Occam's razor
In
This philosophical razor advocates that when presented with competing hypotheses about the same prediction and both theories have equal explanatory power one should prefer the hypothesis that requires the fewest assumptions[4] 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.[5][6]
History
The phrase Occam's razor did not appear until a few centuries after William of Ockham's death in 1347.
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
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.[12] Robert Grosseteste, in Commentary on [Aristotle'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."[13]
The
William of Ockham
While it has been claimed that Occam's razor is not found in any of William's writings,
Nevertheless, the precise words sometimes attributed to William of Ockham, Entia non sunt multiplicanda praeter necessitatem (Entities must not be multiplied beyond necessity),[16] are absent in his extant works;[17] this particular phrasing comes from John Punch,[18] who described the principle as a "common axiom" (axioma vulgare) of the Scholastics.[9] 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[further explanation needed] simply because it pleases God.[12]
This principle is sometimes phrased as Pluralitas non est ponenda sine necessitate ("Plurality should not be posited without necessity").
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."[20][21] In the sentence hypotheses non fingo, 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."[22]
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.[23][24][25]
Another technical approach to Occam's razor is ontological parsimony.[26] Parsimony means spareness and is also referred to as the Rule of Simplicity. This is considered a strong version of Occam's razor.[27][28] 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".[29]
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."[30]
This principle goes back at least as far as Aristotle, who wrote "Nature operates in the shortest way possible."[27] 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."[27]
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. This notion was deeply rooted in the aesthetic value that simplicity holds for human thought and the justifications presented for it often drew from theology.[clarification needed] 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."[31]
Beginning in the 20th century, epistemological justifications based on induction, logic, pragmatism, and especially probability theory have become more popular among philosophers.[7]
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
Testing the razor
This section possibly contains original research. Author of this section cites very few reliable sources, and also consistently conflates simplicity with (logical) truth. Occam's razor is not built to differentiate true hypotheses from false ones. (January 2023) |
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
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.[35]
Practical considerations and pragmatism
Mathematical
One justification of Occam's razor 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 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,[36] where he emphasizes that a prior bias in favor of simpler models is not required.
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).[38]
Other philosophers
Karl Popper
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".[39] 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.[40] He has since rejected this account of simplicity, purportedly because it fails to provide an epistemic 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 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?'"[41]
Richard Swinburne
Richard Swinburne argues for simplicity on logical grounds:
... 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
According to Swinburne, since our choice of theory cannot be determined by data (see Underdetermination and Duhem–Quine thesis), 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 [in the way it judges theories and predictions probable] or the principle of simplicity is a fundamental synthetic a priori truth."[42]
Ludwig Wittgenstein
From the Tractatus Logico-Philosophicus:
- 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
This section possibly contains original research. (May 2021) |
Science and the scientific method
In
In chemistry, Occam's razor is often an important heuristic when developing a model of a reaction mechanism.[47][48] 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.[6] 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."[49][50][51] An often-quoted version of this constraint (which cannot be verified as posited by Einstein himself)[52] reduces this to "Everything should be kept as simple as possible, but not simpler."
In the scientific method, Occam's razor is not considered an irrefutable principle of logic 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.[53][54][55] 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.[5][54] 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.[53][54][55]
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.[clarification needed] 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.[55]
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.[6] 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 and used Occam's razor logic to formulate the quantum hypothesis, even resisting that hypothesis as it became more obvious that it was correct.[6]
Appeals to simplicity were used to argue against the phenomena of meteorites,
In the same way, postulating the aether is more complex than transmission of light through a vacuum. 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's idea of light particles seemed simpler than Christiaan Huygens'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.[54]
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,
Biology
This section has an unclear citation style. (January 2023) |
Biologists or philosophers of biology use Occam's razor in either of two contexts both in
However, more recent biological analyses, such as Richard Dawkins' 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.
It is among the cladists that Occam's razor is applied, through the method of cladistic parsimony. Cladistic parsimony (or
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, a close cousin of likelihood, which uses Occam's razor in the same way. 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.[61]
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 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."[62] This is an ontological critique of parsimony.
In
Religion
In the
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:[65]
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
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, who viewed belief in God as a leap of faith that sometimes directly opposed reason.[66] This is also the doctrine of Gordon Clark's presuppositional apologetics, with the exception that Clark never thought the leap of faith was contrary to reason (see also Fideism).
Various
Another application of the principle is to be found in the work of
Occam's razor may also be recognized in the apocryphal story about an exchange between
Philosophy of mind
In his article "Sensations and Brain Processes" (1959),
Penal ethics
In penal theory and the philosophy of punishment, parsimony refers specifically to taking care in the distribution of punishment in order to avoid excessive punishment. In the utilitarian approach to the philosophy of punishment, Jeremy Bentham'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 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.[70]
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, and using it to come up with criteria for penalizing complexity in statistical inference. Papers[71][72] have suggested a connection between Occam's razor and Kolmogorov complexity.[73]
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
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 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 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.[74] 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 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."[75] 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.[71][72]
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.[24][76]
According to Jürgen Schmidhuber, the appropriate mathematical theory of Occam's razor already exists, namely, Solomonoff's theory of optimal inductive inference[77] and its extensions.[78] See discussions in David L. Dowe's "Foreword re C. S. Wallace"[79] 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"[80] 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".[81]
Mathematical arguments against Occam's razor
This section may be too technical for most readers to understand.(February 2024) |
The no free lunch (NFL) theorems for inductive inference prove that Occam's razor must rely on ultimately arbitrary assumptions concerning the prior probability distribution found in our world.[82] Specifically, suppose one is given two inductive inference algorithms, A and B, where A is a Bayesian procedure based on the choice of some prior distribution motivated by Occam's razor (e.g., the prior might favor hypotheses with smaller Kolmogorov complexity). Suppose that B is the anti-Bayes procedure, which calculates what the Bayesian algorithm. A based on Occam's razor will predict – and then predicts the exact opposite. Then there are just as many actual priors (including those different from the Occam's razor prior assumed by A) in which algorithm B outperforms A as priors in which the procedure A based on Occam's razor comes out on top. In particular, the NFL theorems show that the "Occam factors" Bayesian argument for Occam's razor must make ultimately arbitrary modeling assumptions.[83]
Software development
In software development, the
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.[a] Occam's razor is used to adjudicate between theories that have already passed "theoretical scrutiny" tests and are equally well-supported by evidence.[b] 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.[citation needed]
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.[c] Quine, in a discussion on definition, referred to these two perspectives as "economy of practical expression" and "economy in grammar and vocabulary", respectively.[86]
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 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.
Instances of using Occam's razor to justify belief in less complex and more simple theories have been criticized as using the razor inappropriately. For instance Francis Crick stated that "While Occam's razor is a useful tool in the physical sciences, it can be a very dangerous implement in biology. It is thus very rash to use simplicity and elegance as a guide in biological research."[87]
Anti-razors
Occam's razor has met some opposition from people who consider it too extreme or rash. Walter Chatton (c. 1290–1343) 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 several philosophers who have formulated similar anti-razors since Chatton's time, no one anti-razor has perpetuated as notably as Chatton's anti-razor, although this could be the case of the Late Renaissance Italian motto of unknown attribution Se non è vero, è 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
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 (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 in his story/mock-essay "Tlön, Uqbar, Orbis Tertius". Physicist R. V. Jones contrived Crabtree's Bludgeon, which states that "[n]o set of mutually inconsistent observations can exist for which some human intellect cannot conceive a coherent explanation, however complicated."[89]
Recently, American physicist Igor Mazin argued that because high-profile physics journals prefer publications offering exotic and unusual interpretations, the Occam's razor principle is being replaced by an "Inverse Occam's razor", implying that the simplest possible explanation is usually rejected.[90]
See also
- Chekhov's gun – Dramatic principle
- Duck test – Classification based on observable evidence
- Explanatory power – Ability of a theory to explain a subject
- Hanlon's razor – Adage to assume stupidity over malice
- Hickam's dictum – Medical principle that a patient's symptoms could be caused by several diseases
- Hitchens's razor – General rule rejecting claims made without evidence
- KISS principle – Design principle preferring simplicity
- Minimum description length – Model selection principle
- Minimum message length – Formal information theory restatement of Occam's Razor
- Newton's flaming laser sword– Australian mathematician and philosopher
- Philosophical razor – Principle that allows one to eliminate unlikely explanations
- Philosophy of science – Study of foundations, methods, and implications of science
- Simplicity – State of being simple
Notes
- ^ "Ockham's razor does not say that the more simple a hypothesis, the better."[85]
- ^ "Today, we think of the principle of parsimony as a heuristic device. We don't assume that the simpler theory is correct and the more complex one false. We know from experience that more often than not the theory that requires more complicated machinations is wrong. Until proved otherwise, the more complex theory competing with a simpler explanation should be put on the back burner, but not thrown onto the trash heap of history until proven false."[85]
- ^ "While these two facets of simplicity are frequently conflated, it is important to treat them as distinct. One reason for doing so is that considerations of parsimony and of elegance typically pull in different directions. Postulating extra entities may allow a theory to be formulated more simply, while reducing the ontology of a theory may only be possible at the price of making it syntactically more complex."[53]
References
- ^ Barry, C. M. (27 May 2014). "Who sharpened Occam's Razor?". Irish Philosophy. Archived from the original on 5 October 2022. Retrieved 5 August 2022.
- (PDF) from the original on 9 September 2020. Retrieved 8 August 2019.
- ^ Duignan, Brian. "Occam's Razor". Encyclopedia Britannica. Archived from the original on 25 September 2023. Retrieved 11 May 2021.
- ^ Ball, Philip (11 August 2016). "The Tyranny of Simple Explanations". The Atlantic. Archived from the original on 2 February 2023. Retrieved 2 February 2023.
- ^ ISBN 978-0-521-01708-4.
- ^ a b c d e f Hoffman, Roald; Minkin, Vladimir I.; Carpenter, Barry K. (1997). "Ockham's Razor and Chemistry". Hyle: International Journal for Philosophy of Chemistry. 3: 3–28. Archived from the original on 14 July 2018. Retrieved 30 May 2004.
- ^ ISBN 978-1107692534.
- ^ Roger Ariew, Ockham's Razor: A Historical and Philosophical Analysis of Ockham's Principle of Parsimony, 1976
- ^ a b 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
- ^ Aristotle, Physics 189a15, On the Heavens 271a33. See also Franklin, op cit. note 44 to chap. 9.
- .
- ^ a b c Franklin, James (2001). The Science of Conjecture: Evidence and Probability before Pascal. The Johns Hopkins University Press. Chap 9. p. 241.
- ^ Alistair Cameron Crombie, Robert Grosseteste and the Origins of Experimental Science 1100–1700 (1953) pp. 85–86
- ^ "SUMMA THEOLOGICA: The existence of God (Prima Pars, Q. 2)". Newadvent.org. Archived from the original on 28 April 2013. Retrieved 26 March 2013.
- ^ Vallee, Jacques (11 February 2013). "What Ockham really said". Boing Boing. Archived from the original on 31 March 2013. Retrieved 26 March 2013.
- ^ Bauer, Laurie (2007). The linguistics Student's Handbook. Edinburgh: Edinburgh University Press. p. 155.
- ^ Flew, Antony (1979). A Dictionary of Philosophy. London: Pan Books. p. 253.
- ^ Crombie, Alistair Cameron (1959), Medieval and Early Modern Philosophy, Cambridge, MA: Harvard, Vol. 2, p. 30.
- ^ "Ockham's razor". Encyclopædia Britannica. Encyclopædia Britannica Online. 2010. Archived from the original on 23 August 2010. Retrieved 12 June 2010.
- ]
- ^ Primary source: Newton (2011, p. 387) 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.
- ^ Logical Constructions. Metaphysics Research Lab, Stanford University. 2016. Archived from the original on 26 January 2021. Retrieved 29 March 2011.
- ^ Induction: From Kolmogorov and Solomonoff to De Finetti and Back to Kolmogorov JJ McCall – Metroeconomica, 2004 – Wiley Online Library.
- ^ S2CID 14940740.
- S2CID 2499910.
- ^ Baker, Alan (25 February 2010). "Simplicity". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Summer 2011 Edition). Archived from the original on 24 February 2021. Retrieved 6 April 2013.
- ^ a b c "What is Occam's Razor?". math.ucr.edu. Archived from the original on 6 July 2017.
- ISBN 9781480838024. Archivedfrom the original on 28 October 2023. Retrieved 22 May 2017.
- ISBN 978-0-9818193-0-3.
- JSTOR 2177489.
- ^ Pegis 1945.
- ^ Stanovich, Keith E. (2007). How to Think Straight About Psychology. Boston: Pearson Education, pp. 19–33.
- ^ "ad hoc hypothesis - The Skeptic's Dictionary - Skepdic.com". skepdic.com. Archived from the original on 27 April 2009.
- ^ Swinburne 1997 and Williams, Gareth T, 2008.
- from the original on 8 June 2020. Retrieved 22 January 2019.(subscription required)
- Bibcode:2003itil.book.....M. Archived(PDF) from the original on 15 September 2012.
- ^ (PDF) from the original on 4 March 2005. (preprint available as "Sharpening Occam's Razor on a Bayesian Strop").
- ISBN 9781461471370.
- ISBN 978-84-309-0711-3.
- ISBN 978-0-19-824407-3.
- ISBN 978-0-521-80361-8. Archived from the original on 28 October 2023. Retrieved 4 August 2012. Paper as PDF.
- ISBN 978-0-87462-164-8.
- from the original on 21 October 2019. Retrieved 21 October 2019.
- ^ L. Nash, The Nature of the Natural Sciences, Boston: Little, Brown (1963).
- ^ de Maupertuis, P. L. M. (1744). Mémoires de l'Académie Royale (in French). p. 423.
- ^ de Broglie, L. (1925). Annales de Physique (in French). pp. 22–128.
- ^ 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.
- from the original on 22 January 2023. Retrieved 22 January 2023.
- ISBN 978-3-16-146910-7. Archivedfrom the original on 22 January 2023. Retrieved 22 January 2023.
- ISBN 978-1-55553-360-1. Archivedfrom the original on 5 April 2023. Retrieved 10 February 2023.
- ^ "Everything Should Be Made as Simple as Possible, But Not Simpler". 13 May 2011. Archived from the original on 29 May 2012.
- ^ from the original on 26 March 2014. Retrieved 22 January 2005.
- ^ Bibcode:2008arXiv0812.4932C.
- ^ a b c Sober, Elliott (1994). "Let's Razor Occam's Razor". In Knowles, Dudley (ed.). Explanation and Its Limits. Cambridge University Press. pp. 73–93.
- PMID 16368772.
- from the original on 11 November 2020. Retrieved 4 October 2009.
- ISBN 978-0-262-69144-4.
- ^ Wiley, Edward O. (1981). Phylogenetics: the theory and practice of phylogenetic systematics. Wiley and Sons Interscience.
- PMID 34649374.
- ^ Brower &, Schuh (2021). Biological Systematics: Principles and Applications (3rd edn.). Cornell University Press.
- ^ Crick 1988, p. 146.
- ^ "William Ockham". Encyclopedia of Philosophy. Stanford. Archived from the original on 7 October 2019. Retrieved 24 February 2016.
- ISBN 9781570753961.
- ^ "SUMMA THEOLOGICA: The existence of God (Prima Pars, Q. 2)". Newadvent.org. Archived from the original on 28 April 2013. Retrieved 26 March 2013.
- ^ McDonald 2005.
- ISBN 978-0-552-77331-7.
- ISBN 978-0-226-73889-5.
- ^ p. 282, Mémoires du docteur F. Antommarchi, ou les derniers momens de Napoléon Archived 14 May 2016 at the Wayback Machine, vol. 1, 1825, Paris: Barrois L'Ainé
- ^ Tonry, Michael (2005). "Obsolescence and Immanence in Penal Theory and Policy" (PDF). Columbia Law Review. 105: 1233–1275. Archived from the original (PDF) on 23 June 2006.
- ^ a b Chris S. Wallace and David M. Boulton; Computer Journal, Volume 11, Issue 2, 1968 Page(s):185–194, "An information measure for classification."
- ^ a b 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."
- ^ Nannen, Volker. "A short introduction to Model Selection, Kolmogorov Complexity and Minimum Description Length" (PDF). Archived (PDF) from the original on 2 June 2010. Retrieved 3 July 2010.
- ^ "Algorithmic Information Theory". Archived from the original on 24 December 2007.
- ^ Paul M. B. Vitányi and Ming Li; IEEE Transactions on Information Theory, Volume 46, Issue 2, Mar 2000 Page(s):446–464, "Minimum Description Length Induction, Bayesianism and Kolmogorov Complexity."
- S2CID 17143230.
- .
- arXiv:cs.AI/0302012.
- S2CID 5387092.
- ^ David L. Dowe (2010): "MML, hybrid Bayesian network graphical models, statistical consistency, invariance and uniqueness. A formal theory of inductive inference." Handbook of the Philosophy of Science – (HPS Volume 7) Philosophy of Statistics, Elsevier 2010 Page(s):901–982. https://web.archive.org/web/20140204001435/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.185.709&rep=rep1&type=pdf
- ^ Scott Needham and David L. Dowe (2001):" Message Length as an Effective Ockham's Razor in Decision Tree Induction." Proc. 8th International Workshop on Artificial Intelligence and Statistics (AI+STATS 2001), Key West, Florida, U.S.A., January 2001 Page(s): 253–260 "2001 Ockham.pdf" (PDF). Archived (PDF) from the original on 23 September 2015. Retrieved 2 September 2015.
- ^ Adam, S., and Pardalos, P. (2019), No-free lunch Theorem: A review, in "Approximation and Optimization", Springer, 57-82
- ^ Wolpert, D.H (1995), On the Bayesian "Occam Factors" Argument for Occam's Razor, in "Computational Learning Theory and Natural Learning Systems: Selecting Good Models", MIT Press
- ^ Berners-Lee, Tim (4 March 2013). "Principles of Design". World Wide Web Consortium. Archived from the original on 15 June 2022. Retrieved 5 June 2022.
- ^ a b Robert T. Carroll (12 September 2014). "Occam's Razor". The Skeptic's Dictionary. Archived from the original on 1 March 2016. Retrieved 24 February 2016.
- ISBN 978-0-674-32351-3.
- ^ Gross, F. (2019). Occam's Razor in Molecular and Systems Biology. Philosophy of Science, 86(5), 1134-1145. doi:10.1086/705474
- ^ Immanuel Kant (1929). Norman Kemp-Smith transl (ed.). The Critique of Pure Reason. Palgrave Macmillan. p. 92. Archived from the original on 16 May 2012. Retrieved 27 October 2012.
Entium varietates non-temere esse minuendas
- ISBN 978-1-84816-893-0. Archivedfrom the original on 28 October 2023. Retrieved 10 August 2021.
- from the original on 9 July 2023. Retrieved 9 July 2023.
Further reading
- Ariew, Roger (1976). Ockham's Razor: A Historical and Philosophical Analysis of Ockham's Principle of Parsimony. Champaign-Urbana, University of Illinois.
- ISBN 978-0-262-53050-7.
- ISBN 978-0-465-09137-9.
- Dowe, David L.; Steve Gardner; Graham Oppy (December 2007). "Bayes not Bust! Why Simplicity is no Problem for Bayesians" (PDF). British Journal for the Philosophy of Science. 58 (4): 709–754. (PDF) from the original on 9 October 2022.
- Duda, Richard O.; Peter E. Hart; David G. Stork (2000). Pattern Classification (2nd ed.). ISBN 978-0-471-05669-0.
- Epstein, Robert (1984). "The Principle of Parsimony and Some Applications in Psychology". Journal of Mind Behavior. 5: 119–130.
- Hoffmann, Roald; Vladimir I. Minkin; Barry K. Carpenter (1997). "Ockham's Razor and Chemistry". Hyle: International Journal for Philosophy of Chemistry. 3: 3–28. Archived from the original on 14 July 2018. Retrieved 14 April 2006.
- Jacquette, Dale (1994). Philosophy of Mind. Engleswoods Cliffs, New Jersey: ISBN 978-0-13-030933-4.
- ISBN 978-0-521-59271-0. Archivedfrom the original on 24 October 2018. Retrieved 24 November 2003.
- Jefferys, William H.; Berger, James O. (1991). "Ockham's Razor and Bayesian Statistics". American Scientist. 80: 64–72. (Preprint available as "Sharpening Occam's Razor on a Bayesian Strop Archived 4 March 2005 at the Wayback Machine").
- Katz, Jerrold (1998). Realistic Rationalism. MIT Press. ISBN 978-0-262-11229-1.
- Kneale, William; Martha Kneale (1962). The Development of Logic. London: ISBN 978-0-19-824183-6.
- ISBN 978-0-521-64298-9. Archivedfrom the original on 17 February 2016. Retrieved 24 February 2016.
- Maurer, A. (1984). "Ockham's Razor and Chatton's Anti-Razor". Mediaeval Studies. 46: 463–475. .
- McDonald, William (2005). "Søren Kierkegaard". Stanford Encyclopedia of Philosophy. Archived from the original on 25 February 2017. Retrieved 14 April 2006.
- Menger, Karl (1960). "A Counterpart of Ockham's Razor in Pure and Applied Mathematics: Ontological Uses". Synthese. 12 (4): 415–428. S2CID 46962297.
- Morgan, C. Lloyd (1903). "Other Minds than Ours". An Introduction to Comparative Psychology (2nd ed.). London: W. Scott. p. 59. ISBN 978-0-89093-171-4. Archived from the originalon 12 April 2005. Retrieved 15 April 2006.
- ISBN 978-1-60386-435-0.
- Nolan, D. (1997). "Quantitative Parsimony". S2CID 229320568.
- Basic Writings of St. Thomas Aquinas. Translated by Pegis, A. C. New York: Random House. 1945. p. 129. ISBN 978-0-87220-380-8.
- Popper, Karl (1992) [First composed 1934 (Logik der Forschung)]. "7. Simplicity". The Logic of Scientific Discovery (2nd ed.). London: Routledge. pp. 121–132. ISBN 978-84-309-0711-3.
- Rodríguez-Fernández, J. L. (1999). "Ockham's Razor". Endeavour. 23 (3): 121–125. .
- Schmitt, Gavin C. (2005). "Ockham's Razor Suggests Atheism". Archived from the original on 11 February 2007. Retrieved 15 April 2006.
- Smart, J. J. C. (1959). "Sensations and Brain Processes". JSTOR 2182164.
- Sober, Elliott (1975). Simplicity. Oxford: Oxford University Press.
- Sober, Elliott (1981). "The Principle of Parsimony" (PDF). British Journal for the Philosophy of Science. 32 (2): 145–156. S2CID 120916709. Archived from the original(PDF) on 15 December 2011. Retrieved 4 August 2012.
- Sober, Elliott (1990). "Let's Razor Ockham's Razor". In Dudley Knowles (ed.). Explanation and its Limits. Cambridge: Cambridge University Press. pp. 73–94.
- Sober, Elliott (2002). Zellner; et al. (eds.). "What is the Problem of Simplicity?" (PDF). Archived from the original (PDF) on 8 November 2006. Retrieved 4 August 2012.
- Sober, Elliott (2015). Ockham's Razors - A User's Manual. Cambridge, England: ISBN 978-1-107-06849--0.
- Swinburne, Richard (1997). Simplicity as Evidence for Truth. Milwaukee, Wisconsin: ISBN 978-0-87462-164-8.
- Thorburn, W. M. (1918). "The Myth of Occam's Razor". Mind. 27 (107): 345–353. from the original on 5 October 2019. Retrieved 11 July 2009.
- Williams, George C. (1966). Adaptation and natural selection: A Critique of some Current Evolutionary Thought. Princeton, New Jersey: ISBN 978-0-691-02615-2.
External links
- Ockham's Razor, BBC Radio 4 discussion with Sir Anthony Kenny, Marilyn Adams & Richard Cross (In Our Time, 31 May 2007)