In philosophy, Occam's razor (also spelled Ockham's razor or Ocham's razor; ) is the problem-solving principle that recommends searching for explanations constructed with the smallest possible set of elements. It is also known as the principle of parsimony or the law of parsimony (). Attributed to William of Ockham, a 14th-century English philosopher and theologian, it is frequently cited as , which translates as "Entities must not be multiplied beyond necessity", although Occam never used those words. Popularly, the principle is sometimes paraphrased as "of two competing theories, the simpler explanation of an entity is to be preferred".
The philosophical razor advocates that when presented with competing hypotheses about the same prediction and both hypotheses have equal explanatory power, one should prefer the hypothesis that requires 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 heuristic 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, , .
History The phrase Occam's razor did not appear until a few centuries after William of Ockham's death in 1347. Libert Froidmont, in his 1649 Philosophia Christiana de Anima (On Christian Philosophy of the Soul) claimed to have coined the phrase novacula occami when he described Gregory of Rimini, one of Ockham's critics. Ockham did not invent this principle, but its fame – and its association with him – may be due to the frequency and effectiveness with which he used it.Roger Ariew, Ockham's Razor: A Historical and Philosophical Analysis of Ockham's Principle of Parsimony, 1976 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 philosopher John Punch in his 1639 commentary on the works of Duns Scotus.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 (Moses ben-Maimon, 1138–1204), and even Aristotle (384–322 BC).Aristotle, Physics 189a15, On the Heavens 271a33. See also Franklin, op cit. note 44 to chap. 9. Aristotle writes in his Posterior Analytics, "We may assume the superiority [all other things being equal] of the demonstration which derives from fewer postulates or hypotheses." Ptolemy () 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 [Aristotle's] the Posterior Analytics Books (Commentarius in Posteriorum Analyticorum Libros) (), 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."Alistair Cameron Crombie, Robert Grosseteste and the Origins of Experimental Science 1100–1700 (1953) pp. 85–86The Summa Theologica of Thomas Aquinas (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), and specifically, through an argument based on causality. 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).
William of Ockham
William of Ockham () was an English Franciscan friar and theologian, an influential medieval philosopher 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 ("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), p. 155. are absent in his extant works; p. 253. this particular phrasing comes from John Punch,Crombie, Alistair Cameron (1959), Medieval and Early Modern Philosophy, Cambridge, MA: Harvard, Vol. 2, p. 30. who described the principle as a "common axiom" (axioma vulgare) of the Scholastics. William of Ockham himself seems to restrict the operation of this principle in matters pertaining to miracles and God's power, considering a plurality of miracles possible in the Eucharist 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 and Kneale, 1962, p. 243.)
Later formulations To quote Isaac Newton, "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."Primary source: wrote the following two "philosophizing rules" at the beginning of part 3 of the Principia 1726 edition. Regula I. Causas rerum naturalium non-plures admitti debere, quam quæ & veræ sint & earum phænomenis explicandis sufficient. Regula II. Ideoque effectuum naturalium ejusdem generis eædem assignandæ sunt causæ, quatenus fieri potest. In the sentence hypotheses non fingo ("I contrive no hypotheses"), Newton affirms the success of this approach.
Bertrand Russell 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 observations for 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's dictum "When you hear hoofbeats, think of horses not zebras".
Ernst Mach formulated the stronger version of Occam's razor into physics, 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."
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 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 [if] one suffices."Pegis 1945.
Beginning in the 20th century, epistemological justifications based on induction, logic, pragmatism, and especially probability theory 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, excessively complex models are affected by statistical noise (a problem also known as the bias–variance tradeoff), 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.).
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. Ad hoc hypotheses are justifications that prevent theories from being falsified.
For example, if a man, accused of breaking a vase, makes supernatural 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.Swinburne 1997 and Williams, Gareth T, 2008.