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Occam's razor, Ockham's razor, or Ocham's razor ( la, novacula Occami), also known as the principle of parsimony or the law of parsimony ( la, lex parsimoniae), is the problem-solving principle that "entities should not be multiplied beyond necessity". It is generally understood in the sense that with competing theories or explanations, the simpler one, for example a model with fewer parameters, is to be preferred. The idea is frequently attributed to English
Franciscan , image = FrancescoCoA PioM.svg , image_size = 200px , caption = A cross, Christ's arm and Saint Francis's arm, a universal symbol of the Franciscans , abbreviation = OFM , predecessor = , ...
friar A friar is a member of one of the mendicant orders founded in the twelfth or thirteenth century; the term distinguishes the mendicants' itinerant apostolic character, exercised broadly under the jurisdiction of a superior general, from the ...
William of Ockham (), a scholastic philosopher and theologian, although he never used these exact words. This
philosophical razor In philosophy, a razor is a principle or rule of thumb that allows one to eliminate ("shave off") unlikely explanations for a phenomenon, or avoid unnecessary actions. Razors include: *Occam's razor: Simpler explanations are more likely to be corr ...
advocates that when presented with competing
hypotheses A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obse ...
about the same prediction, one should select the solution with the fewest assumptions, and that this is not meant to be a way of choosing between hypotheses that make different predictions. Similarly, in science, Occam's razor is used as an
abductive Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th centur ...
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
in the development of theoretical models rather than as a rigorous arbiter between candidate models.Hugh G. Gauch, ''Scientific Method in Practice, Cambridge University Press'', 2003, , . In the
scientific method The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientifi ...
, Occam's razor is not considered an irrefutable principle of
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
or a scientific result; the preference for simplicity in the scientific method is based on the falsifiability criterion. For each accepted explanation of a phenomenon, there may be an extremely large, perhaps even incomprehensible, number of possible and more complex alternatives. Since failing explanations can always be burdened with ''ad hoc'' hypotheses to prevent them from being falsified, simpler theories are preferable to more complex ones because they tend to be more testable.


History

The phrase ''Occam's razor'' did not appear until a few centuries after William of Ockham's death in 1347.
Libert Froidmont Libert Froidmont (Latin: ''Libertus Fromondus''; 3 September 1587, in Haccourt-Liège – 28 October 1653, in Louvain) a son of Gerard Libert de Froidmont and Marguerite Radoux, was a Liégeois theologian and scientist. He was a close companion to ...
, in his ''On Christian Philosophy of the Soul'', takes credit for the phrase, speaking of "''novacula occami''". Ockham did not invent this principle, but the "razor"—and its association with him—may be due to the frequency and effectiveness with which he used it. Ockham stated the principle in various ways, but the most popular version, "Entities are not to be multiplied without necessity" () was formulated by the Irish
Franciscan , image = FrancescoCoA PioM.svg , image_size = 200px , caption = A cross, Christ's arm and Saint Francis's arm, a universal symbol of the Franciscans , abbreviation = OFM , predecessor = , ...
philosopher John Punch in his 1639 commentary on the works of
Duns Scotus John Duns Scotus ( – 8 November 1308), commonly called Duns Scotus ( ; ; "Duns the Scot"), was a Scottish Catholic priest and Franciscan friar, university professor, philosopher, and theologian. He is one of the four most important ...
.Johannes Poncius's commentary on John Duns Scotus's ''Opus Oxoniense,'' book III, dist. 34, q. 1. in John Duns Scotus ''Opera Omnia'', vol.15, Ed. Luke Wadding, Louvain (1639), reprinted Paris: Vives, (1894) p.483a


Formulations before William of Ockham

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), Robert Grosseteste (1175–1253),
Maimonides Musa ibn Maimon (1138–1204), commonly known as Maimonides (); la, Moses Maimonides and also referred to by the acronym Rambam ( he, רמב״ם), was a Sephardic Jewish philosopher who became one of the most prolific and influential Torah ...
(Moses ben-Maimon, 1138–1204), and even
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
(384–322 BC). Aristotle writes in his ''
Posterior Analytics The ''Posterior Analytics'' ( grc-gre, Ἀναλυτικὰ Ὕστερα; la, Analytica Posteriora) is a text from Aristotle's '' Organon'' that deals with demonstration, definition, and scientific knowledge. The demonstration is distinguis ...
'', "We may assume the superiority ther things being equalof the demonstration which derives from fewer postulates or hypotheses."
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
() stated, "We consider it a good principle to explain the phenomena by the simplest hypothesis possible." Chap 9. p. 241. Phrases such as "It is vain to do with more what can be done with fewer" and "A plurality is not to be posited without necessity" were commonplace in 13th-century scholastic writing. Robert Grosseteste, in ''Commentary on'' ristotle's''the Posterior Analytics Books'' (''Commentarius in Posteriorum Analyticorum Libros'') (c. 1217–1220), declares: "That is better and more valuable which requires fewer, other circumstances being equal... For if one thing were demonstrated from many and another thing from fewer equally known premises, clearly that is better which is from fewer because it makes us know quickly, just as a universal demonstration is better than particular because it produces knowledge from fewer premises. Similarly in natural science, in moral science, and in metaphysics the best is that which needs no premises and the better that which needs the fewer, other circumstances being equal." The '' Summa Theologica'' of
Thomas Aquinas Thomas Aquinas, Dominican Order, OP (; it, Tommaso d'Aquino, lit=Thomas of Aquino, Italy, Aquino; 1225 – 7 March 1274) was an Italian Dominican Order, Dominican friar and Catholic priest, priest who was an influential List of Catholic philo ...
(1225–1274) states that "it is superfluous to suppose that what can be accounted for by a few principles has been produced by many." Aquinas uses this principle to construct an objection to God's existence, an objection that he in turn answers and refutes generally (cf. ''
quinque viae The ''Quinque viæ'' (Latin for "Five Ways") (sometimes called "five proofs") are five logical arguments for the existence of God summarized by the 13th-century Catholic philosopher and theologian Thomas Aquinas in his book '' Summa Theologica ...
''), and specifically, through an argument based on
causality Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
. Hence, Aquinas acknowledges the principle that today is known as Occam's razor, but prefers causal explanations to other simple explanations (cf. also
Correlation does not imply causation The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. The id ...
).


William of Ockham

William of Ockham (''circa'' 1287–1347) was an English Franciscan friar and
theologian Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing the ...
, an influential medieval
philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
and a nominalist. His popular fame as a great logician rests chiefly on the maxim attributed to him and known as Occam's razor. The term ''razor'' refers to distinguishing between two hypotheses either by "shaving away" unnecessary assumptions or cutting apart two similar conclusions. While it has been claimed that Occam's razor is not found in any of William's writings, one can cite statements such as William of Ockham – Wikiquote ("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). Nevertheless, the precise words sometimes attributed to William of Ockham, (Entities must not be multiplied beyond necessity), are absent in his extant works; this particular phrasing comes from John Punch, who described the principle as a "common axiom" (''axioma vulgare'') of the Scholastics. William of Ockham's contribution seems to restrict the operation of this principle in matters pertaining to miracles and God's power; so, in the
Eucharist The Eucharist (; from Greek , , ), also known as Holy Communion and the Lord's Supper, is a Christian rite that is considered a sacrament in most churches, and as an ordinance in others. According to the New Testament, the rite was institu ...
, a plurality of miracles is possible, simply because it pleases God. This principle is sometimes phrased as ("Plurality should not be posited without necessity"). In his ''Summa Totius Logicae'', i. 12, William of Ockham cites the principle of economy, ("It is futile to do with more things that which can be done with fewer"; Thorburn, 1918, pp. 352–53;
Kneale Kneale is a surname of Manx origin. The name is an Anglicisation of the Gaelic ''Mac Néill'', meaning "son of ''Niall''".. For the surname ''Kneale'', this webpage cited: People *Bryan Kneale, (born 1930), Manx, an artist and sculptor *Campbell ...
and Kneale, 1962, p. 243.)


Later formulations

To quote
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
, "We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. Therefore, to the same natural effects we must, as far as possible, assign the same causes." In the sentence hypotheses non fingo, Newton affirms the success of this approach.
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
offers a particular version of Occam's razor: "Whenever possible, substitute constructions out of known entities for inferences to unknown entities." Around 1960, Ray Solomonoff founded the theory of universal inductive inference, the theory of prediction based on observationsfor example, predicting the next symbol based upon a given series of symbols. The only assumption is that the environment follows some unknown but computable probability distribution. This theory is a mathematical formalization of Occam's razor.Induction: From Kolmogorov and Solomonoff to De Finetti and Back to Kolmogorov JJ McCall – Metroeconomica, 2004 – Wiley Online Library. Another technical approach to Occam's razor is ontological parsimony. Parsimony means spareness and is also referred to as the Rule of Simplicity. This is considered a strong version of Occam's razor. A variation used in medicine is called the " Zebra": a physician should reject an exotic medical diagnosis when a more commonplace explanation is more likely, derived from
Theodore Woodward Theodore Englar Woodward (March 22, 1914 – July 11, 2005) was an American medical researcher in the field of medicine at the University of Maryland, Baltimore. In 1948, he received a Nobel Prize nomination for his role in finding cures for typh ...
's dictum "When you hear hoofbeats, think of horses not zebras".
Ernst Mach Ernst Waldfried Josef Wenzel Mach ( , ; 18 February 1838 – 19 February 1916) was a Moravian-born Austrian physicist and philosopher, who contributed to the physics of shock waves. The ratio of one's speed to that of sound is named the Mach n ...
formulated the stronger version of Occam's razor into
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, which he called the Principle of Economy stating: "Scientists must use the simplest means of arriving at their results and exclude everything not perceived by the senses." This principle goes back at least as far as Aristotle, who wrote "Nature operates in the shortest way possible." The idea of parsimony or simplicity in deciding between theories, though not the intent of the original expression of Occam's razor, has been assimilated into common culture as the widespread layman's formulation that "the simplest explanation is usually the correct one."


Justifications


Aesthetic

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.
Thomas Aquinas Thomas Aquinas, Dominican Order, OP (; it, Tommaso d'Aquino, lit=Thomas of Aquino, Italy, Aquino; 1225 – 7 March 1274) was an Italian Dominican Order, Dominican friar and Catholic priest, priest who was an influential List of Catholic philo ...
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 fone suffices." Beginning in the 20th century,
epistemological Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Episte ...
justifications based on induction,
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, pragmatism, and especially
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
have become more popular among philosophers.


Empirical

Occam's razor has gained strong empirical support in helping to converge on better theories (see Uses section below for some examples). In the related concept of
overfitting mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitt ...
, excessively complex models are affected by statistical noise (a problem also known as the bias-variance trade-off), whereas simpler models may capture the underlying structure better and may thus have better
predictive A prediction (Latin ''præ-'', "before," and ''dicere'', "to say"), or forecast, is a statement about a future event or data. They are often, but not always, based upon experience or knowledge. There is no universal agreement about the exac ...
performance. It is, however, often difficult to deduce which part of the data is noise (cf. model selection,
test set In machine learning, a common task is the study and construction of algorithms that can learn from and make predictions on data. Such algorithms function by making data-driven predictions or decisions, through building a mathematical model from ...
, minimum description length,
Bayesian inference Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and ...
, etc.).


Testing the razor

The razor's statement that "other things being equal, simpler explanations are generally better than more complex ones" is amenable to empirical testing. Another interpretation of the razor's statement would be that "simpler hypotheses are generally better than the complex ones". The procedure to test the former interpretation would compare the track records of simple and comparatively complex explanations. If one accepts the first interpretation, the validity of Occam's razor as a tool would then have to be rejected if the more complex explanations were more often correct than the less complex ones (while the converse would lend support to its use). If the latter interpretation is accepted, the validity of Occam's razor as a tool could possibly be accepted if the simpler hypotheses led to correct conclusions more often than not. Even if some increases in complexity are sometimes necessary, there still remains a justified general bias toward the simpler of two competing explanations. To understand why, consider that for each accepted explanation of a phenomenon, there is always an infinite number of possible, more complex, and ultimately incorrect, alternatives. This is so because one can always burden a failing explanation with an
ad hoc hypothesis In science and philosophy, an ''ad hoc'' hypothesis is a hypothesis added to a theory in order to save it from being falsified. Often, ''ad hoc'' hypothesizing is employed to compensate for anomalies not anticipated by the theory in its unmodifi ...
. Ad hoc hypotheses are justifications that prevent theories from being falsified. For example, if a man, accused of breaking a vase, makes
supernatural Supernatural refers to phenomena or entities that are beyond the laws of nature. The term is derived from Medieval Latin , from Latin (above, beyond, or outside of) + (nature) Though the corollary term "nature", has had multiple meanings si ...
claims that leprechauns were responsible for the breakage, a simple explanation might be that the man did it, but ongoing ad hoc justifications (e.g. "... and that's not me breaking it on the film; they tampered with that, too") could successfully prevent complete disproof. This endless supply of elaborate competing explanations, called saving hypotheses, cannot be technically ruled out – except by using Occam's razor.Stanovich, Keith E. (2007). ''How to Think Straight About Psychology''. Boston: Pearson Education, pp. 19–33. Any more complex theory might still possibly be true. A study of the predictive validity of Occam's razor found 32 published papers that included 97 comparisons of economic forecasts from simple and complex forecasting methods. None of the papers provided a balance of evidence that complexity of method improved forecast accuracy. In the 25 papers with quantitative comparisons, complexity increased forecast errors by an average of 27 percent.


Practical considerations and pragmatism


Mathematical

One justification of Occam's razor is a direct result of basic
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
. 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 have also been other attempts to derive Occam's razor from probability theory, including notable attempts made by Harold Jeffreys and E. T. Jaynes. The probabilistic (Bayesian) basis for Occam's razor is elaborated by David J. C. MacKay in chapter 28 of his book ''Information Theory, Inference, and Learning Algorithms'', where he emphasizes that a prior bias in favor of simpler models is not required. William H. Jefferys and James O. Berger (1991) generalize and quantify the original formulation's "assumptions" concept as the degree to which a proposition is unnecessarily accommodating to possible observable data. (preprint available as "Sharpening Occam's Razor on a Bayesian Strop"). They state, "A hypothesis with fewer adjustable parameters will automatically have an enhanced posterior probability, due to the fact that the predictions it makes are sharp." The use of "sharp" here is not only a tongue-in-cheek reference to the idea of a razor, but also indicates that such predictions are more accurate than competing predictions. The model they propose balances the precision of a theory's predictions against their sharpness, preferring theories that sharply make correct predictions over theories that accommodate a wide range of other possible results. This, again, reflects the mathematical relationship between key concepts in
Bayesian inference Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and ...
(namely
marginal probability In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variab ...
, conditional probability, and
posterior probability The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...
). The bias–variance tradeoff is a framework that incorporates the Occam's razor principle in its balance between overfitting (associated with lower bias but higher variance) and underfitting (associated with lower variance but higher bias).


Other philosophers


Karl Popper

Karl Popper Sir Karl Raimund Popper (28 July 1902 – 17 September 1994) was an Austrian-British philosopher, academic and social commentator. One of the 20th century's most influential philosophers of science, Popper is known for his rejection of the ...
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". 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.


Elliott Sober

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, in the sense that it requires less information to a question. He has since rejected this account of simplicity, purportedly because it fails to provide an
epistemic Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Episte ...
justification for simplicity. He now believes 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 ''Sui generis'' ( , ) is a Latin phrase that means "of its/their own kind", "in a class by itself", therefore "unique". A number of disciplines use the term to refer to unique entities. These include: * Biology, for species that do not fit in ...
'' 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 use 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?'"Paper as PDF.
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Richard Swinburne

Richard Swinburne argues for simplicity on logical grounds: According to Swinburne, since our choice of theory cannot be determined by data (see Underdetermination and
Duhem–Quine thesis The Duhem–Quine thesis, also called the Duhem–Quine problem, after Pierre Duhem and Willard Van Orman Quine, is that in science it is impossible to experimentally test a scientific hypothesis in isolation, because an empirical test of the ...
), we must rely on some criterion to determine which theory to use. Since it is absurd to have no logical method for settling on one hypothesis amongst an infinite number of equally data-compliant hypotheses, we should choose the simplest theory: "Either science is irrational n the way it judges theories and predictions probableor the principle of simplicity is a fundamental synthetic a priori truth.".


Ludwig Wittgenstein

From the ''
Tractatus Logico-Philosophicus The ''Tractatus Logico-Philosophicus'' (widely abbreviated and cited as TLP) is a book-length philosophical work by the Austrian philosopher Ludwig Wittgenstein which deals with the relationship between language and reality and aims to define th ...
'': * 3.328 "If a sign is not necessary then it is meaningless. That is the meaning of Occam's Razor." : (If everything in the symbolism works as though a sign had meaning, then it has meaning.) * 4.04 "In the proposition, there must be exactly as many things distinguishable as there are in the state of affairs, which it represents. They must both possess the same logical (mathematical) multiplicity (cf. Hertz's Mechanics, on Dynamic Models)." * 5.47321 "Occam's Razor is, of course, not an arbitrary rule nor one justified by its practical success. It simply says that unnecessary elements in a symbolism mean nothing. Signs which serve one purpose are logically equivalent; signs which serve no purpose are logically meaningless." and on the related concept of "simplicity": * 6.363 "The procedure of induction consists in accepting as true the simplest law that can be reconciled with our experiences."


Uses


Science and the scientific method

In
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...
, Occam's razor is used as a
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
to guide scientists in developing theoretical models rather than as an arbiter between published models. In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, parsimony was an important heuristic in
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
's formulation of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
,L. Nash, The Nature of the Natural Sciences, Boston: Little, Brown (1963). in the development and application of the principle of least action by Pierre Louis Maupertuis and
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
, and in the development of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
by Max Planck,
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...
and Louis de Broglie. In
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
, Occam's razor is often an important heuristic when developing a model of a
reaction mechanism In chemistry, a reaction mechanism is the step by step sequence of elementary reactions by which overall chemical change occurs. A chemical mechanism is a theoretical conjecture that tries to describe in detail what takes place at each stage o ...
.RA Jackson, Mechanism: An Introduction to the Study of Organic Reactions, Clarendon, Oxford, 1972.Carpenter, B. K. (1984). ''Determination of Organic Reaction Mechanism'', New York: Wiley-Interscience. Although it is useful as a heuristic in developing models of reaction mechanisms, it has been shown to fail as a criterion for selecting among some selected published models. In this context, Einstein himself expressed caution when he formulated Einstein's Constraint: "It can scarcely be denied 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". An often-quoted version of this constraint (which cannot be verified as posited by Einstein himself) says "Everything should be kept as simple as possible, but not simpler." In the scientific method, parsimony is an
epistemological Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Episte ...
,
metaphysical Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
or
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
preference, not an irrefutable principle of
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
or 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 support more complex theories than do existing data. Science prefers the simplest explanation that is consistent with the data available at a given time, but the simplest explanation may be ruled out as new data become available. That is, 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 has meaning only in a very specific context of inquiry. Several 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. It has been suggested that Occam's razor is a widely accepted example of extraevidential consideration, even though it is entirely a metaphysical assumption. Most of the time, however, 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 en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = Postal code , postal_code = , timezone = CET , utc_offset = +1 , timezone_DST ...
and
Jeans Jeans are a type of pants or trousers made from denim or dungaree cloth. Often the term "jeans" refers to a particular style of trousers, called "blue jeans", with copper-riveted pockets which were invented by Jacob W. Davis in 1871 and pa ...
radiation laws and used Occam's razor logic to formulate the quantum hypothesis, even resisting that hypothesis as it became more obvious that it was correct. Appeals to simplicity were used to argue against the phenomena of meteorites, ball lightning, continental drift, and
reverse transcriptase A reverse transcriptase (RT) is an enzyme used to generate complementary DNA (cDNA) from an RNA template, a process termed reverse transcription. Reverse transcriptases are used by viruses such as HIV and hepatitis B to replicate their genom ...
. One can argue for atomic building blocks for matter, because it provides a simpler explanation for the observed reversibility of both and chemical reactions as simple separation and rearrangements of atomic building blocks. At the time, however, the
atomic theory Atomic theory is the scientific theory that matter is composed of particles called atoms. Atomic theory traces its origins to an ancient philosophical tradition known as atomism. According to this idea, if one were to take a lump of matter ...
was considered more complex because it implied the existence of invisible particles that had not been directly detected.
Ernst Mach Ernst Waldfried Josef Wenzel Mach ( , ; 18 February 1838 – 19 February 1916) was a Moravian-born Austrian physicist and philosopher, who contributed to the physics of shock waves. The ratio of one's speed to that of sound is named the Mach n ...
and the logical positivists rejected
John Dalton John Dalton (; 5 or 6 September 1766 – 27 July 1844) was an English chemist, physicist and meteorologist. He is best known for introducing the atomic theory into chemistry, and for his research into Color blindness, colour blindness, which ...
's
atomic theory Atomic theory is the scientific theory that matter is composed of particles called atoms. Atomic theory traces its origins to an ancient philosophical tradition known as atomism. According to this idea, if one were to take a lump of matter ...
until the reality of atoms was more evident in
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, as shown by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
. In the same way, postulating the aether is more complex than transmission of light through a
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
. At the time, however, all known waves propagated through a physical medium, and it seemed simpler to postulate the existence of a medium than to theorize about wave propagation without a medium. Likewise,
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
's idea of light particles seemed simpler than
Christiaan Huygens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists o ...
's idea of waves, so many favored it. In this case, as it turned out, neither the wave—nor the particle—explanation alone suffices, as light behaves like waves and like particles. Three axioms presupposed by the scientific method are realism (the existence of objective reality), the existence of natural laws, and the constancy of 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 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. If multiple models of natural law make exactly the same testable predictions, they are equivalent and there is no need for parsimony to choose a preferred one. 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 that make the same testable predictions.


Biology

Biologists or philosophers of biology use Occam's razor in either of two contexts both in
evolutionary biology Evolutionary biology is the subfield of biology that studies the evolutionary processes (natural selection, common descent, speciation) that produced the diversity of life on Earth. It is also defined as the study of the history of life ...
: the units of selection controversy and
systematics Biological systematics is the study of the diversification of living forms, both past and present, and the relationships among living things through time. Relationships are visualized as evolutionary trees (synonyms: cladograms, phylogenetic t ...
. George C. Williams in his book '' Adaptation and Natural Selection'' (1966) argues that the best way to explain
altruism Altruism is the principle and moral practice of concern for the welfare and/or happiness of other human beings or animals, resulting in a quality of life both material and spiritual. It is a traditional virtue in many cultures and a cor ...
among animals is based on low-level (i.e., individual) selection as opposed to high-level group selection. Altruism is defined by some evolutionary biologists (e.g., R. Alexander, 1987; W. D. Hamilton, 1964) as behavior that is beneficial to others (or to the group) at a cost to the individual, and many posit individual selection as the mechanism that explains altruism solely in terms of the behaviors of individual organisms acting in their own self-interest (or in the interest of their genes, via kin selection). Williams was arguing against the perspective of others who propose selection at the level of the group as an evolutionary mechanism that selects for altruistic traits (e.g., D. S. Wilson & E. O. Wilson, 2007). 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
Morgan's Canon Morgan's Canon, also known as Lloyd Morgan's Canon, Morgan's Canon of Interpretation or the principle or law of parsimony, is a fundamental precept of comparative (animal) psychology, coined by 19th-century British psychologist C. Lloyd Morgan. In ...
: "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 Richard Dawkins (born 26 March 1941) is a British evolutionary biologist and author. He is an emeritus fellow of New College, Oxford and was Professor for Public Understanding of Science in the University of Oxford from 1995 to 2008. An ...
' '' The Selfish Gene'', have contended that Morgan's Canon is not the simplest and most basic explanation. Dawkins argues the way evolution works is that the genes propagated in most copies 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 Zoology ()The pronunciation of zoology as is usually regarded as nonstandard, though it is not uncommon. is the branch of biology that studies the animal kingdom, including the structure, embryology, evolution, classification, habits, an ...
provides an example. Muskoxen, when threatened by
wolves The wolf (''Canis lupus''; : wolves), also known as the gray wolf or grey wolf, is a large canine native to Eurasia and North America. More than thirty subspecies of ''Canis lupus'' have been recognized, and gray wolves, as popularly un ...
, form a circle with the males on the outside and the females and young on the inside. This is 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. Another interpretation is kin selection: if the males are protecting their offspring, they are protecting copies of their own alleles. Engaging in this behavior would be favored by individual selection if the cost to the male musk ox is less than half of the benefit received by his calf – which could easily be the case if wolves have an easier time killing calves than adult males. It could also be the case that male musk oxen would be individually less likely to be killed by wolves if they stood in a circle with their horns pointing out, regardless of whether they were protecting the females and offspring. That would be an example of regular natural selection – a phenomenon called "the selfish herd".
Systematics Biological systematics is the study of the diversification of living forms, both past and present, and the relationships among living things through time. Relationships are visualized as evolutionary trees (synonyms: cladograms, phylogenetic t ...
is the branch of
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
that attempts to establish patterns of relationship among biological taxa, today generally thought to reflect evolutionary history. It is also concerned with their classification. There are three primary camps in systematics: cladists, pheneticists, and evolutionary taxonomists. Cladists hold that classification should be based on
synapomorphies In phylogenetics, an apomorphy (or derived trait) is a novel character or character state that has evolved from its ancestral form (or plesiomorphy). A synapomorphy is an apomorphy shared by two or more taxa and is therefore hypothesized to ha ...
(shared, derived character states), pheneticists contend that overall similarity (synapomorphies and complementary symplesiomorphies) is the determining criterion, while evolutionary taxonomists say that both genealogy and similarity count in classification (in a manner determined by the evolutionary taxonomist). It is among the cladists that Occam's razor is applied, through the method of ''cladistic parsimony''. Cladistic parsimony (or
maximum parsimony In phylogenetics, maximum parsimony is an optimality criterion under which the phylogenetic tree that minimizes the total number of character-state changes (or miminizes the cost of differentially weighted character-state changes) is preferred. ...
) is a method of phylogenetic inference that yields phylogenetic trees (more specifically, cladograms). Cladograms are branching, diagrams used to represent hypotheses of relative degree of relationship, based on
synapomorphies In phylogenetics, an apomorphy (or derived trait) is a novel character or character state that has evolved from its ancestral form (or plesiomorphy). A synapomorphy is an apomorphy shared by two or more taxa and is therefore hypothesized to ha ...
. Cladistic parsimony is used to select as the preferred hypothesis of relationships the cladogram that requires the fewest implied character state transformations (or smallest weight, if characters are differentially weighted). Critics of the cladistic approach often observe that for some types of data, parsimony could produce the wrong results, regardless of how much data is collected (this is called statistical inconsistency, or long branch attraction). However, this criticism is also potentially true for any type of phylogenetic inference, unless the model used to estimate the tree reflects the way that evolution actually happened. Because this information is not empirically accessible, the criticism of statistical inconsistency against parsimony holds no force. For a book-length 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 Sober's article "Let's Razor Ockham's Razor" (1990). Other methods for inferring evolutionary relationships use parsimony in a more general way. Likelihood methods for phylogeny use parsimony as they do for all likelihood tests, with hypotheses requiring fewer differing parameters (i.e., numbers or different rates of character change or different frequencies of character state transitions) being treated as null hypotheses relative to hypotheses requiring more 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 Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...
, a close cousin of likelihood, which uses Occam's razor in the same way. Of course, the choice of the "shortest tree" relative to a not-so-short tree under any optimality criterion (smallest distance, fewest steps, or maximum likelihood) is always based on parsimony
Francis Crick Francis Harry Compton Crick (8 June 1916 – 28 July 2004) was an English molecular biologist, biophysicist, and neuroscientist. He, James Watson, Rosalind Franklin, and Maurice Wilkins played crucial roles in deciphering the helical stru ...
has commented on potential limitations of Occam's razor in biology. He advances the argument that because biological systems are the products of (an ongoing) 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." This is an ontological critique of parsimony. In
biogeography Biogeography is the study of the distribution of species and ecosystems in geographic space and through geological time. Organisms and biological communities often vary in a regular fashion along geographic gradients of latitude, elevation, ...
, parsimony is used to infer ancient vicariant events or migrations of
species In biology, a species is the basic unit of classification and a taxonomic rank of an organism, as well as a unit of biodiversity. A species is often defined as the largest group of organisms in which any two individuals of the appropriat ...
or
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction usi ...
s by observing the geographic distribution and relationships of existing
organism In biology, an organism () is any living system that functions as an individual entity. All organisms are composed of cells ( cell theory). Organisms are classified by taxonomy into groups such as multicellular animals, plants, and fu ...
s. Given the phylogenetic tree, ancestral population subdivisions are inferred to be those that require the minimum amount of change.


Religion

In the
philosophy of religion Philosophy of religion is "the philosophical examination of the central themes and concepts involved in religious traditions". Philosophical discussions on such topics date from ancient times, and appear in the earliest known texts concerning p ...
, Occam's razor is sometimes applied to the existence of God. William of Ockham himself was a Christian. He believed in God, and in the authority of Scripture; he writes that "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." Ockham believed that an explanation has no sufficient basis in reality when it does not harmonize with reason, experience, or the Bible. However, unlike many theologians of his time, Ockham did not believe God could be logically proven with arguments. To Ockham, science was a matter of discovery, but theology was a matter of revelation and
faith Faith, derived from Latin ''fides'' and Old French ''feid'', is confidence or trust in a person, thing, or In the context of religion, one can define faith as "belief in God or in the doctrines or teachings of religion". Religious people ofte ...
. He states: "only faith gives us access to theological truths. The ways of God are not open to reason, for God has freely chosen to create a world and establish a way of salvation within it apart from any necessary laws that human logic or rationality can uncover." St. Thomas Aquinas, in the '' Summa Theologica'', uses a formulation of Occam's razor to construct an objection to the idea that God exists, which he refutes directly with a counterargument:
Further, it is superfluous to suppose that what can be accounted for by a few principles has been produced by many. But it seems that everything we see in the world can be accounted for by other principles, supposing God did not exist. For all natural things can be reduced to one principle which is nature; and all voluntary things can be reduced to one principle which is human reason, or will. Therefore there is no need to suppose God's existence.
In turn, Aquinas answers this with the ''
quinque viae The ''Quinque viæ'' (Latin for "Five Ways") (sometimes called "five proofs") are five logical arguments for the existence of God summarized by the 13th-century Catholic philosopher and theologian Thomas Aquinas in his book '' Summa Theologica ...
'', and addresses the particular objection above with the following answer:
Since nature works for a determinate end under the direction of a higher agent, whatever is done by nature must needs be traced back to God, as to its first cause. So also whatever is done voluntarily must also be traced back to some higher cause other than human reason or will, since these can change or fail; for all things that are changeable and capable of defect must be traced back to an immovable and self-necessary first principle, as was shown in the body of the Article.
Rather than argue for the necessity of a god, some theists base their belief upon grounds independent of, or prior to, reason, making Occam's razor irrelevant. This was the stance of
Søren Kierkegaard Søren Aabye Kierkegaard ( , , ; 5 May 1813 – 11 November 1855) was a Danish theologian, philosopher, poet, social critic, and religious author who is widely considered to be the first existentialist philosopher. He wrote critical texts on ...
, who viewed belief in God as a leap of faith that sometimes directly opposed reason. This is also the doctrine of Gordon Clark's
presuppositional apologetics Presuppositionalism is an epistemological school of Christian apologetics that examines the presuppositions on which worldviews are based, and invites comparison and contrast between the results of those presuppositions. It claims that apart from ...
, with the exception that Clark never thought the leap of faith was contrary to reason (see also
Fideism Fideism () is an epistemological theory which maintains that faith is independent of reason, or that reason and faith are hostile to each other and faith is superior at arriving at particular truths (see natural theology). The word ''fideism'' c ...
). Various arguments in favor of God establish God as a useful or even necessary assumption. Contrastingly some anti-theists hold firmly to the belief that assuming the existence of God introduces unnecessary complexity (Schmitt 2005, e.g., the
Ultimate Boeing 747 gambit The Ultimate Boeing 747 gambit is a counter-argument to modern versions of the Teleological argument, argument from design for the existence of God. It was introduced by Richard Dawkins in chapter 4 of his 2006 book ''The God Delusion'', "Why t ...
). Another application of the principle is to be found in the work of
George Berkeley George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley ( Bishop of Cloyne of the Anglican Church of Ireland) – was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immate ...
(1685–1753). Berkeley was an idealist who believed that all of reality could be explained in terms of the mind alone. He invoked Occam's razor against
materialism Materialism is a form of philosophical monism which holds matter to be the fundamental substance in nature, and all things, including mental states and consciousness, are results of material interactions. According to philosophical materialis ...
, stating that matter was not required by his metaphysics and was thus eliminable. One potential problem with this belief is that it's possible, given Berkeley's position, to find solipsism itself more in line with the razor than a God-mediated world beyond a single thinker. Occam's razor may also be recognized in the apocryphal story about an exchange between
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarize ...
and
Napoleon Napoleon Bonaparte ; it, Napoleone Bonaparte, ; co, Napulione Buonaparte. (born Napoleone Buonaparte; 15 August 1769 – 5 May 1821), later known by his regnal name Napoleon I, was a French military commander and political leader wh ...
. It is said that in praising Laplace for one of his recent publications, the emperor asked how it was that the name of God, which featured so frequently in the writings of
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaatheism Atheism, in the broadest sense, is an absence of belief in the existence of deities. Less broadly, atheism is a rejection of the belief that any deities exist. In an even narrower sense, atheism is specifically the position that there no d ...
, more careful consideration suggests that he may instead have intended merely to illustrate the power of
methodological naturalism In philosophy, naturalism is the idea or belief that only natural laws and forces (as opposed to supernatural ones) operate in the universe. According to philosopher Steven Lockwood, naturalism can be separated into an ontological sense and a me ...
, or even simply that the fewer logical premises one assumes, the stronger is one's conclusion.


Philosophy of mind

In his article "Sensations and Brain Processes" (1959), J. J. C. Smart invoked Occam's razor with the aim to justify his preference of the mind-brain identity theory over spirit-body dualism. Dualists state that there are two kinds of substances in the universe: physical (including the body) and spiritual, which is non-physical. In contrast, identity theorists state that everything is physical, including consciousness, and that there is nothing nonphysical. Though it is impossible to appreciate the spiritual when limiting oneself to the physical, Smart maintained that identity theory explains all phenomena by assuming only a physical reality. Subsequently, Smart has been severely criticized for his use (or misuse) of Occam's razor and ultimately retracted his advocacy of it in this context. Paul Churchland (1984) states that by itself Occam's razor is inconclusive regarding duality. In a similar way, Dale Jacquette (1994) stated that Occam's razor has been used in attempts to justify 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.


Penal ethics

In penal theory and the philosophy of punishment, parsimony refers specifically to taking care in the distribution of
punishment Punishment, commonly, is the imposition of an undesirable or unpleasant outcome upon a group or individual, meted out by an authority—in contexts ranging from child discipline to criminal law—as a response and deterrent to a particular ac ...
in order to avoid excessive punishment. In the utilitarian approach to the philosophy of punishment,
Jeremy Bentham Jeremy Bentham (; 15 February 1748 ld Style and New Style dates, O.S. 4 February 1747– 6 June 1832) was an English philosopher, jurist, and social reformer regarded as the founder of modern utilitarianism. Bentham defined as the "fundam ...
's "parsimony principle" states that any punishment greater than is required to achieve its end is unjust. The concept is related but not identical to the legal concept of proportionality. Parsimony is a key consideration of the modern restorative justice, and is a component of utilitarian approaches to punishment, as well as the prison abolition movement. Bentham believed that true parsimony would require punishment to be individualised to take account of the
sensibility Sensibility refers to an acute perception of or responsiveness toward something, such as the emotions of another. This concept emerged in eighteenth-century Britain, and was closely associated with studies of sense perception as the means thr ...
of the individual—an individual more sensitive to punishment should be given a proportionately lesser one, since otherwise needless pain would be inflicted. Later utilitarian writers have tended to abandon this idea, in large part due to the impracticality of determining each alleged criminal's relative sensitivity to specific punishments.


Probability theory and statistics

Marcus Hutter's universal artificial intelligence builds upon Solomonoff's mathematical formalization of the razor to calculate the expected value of an action. There are various papers in scholarly journals deriving formal versions of Occam's razor from probability theory, applying it in
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properti ...
, and using it to come up with criteria for penalizing complexity in statistical inference. PapersChris S. Wallace and David M. Boulton; ''Computer Journal'', Volume 11, Issue 2, 1968 Page(s):185–194, "An information measure for classification."Chris S. Wallace and David L. Dowe; ''Computer Journal'', Volume 42, Issue 4, Sep 1999 Page(s):270–283, "Minimum Message Length and Kolmogorov Complexity." have suggested a connection between Occam's razor and Kolmogorov complexity. One of the problems with the original formulation of the razor is that it only applies to models with the same explanatory power (i.e., it only tells us to prefer the simplest of equally good models). A more general form of the razor can be derived from Bayesian model comparison, which is based on Bayes factors and can be used to compare models that don't fit the observations 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 In statistics, the Bayesian information criterion (BIC) or Schwarz information criterion (also SIC, SBC, SBIC) is a criterion for model selection among a finite set of models; models with lower BIC are generally preferred. It is based, in part, o ...
, Variational Bayesian methods, false discovery rate, and
Laplace's method In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form :\int_a^b e^ \, dx, where f(x) is a twice- differentiable function, ''M'' is a large number, and the endpoints ''a'' ...
are used. Many
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech ...
researchers are now employing such techniques, for instance through work on
Occam Learning In computational learning theory, Occam learning is a model of algorithmic learning where the objective of the learner is to output a succinct representation of received training data. This is closely related to probably approximately correct (P ...
or more generally on the
Free energy principle The free energy principle is a mathematical principle in biophysics and cognitive science that provides a formal account of the representational capacities of physical systems: that is, why things that exist look as if they track properties of the ...
. Statistical versions of Occam's razor have a more rigorous formulation than what philosophical discussions produce. In particular, they must have a specific definition of the term ''simplicity'', and that definition can vary. For example, in the Kolmogorov
Chaitin Gregory John Chaitin ( ; born 25 June 1947) is an Argentine- American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer- ...
minimum description length approach, the subject must pick a
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer alg ...
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 camps: one that believes Occam's razor is objective, and one that believes it is subjective.


Objective razor

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
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
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." Interpreting this as minimising the total length of a two-part message encoding model followed by data given model gives us the minimum message length (MML) principle. 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. According to
Jürgen Schmidhuber Jürgen Schmidhuber (born 17 January 1963) is a German computer scientist most noted for his work in the field of artificial intelligence, deep learning and artificial neural networks. He is a co-director of the Dalle Molle Institute for Artific ...
, the appropriate mathematical theory of Occam's razor already exists, namely, Solomonoff's theory of optimal inductive inference and its extensions. See discussions in David L. Dowe's "Foreword re C. S. Wallace" for the subtle distinctions between the algorithmic probability work of Solomonoff and the MML work of Chris Wallace, and see Dowe's "MML, hybrid Bayesian network graphical models, statistical consistency, invariance and uniqueness" both for such discussions and for (in section 4) discussions of MML and Occam's razor. For a specific example of MML as Occam's razor in the problem of decision tree induction, see Dowe and Needham's "Message Length as an Effective Ockham's Razor in Decision Tree Induction".


Software Development

In software development, the rule of least power argues the correct
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
to use is the one that is simplest while also solving the targeted software problem. In that form the rule is often credited to
Tim Berners-Lee Sir Timothy John Berners-Lee (born 8 June 1955), also known as TimBL, is an English computer scientist best known as the inventor of the World Wide Web. He is a Professorial Fellow of Computer Science at the University of Oxford and a profes ...
since it appeared in his design guidelines for the original
Hypertext Transfer Protocol The Hypertext Transfer Protocol (HTTP) is an application layer protocol in the Internet protocol suite model for distributed, collaborative, hypermedia information systems. HTTP is the foundation of data communication for the World Wide We ...
. Complexity in this context is measured either by placing a language into the Chomsky hierarchy or by listing idiomatic features of the language and comparing according to some agreed to scale of difficulties between idioms. Many languages once thought to be of lower complexity have evolved or later been discovered to be more complex than originally intended; so, in practice this rule is applied to the relative ease of a programmer to obtain the power of the language, rather than the precise theoretical limits of the language.


Controversial aspects

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. Occam's razor is used to adjudicate between theories that have already passed "theoretical scrutiny" tests and are equally well-supported by 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 In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure ( constituenc ...
), while its
ontology In metaphysics, ontology is the philosophy, philosophical study of being, as well as related concepts such as existence, Becoming (philosophy), becoming, and reality. Ontology addresses questions like how entities are grouped into Category ...
(or
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
) becomes simpler, or vice versa. Quine, in a discussion on definition, referred to these two perspectives as "economy of practical expression" and "economy in grammar and vocabulary", respectively.
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He ...
lampooned the ''misuse'' of Occam's razor in his ''
Dialogue Dialogue (sometimes spelled dialog in American and British English spelling differences, American English) is a written or spoken conversational exchange between two or more people, and a literature, literary and theatrical form that depicts suc ...
''. The principle is represented in the dialogue by Simplicio. The telling point that Galileo presented ironically was that if one really wanted to start from a small number of entities, one could always consider the letters of the alphabet as the fundamental entities, since one could construct the whole of human knowledge out of them.


Anti-razors

Occam's razor has met some opposition from people who have considered it too extreme or rash. Walter Chatton () was a contemporary of William of Ockham 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 notability as Chatton's anti-razor, although this could be the case of the Late Renaissance Italian motto of unknown attribution ("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 Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...
(1646–1716),
Immanuel Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and ...
(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 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." 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." A less serious but even more extremist anti-razor is 'Pataphysics, the "science of imaginary solutions" developed by
Alfred Jarry Alfred Jarry (; 8 September 1873 – 1 November 1907) was a French symbolist writer who is best known for his play ''Ubu Roi'' (1896). He also coined the term and philosophical concept of 'pataphysics. Jarry was born in Laval, Mayenne, France, ...
(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 Argentine writer
Jorge Luis Borges Jorge Francisco Isidoro Luis Borges Acevedo (; ; 24 August 1899 – 14 June 1986) was an Argentine short-story writer, essayist, poet and translator, as well as a key figure in Spanish-language and international literature. His best-known b ...
in his story/mock-essay "
Tlön, Uqbar, Orbis Tertius "Tlön, Uqbar, Orbis Tertius" is a short story by the 20th-century Argentinian writer Jorge Luis Borges. The story was first published in the Argentinian journal '' Sur'', May 1940. The "postscript" dated 1947 is intended to be anachronistic, s ...
". Physicist
R. V. Jones Reginald Victor Jones , FRSE, LLD (29 September 1911 – 17 December 1997) was a British physicist and scientific military intelligence expert who played an important role in the defence of Britain in by solving scientific and technical pr ...
contrived Crabtree's Bludgeon, which states that " set of mutually inconsistent observations can exist for which some human intellect cannot conceive a coherent explanation, however complicated."


See also

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Notes


References


Further reading

* * * * * * * * * * (Preprint available as
Sharpening Occam's Razor on a Bayesian Strop
). * * * * * * * * * * * * * * * * * * * * * *


External links


Ockham's Razor
BBC Radio 4 discussion with Sir Anthony Kenny, Marilyn Adams & Richard Cross (''In Our Time'', 31 May 2007) {{DEFAULTSORT:Occam's Razor Adages Concepts in epistemology Concepts in ethics Concepts in metaphysics Concepts in the philosophy of mind Concepts in the philosophy of science Deductive reasoning Epistemological theories Epistemology of science Heuristics History of logic History of science Hypotheses Inductive reasoning Metaphysical theories Occamism Ontology Philosophical analogies Philosophical logic Philosophical problems Philosophy of science Principles Razors (philosophy) Reductionism Scholasticism