occam's razor

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Occam’s razor For the aerial theatre company, see Ockham’s Razor The- atre Company. Occam’s razor (also written as Ockham’s razor and in Andreas Cellarius's illustration of the Copernican system, from the Harmonia Macrocosmica (1708). The motions of the sun, moon and other solar system planets can be calculated using a geocentric model (the earth is at the center) or using a heliocentric model (the sun is at the center). Both work, but the geocen- tric system requires many more assumptions than the heliocen- tric system, which has only seven. This was pointed out in a preface to Copernicus' first edition of De revolutionibus orbium coelestium. Latin lex parsimoniae, which means 'law of parsimony') is a problem-solving principle devised by William of Ock- ham (c. 1287–1347), who was an English Franciscan friar and scholastic philosopher and theologian. The prin- ciple states that among competing hypotheses that predict equally well, the one with the fewest assumptions should be selected. Other, more complicated solutions may ul- timately prove to provide better predictions, but—in the absence of differences in predictive ability—the fewer as- sumptions that are made, the better. The application of the principle can be used to shift the burden of proof in a discussion. However, Alan Baker, who suggests this in the online Stanford Encyclopedia of Philosophy, is careful to point out that his suggestion should not be taken generally, but only as it applies in a particular context, that is: philosophers who argue in op- position to metaphysical theories that involve an allegedly “superfluous ontological apparatus.” [lower-alpha 1] Baker then notices that principles, including Occam’s ra- zor, are often expressed in a way that is unclear regarding which facet of “simplicity”—parsimony or elegance—the principle refers to, and that in a hypothetical formulation the facets of simplicity may work in different directions: a simpler description may refer to a more complex hy- pothesis, and a more complex description may refer to a simpler hypothesis. [lower-alpha 2] Solomonoff’s theory of inductive inference is a mathe- matically formalized Occam’s razor: [2][3][4][5][6][7] shorter computable theories have more weight when calculating the probability of the next observation, using all com- putable theories that perfectly describe previous obser- vations. In science, Occam’s razor is used as a heuristic tech- nique (discovery tool) to guide scientists in the devel- opment of theoretical models, rather than as an arbiter between published models. [8][9] In the scientific method, Occam’s razor is not considered an irrefutable principle of logic or a scientific result; the preference for simplicity in the scientific method is based on the falsifiability cri- terion. For each accepted explanation of a phenomenon, there is always an infinite number of possible and more complex alternatives, because one can always burden failing explanations with ad hoc hypothesis to prevent them from being falsified; therefore, simpler theories are preferable to more complex ones because they are more testable. [1][10][11] 1 History The term Occam’s razor first appeared in 1852 in the works of Sir William Hamilton, 9th Baronet (1788– 1856), centuries after William of Ockham's death in 1347. [12] Ockham did not invent this “razor”—its associ- ation with him may be due to the frequency and effective- ness with which he used it (Ariew 1976). Ockham stated the principle in various ways, but the most popular ver- sion, “Entities must not be multiplied beyond necessity” (Non sunt multiplicanda entia sine necessitate) was formu- lated by the Irish Franciscan philosopher John Punch in his 1639 commentary on the works of Duns Scotus. [13] 1.1 Formulations before Ockham The origins of what has come to be known as Occam’s ra- zor are traceable to the works of earlier philosophers such as John Duns Scotus (1265–1308), Robert Grosseteste (1175-1253), Maimonides (Moses ben-Maimon, 1138– 1204), and even Aristotle (384–322 BC). [14][15] Aristo- 1

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Page 1: Occam's Razor

Occam’s razor

For the aerial theatre company, see Ockham’s Razor The-atre Company.Occam’s razor (also written asOckham’s razor and in

Andreas Cellarius's illustration of the Copernican system, fromthe Harmonia Macrocosmica (1708). The motions of the sun,moon and other solar system planets can be calculated using ageocentric model (the earth is at the center) or using a heliocentricmodel (the sun is at the center). Both work, but the geocen-tric system requires many more assumptions than the heliocen-tric system, which has only seven. This was pointed out in apreface to Copernicus' first edition of De revolutionibus orbiumcoelestium.

Latin lex parsimoniae, whichmeans 'law of parsimony') isa problem-solving principle devised by William of Ock-ham (c. 1287–1347), who was an English Franciscanfriar and scholastic philosopher and theologian. The prin-ciple states that among competing hypotheses that predictequally well, the one with the fewest assumptions shouldbe selected. Other, more complicated solutions may ul-timately prove to provide better predictions, but—in theabsence of differences in predictive ability—the fewer as-sumptions that are made, the better.The application of the principle can be used to shift theburden of proof in a discussion. However, Alan Baker,who suggests this in the online Stanford Encyclopediaof Philosophy, is careful to point out that his suggestionshould not be taken generally, but only as it applies in aparticular context, that is: philosophers who argue in op-position to metaphysical theories that involve an allegedly“superfluous ontological apparatus.”[lower-alpha 1]

Baker then notices that principles, including Occam’s ra-zor, are often expressed in a way that is unclear regardingwhich facet of “simplicity”—parsimony or elegance—the

principle refers to, and that in a hypothetical formulationthe facets of simplicity may work in different directions:a simpler description may refer to a more complex hy-pothesis, and a more complex description may refer to asimpler hypothesis.[lower-alpha 2]

Solomonoff’s theory of inductive inference is a mathe-matically formalized Occam’s razor:[2][3][4][5][6][7] shortercomputable theories have more weight when calculatingthe probability of the next observation, using all com-putable theories that perfectly describe previous obser-vations.In science, Occam’s razor is used as a heuristic tech-nique (discovery tool) to guide scientists in the devel-opment of theoretical models, rather than as an arbiterbetween published models.[8][9] In the scientific method,Occam’s razor is not considered an irrefutable principleof logic or a scientific result; the preference for simplicityin the scientific method is based on the falsifiability cri-terion. For each accepted explanation of a phenomenon,there is always an infinite number of possible and morecomplex alternatives, because one can always burdenfailing explanations with ad hoc hypothesis to preventthem from being falsified; therefore, simpler theories arepreferable to more complex ones because they are moretestable.[1][10][11]

1 History

The term Occam’s razor first appeared in 1852 in theworks of Sir William Hamilton, 9th Baronet (1788–1856), centuries after William of Ockham's death in1347.[12] Ockham did not invent this “razor”—its associ-ation with him may be due to the frequency and effective-ness with which he used it (Ariew 1976). Ockham statedthe principle in various ways, but the most popular ver-sion, “Entities must not be multiplied beyond necessity”(Non sunt multiplicanda entia sine necessitate)was formu-lated by the Irish Franciscan philosopher John Punch inhis 1639 commentary on the works of Duns Scotus.[13]

1.1 Formulations before Ockham

The origins of what has come to be known as Occam’s ra-zor are traceable to the works of earlier philosophers suchas John Duns Scotus (1265–1308), Robert Grosseteste(1175-1253), Maimonides (Moses ben-Maimon, 1138–1204), and even Aristotle (384–322 BC).[14][15] Aristo-

1

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2 1 HISTORY

Part of a page fromDuns Scotus’ bookOrdinatio: "Pluralitas nonest ponenda sine necessitate", i.e., “Plurality is not to be positedwithout necessity”

tle writes in his Posterior Analytics, “We may assume thesuperiority ceteris paribus [other things being equal] ofthe demonstration which derives from fewer postulates orhypotheses.”[16] Ptolemy (c. AD 90 – c. AD 168) stated,“We consider it a good principle to explain the phenom-ena by the simplest hypothesis possible.”[17]

Phrases such as “It is vain to do with more what can bedone with fewer” and “A plurality is not to be positedwithout necessity” were commonplace in 13th-centuryscholastic writing.[17] Robert Grosseteste, in Commen-tary on [Aristotle’s] the Posterior Analytics Books (Com-mentarius in Posteriorum Analyticorum Libros) (c. 1217–1220), declares: “That is better and more valuable whichrequires fewer, other circumstances being equal... Forif one thing were demonstrated from many and anotherthing from fewer equally known premises, clearly thatis better which is from fewer because it makes us knowquickly, just as a universal demonstration is better thanparticular because it produces knowledge from fewerpremises. Similarly in natural science, in moral sci-ence, and in metaphysics the best is that which needsno premises and the better that which needs the fewer,other circumstances being equal.”[18] The Summa Theo-logica of Thomas Aquinas (1225–1274) states that “it issuperfluous to suppose that what can be accounted for bya few principles has been produced by many”. Aquinasuses this principle to construct an objection to God’s ex-istence, an objection that he in turn answers and refutesgenerally (cf. quinque viae), and specifically, throughan argument based on causality.[19] Hence, Aquinas ac-knowledges the principle that today is known as Occam’srazor, but prefers causal explanations to other simple ex-planations (cf. also Correlation does not imply causa-tion).The Indian Hindu philosopherMadhva in verse 400 of hisVishnu-Tattva-Nirnaya says: "dvidhAkalpane kalpanA-gauravamiti" (“To make two suppositions when one isenough is to err by way of excessive supposition”).

1.2 Ockham

William of Ockham (circa 1287–1347) was an EnglishFranciscan friar and theologian, an influential medievalphilosopher and a nominalist. His popular fame as a great

logician rests chiefly on the maxim attributed to him andknown as Ockham’s razor. The term razor refers to dis-tinguishing between two hypotheses either by “shavingaway” unnecessary assumptions or cutting apart two sim-ilar conclusions.While it has been claimed that Ockham’s razor is notfound in any of his writings,[20] one can cite statementssuch as Numquam ponenda est pluralitas sine necessitate[Plurality must never be posited without necessity], whichoccurs in his theological work on the 'Sentences of Pe-ter Lombard' (Quaestiones et decisiones in quattuor librosSententiarum Petri Lombardi (ed. Lugd., 1495), i, dist.27, qu. 2, K).Nevertheless, the precise words sometimes attributedto Ockham, entia non sunt multiplicanda praeter ne-cessitatem (entities must not be multiplied beyondnecessity),[21] are absent in his extant works;[22] this par-ticular phrasing comes from John Punch,[23] who de-scribed the principle as a “common axiom” (axioma vul-gare) of the Scholastics.[13] Ockham’s contribution seemsto be to restrict the operation of this principle in mat-ters pertaining to miracles and God’s power: so, in theEucharist, a plurality of miracles is possible, simply be-cause it pleases God.[17]

This principle is sometimes phrased as pluralitas non estponenda sine necessitate (“plurality should not be positedwithout necessity”).[24] In his Summa Totius Logicae, i.12, Ockham cites the principle of economy, Frustra fitper plura quod potest fieri per pauciora (It is futile to dowith more things that which can be done with fewer”).(Thorburn, 1918, pp. 352–53; Kneale and Kneale, 1962,p. 243.)

1.3 Later formulations

To quote Isaac Newton, “We are to admit no more causesof natural things than such as are both true and sufficientto explain their appearances. Therefore, to the same nat-ural effects we must, as far as possible, assign the samecauses.”[25][26]

Bertrand Russell offers a particular version of Occam’srazor: “Whenever possible, substitute constructions outof known entities for inferences to unknown entities.”[27]

Around 1960, Ray Solomonoff founded the theory ofuniversal inductive inference, the theory of predic-tion based on observations; for example, predicting thenext symbol based upon a given series of symbols.The only assumption is that the environment followssome unknown but computable probability distribution.This theory is a mathematical formalization of Occam’srazor.[2][3][4][5][28]

Another technical approach to Occam’s razor isontological parsimony.[29]

The widespread layperson’s formulation that “the sim-

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2.2 Empirical 3

plest explanation is usually the correct one” appears tohave been derived from Occam’s razor.

2 Justifications

Beginning in the 20th century, epistemological justifi-cations based on induction, logic, pragmatism, and es-pecially probability theory have become more popularamong philosophers.

2.1 Aesthetic

Prior to the 20th century, it was a commonly held beliefthat nature itself was simple and that simpler hypothesesabout nature were thus more likely to be true. This notionwas deeply rooted in the aesthetic value simplicity holdsfor human thought and the justifications presented for itoften drew from theology. Thomas Aquinas made thisargument in the 13th century, writing, “If a thing can bedone adequately by means of one, it is superfluous to do itby means of several; for we observe that nature does notemploy two instruments [if] one suffices.”[30]

2.2 Empirical

Occam’s razor has gained strong empirical support inhelping to converge on better theories (see “Applications”section below for some examples).In the related concept of overfitting, excessively complexmodels are affected by statistical noise (a problem alsoknown as the bias-variance trade-off), whereas simplermodels may capture the underlying structure better andmay thus have better predictive performance. It is, how-ever, often difficult to deduce which part of the data isnoise (cf. model selection, test set, minimum descriptionlength, Bayesian inference, etc.).

2.2.1 Testing the razor

The razor’s statement that “other things being equal, sim-pler explanations are generally better than more complexones” is amenable to empirical testing. Another inter-pretation of the razor’s statement would be that “simplerhypotheses (not conclusions, i.e., explanations) are gen-erally better than the complex ones”. The procedure totest the former interpretation would compare the trackrecords of simple and comparatively complex explana-tions. If one accepts the first interpretation, the validityof Occam’s razor as a tool would then have to be rejectedif the more complex explanations were more often cor-rect than the less complex ones (while the converse wouldlend support to its use). If the latter interpretation is ac-cepted, the validity of Occam’s razor as a tool could pos-sibly be accepted if the simpler hypotheses led to correct

conclusions more often than not.

Possible explanations can become needlessly complex. It is coher-ent, for instance, to add the involvement of leprechauns to anyexplanation, but Occam’s razor would prevent such additions un-less they were necessary.

In the history of competing hypotheses, the simpler hy-potheses have led to mathematically rigorous and em-pirically verifiable theories. In the history of compet-ing explanations, this is not the case—at least not gen-erally. Some increases in complexity are sometimes nec-essary. So there remains a justified general bias towardthe simpler of two competing explanations. To under-stand why, consider that for each accepted explanationof a phenomenon, there is always an infinite number ofpossible, more complex, and ultimately incorrect, alter-natives. This is so because one can always burden failingexplanations with ad hoc hypothesis. Ad hoc hypothesesare justifications that prevent theories from being falsi-fied. Even other empirical criteria, such as consilience,can never truly eliminate such explanations as competi-tion. Each true explanation, then, may have had manyalternatives that were simpler and false, but also an infi-nite number of alternatives that were more complex andfalse. But if an alternate ad hoc hypothesis were indeedjustifiable, its implicit conclusions would be empiricallyverifiable. On a commonly accepted repeatability prin-ciple, these alternate theories have never been observedand continue to escape observation. In addition, one doesnot say an explanation is true if it has not withstood thisprinciple.Put another way, any new, and even more complex, the-ory can still possibly be true. For example, if an indi-vidual makes supernatural claims that leprechauns wereresponsible for breaking a vase, the simpler explanationwould be that he is mistaken, but ongoing ad hoc jus-tifications (e.g., "... and that’s not me on the film; theytampered with that, too.”) successfully prevent outrightfalsification. This endless supply of elaborate competing

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4 2 JUSTIFICATIONS

explanations, called saving hypotheses, cannot be ruledout—but by using Occam’s razor.[31][32][33]

2.3 Practical considerations and pragma-tism

See also: pragmatism and problem of induction

The common form of the razor, used to distinguish be-tween equally explanatory hypotheses, may be supportedby the practical fact that simpler theories are easier to un-derstand.Some argue that Occam’s razor is not an inference-drivenmodel, but a heuristic maxim for choosing among othermodels and instead underlies induction.Alternatively, if one wants to have reasonable discussionone may be practically forced to accept Occam’s razorin the same way one is simply forced to accept the lawsof thought and inductive reasoning (given the problem ofinduction). Philosopher Elliott Sober states that not evenreason itself can be justified on any reasonable grounds,and that we must start with first principles of some kind(otherwise an infinite regress occurs).The pragmatist may go on, as David Hume did on thetopic of induction, that there is no satisfying alternativeto granting this premise. Though one may claim that Oc-cam’s razor is invalid as a premise that helps regulate the-ories, putting this doubt into practice would mean doubt-ing whether every step forward will result in locomotionor a nuclear explosion. In other words: “What’s the alter-native?"

2.4 Mathematical

One justification of Occam’s razor is a direct result of ba-sic probability theory. By definition, all assumptions in-troduce possibilities for error; if an assumption does notimprove the accuracy of a theory, its only effect is to in-crease the probability that the overall theory is wrong.There have also been other attempts to derive Occam’srazor from probability theory, including notable attemptsmade by Harold Jeffreys and E. T. Jaynes. The proba-bilistic (Bayesian) basis for Occam’s razor is elaboratedby David J. C. MacKay in chapter 28 of his book Infor-mation Theory, Inference, and Learning Algorithms,[34]where he emphasises that a prior bias in favour of sim-pler models is not required.William H. Jefferys (no relation to Harold Jeffreys) andJames O. Berger (1991) generalize and quantify the orig-inal formulation’s “assumptions” concept as the degreeto which a proposition is unnecessarily accommodatingto possible observable data.[35] They state, “A hypothesiswith fewer adjustable parameters will automatically havean enhanced posterior probability, due to the fact that

the predictions it makes are sharp.”[35] The model theypropose balances the precision of a theory’s predictionsagainst their sharpness—preferring theories that sharplymake correct predictions over theories that accommo-date a wide range of other possible results. This, again,reflects the mathematical relationship between key con-cepts in Bayesian inference (namely marginal probability,conditional probability, and posterior probability).

2.5 Other philosophers

2.5.1 Karl Popper

Karl Popper argues that a preference for simple theo-ries need not appeal to practical or aesthetic consider-ations. Our preference for simplicity may be justifiedby its falsifiability criterion: we prefer simpler theoriesto more complex ones “because their empirical contentis greater; and because they are better testable” (Popper1992). The idea here is that a simple theory applies tomore cases than a more complex one, and is thus moreeasily falsifiable. This is again comparing a simple the-ory to a more complex theory where both explain the dataequally well.

2.5.2 Elliott Sober

The philosopher of science Elliott Sober once arguedalong the same lines as Popper, tying simplicity with“informativeness": The simplest theory is the more in-formative, in the sense that it requires less informa-tion to a question.[36] He has since rejected this ac-count of simplicity, purportedly because it fails to pro-vide an epistemic justification for simplicity. He now be-lieves that simplicity considerations (and considerationsof parsimony in particular) do not count unless they re-flect something more fundamental. Philosophers, he sug-gests, may have made the error of hypostatizing simplic-ity (i.e., endowed it with a sui generis existence), when ithas meaning only when embedded in a specific context(Sober 1992). If we fail to justify simplicity considera-tions on the basis of the context in which we use them,we may have no non-circular justification: “Just as thequestion 'why be rational?' may have no non-circular an-swer, the same may be true of the question 'why shouldsimplicity be considered in evaluating the plausibility ofhypotheses?'"[37]

2.5.3 Richard Swinburne

Richard Swinburne argues for simplicity on logicalgrounds:

... the simplest hypothesis proposed asan explanation of phenomena is more likelyto be the true one than is any other available

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3.1 Science and the scientific method 5

hypothesis, that its predictions are more likelyto be true than those of any other availablehypothesis, and that it is an ultimate a prioriepistemic principle that simplicity is evidencefor truth.—Swinburne 1997

According to Swinburne, since our choice of theory can-not be determined by data (see Underdetermination andQuine-Duhem thesis), we must rely on some criterion todetermine which theory to use. Since it is absurd to haveno logical method for settling on one hypothesis amongstan infinite number of equally data-compliant hypotheses,we should choose the simplest theory: “Either science isirrational [in the way it judges theories and predictionsprobable] or the principle of simplicity is a fundamentalsynthetic a priori truth.” (Swinburne 1997).

2.5.4 Ludwig Wittgenstein

From the Tractatus Logico-Philosophicus:

• 3.328 If a sign is not necessary then it is meaning-less. That is the meaning of Occam’s Razor.

(If everything in the symbolism works asthough a sign had meaning, then it has mean-ing.)

• 4.04 In the proposition there must be exactly asmany things distinguishable as there are in the stateof affairs which it represents. They must both pos-sess the same logical (mathematical) multiplicity(cf. Hertz’s Mechanics, on Dynamic Models).

• 5.47321 Occam’s Razor is, of course, not an arbi-trary rule nor one justified by its practical success.It simply says that unnecessary elements in a sym-bolism mean nothing. Signs which serve one pur-pose are logically equivalent; signs which serve nopurpose are logically meaningless.

and on the related concept of “simplicity":

• 6.363 The procedure of induction consists in accept-ing as true the simplest law that can be reconciledwith our experiences.

3 Applications

3.1 Science and the scientific method

In science, Occam’s razor is used as a heuristic to guidescientists in developing theoretical models rather than as

an arbiter between published models.[8][9] In physics, par-simony was an important heuristic in Albert Einstein'sformulation of special relativity,[38][39] in the develop-ment and application of the principle of least action byPierre Louis Maupertuis and Leonhard Euler,[40] and inthe development of quantum mechanics by Max Planck,Werner Heisenberg and Louis de Broglie.[9][41]

In chemistry, Occam’s razor is often an important heuris-tic when developing a model of a reaction mecha-nism.[42][43] Although it is useful as a heuristic in devel-oping models of reaction mechanisms, it has been shownto fail as a criterion for selecting among some selectedpublished models.[9] In this context, Einstein himselfexpressed caution when he formulated Einstein’s Con-straint: “It can scarcely be denied that the supreme goalof all theory is to make the irreducible basic elements assimple and as few as possible without having to surrenderthe adequate representation of a single datum of experi-ence”. An often-quoted version of this constraint (whichcannot be verified as posited by Einstein himself)[44] says“Everything should be kept as simple as possible, but nosimpler.”In the scientific method, parsimony is an epistemological,metaphysical or heuristic preference, not an irrefutableprinciple of logic or a scientific result.[1][10][45] As a log-ical principle, Occam’s razor would demand that scien-tists accept the simplest possible theoretical explanationfor existing data. However, science has shown repeat-edly that future data often support more complex theo-ries than do existing data. Science prefers the simplestexplanation that is consistent with the data available ata given time, but the simplest explanation may be ruledout as new data become available.[8][10] That is, science isopen to the possibility that future experiments might sup-port more complex theories than demanded by currentdata and is more interested in designing experiments todiscriminate between competing theories than favoringone theory over another based merely on philosophicalprinciples.[1][10][11]

When scientists use the idea of parsimony, it has meaningonly in a very specific context of inquiry. Several back-ground assumptions are required for parsimony to con-nect with plausibility in a particular research problem.The reasonableness of parsimony in one research contextmay have nothing to do with its reasonableness in another.It is a mistake to think that there is a single global princi-ple that spans diverse subject matter.[11]

It has been suggested that Occam’s razor is a widelyaccepted example of extraevidential consideration, eventhough it is entirely a metaphysical assumption. There islittle empirical evidence that the world is actually simpleor that simple accounts are more likely to be true thancomplex ones.[46]

Most of the time, Occam’s razor is a conservative tool,cutting out crazy, complicated constructions and assuringthat hypotheses are grounded in the science of the day,

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6 3 APPLICATIONS

thus yielding “normal” science: models of explanationand prediction. There are, however, notable exceptionswhere Occam’s razor turns a conservative scientist into areluctant revolutionary. For example, Max Planck inter-polated between the Wien and Jeans radiation laws andused Occam’s razor logic to formulate the quantum hy-pothesis, even resisting that hypothesis as it became moreobvious that it was correct.[9]

Appeals to simplicity were used to argue against the phe-nomena of meteorites, ball lightning, continental drift,and reverse transcriptase. One can argue for atomicbuilding blocks for matter, because it provides a simplerexplanation for the observed reversibility of both mix-ing and chemical reactions as simple separation and rear-rangements of atomic building blocks. At the time, how-ever, the atomic theory was considered more complex be-cause it implied the existence of invisible particles thathad not been directly detected. Ernst Mach and the logi-cal positivists rejected John Dalton's atomic theory untilthe reality of atoms was more evident in Brownian mo-tion, as shown by Albert Einstein.[47]

In the same way, postulating the aether is more complexthan transmission of light through a vacuum. At the time,however, all known waves propagated through a physi-cal medium, and it seemed simpler to postulate the exis-tence of a medium than to theorize about wave propaga-tion without a medium. Likewise, Newton’s idea of lightparticles seemed simpler than Christiaan Huygens’s ideaof waves, so many favored it. In this case, as it turned out,neither the wave—nor the particle—explanation alonesuffices, as light behaves like waves and like particles.Three axioms presupposed by the scientific method arerealism (the existence of objective reality), the existenceof natural laws, and the constancy of natural law. Ratherthan depend on provability of these axioms, science de-pends on the fact that they have not been objectively fal-sified. Occam’s razor and parsimony support, but do notprove, these axioms of science. The general principle ofscience is that theories (or models) of natural law mustbe consistent with repeatable experimental observations.This ultimate arbiter (selection criterion) rests upon theaxioms mentioned above.[10]

There are examples where Occam’s razor would have fa-vored the wrong theory given the available data. Sim-plicity principles are useful philosophical preferences forchoosing a more likely theory from among several possi-bilities that are all consistent with available data. A sin-gle instance of Occam’s razor favoring a wrong theoryfalsifies the razor as a general principle.[10] Michael Leeand others[48] provide cases in which a parsimonious ap-proach does not guarantee a correct conclusion and, ifbased on incorrect working hypotheses or interpretationsof incomplete data, may even strongly support a falseconclusion. Lee states, “When parsimony ceases to bea guideline and is instead elevated to an ex cathedra pro-nouncement, parsimony analysis ceases to be science.”

If multiple models of natural law make exactly the sametestable predictions, they are equivalent and there is noneed for parsimony to choose a preferred one. For ex-ample, Newtonian, Hamiltonian and Lagrangian classi-cal mechanics are equivalent. Physicists have no interestin using Occam’s razor to say the other two are wrong.Likewise, there is no demand for simplicity principles toarbitrate between wave and matrix formulations of quan-tum mechanics. Science often does not demand arbitra-tion or selection criteria between models that make thesame testable predictions.[10]

3.2 Biology

Biologists or philosophers of biology use Occam’s razorin either of two contexts both in evolutionary biology: theunits of selection controversy and systematics. GeorgeC. Williams in his book Adaptation and Natural Selection(1966) argues that the best way to explain altruism amonganimals is based on low-level (i.e., individual) selectionas opposed to high-level group selection. Altruism is de-fined by some evolutionary biologists (e.g., R. Alexander,1987; W. D. Hamilton, 1964) as behavior that is benefi-cial to others (or to the group) at a cost to the individual,andmany posit individual selection as the mechanism thatexplains altruism solely in terms of the behaviors of in-dividual organisms acting in their own self-interest (or inthe interest of their genes, via kin selection). Williamswas arguing against the perspective of others who pro-pose selection at the level of the group as an evolutionarymechanism that selects for altruistic traits (e.g., D. S.Wil-son & E. O. Wilson, 2007). The basis for Williams’ con-tention is that of the two, individual selection is the moreparsimonious theory. In doing so he is invoking a variantof Occam’s razor known as Morgan’s Canon: “In no caseis an animal activity to be interpreted in terms of higherpsychological processes, if it can be fairly interpreted interms of processes which stand lower in the scale of psy-chological evolution and development.” (Morgan 1903).However, more recent biological analyses, such asRichard Dawkins' The Selfish Gene, have contended thatMorgan’s Canon is not the simplest and most basic expla-nation. Dawkins argues the way evolution works is thatthe genes propagated in most copies end up determiningthe development of that particular species, i.e., natural se-lection turns out to select specific genes, and this is reallythe fundamental underlying principle, that automaticallygives individual and group selection as emergent featuresof evolution.Zoology provides an example. Muskoxen, when threat-ened by wolves, form a circle with the males on the out-side and the females and young on the inside. This isan example of a behavior by the males that seems to bealtruistic. The behavior is disadvantageous to them indi-vidually but beneficial to the group as a whole and wasthus seen by some to support the group selection theory.

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3.3 Medicine 7

However, a much better explanation immediately offersitself once one considers that natural selection works ongenes. If the male musk ox runs off leaving his offspringto the wolves, his genes do not propagate. If, however,he fights, his genes may live on in his offspring. Thus,the “stay-and-fight” gene prevails. This is an example ofkin selection. An underlying general principle thus offersa much simpler explanation, without retreating to specialprinciples as group selection.Systematics is the branch of biology that attempts to es-tablish genealogical relationships among organisms. Itis also concerned with their classification. There arethree primary camps in systematics: cladists, pheneti-cists, and evolutionary taxonomists. The cladists hold thatgenealogy alone should determine classification and phe-neticists contend that similarity over propinquity of de-scent is the determining criterion while evolutionary tax-onomists say that both genealogy and similarity count inclassification.[49]

It is among the cladists that Occam’s razor is to be found,although their term for it is cladistic parsimony. Cladis-tic parsimony (or maximum parsimony) is a methodof phylogenetic inference in the construction of typesof phylogenetic trees (more specifically, cladograms).Cladograms are branching, tree-like structures used torepresent lines of descent based on one ormore evolution-ary changes. Cladistic parsimony is used to support thehypotheses that require the fewest evolutionary changes.For some types of tree, it consistently produces the wrongresults, regardless of how much data is collected (this iscalled long branch attraction). For a full treatment ofcladistic parsimony, see Elliott Sober's Reconstructing thePast: Parsimony, Evolution, and Inference (1988). For adiscussion of both uses of Occam’s razor in biology, seeSober’s article “Let’s Razor Ockham’s Razor” (1990).Other methods for inferring evolutionary relationshipsuse parsimony in a more traditional way. Likelihoodmethods for phylogeny use parsimony as they do for alllikelihood tests, with hypotheses requiring few differingparameters (i.e., numbers of different rates of characterchange or different frequencies of character state transi-tions) being treated as null hypotheses relative to hypothe-ses requiring many differing parameters. Thus, complexhypotheses must predict data much better than do sim-ple hypotheses before researchers reject the simple hy-potheses. Recent advances employ information theory, aclose cousin of likelihood, which uses Occam’s razor inthe same way.Francis Crick has commented on potential limitations ofOccam’s razor in biology. He advances the argument thatbecause biological systems are the products of (an ongo-ing) natural selection, the mechanisms are not necessarilyoptimal in an obvious sense. He cautions: “While Ock-ham’s razor is a useful tool in the physical sciences, it canbe a very dangerous implement in biology. It is thus veryrash to use simplicity and elegance as a guide in biological

research.”[50]

In biogeography, parsimony is used to infer ancientmigrations of species or populations by observing thegeographic distribution and relationships of existingorganisms. Given the phylogenetic tree, ancestral migra-tions are inferred to be those that require the minimumamount of total movement.

3.3 Medicine

When discussing Occam’s razor in contemporarymedicine, doctors and philosophers of medicine speak ofdiagnostic parsimony. Diagnostic parsimony advocatesthat when diagnosing a given injury, ailment, illness,or disease a doctor should strive to look for the fewestpossible causes that account for all the symptoms. Thisphilosophy is one of several demonstrated in the popularmedical adage “when you hear hoofbeats behind you,think horses, not zebras". While diagnostic parsimonymight often be beneficial, credence should also be givento the counter-argument modernly known as Hickam’sdictum, which succinctly states that, “Patients can haveas many diseases as they damn well please.” It is oftenstatistically more likely that a patient has several commondiseases rather than a single rarer disease that explainsmyriad symptoms. Also, independently of statisticallikelihood, some patients do in fact turn out to havemultiple diseases, which by common sense nullifies theapproach of insisting to explain any given collection ofsymptoms with one disease.These misgivings emerge from simple probabilitytheory—which is already taken into account in manymodern variations of the razor—and from the fact thatthe loss function is much greater in medicine than inmost of general science. Because misdiagnosis can resultin the loss of a person’s health and potentially life, itis considered better to test and pursue all reasonabletheories even if there is some theory that appears themost likely.Diagnostic parsimony and the counterbalance it finds inHickam’s dictum have very important implications inmedical practice. Any set of symptoms could be indica-tive of a range of possible diseases and disease combi-nations; though at no point is a diagnosis rejected or ac-cepted just on the basis of one disease appearing morelikely than another, the continuous flow of hypothesis for-mulation, testing and modification benefits greatly fromestimates regarding which diseases (or sets of diseases)are relatively more likely responsible for a set of symp-toms, given the patient’s environment, habits, medicalhistory, and so on. For example, if a hypothetical pa-tient’s immediately apparent symptoms include fatigueand cirrhosis and they test negative for hepatitis C, theirdoctor might formulate a working hypothesis that the cir-rhosis was caused by their drinking problem, and thenseek symptoms and perform tests to formulate and rule

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out hypotheses as to what has been causing the fatigue;but if the doctor were to further discover that the patient’sbreath inexplicably smells of garlic and they are sufferingfrom pulmonary edema, they might decide to test for therelatively rare condition of selenium poisoning.

3.4 Religion

Main article: Existence of God

In the philosophy of religion, Occam’s razor is sometimesapplied to the existence of God. William of Ockhamhimself was a Christian. He believed in God, and in theauthority of Scripture; he writes that “nothing ought tobe posited without a reason given, unless it is self-evident(literally, known through itself) or known by experienceor proved by the authority of Sacred Scripture.”[51] Ock-ham believed that an explanation has no sufficient basisin reality when it does not harmonize with reason, experi-ence, or the Bible. However, unlike many theologians ofhis time, Ockham did not believe God could be logicallyproven with arguments. To Ockham, science was a mat-ter of discovery, but theology was a matter of revelationand faith. He states: “only faith gives us access to theo-logical truths. The ways of God are not open to reason,for God has freely chosen to create a world and establisha way of salvation within it apart from any necessary lawsthat human logic or rationality can uncover.”[52]

St. Thomas Aquinas, in the Summa Theologica, uses aformulation of Occam’s razor to construct an objectionto the idea that God exists, which he refutes directly witha counterargument:[53]

Further, it is superfluous to suppose thatwhat can be accounted for by a few princi-ples has been produced by many. But it seemsthat everything we see in the world can be ac-counted for by other principles, supposing Goddid not exist. For all natural things can be re-duced to one principle which is nature; and allvoluntary things can be reduced to one princi-ple which is human reason, or will. Thereforethere is no need to suppose God’s existence.

In turn, Aquinas answers this with the quinque viae, andaddresses the particular objection above with the follow-ing answer:

Since nature works for a determinate endunder the direction of a higher agent, whateveris done by nature must needs be traced backto God, as to its first cause. So also whateveris done voluntarily must also be traced backto some higher cause other than human reasonor will, since these can change or fail; for allthings that are changeable and capable of de-fect must be traced back to an immovable and

self-necessary first principle, as was shown inthe body of the Article.

Rather than argue for the necessity of a god, some the-ists base their belief upon grounds independent of, orprior to, reason, making Occam’s razor irrelevant. Thiswas the stance of Søren Kierkegaard, who viewed be-lief in God as a leap of faith that sometimes directlyopposed reason.[54] This is also the doctrine of GordonClark's presuppositional apologetics, with the exceptionthat Clark never thought the leap of faith was contrary toreason (see also Fideism).Various arguments in favour of God establish Godas a useful or even necessary assumption. Contrast-ingly,some atheists hold firmly to the belief that assumingthe existence of God introduces unnecessary complexity(Schmitt 2005, e.g., the Ultimate Boeing 747 gambit).Taking a nuanced position, philosopher Del Ratzsch[55]suggests that the application of the razor to God may notbe so simple, least of all when we are comparing that hy-pothesis with theories postulating multiple invisible uni-verses.[56]

Another application of the principle is to be found in thework of George Berkeley (1685–1753). Berkeley was anidealist who believed that all of reality could be explainedin terms of the mind alone. He invoked Occam’s razoragainst materialism, stating that matter was not requiredby his metaphysic and was thus eliminable. One potentialproblem with this belief is that it’s possible, given Berke-ley’s position, to find solipsism itself more in line with therazor than a God-mediated world beyond a single thinker.In his article “Sensations and Brain Processes” (1959),J. J. C. Smart invoked Occam’s razor with the aim tojustify his preference of the mind-brain identity theoryover spirit-body dualism. Dualists state that there are twokinds of substances in the universe: physical (includingthe body) and spiritual, which is non-physical. In con-trast, identity theorists state that everything is physical,including consciousness, and that there is nothing non-physical. Though it is impossible to appreciate the spir-itual when limiting oneself to the physical, Smart main-tained that identity theory explains all phenomena by as-suming only a physical reality. Subsequently, Smart hasbeen severely criticized for his use (or misuse of Occam’srazor and ultimately retracted his advocacy of it in thiscontext. Paul Churchland (1984) states that by itself Oc-cam’s razor is inconclusive regarding duality. In a simi-lar way, Dale Jacquette (1994) stated that Occam’s razorhas been used in attempts to justify eliminativism and re-ductionism in the philosophy of mind. Eliminativism isthe thesis that the ontology of folk psychology includingsuch entities as “pain”, “joy”, “desire”, “fear”, etc., areeliminable in favor of an ontology of a completed neuro-science.

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3.5 Penal ethics

In penal theory and the philosophy of punishment, parsi-mony refers specifically to taking care in the distributionof punishment in order to avoid excessive punishment. Inthe utilitarian approach to the philosophy of punishment,Jeremy Bentham's “parsimony principle” states that anypunishment greater than is required to achieve its end isunjust. The concept is related but not identical to the legalconcept of proportionality. Parsimony is a key considera-tion of the modern restorative justice, and is a componentof utilitarian approaches to punishment, as well as theprison abolition movement. Bentham believed that trueparsimony would require punishment to be individualisedto take account of the sensibility of the individual—an in-dividual more sensitive to punishment should be given aproportionately lesser one, since otherwise needless painwould be inflicted. Later utilitarian writers have tended toabandon this idea, in large part due to the impracticalityof determining each alleged criminal’s relative sensitivityto specific punishments.[57]

3.6 Probability theory and statistics

Marcus Hutter’s universal artificial intelligence buildsupon Solomonoff’s mathematical formalization of the ra-zor to calculate the expected value of an action.There are various papers in scholarly journals derivingformal versions of Occam’s razor from probability theory,applying it in statistical inference, and using it to come upwith criteria for penalizing complexity in statistical infer-ence. Papers[58][59] have suggested a connection betweenOccam’s razor and Kolmogorov complexity.[60]

One of the problems with the original formulation of therazor is that it only applies to models with the same ex-planatory power (i.e., it only tells us to prefer the sim-plest of equally good models). A more general formof the razor can be derived from Bayesian model com-parison, which is based on Bayes factors and can beused to compare models that don't fit the data equallywell. These methods can sometimes optimally balancethe complexity and power of a model. Generally, the ex-act Occam factor is intractable, but approximations suchas Akaike information criterion, Bayesian informationcriterion, Variational Bayesian methods, false discoveryrate, and Laplace’s method are used. Many artificial intel-ligence researchers are now employing such techniques,for instance through work on Occam Learning.Statistical versions of Occam’s razor have a more rigor-ous formulation than what philosophical discussions pro-duce. In particular, they must have a specific definition ofthe term simplicity, and that definition can vary. For ex-ample, in the Kolmogorov–Chaitin minimum descriptionlength approach, the subject must pick a Turing machinewhose operations describe the basic operations believedto represent “simplicity” by the subject. However, one

could always choose a Turing machine with a simple op-eration that happened to construct one’s entire theory andwould hence score highly under the razor. This has led totwo opposing camps: one that believes Occam’s razor isobjective, and one that believes it is subjective.

3.6.1 Objective razor

The minimum instruction set of a universal Turing ma-chine requires approximately the same length descrip-tion across different formulations, and is small comparedto the Kolmogorov complexity of most practical theo-ries. Marcus Hutter has used this consistency to definea “natural” Turing machine of small size as the properbasis for excluding arbitrarily complex instruction sets inthe formulation of razors.[61] Describing the program forthe universal program as the “hypothesis”, and the rep-resentation of the evidence as program data, it has beenformally proven under Zermelo–Fraenkel set theory that“the sum of the log universal probability of the modelplus the log of the probability of the data given the modelshould be minimized.”[62] Interpreting this as minimisingthe total length of a two-part message encoding modelfollowed by data given model gives us the minimummes-sage length (MML) principle.[63][64]

One possible conclusion from mixing the concepts ofKolmogorov complexity and Occam’s razor is that anideal data compressor would also be a scientific expla-nation/formulation generator. Some attempts have beenmade to re-derive known laws from considerations ofsimplicity or compressibility.[65][66]

According to Jürgen Schmidhuber, the appropriate math-ematical theory of Occam’s razor already exists, namely,Solomonoff’s theory of optimal inductive inference[67]and its extensions.[68] See discussions in David L. Dowe’s“Foreword re C. S. Wallace”[69] for the subtle distinctionsbetween the algorithmic probability work of Solomonoffand the MML work of Chris Wallace, and see Dowe’s“MML, hybrid Bayesian network graphical models, sta-tistical consistency, invariance and uniqueness”[70] bothfor such discussions and for (in section 4) discussionsof MML and Occam’s razor. For a specific exam-ple of MML as Occam’s razor in the problem of deci-sion tree induction, see Dowe and Needham’s “MessageLength as an Effective Ockham’s Razor in Decision TreeInduction”.[71]

4 Controversial aspects of the ra-zor

Occam’s razor is not an embargo against the positing ofany kind of entity, or a recommendation of the simplesttheory come what may.[lower-alpha 3] Occam’s razor is usedto adjudicate between theories that have already passed“theoretical scrutiny” tests and are equally well-supported

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by evidence.[lower-alpha 4] Furthermore, it may be used toprioritize empirical testing between two equally plausi-ble but unequally testable hypotheses; thereby minimiz-ing costs and wastes while increasing chances of falsifica-tion of the simpler-to-test hypothesis.Another contentious aspect of the razor is that a theorycan become more complex in terms of its structure (orsyntax), while its ontology (or semantics) becomes sim-pler, or vice versa.[lower-alpha 5] Quine, in a discussion ondefinition, referred to these two perspectives as “econ-omy of practical expression” and “economy in grammarand vocabulary”, respectively.[73] The theory of relativ-ity is often given as an example of the proliferation ofcomplex words to describe a simple concept.Galileo Galilei lampooned themisuse of Occam’s razor inhis Dialogue. The principle is represented in the dialogueby Simplicio. The telling point that Galileo presentedironically was that if one really wanted to start from asmall number of entities, one could always consider theletters of the alphabet as the fundamental entities, sinceone could construct the whole of human knowledge outof them.

5 Anti-razors

Occam’s razor has met some opposition from people whohave considered it too extreme or rash. Walter Chatton(c. 1290–1343) was a contemporary of William of Ock-ham (c. 1287–1347) who took exception to Occam’s ra-zor and Ockham’s use of it. In response he devised hisown anti-razor: “If three things are not enough to verifyan affirmative proposition about things, a fourth must beadded, and so on.” Although there have been a numberof philosophers who have formulated similar anti-razorssince Chatton’s time, no one anti-razor has perpetuated inas much notability as Chatton’s anti-razor, although thiscould be the case of the Late Renaissance Italian motto ofunknown attribution Se non è vero, è ben trovato (“Evenif it is not true, it is well conceived”) when referred to aparticularly artful explanation. For further information,see “Ockham’s Razor and Chatton’s Anti-Razor” (1984)by Armand Maurer.Anti-razors have also been created by Gottfried WilhelmLeibniz (1646–1716), Immanuel Kant (1724–1804), andKarl Menger (1902–1985). Leibniz’s version took theform of a principle of plenitude, as Arthur Lovejoy hascalled it: the idea being that God created the most var-ied and populous of possible worlds. Kant felt a need tomoderate the effects of Occam’s razor and thus createdhis own counter-razor: “The variety of beings should notrashly be diminished.”[74]

Karl Menger found mathematicians to be too parsimo-nious with regard to variables, so he formulated his LawAgainst Miserliness, which took one of two forms: “En-tities must not be reduced to the point of inadequacy”

and “It is vain to do with fewer what requires more.” Aless serious but (some might say) even more extremistanti-razor is 'Pataphysics, the “science of imaginary solu-tions” developed by Alfred Jarry (1873–1907). Perhapsthe ultimate in anti-reductionism, "'Pataphysics seeks noless than to view each event in the universe as completelyunique, subject to no laws but its own.” Variations onthis theme were subsequently explored by the Argentinewriter Jorge Luis Borges in his story/mock-essay "Tlön,Uqbar, Orbis Tertius". There is also Crabtree’s Bludgeon,which cynically states that "[n]o set of mutually incon-sistent observations can exist for which some human in-tellect cannot conceive a coherent explanation, howevercomplicated.”

6 See also• Algorithmic information theory

• Chekhov’s gun

• Common sense

• Cladistics

• Eliminative materialism

• Falsifiability

• Framing (social sciences)

• Greedy reductionism

• Hanlon’s razor

• Hitchens’s razor

• Inductive probability

• KISS principle

• Metaphysical naturalism

• Minimum description length

• Minimum message length

• Newton’s flaming laser sword

• Philosophy of science

• Principle of least astonishment

• Pseudoscience

• Rationalism

• Razor (philosophy)

• Regress argument

• Scientific method

• Scientific reductionism

• Scientific skepticism

• Simplicity

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7 Notes[1] “The aim of appeals to simplicity in such contexts seem to

be more about shifting the burden of proof, and less aboutrefuting the less simple theory outright.”[1]

[2] “In analyzing simplicity, it can be difficult to keep its twofacets – elegance and parsimony – apart. Principles suchas Occam’s razor are frequently stated in a way which isambiguous between the two notions ... While these twofacets of simplicity are frequently conflated, it is importantto treat them as distinct. One reason for doing so is thatconsiderations of parsimony and of elegance typically pullin different directions.”[1]

[3] “Ockham’s razor does not say that the more simple a hy-pothesis, the better.”[72]

[4] “Today, we think of the principle of parsimony as aheuristic device. We don't assume that the simpler the-ory is correct and the more complex one false. We knowfrom experience that more often than not the theory thatrequires more complicated machinations is wrong. Un-til proved otherwise, the more complex theory compet-ing with a simpler explanation should be put on the backburner, but not thrown onto the trash heap of history untilproven false.”[72]

[5] “While these two facets of simplicity are frequently con-flated, it is important to treat them as distinct. One reasonfor doing so is that considerations of parsimony and of el-egance typically pull in different directions. Postulatingextra entities may allow a theory to be formulated moresimply, while reducing the ontology of a theory may onlybe possible at the price of making it syntactically morecomplex.”[1]

8 References[1] Alan Baker (2010) [2004]. “Simplicity”. Stanford Ency-

clopedia of Philosophy. California: Stanford University.ISSN 1095-5054.

[2] Induction: From Kolmogorov and Solomonoff to DeFinetti and Back to Kolmogorov JJ McCall - Metroeco-nomica, 2004 - Wiley Online Library.

[3] Foundations of Occam’s Razor and parsimony in learningfrom ricoh.comD Stork - NIPS 2001 Workshop, 2001.

[4] A.N. Soklakov (2002). “Occam’s Razor as a formal ba-sis for a physical theory”. Foundations of Physics Letters(Springer).

[5] J. HERNANDEZ-ORALLO (2000). “Beyond the TuringTest” (PDF). Journal of Logic, Language, and ...

[6] M. Hutter (2003). “On the existence and convergence ofcomputable universal priors”. Springer.

[7] Samuel Rathmanner; Marcus Hutter (2011). “A philo-sophical treatise of universal induction”. Entropy 13 (6):1076–1136. doi:10.3390/e13061076.

[8] Hugh G. Gauch, Scientific Method in Practice, CambridgeUniversity Press, 2003, ISBN 0-521-01708-4, ISBN 978-0-521-01708-4.

[9] Roald Hoffmann, Vladimir I. Minkin, Barry K. Car-penter, Ockham’s Razor and Chemistry, HYLE—International Journal for Philosophy of Chemistry, Vol.3, pp. 3–28, (1997).

[10] Courtney A, Courtney M (2008). “Comments Regarding“On the Nature Of Science"" (PDF). Physics in Canada64 (3): 7–8. Retrieved 1 August 2012.

[11] Elliott Sober, Let’s Razor Occam’s Razor, pp. 73–93,from Dudley Knowles (ed.) Explanation and Its Limits,Cambridge University Press (1994).

[12] Vogel Carey, Toni (Oct 2010). Lewis, Rick, ed.“Parsimony (In as few words as possible)". PhilosophyNow (UK) (81). Retrieved 27 October 2012.

[13] Johannes Poncius’s commentary on John Duns Scotus’sOpus Oxoniense, book III, dist. 34, q. 1. in John DunsScotusOpera Omnia, vol.15, Ed. LukeWadding, Louvain(1639), reprinted Paris: Vives, (1894) p.483a

[14] Aristotle, Physics 189a15, On the Heavens 271a33. Seealso Franklin, op cit. note 44 to chap. 9.

[15] Charlesworth, M. J. (1956). “Aristotle’s Razor”. Philo-sophical Studies (Ireland)

[16] Wikipedians, Complexity and Dynamics citing RichardMcKeon (tr.) Aristotle’s Posterior Analytics (1963) p.150

[17] James Franklin (2001). The Science of Conjecture: Evi-dence and Probability before Pascal. The Johns HopkinsUniversity Press. Chap 9. p. 241.

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[68] J. Schmidhuber (2006) “The New AI: General & Sound& Relevant for Physics.” In B. Goertzel and C. Pennachin,eds.: Artificial General Intelligence, pp. 177–200 http://arxiv.org/abs/cs.AI/0302012

[69] David L. Dowe (2008): Foreword re C. S. Wallace; Com-puter Journal, Volume 51, Issue 5, Sept 2008 Pages:523–560.

[70] David L. Dowe (2010): “MML, hybrid Bayesian net-work graphical models, statistical consistency, invari-ance and uniqueness. A formal theory of inductiveinference.” Handbook of the Philosophy of Science –(HPS Volume 7) Philosophy of Statistics, Elsevier 2010Page(s):901–982. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.185.709&rep=rep1&type=pdf

[71] Scott Needham and David L. Dowe (2001):" MessageLength as an Effective Ockham’s Razor in DecisionTree Induction.” Proc. 8th International Workshop onArtificial Intelligence and Statistics (AI+STATS 2001),Key West, Florida, U.S.A., Jan. 2001 Page(s): 253–260http://www.csse.monash.edu.au/~{}dld/Publications/2001/Needham+Dowe2001_Ockham.pdf

[72] Robert T. Carroll. “Occam’s Razor”. The Skeptic’s Dictio-nary Last updated 18 February 2012

[73] Quine, W V O (1961). “Two dogmas of empiricism”.From a logical point of view. Cambridge: Harvard Uni-versity Press. pp. 20–46. ISBN 0-674-32351-3.

[74] Immanuel Kant (1929). Norman Kemp-Smith transl, ed.The Critique of Pure Reason. Palgrave Macmillan. p. 92.Retrieved 27October 2012. Entium varietates non temereesse minuendas

9 Further reading• Ariew, Roger (1976). Ockham’s Razor: A Historicaland Philosophical Analysis of Ockham’s Principle ofParsimony. Champaign-Urbana, University of Illi-nois.

• Charlesworth, M. J. (1956). “Aristotle’s Ra-zor”. Philosophical Studies (Ireland) 6: 105–112.doi:10.5840/philstudies1956606.

• Churchland, Paul M. (1984). Matter and Conscious-ness. Cambridge, Massachusetts: MIT Press. ISBN0-262-53050-3. ISBN.

• Crick, Francis H. C. (1988). What Mad Pursuit: APersonal View of Scientific Discovery. New York,New York: Basic Books. ISBN 0-465-09137-7.ISBN.

• Dowe, David L.; Steve Gardner; Graham Oppy(December 2007). “Bayes not Bust! Why Sim-plicity is no Problem for Bayesians”. British J.for the Philosophy of Science 58 (4): 709–754.doi:10.1093/bjps/axm033. Retrieved 2007-09-24.

• Duda, Richard O.; Peter E. Hart; David G. Stork(2000). Pattern Classification (2nd ed.). Wiley-Interscience. pp. 487–489. ISBN 0-471-05669-3.ISBN.

• Epstein, Robert (1984). “The Principle of Parsi-mony and Some Applications in Psychology”. Jour-nal of Mind Behavior 5: 119–130.

• Hoffmann, Roald; Vladimir I. Minkin; Barry K.Carpenter (1997). “Ockham’s Razor and Chem-istry”. HYLE—International Journal for the Philos-ophy of Chemistry 3: 3–28. Retrieved 2006-04-14.

• Jacquette, Dale (1994). Philosophy of Mind. En-gleswoods Cliffs, New Jersey: Prentice Hall. pp.34–36. ISBN 0-13-030933-8. ISBN.

• Jaynes, Edwin Thompson (1994). “Model Com-parison and Robustness”. Probability Theory: TheLogic of Science. ISBN 0-521-59271-2.

• Jefferys, William H.; Berger, James O. (1991).“Ockham’s Razor and Bayesian Statistics (Preprintavailable as “Sharpening Occam’s Razor on aBayesian Strop)",” (PDF). American Scientist 80:64–72.

• Katz, Jerrold (1998). Realistic Rationalism. MITPress. ISBN 0-262-11229-9.

• Kneale, William; Martha Kneale (1962). The De-velopment of Logic. London: Oxford UniversityPress. p. 243. ISBN 0-19-824183-6. ISBN.

• MacKay, David J. C. (2003). Information The-ory, Inference and Learning Algorithms. CambridgeUniversity Press. ISBN 0-521-64298-1. ISBN.

• Maurer, A. (1984). “Ockham’s Razor and Chatton’sAnti-Razor”. Medieval Studies 46: 463–475.

• McDonald, William (2005). “Søren Kierkegaard”.Stanford Encyclopedia of Philosophy. Retrieved2006-04-14.

• Menger, Karl (1960). “A Counterpart of Ock-ham’s Razor in Pure and Applied Mathemat-ics: Ontological Uses”. Synthese 12 (4): 415.doi:10.1007/BF00485426.

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14 10 EXTERNAL LINKS

• Morgan, C. Lloyd (1903). “Other Minds thanOurs”. An Introduction to Comparative Psychology(2nd ed.). London: W. Scott. p. 59. ISBN 0-89093-171-2. Retrieved 2006-04-15.

• Newton, Isaac (2011) [1726]. PhilosophiæNaturalisPrincipia Mathematica (3rd ed.). London: HenryPemberton. ISBN 978-1-60386-435-0.

• Nolan, D. (1997). “Quantitative Parsimony”.British Journal for the Philosophy of Science 48 (3):329–343. doi:10.1093/bjps/48.3.329.

• Pegis, A. C., translator (1945). Basic Writings of St.Thomas Aquinas. New York: Random House. p.129. ISBN 0-87220-380-8.

• Popper, Karl (1992). “7. Simplicity”. The Logic ofScientific Discovery (2nd ed.). London: Routledge.pp. 121–132. ISBN 84-309-0711-4.

• Rodríguez-Fernández, J. L. (1999). “Ock-ham’s Razor”. Endeavour 23 (3): 121–125.doi:10.1016/S0160-9327(99)01199-0.

• Schmitt, Gavin C. (2005). “Ockham’s Razor Sug-gests Atheism”. Archived from the original on2007-02-11. Retrieved 2006-04-15.

• Smart, J. J. C. (1959). “Sensations and BrainProcesses”. Philosophical Review (The Philosoph-ical Review, Vol. 68, No. 2) 68 (2): 141–156.doi:10.2307/2182164. JSTOR 2182164.

• Sober, Elliott (1975). Simplicity. Oxford: OxfordUniversity Press.

• Sober, Elliott (1981). “The Principle of Parsimony”(PDF). British Journal for the Philosophy of Science32 (2): 145–156. doi:10.1093/bjps/32.2.145. Re-trieved 4 August 2012.

• Sober, Elliott (1990). “Let’s Razor Ockham’s Ra-zor”. In Dudley Knowles. Explanation and its Lim-its. Cambridge: Cambridge University Press. pp.73–94. ISBN.

• Sober, Elliott (2002). Zellner et al., eds. “What isthe Problem of Simplicity?" (PDF). Retrieved 4 Au-gust 2012.

• Swinburne, Richard (1997). Simplicity as Evidencefor Truth. Milwaukee, Wisconsin: Marquette Uni-versity Press. ISBN 0-87462-164-X.

• Thorburn, W. M. (1918). “The Myth of Oc-cam’s Razor”. Mind 27 (107): 345–353.doi:10.1093/mind/XXVII.3.345.

• Williams, George C. (1966). Adaptation and nat-ural selection: A Critique of some Current Evolu-tionary Thought. Princeton, New Jersey: PrincetonUniversity Press. ISBN 0-691-02615-7. ISBN.

10 External links• What is Occam’s Razor? This essay distinguishesOccam’s Razor (used for theories with identical pre-dictions) from the Principle of Parsimony (whichcan be applied to theories with different predic-tions).

• Skeptic’s Dictionary: Occam’s Razor

• Ockham’s Razor, an essay at The Galilean Libraryon the historical and philosophical implications byPaul Newall.

• The Razor in the Toolbox: The history, use, andabuse of Occam’s razor, by Robert Novella

• NIPS 2001Workshop “Foundations of Occam’s Ra-zor and parsimony in learning”

• Simplicity at Stanford Encyclopedia of Philosophy

• Occam’s Razor at PlanetMath.org.

• Disproof of parsimony as a general principle in sci-ence

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15

11 Text and image sources, contributors, and licenses

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