William of Ockham (c. 1285–1349) is remembered as an influential nominalist but his popular fame as a great logician rests chiefly on the maxim attributed to him and known as Ockham's razor: Entia non sunt multiplicanda praeter necessitatem or "Entities should not be multiplied unnecessarily." The term razor refers to the act of shaving away unnecessary assumptions to get to the simplest explanation. No doubt this maxim 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 [Plurality must never be posited without necessity], which occurs in his theological work on the Sentences of Peter Lombard (Quaestiones et decisiones in quattuor libros Sententiarum Petri Lombardi (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 [It is futile to do with more things that which can be done with fewer].
The origins of what has come to be known as Occam's razor are traceable to the works of earlier philosophers such as Maimonides (1138–1204), John Duns Scotus (1265–1308), Thomas Aquinas (c. 1225–1274) and even Aristotle (384–322 BC) (Charlesworth 1956). The term "Ockham's razor" first appeared in 1852 in the works of Sir William Hamilton, 9th Baronet (1788–1856), centuries after Ockham's death. Ockham did not invent this "razor", so its association with him may be due to the frequency and effectiveness with which he used it (Ariew 1976). Though Ockham stated the principle in various ways, the most popular version was written not by him, but by John Ponce from Cork Ireland in 1639 (Meyer 1957).
The version of the Razor most often found in Ockham's work is Numquam ponenda est pluralitas sine necessitate, –For nothing ought to be posited without a reason given, unless it is self-evident (literally, known through itself) or known by experience or proved by the authority of Sacred Scripture.– For Ockham, the only truly necessary entity is God; everything else, the whole of creation, is radically contingent through and through. In short, Ockham does not accept the Principle of Sufficient Reason.
Prior to the 20th century, it was a commonly-held belief that nature itself was simple and that simpler hypotheses about nature were thus more likely to be true. This notion was deeply rooted in the aesthetic value simplicity holds for human thought and the justifications presented for it often drew from theology. 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."
The common form of the razor, used to distinguish between equally explanatory hypotheses, can be supported by appeals to the practical value of simplicity. Hypotheses 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 hypotheses match the observations equally well, and neither appears more probable than the other, the simple one is to be preferred over the complicated one, because simplicity is practical.
One way a theory or a principle could be justified is empirically; that is to say, if simpler theories were to have a better record of turning out to be correct than more complex ones, that would corroborate Occam's razor.
Some argue that Occam's razor is not a theory in the classic sense of being a model that explains physical observations, relying on induction; rather, it is a heuristic maxim for choosing among such theories and underlies induction.
There are many different ways of making inductive inferences from past data concerning the success of different theories throughout the history of science, and inferring that "simpler theories are, other things being equal, generally better than more complex ones" is just one way of many–which only seems more plausible to us because we are already assuming the razor to be true (see e.g. Swinburne 1997 and Williams, Gareth T, 2008). This, however, does not exclude legitimate attempts at a deductive justification of the razor (and indeed these are inherent to many of its modern derivatives). Failing even that, the razor may be accepted a priori on pragmatist grounds.
One should note the related concept of overfitting, where excessively complex models are affected by statistical noise, whereas simpler models may capture the underlying structure better and may thus have better predictive performance. It is, however, often difficult to deduce which part of the data is noise (cf. model selection, test set, minimum description length, Bayesian inference, etc.).
Karl Popper argues that a preference for simple theories need not appeal to practical or aesthetic considerations. Our preference for simplicity may be justified by its falsifiability criterion: We prefer simpler theories to more complex ones "because their empirical content is greater; and because they are better testable" (Popper 1992). The idea here is that a simple theory applies to more cases than a more complex one, and is thus more easily falsifiable. This is again comparing a simple theory to a more complex theory where both explain the data equally well.
The philosopher of science Elliott Sober once argued along the same lines as Popper, tying simplicity with "informativeness": The simplest theory is the more informative one, in the sense that less information is required in order to answer one's questions (Sober 1975). He has since rejected this account of simplicity, purportedly because it fails to provide an epistemic justification for simplicity. He now expresses views to the effect that simplicity considerations (and considerations of parsimony in particular) do not count unless they reflect something more fundamental. Philosophers, he suggests, may have made the error of hypostatizing simplicity (i.e. endowed it with a sui generis existence), when it has meaning only when embedded in a specific context (Sober 1992). If we fail to justify simplicity considerations on the basis of the context in which we make use of them, we may have no non-circular justification: "just as the question 'why be rational?' may have no non-circular answer, the same may be true of the question 'why should simplicity be considered in evaluating the plausibility of hypotheses?'" (Sober 2001)
Richard Swinburne argues for simplicity on logical grounds: "...other things being equal...the simplest hypothesis proposed as an explanation of phenomena is more likely to be the true one than is any other available hypothesis, that its predictions are more likely to be true than those of any other available hypothesis, and that it is an ultimate a priori epistemic principle that simplicity is evidence for truth" (Swinburne 1997).
He maintains that we have an innate bias towards simplicity and that simplicity considerations are part and parcel of common sense. Since our choice of theory cannot be determined by data (see Underdetermination and Quine-Duhem thesis), we must rely on some criterion to determine which theory to use. Since it is absurd to have no logical method by which to settle on one hypothesis amongst an infinite number of equally data-compliant hypotheses, we should choose the simplest theory: "...either science is irrational [in the way it judges theories and predictions probable] or the principle of simplicity is a fundamental synthetic a priori truth" (Swinburne 1997).
In science, Occam–s razor is used as a heuristic (rule of thumb) to guide scientists in the development of theoretical models rather than as an arbiter between published models. In physics, parsimony was an important heuristic in the formulation of special relativity by Albert Einstein, the development and application of the principle of least action by Pierre Louis Maupertuis and Leonhard Euler, and the development of quantum mechanics by Ludwig Boltzmann, Max Planck, Werner Heisenberg and Louis de Broglie. In chemistry, Occam–s razor is often an important heuristic when developing a model of a reaction mechanism. However, while it is useful as a heuristic in developing models of reaction mechanisms, it has been shown to fail as a criterion for selecting among published models. In this context, Einstein himself expressed a certain caution when he formulated Einstein's Constraint: 'Everything should be kept as simple as possible, but no simpler.' Elsewhere, Einstein harks back to the theological roots of the Razor, with his famous put-down: 'The Good Lord may be subtle, but he is not malicious.'
In the scientific method, parsimony is an epistemological, metaphysical or heuristic preference, not an irrefutable principle of logic, and certainly not a scientific result. As a logical principle, Occam's razor would demand that scientists accept the simplest possible theoretical explanation for existing data. However, science has shown repeatedly that future data often supports more complex theories than existing data. Science tends to prefer the simplest explanation that is consistent with the data available at a given time, but history shows that these simplest explanations often yield to complexities as new data become available. Science is open to the possibility that future experiments might support more complex theories than demanded by current data and is more interested in designing experiments to discriminate between competing theories than favoring one theory over another based merely on philosophical principles.
When scientists use the idea of parsimony, it only has meaning in a very specific context of inquiry. A number of background assumptions are required for parsimony to connect with plausibility in a particular research problem. The reasonableness of parsimony in one research context may have nothing to do with its reasonableness in another. It is a mistake to think that there is a single global principle that spans diverse subject matter.
As a methodological principle, the demand for simplicity suggested by Occam–s razor cannot be generally sustained. Occam–s razor cannot help toward a rational decision between competing explanations of the same empirical facts. One problem in formulating an explicit general principle is that complexity and simplicity are perspective notions whose meaning depends on the context of application and the user–s prior understanding. In the absence of an objective criterion for simplicity and complexity, Occam–s razor itself does not support an objective epistemology.
The problem of deciding between competing explanations for empirical facts cannot be solved by formal tools. Simplicity principles can be useful heuristics in formulating hypotheses, but they do not make a contribution to the selection of theories. A theory that is compatible with one person–s world view will be considered simple, clear, logical, and evident, whereas what is contrary to that world view will quickly be rejected as an overly complex explanation with senseless additional hypotheses. Occam–s razor, in this way, becomes a –mirror of prejudice.–
It has been suggested that Occam–s razor is a widely accepted example of extraevidential consideration, even though it is entirely a metaphysical assumption. There is little empirical evidence that the world is actually simple or that simple accounts are more likely than complex ones to be true.
Most of the time, Occam–s razor is a conservative tool, cutting out crazy, complicated constructions and assuring that hypotheses are grounded in the science of the day, thus yielding –normal– science: models of explanation and prediction. There are, however, notable exceptions where Occam–s razor turns a conservative scientist into a reluctant revolutionary. For example, Max Planck interpolated between the Wien and Jeans radiation laws used an Occam–s razor logic to formulate the quantum hypothesis, and even resisting that hypothesis as it became more obvious that it was correct.
However, on many occasions Occam's razor has stifled or delayed scientific progress. For example, appeals to simplicity were used to deny the phenomena of meteorites, ball lightning, continental drift, and reverse transcriptase. It originally rejected DNA as the carrier of genetic information in favor of proteins, since proteins provided the simpler explanation. Theories that reach far beyond the available data are rare, but general relativity provides one example.
In hindsight, one can argue that it is simpler to consider DNA as the carrier of genetic information, because it uses a smaller number of building blocks (four nitrogenous bases). However, during the time that proteins were the favored genetic medium, it seemed like a more complex hypothesis to confer genetic information in DNA rather than proteins.
One can also argue (also in hindsight) for atomic building blocks for matter, because it provides a simpler explanation for the observed reversibility of both mixing and chemical reactions as simple separation and re-arrangements of the atomic building blocks. However, at the time, the atomic theory was considered more complex because it inferred the existence of invisible particles which had not been directly detected. Ernst Mach and the logical positivists rejected the atomic theory of John Dalton, until the reality of atoms was more evident in Brownian motion, as explained by Albert Einstein.
In the same way, hindsight argues that postulating the aether is more complex than transmission of light through a vacuum. However, at the time, all known waves propagated through a physical medium, and it seemed simpler to postulate the existence of a medium rather than theorize about wave propagation without a medium. Likewise, Newton's idea of light particles seemed simpler than Young's idea of waves, so many favored it; however in this case, as it turned out, neither the wave- nor the particle-explanation alone suffices, since light behaves like waves as well as like particles (wave–particle duality).
Three axioms presupposed by the scientific method are realism (the existence of objective reality), the existence of observable natural laws, and the constancy of observable natural law. Rather than depend on provability of these axioms, science depends on the fact that they have not been objectively falsified. Occam–s razor and parsimony support, but do not prove these general axioms of science. The general principle of science is that theories (or models) of natural law must be consistent with repeatable experimental observations. This ultimate arbiter (selection criterion) rests upon the axioms mentioned above.
There are many examples where Occam–s razor would have picked the wrong theory given the available data. Simplicity principles are useful philosophical preferences for choosing a more likely theory from among several possibilities that are each consistent with available data. A single instance of Occam–s razor picking a wrong theory falsifies the razor as a general principle.
If multiple models of natural law make exactly the same testable predictions, they are equivalent and there is no need for parsimony to choose one that is preferred. For example, Newtonian, Hamiltonian, and Lagrangian classical mechanics are equivalent. Physicists have no interest in using Occam–s razor to say the other two are wrong. Likewise, there is no demand for simplicity principles to arbitrate between wave and matrix formulations of quantum mechanics. Science often does not demand arbitration or selection criteria between models which make the same testable predictions.
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 biological analyses, such as Richard Dawkins's The Selfish Gene, have contended 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 turns out 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. Muskoxen, 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).
Other methods for inferring evolutionary relationships use parsimony in a more traditional way. Likelihood methods for phylogeny use parsimony as they do for all likelihood tests, with hypotheses requiring few differing parameters (i.e., numbers of different rates of character change or different frequencies of character state transitions) being treated as null hypotheses relative to hypotheses requiring many differing parameters. Thus, complex hypotheses must predict data much better than do simple hypotheses before researchers reject the simple hypotheses. Recent advances employ information theory, a close cousin of likelihood, which uses Occam's Razor in the same way.
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 Ockham'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." 
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 philosophy is one of several demonstrated in the popular medical adage "when you hear hoofbeats, think horses, not zebras". While diagnostic parsimony might often be beneficial, credence should also be given to the counter-argument modernly known as Hickam's dictum, which succinctly states that "patients can have as many diseases as they damn well please". It is often statistically more likely that a patient has several common diseases, rather than having a single rarer disease which explains their myriad symptoms. Also, independently of statistical likelihood, some patients do in fact turn out to have multiple diseases, which by common sense nullifies the approach of insisting to explain any given collection of symptoms with one disease. These misgivings emerge from simple probability theory–which is already taken into account in many modern variations of the razor–and from the fact that the loss function is much greater in medicine than in most of general science. Because misdiagnosis can result in the loss of a person's health and potentially life, it is considered better to test and pursue all reasonable theories even if there is some theory that appears the most likely.
Diagnostic parsimony and the counter-balance it finds in Hickam's dictum have very important implications in medical practice. Any set of symptoms could be indicative of a range of possible diseases and disease combinations; though at no point is a diagnosis rejected or accepted just on the basis of one disease appearing more likely than another, the continuous flow of hypothesis formulation, testing and modification benefits greatly from estimates regarding which diseases (or sets of diseases) are relatively more likely to be responsible for a set of symptoms, given the patient's environment, habits, medical history and so on. For example, if a hypothetical patient's immediately apparent symptoms include fatigue and cirrhosis and they test negative for Hepatitis C, their doctor might formulate a working hypothesis that the cirrhosis was caused by their drinking problem, and then seek symptoms and perform tests to formulate and rule out hypotheses as to what has been causing the fatigue; but if the doctor were to further discover that the patient's breath inexplicably smells of garlic and they are suffering from pulmonary edema, they might decide to test for the relatively rare condition of Selenium poisoning.
Prior to effective anti-retroviral therapy for HIV it was frequently stated that the most obvious implication of Occam's razor, that of cutting down the number of postulated diseases to a minimum, does not apply to patients with AIDS, as they frequently did have multiple infectious processes going on at the same time. While the probability of multiple diseases being higher certainly reduces the degree to which this kind of analysis is useful, it does not go all the way to invalidating it altogether; even in such a patient, it would make more sense to first test a theory postulating three diseases to be the cause of the symptoms than a theory postulating seven.
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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). It is argued to imply that, in the absence of compelling reasons to believe in God, disbelief should be preferred. Such arguments are based on the assertion that belief in God requires more complex assumptions to explain the universe than non-belief.
The history of theistic thought has produced many arguments attempting to show that this is not the case – that the difficulties encountered by a theory without God are equal to or greater than those encountered by a theory postulating one. The cosmological argument, for example, states that the universe must be the result of a "first cause" and that that first cause can be thought of as God. Similarly, the teleological argument credits the design and order in the universe to supernatural intelligence. Many people believe in miracles or have what they call religious experiences, and creationists consider divine design to be more believable than naturalistic explanations for the diversity and history of life on earth.
Many scientists generally do not accept these arguments, and prefer to rely on explanations that deal with the same phenomena within the confines of existing scientific models. Among leading scientists defined as members of the National Academy of Sciences, in the United States, 72.2% expressed disbelief and 93% expressed disbelief or doubt in the existence of a personal god in a survey conducted in 1998 (an ongoing survey being conducted by Elaine Ecklund of Rice University since 2004 indicates that this figure drops to as low as 38% when social scientists are included and the definition of "God" is expanded to allow a non-personal god as per Pantheism or Deism). The typical scientific view challenges the validity of the teleological argument 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 Sren 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.
Ockham himself was a theist. He considered some Christian sources to be valid sources of factual data, equal to both logic and sense perception. He wrote, "No plurality should be assumed unless it can be proved (a) by reason, or (b) by experience, or (c) by some infallible authority"; referring in the last clause "to the Bible, the Saints and certain pronouncements of the Church" (Hoffmann 1997). In Ockham's view, an explanation which does not harmonize with reason, experience or the aforementioned sources cannot be considered valid.
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.
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.
Additionally, Occam's razor applied to the demands placed on physical matter by Newton's 3rd Law of Motion (every action has an equal and opposite reaction) to explain both the dual-existentialism of matter-in-motion as observed today (motion is always considered a reaction) requiring both a metaphysical initiator of physical matter's existentialism, as well as the metaphysical initiator to initiate the existentialism of the motion of physical matter itself. Occam's razor which would favor the concept of a single (though multi-purposed) metaphysical initiator serving as both the initiator of the temporary time-bounded physical matter's existence as well as the initiator of the existence of matter's time-bound motion itself. Conversely, the assumption of the need for compound or multiple metaphysical entities (call them gods, coincidences or entities), whether non-purposed/purposed entities, causing the dual-existentialism of both matter and its motion, would be contrary to Occam's razor. This relevance is this application of Occam's razor is of course predicated on the acceptance of Newton's Laws of Motion to be factual.
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One intuitive justification of Occam's Razor's admonition against unnecessary hypotheses is a direct result of basic probability theory. By definition, all assumptions introduce possibilities for error; If an assumption does not improve the accuracy of a theory, its only effect is to increase the probability that the overall theory is wrong.
There are various papers in scholarly journals deriving formal 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 Ockham factor is intractable but approximations such as Akaike Information Criterion, Bayesian Information Criterion, Variational Bayes, False discovery rate and Laplace approximation are used. Many artificial intelligence researchers are now employing such techniques.
William H. Jefferys and James O. Berger (1991) generalise and quantify the original formulation's "assumptions" concept as the degree to which a proposition is unnecessarily accommodating to possible observable data. The model they propose balances the precision of a theory's predictions against their sharpness; theories which sharply made their correct predictions are preferred over theories which would have accommodated a wide range of other possible results. This, again, reflects the mathematical relationship between key concepts in Bayesian inference (namely marginal probability, conditional probability and posterior probability).
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. This has led to two opposing views of the objectivity of Occam's razor.
The Turing machine can be thought of as embodying 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. Thus it might be argued that one's choice of exactly which razor to use is culturally relative.
The minimum instruction set of a Universal Turing machine requires approximately the same length description across different formulations, and is small compared to the Kolmogorov complexity of most practical theories. Marcus Hutter has used this consistency to define a "natural" Turing machine of small size as the proper basis for excluding arbitrarily complex instruction sets in the formulation of razors. Describing the program for the universal program as the "hypothesis", and the representation of the evidence as program data, it has been formally proven under ZF that "the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized." 
One possible conclusion from mixing the concepts of Kolmogorov complexity and Occam's Razor is that an ideal data compressor would also be a scientific explanation/formulation generator. Some attempts have been made to re-derive known laws from considerations of simplicity or compressibility.
Occam's razor has been recommended as a measure of how good the plot of a novel is. Simple and logical plots are easy to explain and this enhances the experience of the reader. The writer is also less likely to make an error while explaining the plot to the reader.
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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. 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).
Other common restatements are:
Entities are not to be multiplied without necessity.
The simplest answer is usually the correct answer.
A restatement 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 critical 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.
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.
Another related quote is attributed to Albert Einstein
Make everything as simple as possible, but not simpler.
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 is usually the best.
Concepts are not to be multiplied beyond necessity–the corollary of which is: nor are they to be integrated in disregard of necessity.
Occam's razor is not an embargo against the positing of any kind of entity, or a recommendation of the simplest theory come what may (note that simplest theory is something like "only I exist" or "nothing exists").
The other things in question are the evidential support for the theory. Therefore, according to the principle, a simpler but less correct theory should not be preferred over a more complex but more correct one. It is this fact which gives the lie to the common misinterpretation of Occam's Razor that "the simplest" one is usually the correct one.
For instance, classical physics is simpler than more recent theories; nonetheless it should not be preferred over them, because it is demonstrably wrong in certain respects.
Occam's razor is used to adjudicate between theories that have already passed 'theoretical scrutiny' tests, and which are equally well-supported by the evidence. Furthermore, it may be used to prioritize empirical testing between two equally plausible but unequally testable hypotheses; thereby minimizing costs and wastes while increasing chances of falsification of the simpler-to-test hypothesis.
Another contentious aspect of the Razor is that a theory can become more complex in terms of its structure (or syntax), while its ontology (or semantics) becomes simpler, or vice versa. The theory of relativity is often given as an example of the proliferation of complex words to describe a simple concept.
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.
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, no one anti-razor has perpetuated in as much notoriety as Occam's razor, although this could be the case of the Late Renaissance Italian motto of unknown attribution Se non vero, ben trovato ("Even if it is not true, it is well conceived") when referred to a particularly artful explanation.
Anti-razors have also been created by Gottfried Wilhelm Leibniz (1646–1716), Immanuel Kant (1724–1804), and Karl Menger. Leibniz's version took the form of a principle of plenitude, as Arthur Lovejoy has called it, the idea being that God created the most varied and populous of possible worlds. 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." Einstein supposedly remarked, "Everything should be made as simple as possible, but not simpler."
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 Argentinean writer Jorge Luis Borges in his story/mock-essay Tln, 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.'
While not technically contradicting the razor's notion that (other things being equal) "the simplest explanation is always the best", the reverse corollary – that the best explanation is not always the simplest – is well expressed by the Sir Arthur Conan Doyle character, Sherlock Holmes, especially in the following famous quote from The Sign of the Four: "When you have eliminated the impossible, whatever remains, however improbable, must be the truth."
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