Occam's Razor

Occam's Razor (also Ockham's Razor or any of several other spellings), is a principle attributed to the 14th century English logician and Franciscan friar, William of Ockham that forms the basis of methodological reductionism, also called the principle of parsimony or law of economy.

In its simplest form, Occam's Razor states that one should make no more assumptions than needed. When multiple explanations are available for a phenomenon, the simplest version is preferred. For example, a charred tree on the ground could be caused by a landing alien ship or a lightning strike. According to Occam's Razor, the lightning strike is the preferred explanation as it requires the fewest assumptions.

Contents

1 Variations 2 History 3 Justifications 4 Chatton's Anti-razor 5 In science 6 In biology 7 In medicine 8 In philosophy of mind 9 In religion 10 In statistics 11 See also 12 References 13 External links

Variations

The principle is most often expressed as Entia non sunt multiplicanda praeter necessitatem, or "Entities should not be multiplied beyond necessity", but this sentence was written by later authors and is not found in Ockham's surviving writings. William wrote, in Latin, Pluralitas non est ponenda sine necessitate, which translates literally into English as "Plurality should not be posited without necessity".

This can be interpreted in two subtly different ways. One is a preference for the simplest theory that adequately accounts for the data. Another is a preference for the simplest subset of any given theory which accounts for the data. The difference is simply that it is possible for two different theories to explain the data equally well, but have no relation to one another. They share none of the same elements. Some would argue that in this case Occam's Razor does not suggest a preference. Rather Occam's Razor only comes into practice when a sufficient theory has something added to it which does not improve its predictive power. Occam's Razor neatly cuts these additional theoretical elements away.

The principle of Occam's Razor has inspired numerous expressions including: "parsimony of postulates", the "principle of simplicity", the "KISS principle" (keep it simple, stupid), and in some medical schools "When you hear hoofbeats, think horses, not zebras".

A re-statement of Occam's Razor, in more formal terms, is provided by information theory in the form of minimum message length.

"When deciding between two models which make equivalent predictions, choose the simpler one," makes the point that a simpler model that doesn't make equivalent predictions is not among the models that this criterion applies to in the first place. [1] (http://www.cs.mcgill.ca/~jacob/occams-razor/interpretations.html)

Another variant of this law is Thargola's Sword from Nightfall, (originally a short story by Isaac Asimov and later expanded to a novel in conjunction with Robert Silverberg):

We must drive a sword through any hypothesis that is not strictly necessary.

Leonardo da Vinci (1452–1519) lived after Ockham's time and has a variant of Occam's razor. His variant short-circuits the need for sophistication by equating it to simplicity.

Simplicity is the ultimate sophistication.

Occam's Razor is now usually stated as follows:

Of two equivalent theories or explanations, all other things being equal, the simpler one is to be preferred.

As this is ambiguous, Isaac Newton's version may be better:

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

In the spirit of Occam's Razor itself, the rule is sometimes stated as:

The simplest explanation is usually the best.

History

William of Ockham (ca. 1285–1349) is usually credited with formulating the razor that bears his name, which is typically phrased "entities are not to be multiplied beyond necessity." In Latin, "entia non sunt multiplicanda preaeter necessitatem". However this phrase does not appear in any of his extant writings. It is not until 1639 that this phrasing was coined by John Ponce of Cork. There are a variety of similar phrases such as "frustra fit per plura quod potest fieri per pauciora", "non est ponenda pluritas sine necessitate", and "si duae res sufficient ad ejus veritatem, superfluum est ponere aliam (tertiam) rem". These translate as "in vain we do by many which can be done by means of fewer", "pluralities ought not be supposed without necessity", and "if two things are sufficient for the purpose of truth, it is superfluous to suppose another", respectively. The origins of what has come to be known as Occam's razor are traceable to the works of earlier philosophers such as John Duns Scotus (1265–1308) and even as early as Aristotle (384–322 BC) (Charlesworth, 1956). Even the name 'Occam's Razor' was unknown to William. This phrase does not appear until the 19th century in the works of Sir William Hamilton (1805–1865). It is perhaps how often and effectively he used it that accounts for its association with Ockham. See Roger Ariew's dissertation of 1976, Ockham's Razor: A Historical and Philosophical Analysis of Ockham's Principle of Parsimony and W. M. Thornburn's The Myth of Occam's Razor.

Justifications

Occam's Razor is known by several different names including the Principle of Parsimony, the Principle of Simplicity, and the Principle of Economy. The reason for these alternate names can be explained by the association of simplicity and parsimony with Occam's Razor. Prior to the 20th century it was believed that the metaphysical justification for Occam's Razor was simplicity. It was thought that nature was in some sense simple and that our theories about nature should reflect that simplicity. With such a metaphysical justification came the implication that Occam's Razor is a metaphysical principle. From the beginning of the 20th century, these views fell out of favor as scientists presented an increasingly complex world view. In response, philosophers turned away from metaphysical justifications for Occam's Razor to epistemological ones including inductive, pragmatic, likelihood and probabilistic justifications, which is where things stand today. Thus, Occam's Razor is currently conceived of as a methodological principle. Elliott Sober has expressed dissatisfaction with epistemological justifications for Occam's Razor. He thinks that there must be a metaphysical presupposition for Occam's Razor, but offers no possibilities (Sober, 1990). For a summary of epistemological justifications for Occam's Razor see Roger Ariew's dissertation of 1976 "Ockham's Razor: A Historical and Philosophical Analysis of Ockham's Principle of Parsimony".

Chatton's Anti-razor

Walter of Chatton was a contemporary of William of Ockham (1287-1347) who took exception to Occam's Razor and Ockham's use of it. In response he devised his own anti-razor: "If three things are not enough to verify an affirmative proposition about things, a fourth must be added, and so on". Although there have been a number of philosophers who have formulated similar anti-razors since Chatton's time, Chatton's anti-razor has not known anything like the success of Occam's Razor. Among those who have coined their own anti-razors are Gottfried Wilhelm Leibniz (1646-1716), Immanuel Kant (1724-1804), and Karl Menger (20th century). Leibniz's version took the form of a principle of plenitude as Arthur Lovejoy has called it. The idea behind the principle was that God created the world with the most possible creatures. Kant felt a need to moderate the effects of Occam's Razor and thus created his own counter razor: "The variety of beings should not rashly be diminished." Karl Menger found mathematicians to be too parsimonious with regard to variables so he formulated his Law Against Miserliness which took one of two forms: "Entities must not be reduced to the point of inadequacy" and "It is vain to do with fewer what requires more". See Ockham's Razor and Chatton's Anti-Razor (1984) by Armand Maurer. A less serious, but (some might say) even more extremist anti-razor is Pataphysics, the "science of imaginary solutions" invented by Alfred Jarry (1873-1907). Perhaps the ultimate in anti-reductionism, Pataphysics seeks no less than to view each event in the universe as completely unique, subject to no laws but its own. Variations on this theme were subsequently explored by the Argentinian writer Jorge Luis Borges in his story/mock-essay Tlön, Uqbar, Orbis Tertius.

In science

Occam's Razor has become a basic perspective for those who follow the scientific method. It is important to note that it is a heuristic argument that does not necessarily give correct answers; it is a loose guide to choosing the scientific hypothesis which (currently) contains the least number of unproven assumptions. Often, several hypotheses are equally "simple" and Occam's Razor does not express any preference in such cases.

At the same time, without the principle of Occam's Razor science does not exist. The primary activity of science, formulating theories and selecting the most promising theory based on analysis of collected evidence, is not possible without some method of selecting between theories which do fit the evidence. This is because, for every set of data, there are an infinite number of theories which are consistent with those data (this is known as the Underdetermination Problem). As an example, perhaps you are investigating Newton's famous theory that every action has an equal and opposite reaction. It's easy to think of alternative theories which fit the data equally well. One such theory would be that for every action there is an opposite action of half intensity, but benevolent indetectable creatures magnify the opposing action with input of their own energy so it appears to be equal. These creatures will all die in the year 2055, and at that point the observable nature of the universe will instantly shift. This is an alternative theory which fits currently observable evidence just as well as Newton's theory. Furthermore we are currently unable to collect any evidence that one theory is superior to the other. Because the second theory states these creatures are undetectable, we cannot have any evidence to distinguish between the two theories until 2055. Furthermore, each theory has profoundly different implications for what we should expect of the future (for example, we may choose to live our lives differently if we know that life as we know it will cease in the year 2055). And finally, it can easily be seen that there are an infinite number of competing theories by uncreatively incrementing the year. 2056 is another theory. 2057 is another theory, and so on. Because there are an infinite number of theories which fit any body of evidence equally well, and all make radically different predictions, if science cannot choose between them, then science can never determine any useful theories. So far the only known way to usefully choose between the infinite number of theories which fit a body of evidence is Occam's Razor. For this reason Occam's Razor is seen as an indispensable aspect of science, without which science ceases to function entirely.

For example, after a storm you notice that a tree has fallen. Based on the evidence of "a storm" and "a fallen tree" a reasonable hypothesis would be that "the storm blew down the tree" – a hypothesis that requires only one assumption – that it was, in fact, a strong wind that knocked over the tree, rather than a meteor or an elephant. The hypothesis that "the tree was knocked over by marauding 200 meter tall space aliens" requires several additional assumptions (concerning the very existence of aliens, their ability and desire to travel interstellar distances and the alien biology that allows them to be 200 meters tall in terrestrial gravity) and is therefore less preferable.

Occam's Razor is not equivalent to the idea that "perfection is simplicity". Albert Einstein probably had this in mind when he wrote in 1933 that "The supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience" often paraphrased as "Theories should be as simple as possible, but no simpler." It often happens that the best explanation is much more complicated than the simplest possible explanation because it requires fewer assumptions. Some people have oversimplified Occam's Razor as "The simplest explanation is the best (or true) one".

There are two senses in which Occam's Razor can be seen at work in the history of science. One is ontological reduction by elimination and the other is by intertheoretic competition. In the former case the following are examples of reduction by elimination: The impetus of Aristotelian Physics, the angelic motors of medieval celestial mechanics, the four humors of ancient and medieval medicine, demonic possession as an explanation of mental illness, Phlogiston from premodern chemistry, and vital spirits of premodern Biology.

In the latter case there are three examples from the history of science where the simpler of two competing theories each of which explains all the observed phenomena has been chosen over its ontologically bloated competitor: the Copernican heliocentric model of celestial mechanics over the Ptolemaic geocentric model, the mechanical theory of heat over the Caloric theory, and the Einsteinian theory of electromagnetism over the luminiferous aether theory. In the first example, the Copernican model is said to have been chosen over the Ptolemaic due to its greater simplicity. The Ptolemaic model, in order to explain the apparent retrograde motion of Mercury relative to Venus, posited the existence of epicycles within the orbit of Mercury. The Copernican model (as expanded by Kepler) was able to account for this motion by displacing the Earth from the center of the solar system and replacing it with the sun as the orbital focus of planetary motions while simultaneously replacing the circular orbits of the Ptolemaic model with elliptical ones. In addition the Copernican model excluded any mention of the crystaline spheres that the planets were thought to be embedded in according the Ptolemaic model. In a single stroke the Copernican model reduced by a factor of two the ontology of Astronomy. According to the Caloric theory of heat, heat is a weightless substance that can travel from one object to another. This theory arose from the study of cannon boring and the invention of the steam engine. It was while studying cannon boring that Count Rumford made observations that conflicted with the Caloric theory and he formulated his mechanical theory to replace it. The Mechanical theory eliminated the Caloric and was thus ontologically simpler than its predecessor. During the 19th century Physicists believed that light required a medium of transmission much as sound waves do. It was hypothesized that a universal aether was such a medium and much effort was expended to detect it. In one of the most famous negative experiments in the history of science, the Michelson-Morley experiment failed to find any evidence of its existence. Einstein capitalized on this finding and constructed his theory without any reference to the Aether, thus providing another example of a theory chosen in part for its greater ontological simplicity.

In biology

Biologists or philosophers of biology use Occam's Razor in either of two contexts both in evolutionary biology: the units of selection controversy and Systematics. George C. Williams in his book Adaptation and Natural Selection (1966) argues that the best way to explain altruism among animals is based on low level (i.e. individual) selection as opposed to high level group selection. Altruism is defined as behavior that is beneficial to the group but not to the individual, and group selection is thought by some to be the evolutionary mechanism that selects for altruistic traits. Others posit individual selection as the mechanism which explains altruism solely in terms of the behaviors of individual organisms acting in their own self interest without regard to the group. The basis for Williams' contention is that of the two, individual selection is the more parsimonious theory. In doing so he is invoking a variant of Occam's Razor known as Lloyd Morgan's Canon.

However, more recent work by biologists, such as Richard Dawkins' The Selfish Gene, have revealed that Williams' view is not the simplest and most basic. The way evolution works is that the genes that are propagated in most copies will end up determining the development of that particular species, i.e., natural selection acts to select specific genes, and this is really the fundamental underlying principle, that automatically gives individual and group selection as emergent features of evolution.

Zoology provides an example. Musk oxen, when threatened by wolves, will form a circle with the males on the outside and the females and young on the inside. This as an example of a behavior by the males that seems to be altruistic. The behavior is disadvantageous to them individually but beneficial to the group as a whole and was thus seen by some to support the group selection theory.

However, a much better explanation immediately offers itself, once you realise that natural selection works on genes. If the male musk ox runs off, leaving his offspring to the wolves, his genes will not be propagated. If however he takes up the fight his genes will live on in his offspring. And thus the "stay-and-fight" gene prevails. This an example of kin selection. An underlying general principle thus offers a much simpler explanation, without retreating to special principles as group selection.

Systematics is the branch of biology that attempts to establish genealogical relationships among organisms. It is also concerned with their classification. There are three primary camps in systematics; cladists, pheneticists, and evolutionary taxonomists. The cladists hold that genealogy alone should determine classification and pheneticists contend that similarity over propinquity of descent is the determining criterion while evolutionary taxonomists claim that both genealogy and similarity count in classification.

It is among the cladists that Occam's razor is to be found, although their term for it is cladistic parsimony. Cladistic parsimony (or maximum parsimony) is a method of phylogenetic inference in the construction of cladograms. Cladograms are branching tree like structures used to represent lines of descent based on one or more evolutionary change(s). Cladistic parsimony is used to support the hypothesis(es) that require the fewest evolutionary changes. It should be noted that for some types of tree, it will consistently produce the wrong results regardless of how much data is collected (This is called long-branch attraction). For a full treatment of cladistic parsimony see Elliott Sober's Reconstructing the Past: Parsimony, Evolution, and Inference (1988). For a discussion of both uses of Occam's Razor in Biology see Elliott Sober's article Let's Razor Occam's Razor (1990).

In medicine

When discussing Occam's Razor in contemporary medicine, doctors and philosophers of medicine speak of diagnostic parsimony. Diagnostic parsimony advocates that when diagnosing a given injury, ailment, illness, or disease a doctor should strive to look for the fewest possible causes that will account for all the symptoms.

In philosophy of mind

Probably the first person to make use of the principle was Ockham himself. He writes "The source of many errors in philosophy is the claim that a distinct signified thing always corresponds to a distinct word in such a way that there are as many distinct entities being signified as there are distinct names or words doing the signifying. (Summula Philosophiae Naturalis III, chap. 7, see also Summa Totus Logicae Bk I, C.51). We are apt to suppose that a word like "paternity" signifies some "distinct entity", because we suppose that each distinct word signifies a distinct entity. This leads to all sorts of absurdities, such as "a column is to the right by to-the-rightness", "God is creating by creation, is good by goodness, is just by justice, is powerful by power", "an accident inheres by inherence", "a subject is subjected by subjection", "a suitable thing is suitable by suitability", "a chimera is nothing by nothingness", "a blind thing is blind by blindness", " a body is mobile by mobility". We should say instead that a man is a father because he has a son (Summa C.51).

Another application of the principle is to be found in the work of George Berkeley(1685-1753). Berkeley was an idealist believing that all of reality could be explained in terms of the mind alone. He famously invoked Occam's Razor against Idealism's metaphysical competitor materialism claiming that matter was not required by his metaphysic and was thus eliminable. Idealism has few adherents today and Berkeley's arguments find few sympathetic ears.

In the 20th century Philosophy of Mind, Occam's Razor found a champion in J.J.C. Smart who in his article "Sensations and Brain Processes"(1959) claimed Occam's Razor as the basis for his preference of the mind-brain identity theory over mind body dualism. Dualists claim that there are two kinds of substances in the universe: physical (including the body) and mental, which is nonphysical. In contrast identity theorists claim that everything is physical, including consciousness, and that there is nothing nonphysical. The basis for the materialist claim is that of the two competing theories, dualism and mind-brain identity, the identity theory is the simpler since it commits to fewer entities. Smart was criticized for his use of the razor and ultimately retracted his advocacy of it in this context.

Paul Churchland (1984) cites Occam's Razor as the first line of attack against dualism, but admits that by itself it is inconclusive. The deciding factor for Churchland is the greater explanatory prowess of a materialist position in the Philosophy of Mind as informed by findings in neurobiology.

Dale Jacquette (1994) claims that Occam's Razor is the rationale behind eliminativism and reductionism in the philosophy of mind. Eliminativism is the thesis that the ontology of folk psychology including such entities as "pain", "joy", "desire", "fear", etc., are eliminable in favor of an ontology of a completed neuroscience.

In religion

In the philosophy of religion Occam's Razor is sometimes used to challenge arguments for the existence of God. None of these applications has been considered definitive because the competing assumptions are not (and perhaps cannot be) precisely defined. Also, it should be added that the principle is only a guide to the best theory based on current knowledge, not to the "truth".

Galileo Galilei lampooned the misuse of Occam's Razor in his Dialogue. The principle is represented in the dialogue by Simplicio. The telling point that Galileo presented ironically was that if you really wanted to start from a small number of entities, you could always consider the letters of the alphabet as the fundamental entities, since you could certainly construct the whole of human knowledge out of them (a view that Abraham Abulafia presented much more expansively).

Adding another layer of irony, many modern scientists and mathematicians seriously propose that the basic "entities" of reality may be "bits of information", for example, the digits of binary code, in which case the entities of William of Ockham might be seen as foreshadowing the logic of George Boole and modern computing.

It is argued that Ockham was an intellectual forefather of the Scientific Method because he argued for a degree of intellectual freedom in a time of dogmatic belief. He can also, however, be seen as an apologist for Divine Omnipotence, since he was concerned to demonstrate that creation was contingent and the Creator free to change the rules at will. Thus, if God is free to make an infinity of worlds with completely different rules from those which prevail in our world, then we are free to imagine such worlds and their logical and practical consequences.

Perhaps the best formulation of Occam's Razor is the one which states that, of equally good explanations for a phenomenon, the best one is the simplest explanation that accounts for all the facts.

Creationists sometimes attempt to apply Occam's Razor in defence of geogenesis, claiming that theirs is the simpler theory as compared with evolution. However, there is the question of whether a creator with sufficient intelligence and forethought is simpler than a model in which complexity spontaneously generates over geological timescales. In addition there is the question of whether Creationism actually accounts for the physical evidence; the fossil record and evidence of the age of the Earth and solar system appear to falsify the story of divine creation (as described in the book of Genesis). Some people claim that attempting to apply Occam's Razor to the past where it can only be applied retrospectively is erroneous; however, Occam's razor is used extensively and successfully in data sets collected in both archeology and astronomy.

In statistics

There are various papers in scholarly journals deriving versions of Occam's Razor from probability theory and applying it in statistical inference, and also of various criteria for penalizing complexity in statistical inference. Recent papers have suggested a connection between Occam's Razor and Kolmogorov complexity.

One of the problems with the original formulation of the principle is that it only applies to models with the same explanatory power (i.e. prefer the simplest of equally good models). A more general form of Occam's Razor can be derived from Bayesian model comparison and Bayes factors, which can be used to compare models that don't fit the data equally well. These methods can sometimes optimally balance the complexity and power of a model.

Many artificial intelligence researchers are now employing strongly probabilistic Bayesian techniques.

See also

• Bayesian inference
• Buridan's ass
• cladistics
• curve fitting
• eliminative materialism
• falsifiability
• Hanlon's razor
• Kolmogorov complexity
• Morgan's Canon
• philosophy of science
• rationalism
• scientific reductionism
• simplicity
• Scientific laws named after people

References

• Ariew, R.(1976) Ockham's Razor: A Historical and Philosophical Analysis of Ockham's Principle of Parsimony. Philosophy. Champaign-Urbana, University of Illinois.
• Charlesworth, M. J. (1956). "Aristotle's Razor." Philosophical Studies (Ireland) 6: 105-112.
• Churchland, P. (1984) Matter and Consciousness. Cambridge, Massachusetts, The MIT Press.
• Dawkins, R. (1990) The Selfish Gene. Oxford University Press, 1976; 2nd edition, December 1989, hardcover, 352 pages, ISBN 0192177737; April 1992, ISBN 019857519X; trade paperback, September, 1990, 352 pages, ISBN 0192860925
• Epstein, R. (1984). The Principle of Parsimony and Some Applications in Psychology. Journal of Mind Behavior 5:119-130
• Hoffmann, Ronald, Vladimir I. Minkin, Barry K. Carpenter. Ockham's Razor and Chemistery. International Journal for the Philosophy of Chemistry. Vol. 3 (1997), pp. 3-28
• Jacquette, D. (1994). Ockham's Razor. Philosophy of Mind. Engleswoods Cliffs, N.J., Prentice Hall:34-36.
• Maurer, A. (1984). Ockham's Razor and Chatton's Anti-Razor. Medieval Studies 46:463-475.
• Menger, Karl. A Counterpart of Ockham's Razor in Pure and Applied Mathematics: Ontological Uses. Synthese 12 (1960) 415
• Morgan, L. C. (1898). An Introduction to Comparative Psychology. London, W. Scott.
• Nolan, D. (1997). Quantitative Parsimony. British Journal for the Philosophy of Science 48(3):329-343.
• Smart, J. J. C. (1959). Sensations and Brain Processes. Philosophical Review68:141-156.
• Sober, E. (1981). The Principle of Parsimony. British Journal for the Philosophy of Science 32:145-156.
• Sober, E. (1990). Let's Razor Ockham's Razor. Philosophy supp:73-93.
• Thornburn, W. M. (1918). The Myth of Occam's Razor. Mind: 345-353.
• Williams, G. C. (1966). Adaptation and Natural Selection, Princeton University Press.
• Richard O. Duda, Peter E. Hart, David G. Stork (2000) Pattern classification (2nd edition), Section 9.6.5, p. 487-489, Wiley, ISBN 0471056693
• Chapter 24 in Probability Theory - The logic of science (http://omega.math.albany.edu:8008/JaynesBook.html) by E. T. Jaynes, 1994.
• David J.C. MacKay (2003) Information theory, inference and learning algorithms, CUP, ISBN 0521642981, (also available online (http://www.inference.phy.cam.ac.uk/mackay/itila/book.html))

External links

• Ockham's Razor and Chemistry (http://www.hyle.org/journal/issues/3/hoffman.htm)
• What is Occam's Razor? (http://www.physics.adelaide.edu.au/~dkoks/Faq/General/occam.html)
• Skeptic's Dictionary: Occam's Razor (http://skepdic.com/occam.html)
• Ockham's Razor (http://www.galilean-library.org/or.html), an essay at The Galilean Library on the historical and philosophical implications by Paul Newall.
• NIPS 2001 Workshop "Foundations of Occam's Razor and parsimony in learning" (http://rii.ricoh.com/~stork/OccamWorkshop.html)
• "We Must Choose The Simplest Physical Theory: Levin-Li-Vitányi Theorem And Its Potential Physical Applications" (http://citeseer.nj.nec.com/fox98we.html)
• "Sharpening Ockham's razor on a Bayesian strop" (http://quasar.as.utexas.edu/papers/ockham.pdf) (pdf), by William H. Jefferys and James O. Berger; gives an objective quantification of Occam's razor within Bayesian statistics (with scientific applications)
• Information Theory, Inference, and Learning Algorithms (http://www.inference.phy.cam.ac.uk/mackay/itila/book.html), by David J.C. MacKay, includes an introductory chapter on the automatic Occam's razor that is embodied by Bayesian model comparison.
• "Message Length as an Effective Ockham's Razor in Decision Tree Induction" (http://www.ai.mit.edu/conferences/aistats2001/files/needham122.ps), by S. Needham and D. Dowe (http://www.csse.monash.edu.au/~dld), Proc. 8th International Workshop on AI and Statistics (http://www.ai.mit.edu/conferences/aistats2001) (2001), pp253-260 (http://www.csse.monash.edu.au/~dld/Publications/2001/Needham+Dowe2001.ref). (Shows how Ockham's razor (http://www.csse.monash.edu.au/~dld/Ockham.html) works fine when interpreted as MML.)
• Lloyd's MML pages (http://www.csse.monash.edu.au/~lloyd/tildeMML/) describe how Minimum Message Length induction extends Ockham's razor (http://www.csse.monash.edu.au/~lloyd/tildeImages/People/Ockham/) for differing hypotheses. (MML (http://www.csse.monash.edu.au/~dld/MML.html) is a scale-invariant Bayesian model selection method.)
• An extensive bibliography (http://occamssword.com/CORE%20READINGS%20FOR%20OCCAMS%20RAZOR.htm) of publications related to Occam's Razor.
• Occam's sword (http://www.internet-encyclopedia.org/wiki.php?title=Occam%27s_sword) at wikinfo
• Simplicity at Stanford Encyclopedia of Philosophy (http://plato.stanford.edu/entries/simplicity/)
• Template:Planetmath referenceca:Navalla d'Occam

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