Ockham's razor

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William of Ockham

Occam's razor (also spelled Ockham's razor) is a principle attributed to the 14th-century English logician and Franciscan friar William of Ockham. Originally a tenet of the reductionist philosophy of nominalism, it is more often taken today as a heuristic maxim that advises economy, parsimony, or simplicity in scientific theories. Occam's razor states that the explanation of any phenomenon should make as few assumptions as possible, eliminating those that make no difference in the observable predictions of the explanatory hypothesis or theory. The principle is often expressed in Latin as the lex parsimoniae (law of succinctness):

entia non sunt multiplicanda praeter necessitatem,

which translates to:

entities should not be multiplied beyond necessity.

Furthermore, when multiple competing theories have equal predictive powers, the principle recommends selecting those that introduce the fewest assumptions and postulate the fewest hypothetical entities. It is in this sense that Occam's razor is usually understood.

History

William Ockham (c. 1295–1349) … is remembered as an influential nominalist, but his popular fame as a great logician rests chiefly on the maxim known as Ockham's razor: Entia non sunt multiplicanda praeter necessitatem. No doubt this represents correctly the general tendency of his philosophy, but it has not so far been found in any of his writings. His nearest pronouncement seems to be Numquam ponenda est pluralitas sine necessitate, which occurs in his theological work on the Sentences of Peter Lombard (Super Quattuor Libros Sententiarum (ed. Lugd., 1495), i, dist. 27, qu. 2, K). In his Summa Totius Logicae, i. 12, Ockham cites the principle of economy, Frustra fit per plura quod potest fieri per pauciora. (Kneale and Kneale, 1962, p. 243).

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), Thomas Aquinas (c. 1225–1274), and even Aristotle (384–322 B.C.E.) (Charlesworth 1956). The term "Occam's razor" first appeared in 1852 in the works of Sir William Rowan Hamilton (1805–1865), long after Ockham's death circa 1349. Ockham did not invent the razor, so its association with him may be due to the frequency and effectiveness with which he used it (Ariew 1976). And though he stated the principle in various ways, the most popular version was written not by himself but by John Ponce of Cork in 1639 (Thorburn 1918).

Justifications

Occam's razor has always been associated with the aesthetic concept of simplicity. Prior to the 20th century, it was believed that nature itself was simple and that simpler theories about nature were thus more likely to be true. Thomas Aquinas made this argument in the 13th century, writing, "If a thing can be done adequately by means of one, it is superfluous to do it by means of several; for we observe that nature does not employ two instruments where one suffices" (Pegis 1945). Beginning in the 20th century, however, epistemological justifications based on induction, pragmatism, and probability theory have become more popular among philosophers. (See Roger Ariew's 1976 dissertation, "Ockham's Razor: A Historical and Philosophical Analysis of Ockham's Principle of Parsimony".)

The razor's strict form, which prohibits irrelevant assumptions in a given theory, is justified by the fact that all assumptions introduce possibilities for error. If an assumption does not improve the accuracy of a theory, its only effect is to make the theory more error-prone, and since error is undesirable in any theory, unnecessary assumptions should be avoided.

The common form of the razor, used to distinguish between equally explanatory theories, can be supported by appeals to the practical value of simplicity. Theories exist to give accurate explanations of phenomena, and simplicity is a valuable aspect of an explanation because it makes the explanation easier to understand and work with. Thus, if two theories are equally accurate and neither appears more probable than the other, the simple one is to be preferred over the complicated one, because simplicity is valuable.

Applications

Science by Razor alone?

When it is proposed as a maxim of science, Ockham's razor is construed as a decision procedure for choosing among competing systems of hypotheses. In this context a system of hypotheses, together with its supporting definitions and its logical consequences, is commonly described as a theory. To evaluate the utility of a radular (razor-like) tool in this setting, it is necessary to establish both the ground rules of scientific procedure and the operational definition of a particular brand of razor with a significant degree of formal precision.

Occam's razor has become a basic tool for those who follow the scientific method. The primary activity of science — formulating theories and selecting the most promising ones — is impossible without a way of choosing from among the theories which fit the evidence equally well, the number of which can be arbitrarily large (see underdetermination).

William H. Jefferys and James O. Berger (1991) quantify this undesirable factor that in its extremity manifests as unnecessary assumptions into the degree to which a proposition is unnecessarily accommodating to possible observable data. Theories which specifically, logically entail the observed set of data or are similarly entailed by it are preferred over theories which are trivially consistent with it by mere virtue of being consistent with a wide range of possible data a priori or ad-hoc adjustment that is otherwise unjustified (see also Bayesian inference and falsifiability).

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. In the light of this, the popular rephrasing of the razor - that "The simplest explanation is the best one" - can lead to a gross oversimplification when the word simple is taken at face value.

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 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. Then when Einstein constructed his theory of special relativity without any reference to the Aether this subsequently became the accepted view, thus providing another example of a theory chosen in part for its greater ontological simplicity.

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's 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: "In no case is an animal activity to be interpreted in terms of higher psychological processes, if it can be fairly interpreted in terms of processes which stand lower in the scale of psychological evolution and development" (Morgan 1903).

However, more recent work by biologists, such as Richard Dawkins's The Selfish Gene, has revealed that Williams's view is not the simplest and most basic. Dawkins argues 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 one considers 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 is 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. 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 Ockham's Razor (1990).

Francis Crick has commented on potential limitations of Occam's razor in biology. He advances the argument that because biological systems are the products of (an on-going) natural selection, the mechanisms are not necessarily optimal in an obvious sense. He cautions: "While Occam's razor is a useful tool in the physical sciences, it can be a very dangerous implement in biology. It is thus very rash to use simplicity and elegance as a guide in biological research."

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.

This principle has very important limitations in medical practice. Among the reasons is that it is more likely for a patient to have several common diseases, rather than having a single rarer disease which explains all their myriad of symptoms. A variant of Occam's Razor typically taught to all medical students learning how to make diagnoses, is the expression, "When you hear hoofbeats, think horses not zebras." This is to impress upon the future doctors that the more common ailment is the most likely. The actual process that occurs when diagnosing a patient is a continuous flow of hypothesis and testing of that hypothesis, then modifying the hypothesis and so on. At no stage can a diagnosis properly be made or excluded because it doesn't immediately appear to fit the principles of Occam's razor. The principle of Occam's razor does not demand that the diagnostician necessarily opt for the simplest explanation, but instead guides the medical practitioner to seek explanations, without unnecessary additional assumptions, which are capable of accounting for all relevant evidence.

"Hickam's dictum" is a modern counterargument to the use of Occam's razor in the medical profession. Put succinctly it states: "Patients can have as many diseases as they like!".

Religion

In the philosophy of religion, Occam's razor is sometimes applied to the existence of God; if the concept of God does not help to explain the universe, it is argued, God is irrelevant and should be cut away (Schmitt 2005). While Occam's razor cannot prove God's nonexistence, it does imply that, in the absence of compelling reasons to believe in God, unbelief should be preferred.

There is much controversy over whether such compelling reasons exist or not. The history of theistic thought is rife with attempts at formulating them: the cosmological argument, for example, states that the universe must be the result of a "first cause" and that that first cause must be God. Similarly, the teleological argument credits the appearance of design and order in the universe to supernatural intelligence. Many people believe in miracles or have what they call religious experiences, and some theists consider creationism to be more believable than naturalistic explanations for the diversity and history of life on earth.

Others maintain that these arguments fail to necessitate the inclusion of the God hypothesis in the world model, instead preferring explanations that deal with the same phenomena within the confines of existing scientific models. The necessity of a God in the teleological argument is challenged by the effects of emergence, leading to the creation-evolution controversy; likewise, religious experiences have naturalistic explanations in the psychology of religion. Other theistic arguments, such as the argument from miracles, are sometimes pejoratively said to be arguing for a mere God of the gaps. Whether or not God actually works miracles, any explanation that "God did it" must fit the facts and make accurate predictions better than more parsimonious guesses like "something did it", or else Occam's razor still cuts God out.

Rather than argue for the necessity of God, some theists consider their belief to be based on grounds independent of, or prior to, reason, making Occam's razor irrelevant. This was the stance of Søren Kierkegaard, who viewed belief in God as a leap of faith which sometimes directly opposed reason (McDonald 2005); this is also the same basic view of Clarkian Presuppositional apologetics, with the exception that Clark never thought the leap of faith was contrary to reason. (See also: Fideism). In a different vein, Alvin Plantinga and others have argued for reformed epistemology, the view that God's existence can properly be assumed as part of a Christian's epistemological structure. (See also: Basic beliefs). Yet another school of thought, Van Tillian Presuppositional apologetics, claims that God's existence is the transcendentally necessary prior condition to the intelligibility of all human experience and thought. In other words, proponents of this view hold that there is no other viable option to ultimately explain any fact of human experience or knowledge, let alone a simpler one.

Considering that the razor is often wielded against theism, it is somewhat ironic that Ockham himself believed in God. He apparently considered Christianity to be outside the scope of his rule, once writing, "No plurality should be assumed unless it can be proved (a) by reason, or (b) by experience, or (c) by some infallible authority." The last clause "refers to the Bible, the Saints and certain pronouncements of the Church" (Hoffmann 1997).

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 who believed 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.

Many scientists, however, claim that this is exactly reversed. Erwin Schrödinger wrote that "Consciousness is the singular for which there is no plural," thus placing consciousness first and everything, including the physical universe, within the realm of consciousness. Dr. Amit Goswami, a physics teacher and author of numerous books, including The Self Aware Universe: How Consciousness Creates the Material World, argues that "consciousness is the ground of all being."

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.

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. Generally the exact Occam factor is intractable but approximations such as Akaike Information Criterion, Bayesian Information Criterion, Variational Bayes and Laplace Approximation are used. Many artificial intelligence researchers are now employing such techniques.

The statistical view leads to a more rigorous formulation of the razor than previous philosophical discussions. In particular, it shows that 'simplicity' must first be defined in some way before the razor may be used, and that this definition will always be subjective. For example, in the Kolmogorov-Chaitin Minimum description length approach, the subject must pick a Turing machine whose operations describe the basic operations believed to represent 'simplicity' by the subject. However one could always choose a Turing machine with a simple operation that happened to construct one's entire theory and would hence score highly under the razor. The Turing machine simplicity can be thought of as a Bayesian prior belief over the space of rival theories. Hence Occam's razor is not an objective comparison method, and merely reflects the subject's prior beliefs. One's choice of exactly which razor to use is culturally relative.

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 Occam's surviving writings. This also applies to non est ponenda pluritas sine necessitate, which translates literally into English as "pluralities ought not be posited without necessity". It 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".

Other common restatments are:

Entities are not to be multiplied without necessity.

and

The simplest answer is usually the correct answer.

Or, as Einstein put it "As simple as possible, but no simpler"

A re-statement of Occam's razor, in more formal terms, is provided by information theory in the form of minimum message length (MML). Tests of Occam's razor on decision tree models which initially appeared criticial have been shown to actually work fine when re-visited using MML. Other criticisms of Occam's razor and MML (e.g., a binary cut-point segmentation problem) have again been rectified when - crucially - an inefficient coding scheme is made more efficient.

"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]

Leonardo da Vinci (1452–1519) lived after Occam'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.

Another common statement of it is :

The simplest explanation that covers all the facts.

This is an over-simplification, or at least a little misleading. See above, "In science".

This rephrasing has several faults, the worst being that Occam's razor is only supposed to be used to choose between two scientific theories which are otherwise equally predictive. The second problem with the "simplest is best" equation is that Occam's razor never claims to choose the 'best' theory, but only proposes simplicity as the deciding factor in choosing between two otherwise equal theories. It's possible that, given more information, the more complex theory might turn out to be correct the majority of the time. Occam's razor makes no explicit claims as to whether or not this will happen, but prompts us to use the simpler theory until we have reason to do otherwise.

The earliest versions of the razor clearly imply that if a more complex theory is "necessary" then it need not be invalid. Perhaps a better way to state it is: "a correct theory of phenomena is only as complex as is necessary — and no more so — to explain said phenomena."

Anti-razors

Occam's razor has met some opposition from people who have considered it too extreme or rash. 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.

Anti-razors have also been created by Gottfried Wilhelm Leibniz (1646–1716), Immanuel Kant (1724–1804), and Karl Menger (1902-1985). Leibniz's version took the form of a principle of plenitude, as Arthur Lovejoy has called it, the idea being 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. There is also Crabtree's Bludgeon, which takes a cynical view that 'No set of mutually inconsistent observations can exist for which some human intellect cannot conceive a coherent explanation, however complicated.'

Trivia

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).

References

ISBN links support NWE through referral fees

• Ariew, Roger (1976). Ockham's Razor: A Historical and Philosophical Analysis of Ockham's Principle of Parsimony. Champaign-Urbana, University of Illinois.

Charlesworth, M. J. (1956). "Aristotle's Razor". Philosophical Studies (Ireland) 6: 105–112.

• Churchland, Paul M. (1984). Matter and Consciousness. Cambridge, Massachusetts: MIT Press. ISBN 0262530503.

• Crick, Francis H. C. (1988). What Mad Pursuit: A Personal View of Scientific Discovery. New York, New York: Basic Books. ISBN 0465091385.

• Dawkins, Richard (1990). The Selfish Gene. Oxford University Press. ISBN 0465091385.

• Duda, Richard O.; Peter E. Hart, David G. Stork (2000). Pattern Classification, 2nd edition, Wiley-Interscience. ISBN 0471056693, 487-489.

• Epstein, Robert (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 (1997). "Ockham's Razor and Chemistry". HYLE—International Journal for the Philosophy of Chemistry 3: 3–28. Retrieved on 2006-04-14.

• Jacquette, Dale (1994). Philosophy of Mind. Engleswoods Cliffs, New Jersey: Prentice Hall. ISBN 0130309338, 34–36.

• Jaynes, Edwin Thompson (1994). “Model Comparison and Robustness”, Probability Theory: The Logic of Science.

• Jefferys, William H. and Berger, James O. (1991). "Sharpening Ockham's Razor on a Bayesian Strop". Purdue University.

• Kneale, William; Martha Kneale (1962). The Development of Logic. London: Oxford University Press. ISBN 0198241836, 243.

• MacKay, David J. C. (2003). Information Theory, Inference and Learning Algorithms. Cambridge University Press. ISBN 0521642981.

• Maurer, A. (1984). "Ockham's Razor and Chatton's Anti-Razor". Medieval Studies 46: 463–475.

• McDonald, William (2005). Søren Kierkegaard. Stanford Encyclopedia of Philosophy. Retrieved on 2006-04-14.

• Menger, Karl (1960). "A Counterpart of Ockham's Razor in Pure and Applied Mathematics: Ontological Uses". Synthese 12: 415.

• Morgan, C. Lloyd (1903). “Other Minds than Ours”, An Introduction to Comparative Psychology, 2nd edition, London: W. Scott, 59. Retrieved on 2006-04-15.

• Nolan, D. (1997). "Quantitative Parsimony". British Journal for the Philosophy of Science 48 (3): 329–343.

• Schmitt, Gavin C. (2005). Ockham's Razor Suggests Atheism. Retrieved on 2006-04-15.

Smart, J. J. C. (1959). "Sensations and Brain Processes". Philosophical Review 68: 141–156.

• Sober, Elliott (1981). "The Principle of Parsimony". British Journal for the Philosophy of Science 32: 145–156.

• Sober, Elliott (1990). “Let's Razor Ockham's Razor”, Dudley Knowles Explanation and its Limits. Cambridge: Cambridge University Press. ISBN 0521395984, 73-94.

• Thorburn, W. M. (1918). "The Myth of Occam's Razor". Mind 27 (107): 345-353.

• Williams, George C. (1966). Adaptation and natural selection: A Critique of some Current Evolutionary Thought. Princeton, New Jersey: Princeton University Press. ISBN 0691023573.

See also

Bayesian inference Buridan's ass Cladistics Curve fitting Eliminative materialism Falsifiability

Hanlon's razor Kolmogorov complexity Minimum message length Morgan's canon Occam programming language Philosophy of science

Poverty of the stimulus Principle of least astonishment Rationalism Reference class problem Scientific method Scientific reductionism Simplicity

External links

• What is Occam's Razor? This essay distinguishes Occam's Razor (used for theories with identical predictions) from the Principle of Parsimony (which can be applied to theories with different predictions).
• Skeptic's Dictionary: Occam's Razor
• Ockham's Razor, 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"
• "We Must Choose The Simplest Physical Theory: Levin-Li-Vitányi Theorem And Its Potential Physical Applications"
• Information Theory, Inference, and Learning Algorithms, 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", by S. Needham and D. Dowe, Proc. 8th International Workshop on AI and Statistics (2001), pp253-260. (Shows how Ockham's razor works fine when interpreted as Minimum Message Length (MML).) Re efficiency and reliability of coding schemes, see also pp272-273 of (Comley and Dowe, Chapter 11, MIT Press, 2005).
• Lloyd's MML pages describe how Minimum Message Length induction extends Ockham's razor for differing hypotheses.
• An extensive bibliography of publications related to Occam's Razor.
• Occam's sword at wikinfo
• Simplicity at Stanford Encyclopedia of Philosophy
• Occam's Razor. PlanetMath
• [2] ABC Radio National program in which speakers are allowed to explicate at length on topics without the moderation of an interviewer. (Podcast available)

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