason which underlies scientists' theorizing about nature
(Kant, 1781/1787, pp. 538-9). Both Galileo and Newton accepted
versions of Occam's Razor. Indeed Newton includes a principle of
parsimony as one of his three 'Rules of Reasoning in
Philosophy' at the beginning of Book III of *Principia
Mathematica* (1687):



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



Newton goes on to remark that "Nature is pleased with
simplicity, and affects not the pomp of superfluous causes"
(Newton 1687, p. 398). Galileo, in the course of making a detailed
comparison 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 her
effects; that she does nothing in vain, and the like" (Galileo
1632, p. 397). Nor are scientific advocates of simplicity principles
restricted to the ranks of physicists and astronomers. Here is the
chemist Lavoisier writing in the late 18th Century



> 
> If all of chemistry can be explained in a satisfactory manner without
> the help of phlogiston, that is enough to render it infinitely likely
> that the principle does not exist, that it is a hypothetical
> substance, a gratuitous supposition. It is, after all, a principle of
> logic not to multiply entities unnecessarily (Lavoisier 1862, pp.
> 623-4).
> 



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



> 
> [T]he grand aim of all science...is to cover the greatest
> possible number of empirical facts by logical deductions from the
> smallest possible number of hypotheses or axioms (Einstein, quoted in
> Nash 1963, p. 173).
> 



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



Within philosophy, Occam's Razor (OR) is often wielded against
metaphysical theories which involve allegedly superfluous ontological
apparatus. Thus materialists about the mind may use OR against
dualism, on the grounds that dualism postulates an extra ontological
category for mental phenomena. Similarly, nominalists about abstract
objects may use OR against their platonist opponents, taking them to
task for committing to an uncountably vast realm of abstract
mathematical entities. The aim of appeals to simplicity in such
contexts seem to be more about shifting the burden of proof, and less
about refuting the less simple theory outright.



The philosophical issues surrounding the notion of simplicity are
numerous and somewhat tangled. The topic has been studied in piecemeal
fashion by scientists, philosophers, and statisticians (though for an
invaluable book-length philosophical treatment see Sober 2015). The
apparent familiarity of the notion of simplicity means that it is
often left unanalyzed, while its vagueness and multiplicity of
meanings contributes to the challenge of pinning the notion down
 precisely.[1]
 A distinction is often made between two fundamentally distinct senses
of simplicity: syntactic simplicity (roughly, the number and
complexity of hypotheses), and ontological simplicity (roughly, the
number and complexity of things
 postulated).[2]
 These two facets of simplicity are often referred to as
*elegance* and *parsimony* respectively. For the
purposes of the present overview we shall follow this usage and
reserve 'parsimony' specifically for simplicity in the
ontological sense. It should be noted, however, that the terms
'parsimony' and 'simplicity' are used
virtually interchangeably in much of the philosophical literature.



Philosophical interest in these two notions of simplicity may be
organized 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 of
inquiry? **[Usage]**
(iii)
Is there a rational justification for such simplicity principles?
**[Justification]**


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



With respect to question (ii), there is an important distinction to be
made between two sorts of simplicity principle. Occam's Razor may be
formulated as an *epistemic* principle: if theory \(T\) is
simpler than theory \(T^\*\), then it is rational (other things
being equal) to believe \(T\) rather than \(T^\*\). Or it may
be formulated as a *methodological* principle: if \(T\) is
simpler than \(T^\*\) then it is rational to adopt \(T\) as
one's working theory for scientific purposes. These two conceptions of
Occam's Razor require different sorts of justification in answer to
question (iii).



In analyzing simplicity, it can be difficult to keep its two
facets--elegance and parsimony--apart. Principles such as
Occam's Razor are frequently stated in a way which is ambiguous
between the two notions, for example, "Don't multiply
postulations beyond necessity." Here it is unclear whether
'postulation' refers to the entities being postulated, or
the hypotheses which are doing the postulating, or both. The first
reading corresponds to parsimony, the second to elegance. Examples of
both sorts of simplicity principle can be found in the quotations
given earlier in this section.



While these two facets of simplicity are frequently conflated, it is
important to treat them as distinct. One reason for doing so is that
considerations of parsimony and of elegance typically pull in
different directions. Postulating extra entities may allow a theory to
be formulated more simply, while reducing the ontology of a theory may
only be possible at the price of making it syntactically more complex.
For example the postulation of Neptune, at the time not directly
observable, allowed the perturbations in the orbits of other observed
planets to be explained without complicating the laws of celestial
mechanics. There is typically a trade-off between ontology and
ideology--to use the terminology favored by Quine--in which
contraction in one domain requires expansion in the other. This points
to 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 be
interpreted in terms of minimization. In the (atypical) case of
theoretically idle entities, both forms of minimization pull in the
same direction; postulating the existence of such entities makes both
our theories (of the world) and the world (as represented by our
theories) less simple than they might be.


## 2. Ontological Parsimony



Perhaps the most common formulation of the ontological form of Occam's
Razor is the following:



(OR)
Entities are not to be multiplied beyond necessity.


It should be noted that modern formulations of Occam's Razor are
connected only very tenuously to the 14th-century figure
William of Ockham. We are not here interested in the exegetical
question 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 a
principle of theory choice: OR implies that--other things being
equal--it is rational to prefer theories which commit us to
smaller ontologies. This suggests the following paraphrase of OR:



(OR1)
Other things being equal, if \(T\_1\) is more
ontologically parsimonious than \(T\_2\) then it is
rational to prefer \(T\_1\) to \(T\_2\).


What does it mean to say that one theory is more ontologically
parsimonious than another? The basic notion of ontological parsimony
is quite straightforward, and is standardly cashed out in terms of
Quine's concept of *ontological 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 except
that \(T\_2\) is ontologically committed to \(F\)s
and \(T\_1\) is not, then \(T\_1\) is more
parsimonious than \(T\_2\). More generally, a sufficient
condition 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 considerably
weaker than the informal version of Occam's Razor, OR, with which we
started. OR stipulates only that entities should not be multiplied
*beyond necessity*. OR1, by contrast, states that
entities should not be multiplied *other things being equal*,
and this is compatible with parsimony being a comparatively weak
theoretical virtue.



One 'easy' case where OR1 can be
straightforwardly applied is when a theory, \(T\), postulates
entities which are explanatorily idle. Excising these entities from
\(T\) produces a second theory, \(T^\*\), which has the same
theoretical virtues as \(T\) but a smaller set of ontological
commitments. Hence, according to OR1, it is rational to
pick \(T^\*\) over \(T\). (As previously noted, terminology
such as 'pick' and 'prefer' is crucially
ambiguous between epistemic and methodological versions of Occam's
Razor. For the purposes of defining ontological parsimony, it is not
necessary to resolve this ambiguity.) However, such cases are
presumably rare, and this points to a more general worry concerning
the narrowness of application of OR1. First, how often does
it actually happen that we have two (or more) competing theories for
which 'other things are equal'? As biologist Kent
Holsinger remarks,



> 
> Since Occam's Razor ought to be invoked only when several hypotheses
> explain the same set of facts equally well, in practice its domain
> will be very limited...[C]ases where competing hypotheses explain
> a phenomenon equally well are comparatively rare (Holsinger 1980, pp.
> 144-5).
> 



Second, how often are one candidate theory's ontological commitments a
proper subset of another's? Much more common are situations where
ontologies of competing theories overlap, but each theory has
postulates which are not made by the other. Straightforward
comparisons of ontological parsimony are not possible in such
cases.



Before setting aside the definitional question for ontological
parsimony, one further distinction should be mentioned. This
distinction is between *qualitative* parsimony (roughly, the
number of types (or kinds) of thing postulated) and
*quantitative* parsimony (roughly, the number of individual
things
 postulated).[5]
 The default reading of Occam's Razor in the bulk of the philosophical
literature is as a principle of qualitative parsimony. Thus Cartesian
dualism, for example, is less qualitatively parsimonious than
materialism because it is committed to two broad kinds of entity
(mental and physical) rather than one. Section 6.1 contains a brief
discussion of quantitative parsimony; apart from this the focus will
be on the qualitative notion. It should be noted that interpreting
Occam's Razor in terms of *kinds* of entity brings with it some
extra philosophical baggage of its own. In particular, judgments of
parsimony become dependent on how the world is sliced up into kinds.
Nor is guidance from extra-philosophical usage--and in particular
from science--always clearcut. For example, is a previously
undiscovered subatomic particle made up of a novel rearrangement of
already discovered sub-particles a new 'kind'? What about
a biological species, which presumably does not contain any novel
basic constituents? Also, ought more weight to be given to broad and
seemingly fundamental divisions of kind--for example between the
mental and physical--than between more parochial divisions?
Intuitively, the postulation of a new kind of matter would seem to
require much more extensive and solid justification than the
postulation of a new sub-species of
 spider.[6]



The third and final question from Section 1 concerns potential
justifications for principles of ontological parsimony such as Occam's
Razor. The demand for justification of such principles can be
understood in two importantly distinct ways, corresponding to the
distinction between epistemic principles and methodological principles
made at the end of Section 1. Justifying an epistemic principle
requires answering an epistemic question: why are parsimonious
theories more likely to be true? Justifying a methodological principle
requires answering a pragmatic question: why does it make practical
sense for theorists to adopt parsimonious
 theories?[7]
 Most attention in the literature has centered on the first, epistemic
question. It is easy to see how syntactic elegance in a theory can
bring with it pragmatic advantages such as being more perspicuous,
being easier to use and manipulate, and so on. But the case is more
difficult to make for ontological
 parsimony.[8]
 It is unclear what particular pragmatic disadvantages accrue to
theories which postulate extra kinds of entities; indeed--as was
mentioned in the previous section--such postulations can often
bring with them striking syntactic simplification.



Before looking at approaches to answering the epistemic justification
question, mention should be made of two positions in the literature
which do not fall squarely into either the pragmatic or epistemic
camp. The first position, associated primarily with Quine, argues that
parsimony carries with it pragmatic advantages and that pragmatic
considerations themselves provide rational grounds for discriminating
between competing theories (Quine 1966, Walsh 1979). The Quinean
position bases an answer to the second question on the answer to the
first, thus blurring the boundary between pragmatic and epistemic
justification. The second position, due to Sober, rejects the implicit
assumption in both the above questions that some global justification
of parsimony can be found (Sober 1988, 1994). Instead Sober argues
that appeals to parsimony always depend on local background
assumptions for their rational justification. Thus Sober writes:



> 
> The legitimacy of parsimony stands or falls, in a particular research
> context, on subject matter specific (and *a posteriori*)
> considerations. [...] What makes parsimony reasonable in one
> context may have nothing in common with why it matters in another
> (Sober 1994).
> 



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



(A)
*A priori* philosophical, metaphysical, or theological
justifications.
(B)
Naturalistic justifications, based on appeal to scientific
practice.


As we shall see, the contrast between these two sorts of approach
mirrors a broader divide between the rival traditions of rationalism
and empiricism in philosophy as a whole.



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



(C)
Justifications based on results from probability theory and/or
statistics.


The next three sections examine these three modes of justification of
simplicity principles. The *a priori* justifications in
category (A) concern simplicity in both its parsimony and elegance
forms. The justifications falling under category (B) pertain mostly to
parsimony, while those falling under category (C) pertain mostly to
elegance.


## 3. *A Priori* Justifications of Simplicity



The role of simplicity as a theoretical virtue seems so widespread,
fundamental, and implicit that many philosophers, scientists, and
theologians have sought a justification for principles such as Occam's
Razor on similarly broad and basic grounds. This rationalist approach
is connected to the view that making *a priori* simplicity
assumptions is the only way to get around the underdetermination of
theory by data. Until the second half of the 20th Century
this was probably the predominant approach to the issue of simplicity.
More recently, the rise of empiricism within analytic philosophy led
many philosophers to argue disparagingly that *a priori*
justifications keep simplicity in the realm of metaphysics (see
Zellner et al. 2001, p.1). Despite its changing fortunes, the
rationalist approach to simplicity still has its adherents. For
example, Richard Swinburne writes:



> 
> I seek...to show that--other things being equal--the
> simplest hypothesis proposed as an explanation of phenomena is more
> likely to be the true one than is any other available hypothesis, that
> its predictions are more likely to be true than those of any other
> available hypothesis, and that it is an ultimate *a priori*
> epistemic principle that simplicity is evidence for truth (Swinburne
> 1997, p. 1).
> 


### 3.1 Theological Justifications



The post-medieval period coincided with a gradual transition from
theology to science as the predominant means of revealing the workings
of nature. In many cases, espoused principles of parsimony continued
to wear their theological origins on their sleeves, as with Leibniz's
thesis that God has created the best and most complete of all possible
worlds, and his linking of this thesis to simplifying principles such
as light always taking the (time-wise) shortest path. A similar
attitude--and rhetoric--is shared by scientists through the
early modern and modern period, including Kepler, Newton, and
Maxwell.



Some of this rhetoric has survived to the present day, especially
among theoretical physicists and cosmologists such as Einstein and
 Hawking.[9]
 Yet there are clear dangers with relying on a theological
justification of simplicity principles. Firstly, many--probably
most--contemporary scientists are reluctant to link
methodological principles to religious belief in this way. Secondly,
even those scientists who do talk of 'God' often turn out
to be using the term metaphorically, and not necessarily as referring
to the personal and intentional Being of monotheistic religions.
Thirdly, even if there is a tendency to justify simplicity principles
via some literal belief in the existence of God, such justification is
only rational to the extent that rational arguments can be given for
the existence of
 God.[10]



For these reasons, few philosophers today are content to rest with a
theological justification of simplicity principles. Yet there is no
doubting the influence such justifications have had on past and
present attitudes to simplicity. As Smart (1994) writes:



> 
> There is a tendency...for us to take simplicity...as a guide
> to metaphysical truth. Perhaps this tendency derives from earlier
> theological notions: we expect God to have created a beautiful
> universe (Smart 1984, p. 121).
> 


### 3.2 Metaphysical Justifications



One approach to justifying simplicity principles is to embed such
principles in some more general metaphysical framework. Perhaps the
clearest historical example of systematic metaphysics of this sort is
the work of Leibniz. The leading contemporary example of this
approach--and in one sense a direct descendent of Leibniz's
methodology--is the possible worlds framework of David Lewis. In
one of his earlier works, Lewis writes,



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



Lewis has been attacked for not saying more about what exactly he
takes simplicity to be (see Woodward 2003). However, what is clear is
that simplicity plays a key role in underpinning his metaphysical
framework, and is also taken to be a *prima facie* theoretical
virtue.



Though Occam's Razor has arguably been a longstanding and important
tool in the rise of analytic metaphysics, it has only been
comparatively recently that there has been much debate among
metaphysicians concerning the principle itself. Cameron (2010),
Schaffer (2010), and Sider (2013) each argue for a version of Occam's
Razor that focuses specifically on *fundamental* entities.
Schaffer (2015, p. 647) dubs this version "The Laser" and formulates
it as an injunction not to multiply fundamental entities beyond
necessity, together with the implicit understanding that there is no
such injunction against multiplying derivative entities. Baron and
Tallant (2018) attack 'razor-revisers' such as Schaffer, arguing that
principles such as The Laser fail to mesh with actual patterns of
theory-choice in science and are also not vindicated by some of the
lines of justification for Occam's Razor.


### 3.3 'Intrinsic Value' Justifications



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



> 
> Just as the question 'why be rational?' may have no
> non-circular answer, the same may be true of the question 'why
> should simplicity be considered in evaluating the plausibility of
> hypotheses?' (Sober 2001, p. 19).
> 



Such intrinsic value may be 'primitive' in some sense, or
it may be analyzable as one aspect of some broader value. For those
who favor the second approach, a popular candidate for this broader
value is aesthetic. Derkse (1992) is a book-length development of this
idea, and echoes can be found in Quine's remarks--in connection
with his defense of Occam's Razor--concerning his taste for
"clear skies" and "desert landscapes." In
general, forging a connection between aesthetic virtue and simplicity
principles seems better suited to defending methodological rather than
epistemic principles.


### 3.4 Justifications via Principles of Rationality



Another approach is to try to show how simplicity principles follow
from 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 of
theoretical virtues is (or ought to be) characteristic of rational
inquiry. A more substantive alternative is to link simplicity to some
particular theoretical goal, for example unification (see Friedman
1983). While this approach might work for elegance, it is less clear
how it can be maintained for ontological parsimony. Conversely, a line
of argument which seems better suited to defending parsimony than to
defending elegance is to appeal to a principle of epistemological
conservatism. Parsimony in a theory can be viewed as minimizing the
number of 'new' kinds of entities and mechanisms which are
postulated. This preference for old mechanisms may in turn be
justified 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 rationalist
and an empiricist gloss. If unification, or epistemological
conservatism, are themselves *a priori* rational principles,
then simplicity principles stand to inherit this feature if this
approach can be carried out successfully. However, philosophers with
empiricist sympathies may also pursue analysis of this sort, and then
justify the base principles either inductively from past success or
naturalistically from the fact that such principles are in fact used
in science.



To summarize, the main problem with *a priori* justifications
of simplicity principles is that it can be difficult to distinguish
between an *a priori* defense and *no* defense(!).
Sometimes the theoretical virtue of simplicity is invoked as a
primitive, self-evident proposition that cannot be further justified
or elaborated upon. (One example is the beginning of Goodman and
Quine's 1947 paper, where they state that their refusal to admit
abstract objects into their ontology is "based on a
philosophical intuition that cannot be justified by appeal to anything
more ultimate.") (Goodman & Quine 1947, p. 174). It is
unclear where leverage for persuading skeptics of the validity of such
principles can come from, especially if the grounds provided are not
themselves to beg further questions. Misgivings of this sort have led
to a shift away from justifications rooted in 'first
philosophy' towards approaches which engage to a greater degree
with 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 analytic
philosophy in the second half of the 20th Century has
largely sidelined the rationalist style of approach. From the
naturalistic perspective, philosophy is conceived of as continuous
with science, and not as having some independently privileged status.
The perspective of the naturalistic philosopher may be broader, but
her concerns and methods are not fundamentally different from those of
the scientist. The conclusion is that science neither needs--nor
can legitimately be given--external philosophical justification.
It is against this broadly naturalistic background that some
philosophers have sought to provide an epistemic justification of
simplicity principles, and in particular principles of ontological
parsimony such as Occam's Razor.



The main empirical evidence bearing on this issue consists of the
patterns of acceptance and rejection of competing theories by working
scientists. Einstein's development of Special Relativity--and its
impact on the hypothesis of the existence of the electromagnetic
ether--is one of the episodes most often cited (by both
philosophers and scientists) as an example of Occam's Razor in action
(see Sober 1981, p. 153). The ether is by hypothesis a fixed medium
and reference frame for the propagation of light (and other
electromagnetic waves). The Special Theory of Relativity includes the
radical postulate that the speed of a light ray through a vacuum is
constant relative to an observer no matter what the state of motion of
the observer. Given this assumption, the notion of a universal
reference frame is incoherent. Hence Special Relativity implies that
the ether does not exist.



This episode can be viewed as the replacement of an empirically
adequate theory (the Lorentz-Poincare theory) by a more
ontologically parsimonious alternative (Special Relativity). Hence it
is often taken to be an example of Occam's Razor in action. The
problem with using this example as evidence for Occam's Razor is that
Special Relativity (SR) has several other theoretical advantages over
the Lorentz-Poincare (LP) theory in addition to being more
ontologically parsimonious. Firstly, SR is a simpler and more unified
theory than LP, since in order to 'save the phenomena' a
number of *ad hoc* and physically unmotivated patches had been
added to LP. Secondly, LP raises doubts about the physical meaning of
distance measurements. According to LP, a rod moving with velocity,
\(v\), contracts by a factor of \((1 - v^2 /c^2)^{1/2}\). Thus
only distance measurements that are made in a frame at rest relative
to the ether are valid without modification by a correction factor.
However, LP also implies that motion relative to the ether is in
principle undetectable. So how is distance to be measured? In other
words, the issue here is complicated by the fact that--according
to LP--the ether is not just an extra piece of ontology but an
*undetectable* extra piece. Given these advantages of SR over
LP, it seems clear that the ether example is not merely a case of
ontological parsimony making up for an otherwise inferior theory.



A genuine test-case for Occam's Razor must involve an ontologically
parsimonious theory which is not clearly superior to its rivals in
other respects. An instructive example is the following historical
episode from biogeography, a scientific subdiscipline which originated
towards the end of the 18th Century, and whose central
purpose was to explain the geographical distribution of plant and
animal
 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 remote
islands which share (so-called) 'cosmopolitan' species
with continental regions a large distance away.



Two rival theories were developed to explain Buffon's Law and its
occasional exceptions. According to the first theory, due to Darwin
and Wallace, both facts can be explained by the combined effects of
two causal mechanisms--dispersal, and evolution by natural
selection. The explanation for Buffon's Law is as follows. Species
gradually migrate into new areas, a process which Darwin calls
"dispersal." As natural selection acts over time on the
contingent initial distribution of species in different areas,
completely distinct species eventually evolve. The existence of
cosmopolitan species is explained by "improbable
dispersal," Darwin's term for dispersal across seemingly
impenetrable barriers by "occasional means of transport"
such as ocean currents, winds, and floating ice. Cosmopolitan species
are explained as the result of improbable dispersal in the relatively
recent past.



In the 1950's, Croizat proposed an alternative to the Darwin-Wallace
theory which rejects their presupposition of geographical stability.
Croizat argues that tectonic change, not dispersal, is the principal
causal mechanism which underlies Buffon's Law. Forces such as
continental drift, the submerging of ocean floors, and the formation
of mountain ranges have acted within the time frame of evolutionary
history to create natural barriers between species where at previous
times there were none. Croizat's theory was the sophisticated
culmination of a theoretical tradition which stretched back to the
late 17th Century. Followers of this so-called
"extensionist" tradition had postulated the existence of
ancient land bridges to account for anomalies in the geographical
distribution of plants and
 animals.[13]



Extensionist theories are clearly less ontologically parsimonious than
Dispersal Theories, since the former are committed to extra entities
such as land bridges or movable tectonic plates. Moreover,
Extensionist theories were (given the evidence then available) not
manifestly superior in other respects. Darwin was an early critic of
Extensionist theories, arguing that they went beyond the
"legitimate deductions of science." Another critic of
Extensionist theories pointed to their "dependence on ad hoc
hypotheses, such as land bridges and continental extensions of vast
extent, to meet each new distributional anomaly" (Fichman 1977,
p. 62) The debate over the more parsimonious Dispersal theories
centered on whether the mechanism of dispersal is sufficient on its
own to explain the known facts about species distribution, without
postulating any extra geographical or tectonic entities.



The criticisms leveled at the Extensionist and Dispersal theories
follow a pattern that is characteristic of situations in which one
theory is more ontologically parsimonious than its rivals. In such
situations the debate is typically over whether the extra ontology is
really necessary in order to explain the observed phenomena. The less
parsimonious theories are condemned for profligacy, and lack of direct
evidential support. The more parsimonious theories are condemned for
their inadequacy to explain the observed facts. This illustrates a
recurring theme in discussions of simplicity--both inside and
outside philosophy--namely, how the correct balance between
simplicity and goodness of fit ought to be struck. This theme takes
center stage in the statistical approaches to simplicity discussed in
Section 5.



Less work has been done on describing episodes in science where
elegance--as opposed to parsimony--has been (or may have
been) the crucial factor. This may just reflect the fact that
considerations linked to elegance are so pervasive in scientific
theory choice as to be unremarkable as a topic for special study. A
notable exception to this general neglect is the area of celestial
mechanics, where the transition from Ptolemy to Copernicus to Kepler
to Newton is an oft-cited example of simplicity considerations in
action, and a case study which makes much more sense when seen through
the lens of elegance rather than of
 parsimony.[14]



Naturalism depends on a number of presuppositions which are open to
debate. But even if these presuppositions are granted, the
naturalistic project of looking to science for methodological guidance
within philosophy faces a major difficulty, namely how to 'read
off' from actual scientific practice what the underlying
methodological principles are supposed to be. Burgess, for example,
argues that what the patterns of scientific behavior show is not a
concern with multiplying entities *per se*, but a concern more
specifically with multiplying 'causal mechanisms' (Burgess
1998). And Sober considers the debate in psychology over psychological
egoism versus motivational pluralism, arguing that the former theory
postulates fewer types of ultimate desire but a larger number of
causal beliefs, and hence that comparing the parsimony of these two
theories depends on what is counted and how (Sober 2001, pp.
14-5). Some of the concerns raised in Sections 1 and 2 also
reappear in this context; for example, how the world is sliced up into
kinds effects the extent to which a given theory
'multiplies' kinds of entity. Justifying a particular way
of slicing becomes more difficult once the epistemological naturalist
leaves behind the *a priori*, metaphysical presuppositions of
the rationalist approach.



One philosophical debate where these worries over naturalism become
particularly acute is the issue of the application of parsimony
principles to *abstract* objects. The scientific data
is--in an important sense--ambiguous. Applications of
Occam's Razor in science are always to concrete, causally efficacious
entities, whether land-bridges, unicorns, or the luminiferous ether.
Perhaps scientists apply an unrestricted version of Occam's Razor to
that portion of reality in which they are interested, namely the
concrete, causal, spatiotemporal world. Or perhaps scientists apply a
'concretized' version of Occam's Razor unrestrictedly.
Which is the case? The answer determines which general philosophical
principle we end up with: ought we to avoid the multiplication of
objects of whatever kind, or merely the multiplication of concrete
objects? The distinction here is crucial for a number of central
philosophical debates. Unrestricted Occam's Razor favors monism over
dualism, and nominalism over platonism. By contrast,
'concretized' Occam's Razor has no bearing on these
debates, 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--*a
priori* rationalism and naturalized empiricism--are both in
some sense extreme. Simplicity principles are taken either to have no
empirical grounding, or to have solely empirical grounding. Perhaps as
a result, both these approaches yield vague answers to certain key
questions about simplicity. In particular, neither seems equipped to
answer how exactly simplicity ought to be balanced against empirical
adequacy. Simple but wildly inaccurate theories are not hard to come
up with. Nor are accurate theories which are highly complex. But how
much accuracy should be sacrificed for a gain in simplicity? The
black-and-white boundaries of the rationalism/empiricism divide may
not provide appropriate tools for analyzing this question. In
response, philosophers have recently turned to the mathematical
framework of probability theory and statistics, hoping in the process
to combine sensitivity to actual practice with the
'trans-empirical' strength of mathematics.



Philosophically influential early work in this direction was done by
Jeffreys and by Popper, both of whom tried to analyze simplicity in
probabilistic terms. Jeffreys argued that "the simpler laws have
the greater prior probability," and went on to provide an
operational measure of simplicity, according to which the prior
probability 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 at
specific equations, but at *families* of equations. For
example, one might compare the family, LIN, of linear equations (of
the form \(y = a + bx)\) with the family, PAR,
of parabolic equations (of the form \(y = a + bx + cx^2)\). Since PAR is of higher degree
than LIN, Jeffreys' proposal assigns higher probability to LIN. Laws
of this form are intuitively simpler (in the sense of being more
elegant).



Popper (1959) points out that Jeffreys' proposal, as it stands,
contradicts the axioms of probability. Every member of LIN is also a
member 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 assigns
higher probability to the former than the latter. But it follows from
the axioms of probability that when \(A\) entails \(B\), the
probability of \(B\) is greater than or equal to the probability
of \(A\). Popper argues, in contrast to Jeffreys, that LIN has
*lower* prior probability than PAR. Hence LIN is--in
Popper's sense--more falsifiable, and hence should be preferred
as the default hypothesis. One response to Popper's objection is to
amend Jeffrey's proposal and restrict members of PAR to equations
where \(c \ne 0\).



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



The standard case in the statistical literature on parsimony concerns
 *curve-fitting*.[15]
 We imagine a situation in which we have a set of discrete data points
and are looking for the curve (i.e. function) which has generated
them. The issue of what family of curves the answer belongs in (e.g.
in LIN or in PAR) is often referred to as *model-selection*.
The basic idea is that there are two competing criteria for model
selection--parsimony and goodness of fit. The possibility of
measurement error and 'noise' in the data means that the
correct curve may not go through every data point. Indeed, if goodness
of fit were the only criterion then there would be a danger of
'overfitting' the model to accidental discrepancies
unrepresentative of the broader regularity. Parsimony acts as a
counterbalance to such overfitting, since a curve passing through
every data point is likely to be very convoluted and hence have many
adjusted parameters.



If proponents of the statistical approach are in general agreement
that simplicity should be cashed out in terms of number of parameters,
there is less unanimity over what the goal of simplicity principles
ought to be. This is partly because the goal is often not made
explicit. (An analogous issue arises in the case of Occam's Razor.
'Entities are not to be multiplied beyond necessity.' But
necessity for *what*, exactly?) Forster distinguishes two
potential goals of model selection, namely probable truth and
predictive accuracy, and claims that these are importantly distinct
(Forster 2001, p. 95). Forster argues that predictive accuracy tends
to be what scientists care about most. They care less about the
probability of an hypothesis being exactly right than they do about it
having a high degree of accuracy.



One reason for investigating statistical approaches to simplicity is a
dissatisfaction with the vagaries of the *a priori* and
naturalistic approaches. Statisticians have come up with a variety of
numerically specific proposals for the trade-off between simplicity
and goodness of fit. However, these alternative proposals disagree
about the 'cost' associated with more complex hypotheses.
Two leading contenders in the recent literature on model selection are
the Akaike Information Criterion [AIC] and the Bayesian Information
Criterion [BIC]. AIC directs theorists to choose the model with the
highest 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. By
contrast, BIC maximizes the value of \(\{\log L(\Theta\_k )/n\} -
k\log[n]/2n.\) In effect, BIC gives an extra positive weighting to
simplicity by a factor of \(\log[n]/2\) (where \(n\) is the size of
the sample).[16]



Extreme answers to the trade-off problem seem to be obviously
inadequate. 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. Always
picking the simplest model, regardless of its fit to the data, cuts
the model free from any link to observation or experiment. Forster
associates the 'Always Complex' and the 'Always
Simple' rule with empiricism and rationalism
 respectively (*ibid.*).
 All the candidate rules that are seriously discussed by statisticians
fall in between these two extremes. Yet they differ in their answers
over how much weight to give simplicity in its trade-off against
goodness of fit. In addition to AIC and BIC, other rules include
Neyman-Pearson hypothesis testing, and the minimum description length
(MDL) criterion.



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



Aside from this issue of conflicting criteria, there are other
problems with the statistical approach to simplicity. One problem,
which afflicts any approach emphasizing the elegance aspect of
simplicity, is language relativity. Crudely put, hypotheses which are
syntactically very complex in one language may be syntactically very
simple in another. The traditional philosophical illustration of this
problem is Goodman's 'grue' challenge to induction. Are
statistical approaches to the measurement of simplicity similarly
language relative, and--if so--what justifies choosing one
language over another? It turns out that the statistical approach has
the resources to at least partially deflect the charge of language
relativity. Borrowing techniques from information theory, it can be
shown that certain syntactic measures of simplicity are asymptotically
independent of choice of measurement
 language.[17]



A second problem for the statistical approach is whether it can
account not only for our preference for small numbers over large
numbers (when it comes to picking values for coefficients or exponents
in model equations), but also our preference for whole numbers and
simple fractions over other values. In Gregor Mendel's original
experiments on the hybridization of garden peas, he crossed pea
varieties with different specific traits, such as tall versus short or
green seeds versus yellow seeds, and then self-pollinated the hybrids
for one or more
 generations.[18]
 In each case one trait was present in all the first-generation
hybrids, but both traits were present in subsequent generations.
Across his experiments with seven different such traits, the ratio of
dominant trait to recessive trait averaged 2.98 : 1. On this
basis, Mendel hypothesized that the true ratio is 3 : 1.
This 'rounding' was made prior to the formulation of any
explanatory model, hence it cannot have been driven by any
theory-specific consideration. This raises two related questions.
First, in what sense is the 3 : 1 ratio hypothesis
*simpler* than the 2.98 : 1 ratio hypothesis? Second,
can this choice be justified within the framework of the statistical
approach to simplicity? The more general worry lying behind these
questions is whether the statistical approach, in defining simplicity
in terms of number of adjustable parameters, is replacing the broad
issue of simplicity with a more narrowly--and perhaps
arbitrarily--defined set of issues.



A third problem with the statistical approach concerns whether it can
shed any light on the specific issue of ontological parsimony. At
first glance, one might think that the postulation of extra entities
can be attacked on probabilistic grounds. For example, quantum
mechanics together with the postulation 'There exist
unicorns' is less probable than quantum mechanics alone, since
the former logically entails the latter. However, as Sober has pointed
out, it is important here to distinguish between *agnostic*
Occam's Razor and *atheistic* Occam's Razor. Atheistic OR
directs theorists to claim that unicorns do *not* exist, in the
absence of any compelling evidence in their favor. And there is no
relation of logical entailment between \(\{QM +\) there exist
unicorns\(\}\) and \(\{QM +\) there do not exist unicorns\(\}\). This also
links back to the terminological issue. Models involving circular
orbits are more parsimonious--in the statisticians' sense of
'parsimonious'--than models involving elliptical
orbits, but the latter models do not postulate the existence of any
more things in the world.


## 6. Other Issues Concerning Simplicity



This section addresses three distinct issues concerning simplicity and
its relation to other methodological issues. These issues concern
quantitative parsimony, plenitude, and induction.


### 6.1 Quantitative Parsimony



Theorists tend to be frugal in their postulation of new entities. When
a trace is observed in a cloud-chamber, physicists may seek to explain
it in terms of the influence of a hitherto unobserved particle. But,
if possible, they will postulate one such unobserved particle, not
two, or twenty, or 207 of them. This desire to minimize the number of
individual new entities postulated is often referred to as
*quantitative parsimony*. (For arguments that the qualitative
parsimony / quantitative parsimony distinction is not so clear, see
Sendlak 2018.) David Lewis articulates the attitude of many
philosophers when he writes:



> 
> I subscribe to the general view that qualitative parsimony is good in
> a philosophical or empirical hypothesis; but I recognize no
> presumption whatever in favour of quantitative parsimony (Lewis 1973,
> p. 87).
> 



Is the initial assumption that one particle is acting to cause the
observed trace more rational than the assumption that 207 particles
are 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 discovery
of the
 neutrino.[19]
 Physicists in the 1930's were puzzled by certain anomalies arising
from experiments in which radioactive atoms emit electrons during
so-called Beta decay. In these experiments the total spin of the
particles in the system before decay exceeds by
1/2 the total spin of the (observed) emitted
particles. Physicists' response was to posit a 'new'
fundamental particle, the neutrino, with spin
1/2 and to hypothesize that exactly one neutrino
is emitted by each electron during Beta decay.



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



(H1)
1 neutrino with a spin of 1/2 is emitted in
each case of Beta decay.
(H2)
2 neutrinos, each with a spin of 1/4 are
emitted in each case of Beta decay.


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



(H\(n\))
\(n\) neutrinos, each with a spin of
1/2 are emitted in each case of Beta
decay.


Each of these hypotheses adequately explains the observation of a
missing 1/2-spin following Beta decay. Yet the
most quantitatively parsimonious hypothesis, H1, is the
obvious default
 choice.[20]



One promising approach is to focus on the relative explanatory power
of the alternative hypotheses, H1, H2, ...
Hn. When neutrinos were first postulated in the 1930's,
numerous experimental set-ups were being devised to explore the
products of various kinds of particle decay. In none of these
experiments had cases of 'missing'
&frac13;-spin, or 1/4-spin, or
\(^1 /\_{100}\)-spin been found. The absence of these
smaller fractional spins was a phenomenon which competing neutrino
hypotheses might potentially help to explain.



Consider the following two competing neutrino hypotheses:



(H1)
1 neutrino with a spin of 1/2 is emitted in
each case of Beta decay.
(H10)
10 neutrinos, each with a spin of \(^1 /\_{20}\), are
emitted in each case of Beta decay.


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



One distinctive feature of the neutrino example is that it is
'additive.' It involves postulating the existence of a
collection of qualitatively identical objects which collectively
explain the observed phenomenon. The explanation is additive in the
sense that the overall phenomenon is explained by summing the
individual positive contributions of each
 object.[21]
 Whether the above approach can be extended to non-additive cases
involving quantitative parsimony is an interesting question. Jansson
and Tallant (2017) argue that it can, and they offer a probabilistic
analysis that aims to bring together a variety of different cases
where quantitative parsimony plays a role in hypothesis selection.
Consider a case in which the aberrations of a planet's orbit can be
explained by postulating a single unobserved planet, or it can be
explained by postulating two or more unobserved planets. In order for
the latter situation to be actual, the multiple planets must orbit in
certain restricted ways so as to match the effects of a single planet.
*Prima facie* this is unlikely, and this counts against the
less quantitatively parsimonious hypothesis.


### 6.2 Principles of Plenitude



Ranged against the principles of parsimony discussed in previous
sections 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 medieval
controversies that spawned Occam's Razor. Ockham's contemporary,
Walter of Chatton, proposed the following counter-principle to Occam's
Razor:



> 
> [I]f three things are not enough to verify an affirmative proposition
> about things, a fourth must be added, and so on (quoted in Maurer
> 1984, p. 464).
> 



A related counter-principle was later defended by Kant:



> 
> The variety of entities should not be rashly diminished (Kant
> 1781/1787, p. 541).
>   
> 
> *Entium varietates non temere esse minuendas*.
> 



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



There is, however, a second family of principles which do appear
directly to conflict with Occam's Razor. These are so-called
'principles of plenitude.' Perhaps the best-known version
is associated with Leibniz, according to whom God created the best of
all possible worlds with the greatest number of possible entities.
More generally, a principle of plenitude claims that if it is
*possible* for an object to exist then that object
*actually* exists. Principles of plenitude conflict with
Occam's Razor over the existence of physically possible but
explanatorily idle objects. Our best current theories presumably do
not rule out the existence of unicorns, but nor do they provide any
support for their existence. According to Occam's Razor we ought
*not* to postulate the existence of unicorns. According to a
principle of plenitude we *ought* to postulate their
existence.



The rise of particle physics and quantum mechanics in the
20th Century led to various principles of plenitude being
appealed to by scientists as an integral part of their theoretical
framework. A particularly clear-cut example of such an appeal is the
case of magnetic
 monopoles.[24]
 The 19th-century theory of electromagnetism postulated
numerous analogies between electric charge and magnetic charge. One
theoretical difference is that magnetic charges must always come in
oppositely-charged pairs, called "dipoles" (as in the
North and South poles of a bar magnet), whereas single electric
charges, or "monopoles," can exist in isolation. However,
no actual magnetic monopole had ever been observed. Physicists began
to wonder whether there was some theoretical reason why monopoles
could not exist. It was initially thought that the newly developed
theory of quantum mechanics ruled out the possibility of magnetic
monopoles, and this is why none had ever been detected. However, in
1931 the physicist Paul Dirac showed that the existence of monopoles
is consistent with quantum mechanics, although it is not required by
it. Dirac went on to assert the existence of monopoles, arguing that
their existence is not ruled out by theory and that "under these
circumstances one would be surprised if Nature had made no use of
it" (Dirac 1930, p. 71, note 5). This appeal to plenitude was
widely--though not universally--accepted by other
physicists.



> 
> One of the elementary rules of nature is that, in the absence of laws
> prohibiting an event or phenomenon it is bound to occur with some
> degree of probability. To put it simply and crudely: anything that
> *can* happen *does* happen. Hence physicists must assume
> that the magnetic monopole exists unless they can find a law barring
> its 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 of
> magnetic monopoles, does not differ from 18th-century
> arguments in favour of mermaids...[A]s the notion of mermaids was
> neither intrinsically contradictory nor colliding with current
> biological laws, these creatures were assumed to
>  exist.[25]
> 



It is difficult to know how to interpret these principles of
plenitude. Quantum mechanics diverges from classical physics by
replacing of a deterministic model of the universe with a model based
on objective probabilities. According to this probabilistic model,
there are numerous ways the universe could have evolved from its
initial state, each with a certain probability of occurring that is
fixed by the laws of nature. Consider some kind of object, say
unicorns, whose existence is not ruled out by the initial conditions
plus the laws of nature. Then one can distinguish between a weak and a
strong version of the principle of plenitude. According to the weak
principle, if there is a small finite probability of unicorns existing
then given enough time and space unicorns will exist. According to the
strong principle, it follows from the theory of quantum mechanics that
if it is possible for unicorns to exist then they do exist. One way in
which 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 every
possible 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 the
inductive problem of predicting future outcomes from observed data.
Less obviously, Schulte (1999) argues for a connection between
induction and ontological parsimony. Schulte frames the problem of
induction in information-theoretic terms: given a data-stream of
observations of non-unicorns (for example), what general conclusion
should be drawn? He argues for two constraints on potential rules.
First, the rule should converge on the truth in the long run (so if no
unicorns exist then it should yield this conclusion). Second, the rule
should minimize the maximum number of changes of hypothesis, given
different possible future observations. Schulte argues that the
'Occam Rule'--conjecture that \(\Omega\) does not exist
until it has been detected in an experiment--is optimal relative
to these constraints. An alternative rule--for example,
conjecturing that \(\Omega\) exists until 1 million negative results have
been obtained--may result in two changes of hypothesis if, say,
\(\Omega\)'s are not detected until the 2 millionth experiment. The Occam
Rule leads to at most one change of hypothesis (when an \(\Omega\) is
first detected). (See also Kelly 2004, 2007.) Schulte (2008) applies
this approach to the problem of discovering conservation laws in
particle physics. The analysis has been criticized by Fitzpatrick
(2013), who raises doubts about why long-run convergence to the truth
should matter when it comes to predicting the outcome of the very next
experiment, and the debate is continued in Kelly et al. (2016).



With respect to the justification question, arguments have been made
in both directions. Scientists are often inclined to justify
simplicity principles on broadly inductive grounds. According to this
argument, scientists select new hypotheses based partly on criteria
that have been generated inductively from previous cases of theory
choice. Choosing the most parsimonious of the acceptable alternative
hypotheses has tended to work in the past. Hence scientists continue
to use this as a rule of thumb, and are justified in so doing on
inductive grounds. One might try to bolster this point of view by
considering a counterfactual world in which all the fundamental
constituents of the universe exist in pairs. In such a
'pairwise' world, scientists might well prefer pairwise
hypotheses in general to their more parsimonious rivals. This line of
argument has a couple of significant weaknesses. Firstly, one might
legitimately wonder just how successful the choice of parsimonious
hypotheses has been; examples from chemistry spring to mind, such as
oxygen molecules containing two atoms rather than one. Secondly, and
more importantly, there remains the issue of explaining *why*
the preference for parsimonious hypotheses in science has been as
successful as it has been.



Making the justificatory argument in the reverse direction, from
simplicity to induction, has a strong historical precedent in
philosophical approaches to the problem of induction, from Hume
onwards. Justifying the 'straight rule' of induction by
appeal to some general Principle of Uniformity is an initially
appealing response to the skeptical challenge. However, in the absence
of a defense of the underlying Principle itself (and one which does
not, on pain of circularity, depend inductively on past success), it
is unclear how much progress this represents. There have also been
attempts (see e.g. Steel 2009) to use simplicity considerations to
respond to Nelson Goodman's 'new riddle of induction.'