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Parsimony

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Parsimony

For the aerial theatre company, see Ockham's Razor Theatre Company.

Occam's razor (also written as Ockham's razor from William of Ockham (c. 1287 – 1347), and in Latin lex parsimoniae) is a principle of parsimony, economy, or succinctness used in logic and problem-solving. It states that among competing hypotheses, the hypothesis with the fewest assumptions should be selected.

The application of the principle often shifts the burden of proof in a discussion.[lower-alpha 1] The razor states that one should proceed to simpler theories until simplicity can be traded for greater explanatory power. The simplest available theory need not be most accurate. Philosophers also point out that the exact meaning of simplest may be nuanced.[lower-alpha 2]

Solomonoff's inductive inference is a mathematically formalized Occam's razor:[2][3][4][5][6][7] shorter computable theories have more weight when calculating the probability of the next observation, using all computable theories which perfectly describe previous observations.

In science, Occam's razor is used as a heuristic (general guiding rule or an observation) to guide scientists in the development of theoretical models rather than as an arbiter between published models.[8][9] In the scientific method, Occam's razor is not considered an irrefutable principle of logic or a scientific result.[1][10][11]

History

The term "Occam's razor" first appeared in 1852 in the works of Sir William Hamilton, 9th Baronet (1788–1856), centuries after William of Ockham's death in 1347.[12] Ockham did not invent this "razor"; its association with him may be due to the frequency and effectiveness with which he used it (Ariew 1976). Ockham stated the principle in various ways, but the most popular version "entities must not be multiplied beyond necessity" was written by John Punch from Cork in 1639 (Meyer 1957).[13]

Formulations before 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), Maimonides (Moses ben-Maimon, 1138–1204), and even Aristotle (384–322 BC)[14] (Charlesworth 1956). Ptolemy (c. AD 90 – c. AD 168) stated, "We consider it a good principle to explain the phenomena by the simplest hypothesis possible."[15] 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.[15]

For instance, the 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 which he in turn answers and refutes generally (cf. quinque viae), and specifically, through an argument based on causality.[16] Hence, Aquinas acknowledges the principle which today is known as Occam's Razor, but prefers causal explanations to other simple explanations (cf. also Correlation does not imply causation).

Ockham

William of Ockham (c. 1287–1347) is remembered as an influential medieval philosopher and nominalist, though his popular fame as a great logician rests chiefly on the maxim attributed to him and known as Ockham's razor. The term razor refers to distinguishing between two hypotheses either by "shaving away" unnecessary assumptions or cutting apart two similar conclusions.

This maxim seems to represent the general tendency of Ockham's philosophy, but it has not been found in any of his writings.[17] His nearest pronouncement seems to be Numquam ponenda est pluralitas sine necessitate [Plurality must never be posited without necessity], which occurs in his theological work on the 'Sentences of Peter Lombard' (Quaestiones et decisiones in quattuor libros Sententiarum Petri Lombardi (ed. Lugd., 1495), i, dist. 27, qu. 2, K).

The words attributed to Ockham, entia non sunt multiplicanda praeter necessitatem (entities must not be multiplied beyond necessity), are absent in his extant works;[18] this particular phrasing owes more to John Punch.[19] Indeed, Ockham's contribution seems to be to restrict the operation of this principle in matters pertaining to miracles and God's power: so, in the Eucharist, a plurality of miracles is possible, simply because it pleases God.[15]

This principle is sometimes phrased as pluralitas non est ponenda sine necessitate ("plurality should not be posited without necessity").[20] In his Summa Totius Logicae, i. 12, Ockham cites the principle of economy, Frustra fit per plura quod potest fieri per pauciora [It is futile to do with more things that which can be done with fewer]. (Thorburn, 1918, pp. 352–3; 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, so far as possible, assign the same causes."[21]

Bertrand Russell offers a particular version of Occam's Razor: "Whenever possible, substitute constructions out of known entities for inferences to unknown entities."[22]

Solomonoff's inductive inference is a mathematical proof[2][3][4][5][23] of a statement akin to Occam's razor, under the assumption that the environment follows some unknown but computable distribution of probability.

Another technical approach to Occam's razor is ontological parsimony.[24]

Justifications

Beginning in the 20th century, epistemological justifications based on induction, logic, pragmatism, and especially probability theory have become more popular among philosophers.

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 simplicity holds for human thought and the justifications presented for it often drew from theology. Thomas Aquinas made this argument in the 13th century, writing, "If a thing can be done adequately by means of one, it is superfluous to do it by means of several; for we observe that nature does not employ two instruments [if] one suffices."[25]

Empirical

Occam's razor has gained strong empirical support as far as helping to converge on better theories (see "Applications" 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 trade-off), 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 "simpler explanations are, other things being equal, generally better than more complex ones" is amenable to empirical testing. Although, another interpretation of the razor's statement would be that "simpler hypotheses (not conclusions i.e. explanations) 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 you accept 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.

In the history of competing hypotheses, it is the case that the simpler hypotheses have led to mathematically rigorous and empirically verifiable theories. In the history of competing explanations this is not the case. At least, not generally (some increases in complexity are sometimes necessary), and so there remains a justified general bias towards 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 failing explanations with ad hoc hypothesis. Ad hoc hypotheses are justifications that prevent theories from being falsified. Even other empirical criteria like consilience can never truly eliminate such explanations as competition. Each true explanation, then, may have had many alternatives that were simpler and false, but also an infinite number of alternatives that were more complex and false. However, if an alternate ad hoc hypothesis was indeed justifiable, its implicit conclusions would be empirically verifiable. On a commonly accepted repeatability principle, these alternate theories have never been observed and continue to not be observed. In addition, we do not say an explanation is true if it has not withstood this principle.

Put another way, any new, and even more complex theory can still possibly be true. For example: If an individual makes supernatural claims that Leprechauns were responsible for breaking a vase, the simpler explanation would be that he is mistaken, but ongoing ad hoc justifications (e.g. "And, that's not me on film, they tampered with that too") successfully prevent outright falsification. This endless supply of elaborate competing explanations, called saving hypotheses, cannot be ruled out—but by using Occam's Razor.[26][27][28]

Practical considerations and pragmatism

The common form of the razor, used to distinguish between equally explanatory hypotheses, may be supported by the practical fact that simpler theories are easier to understand.

Some argue that Occam's razor is not an inference-driven model, but a heuristic maxim for choosing among other models and instead underlies induction.

Alternatively, if we want to have reasonable discussion we may be practically forced to accept Occam's razor in the same way we are simply forced to accept the laws of thought and inductive reasoning (given the problem of induction). Philosopher Elliott Sober states that not even reason itself can be justified on any reasonable grounds, and that we must start with first principles of some kind (otherwise an infinite regress occurs).

The pragmatist may go on, as David Hume did on the topic induction, that there is no satisfying alternative to granting this premise. Though one may claim that Occam's razor is invalid as a premise helping to regulate theories, putting this doubt into practice would mean doubting whether every step forward will result in locomotion or a nuclear explosion. In other words still: "What's the alternative?"

Mathematical

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. Marcus Hutter's universal artificial intelligence builds upon this to calculate the expected value of an action.

There have been other attempts to derive Occam's Razor from probability theory, notable attempts made by Harold Jeffreys and E. T. Jaynes. William H. Jeffreys and Berger stated that "as a consequence of the fact that a hypothesis with fewer adjustable parameters will automatically have an enhanced posterior probability, due to the fact that the predictions it makes are sharp..."[29] 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[30] where he emphasises that a prior bias in favour of simpler models is not required.

Other views

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" (Popper 1992). The idea here is that a simple theory applies to more cases than a more complex one, and is thus more easily falsifiable. This is again comparing a simple theory to a more complex theory where both explain the data equally well.

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 one, in the sense that less information is required in order to answer one's questions.[31] He has since rejected this account of simplicity, purportedly because it fails to provide an epistemic justification for simplicity. He now expresses views to the effect that simplicity considerations (and considerations of parsimony in particular) do not count unless they reflect something more fundamental. Philosophers, he suggests, may have made the error of hypostatizing simplicity (i.e. endowed it with a sui generis existence), when it has meaning only when embedded in a specific context (Sober 1992). If we fail to justify simplicity considerations on the basis of the context in which we make use of them, we may have no non-circular justification: "just as the question 'why be rational?' may have no non-circular answer, the same may be true of the question 'why should simplicity be considered in evaluating the plausibility of hypotheses?'".[32]

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 Quine-Duhem thesis), we must rely on some criterion to determine which theory to use. Since it is absurd to have no logical method by which to settle on one hypothesis amongst an infinite number of equally data-compliant hypotheses, we should choose the simplest theory: "either science is irrational [in the way it judges theories and predictions probable] or the principle of simplicity is a fundamental synthetic a priori truth" (Swinburne 1997).

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.

Applications

Science and the scientific method

In science, Occam's razor is used as a heuristic (rule of thumb) to guide scientists in the development of theoretical models rather than as an arbiter between published models.[8][9] In physics, parsimony was an important heuristic in the formulation of special relativity by Albert Einstein,[33][34] the development and application of the principle of least action by Pierre Louis Maupertuis and Leonhard Euler,[35] and the development of quantum mechanics by Max Planck, Werner Heisenberg and Louis de Broglie.[9][36] In chemistry, Occam's razor is often an important heuristic when developing a model of a reaction mechanism.[37][38] However, while it is useful as a heuristic in developing models of reaction mechanisms, it has been shown to fail as a criterion for selecting among some selected published models.[9] 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 (that cannot be verified as being posited by Einstein himself)[39] says "Everything should be kept as simple as possible, but no simpler."

In the scientific method, parsimony is an epistemological, metaphysical or heuristic preference, not an irrefutable principle of logic or a scientific result.[1][10][40] As a logical principle, Occam's razor would demand that scientists accept the simplest possible theoretical explanation for existing data. However, science has shown repeatedly that future data often supports more complex theories than existing data. Science 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.[8][10] 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.[1][10][11]

When scientists use the idea of parsimony, it only has meaning in a very specific context of inquiry. A number of background assumptions are required for parsimony to connect with plausibility in a particular research problem. The reasonableness of parsimony in one research context may have nothing to do with its reasonableness in another. It is a mistake to think that there is a single global principle that spans diverse subject matter.[11] It has been suggested that Occam's razor is a widely accepted example of extraevidential consideration, even though it is entirely a metaphysical assumption. There is little empirical evidence that the world is actually simple or that simple accounts are more likely than complex ones to be true.[41] Most of the time, Occam's razor is a conservative tool, cutting out crazy, complicated constructions and assuring that hypotheses are grounded in the science of the day, thus yielding "normal" science: models of explanation and prediction. There are, however, notable exceptions where Occam's razor turns a conservative scientist into a reluctant revolutionary. For example, Max Planck interpolated between the Wien and Jeans radiation laws and used an Occam's razor logic to formulate the quantum hypothesis, and even resisting that hypothesis as it became more obvious that it was correct.[9]

However, appeals to simplicity were used to argue against the phenomena of meteorites, ball lightning, continental drift, and reverse transcriptase. It originally rejected DNA as the carrier of genetic information in favor of proteins, since proteins provided the simpler explanation. One can also argue for atomic building blocks for matter, because it provides a simpler explanation for the observed reversibility of both mixing and chemical reactions as simple separation and re-arrangements of the atomic building blocks. However, at the time, the atomic theory was considered more complex because it implied the existence of invisible particles which had not been directly detected. Ernst Mach and the logical positivists rejected the atomic theory of John Dalton, until the reality of atoms was more evident in Brownian motion, as shown by Albert Einstein.[42] In the same way, postulating the aether is more complex than transmission of light through a vacuum. However, at the time, all known waves propagated through a physical medium, and it seemed simpler to postulate the existence of a medium rather than theorize about wave propagation without a medium. Likewise, Newton's idea of light particles seemed simpler than Christiaan Huygens's idea of waves, so many favored it; however in this case, as it turned out, neither the wave- nor the particle-explanation alone suffices, since light behaves like waves as well as like particles.

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 general axioms of science. The general principle of science is that theories (or models) of natural law must be consistent with repeatable experimental observations. This ultimate arbiter (selection criterion) rests upon the axioms mentioned above.[10]

There are examples where Occam's razor would have picked the wrong theory given the available data. Simplicity principles are useful philosophical preferences for choosing a more likely theory from among several possibilities that are each consistent with available data. A single instance of Occam's razor picking a wrong theory falsifies the razor as a general principle.[10] Michael Lee and others[43] provide cases where a parsimonious approach does not guarantee a correct conclusion and, if based on incorrect working hypotheses or interpretations of incomplete data, may even strongly support a false conclusion. He states, "When parsimony ceases to be a guideline and is instead elevated to an ex cathedra pronouncement, parsimony analysis ceases to be science."

If multiple models of natural law make exactly the same testable predictions, they are equivalent and there is no need for parsimony to choose one that is preferred. For example, Newtonian, Hamiltonian, and Lagrangian classical mechanics are equivalent. Physicists have no interest in using Occam's razor to say the other two are wrong. Likewise, there is no demand for simplicity principles to arbitrate between wave and matrix formulations of quantum mechanics. Science often does not demand arbitration or selection criteria between models which make the same testable predictions.[10]

Biology

Biologists or philosophers of biology use Occam's razor in either of two contexts both in evolutionary biology: the units of selection controversy and systematics. George C. Williams in his book Adaptation and Natural Selection (1966) argues that the best way to explain altruism among animals is based on low level (i.e. individual) selection as opposed to high level group selection. Altruism is defined[by whom?] as behavior that is beneficial to the group but not to the individual, and group selection is thought by some[by whom?] to be the evolutionary mechanism that selects for altruistic traits. Others posit individual selection as the mechanism which explains altruism solely in terms of the behaviors of individual organisms acting in their own self-interest without regard to the group. The basis for Williams's contention is that of the two, individual selection is the more parsimonious theory. In doing so he is invoking a variant of Occam's razor known as Lloyd Morgan's Canon: "In no case is an animal activity to be interpreted in terms of higher psychological processes, if it can be fairly interpreted in terms of processes which stand lower in the scale of psychological evolution and development" (Morgan 1903).

However, more recent biological analyses, such as Richard Dawkins's The Selfish Gene, have contended that Occams's view is not the simplest and most basic. Dawkins argues the way evolution works is that the genes propagated in most copies will end up determining the development of that particular species, i.e., natural selection turns out to select specific genes, and this is really the fundamental underlying principle, that automatically gives individual and group selection as emergent features of evolution.

Zoology provides an example. Muskoxen, when threatened by wolves, will form a circle with the males on the outside and the females and young on the inside. This as an example of a behavior by the males that seems to be altruistic. The behavior is disadvantageous to them individually but beneficial to the group as a whole and was thus seen by some to support the group selection theory.

However, a much better explanation immediately offers itself once one considers that natural selection works on genes. If the male musk ox runs off, leaving his offspring to the wolves, his genes will not be propagated. If however he takes up the fight his genes will live on in his offspring. And thus the "stay-and-fight" gene prevails. This is an example of kin selection. An underlying general principle thus offers a much simpler explanation, without retreating to special principles as group selection.

Systematics is the branch of biology that attempts to establish genealogical relationships among organisms. It is also concerned with their classification. There are three primary camps in systematics; cladists, pheneticists, and evolutionary taxonomists. The cladists hold that genealogy alone should determine classification and pheneticists contend that similarity over propinquity of descent is the determining criterion while evolutionary taxonomists say that both genealogy and similarity count in classification.

It is among the cladists that Occam's razor is to be found, although their term for it is cladistic parsimony. Cladistic parsimony (or maximum parsimony) is a method of phylogenetic inference in the construction of types of phylogenetic trees (more specifically, cladograms). Cladograms are branching, tree-like structures used to represent lines of descent based on one or more evolutionary change (s). Cladistic parsimony is used to support the hypothesis (es) that require the fewest evolutionary changes. For some types of tree, it will consistently produce the wrong results regardless of how much data is collected (this is called long branch attraction). For a full treatment of cladistic parsimony, see Elliott Sober's Reconstructing the Past: Parsimony, Evolution, and Inference (1988). For a discussion of both uses of Occam's razor in Biology see Elliott Sober's article Let's Razor Ockham's Razor (1990).

Other methods for inferring evolutionary relationships use parsimony in a more traditional way. Likelihood methods for phylogeny use parsimony as they do for all likelihood tests, with hypotheses requiring few differing parameters (i.e., numbers of different rates of character change or different frequencies of character state transitions) being treated as null hypotheses relative to hypotheses requiring many differing parameters. Thus, complex hypotheses must predict data much better than do simple hypotheses before researchers reject the simple hypotheses. Recent advances employ information theory, a close cousin of likelihood, which uses Occam's razor in the same way.

Francis Crick has commented on potential limitations of Occam's razor in biology. He advances the argument that because biological systems are the products of (an on-going) natural selection, the mechanisms are not necessarily optimal in an obvious sense. He cautions: "While Ockham's razor is a useful tool in the physical sciences, it can be a very dangerous implement in biology. It is thus very rash to use simplicity and elegance as a guide in biological research."[44]

In biogeography, parsimony is used to infer ancient migrations of species or populations by observing the geographic distribution and relationships of existing organisms. Given the phylogenetic tree, ancestral migrations are inferred to be those that require the minimum amount of total movement.

Medicine

When discussing Occam's razor in contemporary medicine, doctors and philosophers of medicine speak of diagnostic parsimony. Diagnostic parsimony advocates that when diagnosing a given injury, ailment, illness, or disease a doctor should strive to look for the fewest possible causes that will account for all the symptoms. This philosophy is one of several demonstrated in the popular medical adage "when you hear hoofbeats behind you, think horses, not zebras". While diagnostic parsimony might often be beneficial, credence should also be given to the counter-argument modernly known as Hickam's dictum, which succinctly states that "patients can have as many diseases as they damn well please". It is often statistically more likely that a patient has several common diseases, rather than having a single rarer disease which explains their myriad symptoms. Also, independently of statistical likelihood, some patients do in fact turn out to have multiple diseases, which by common sense nullifies the approach of insisting to explain any given collection of symptoms with one disease. These misgivings emerge from simple probability theory—which is already taken into account in many modern variations of the razor—and from the fact that the loss function is much greater in medicine than in most of general science. Because misdiagnosis can result in the loss of a person's health and potentially life, it is considered better to test and pursue all reasonable theories even if there is some theory that appears the most likely.

Diagnostic parsimony and the counterbalance it finds in Hickam's dictum have very important implications in medical practice. Any set of symptoms could be indicative of a range of possible diseases and disease combinations; though at no point is a diagnosis rejected or accepted just on the basis of one disease appearing more likely than another, the continuous flow of hypothesis formulation, testing and modification benefits greatly from estimates regarding which diseases (or sets of diseases) are relatively more likely to be responsible for a set of symptoms, given the patient's environment, habits, medical history and so on. For example, if a hypothetical patient's immediately apparent symptoms include fatigue and cirrhosis and they test negative for Hepatitis C, their doctor might formulate a working hypothesis that the cirrhosis was caused by their drinking problem, and then seek symptoms and perform tests to formulate and rule out hypotheses as to what has been causing the fatigue; but if the doctor were to further discover that the patient's breath inexplicably smells of garlic and they are suffering from pulmonary edema, they might decide to test for the relatively rare condition of Selenium poisoning.

Religion

In the philosophy of religion, Occam's razor is sometimes applied to the existence of God; if the concept of God does not help to explain the universe better, then the idea is that atheism should be preferred (Schmitt 2005). Some such arguments are based on the assertion that belief in God requires more complex assumptions to explain the universe than non-belief (e.g. the Ultimate Boeing 747 gambit). On the other hand, there are various arguments in favour of a God which attempt to establish a God as a useful explanation. Philosopher Del Ratzsch[45] suggests that the application of the razor to God may not be so simple, least of all when we are comparing that hypothesis with theories postulating multiple invisible universes.[46]

In speaking on religion in God Is Not Great, Christopher Hitchens espoused his variation named Hitchens' razor, which states "What can be asserted without evidence can be dismissed without evidence."

Relation to proofs of God's existence

Main article: Existence of God

Thomas Aquinas, in the Summa Theologica, uses a formulation of Occam's Razor to construct an objection to the idea that God exists:[47]

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, 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 God, some theists consider their belief to be based on grounds independent of, or prior to, reason, making Occam's razor irrelevant. This was the stance of Søren Kierkegaard, who viewed belief in God as a leap of faith which sometimes directly opposed reason.[48] This is also the same basic view of Clarkian Presuppositional apologetics, with the exception that Clark never thought the leap of faith was contrary to reason. (See also: Fideism).

William of Ockham himself was a theist. He believed in God, and in some validity 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."[49] In Ockham's view, an explanation which does not harmonize with reason, experience or the aforementioned sources cannot be considered valid. 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 (e.g. some sort of Non-overlapping magisteria).[50] 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."[51]

Scriptures written in the past by people may represent historical accounts, imaginations and observations that are equivalent in credence for their time as to what is written today by people, regardless of the source of their inspiration or veracity. Occam's razor applied to selection and survival of ancient texts may indicate ancient writings were considered valuable or were just lucky to survive to modern times. Divine intervention in the survival of ancient writings is a more complex explanation.

Another application of the principle is to be found in the work of George Berkeley (1685–1753). Berkeley was an idealist who believed that all of reality could be explained in terms of the mind alone. He invoked Occam's razor against materialism, stating that matter was not required by his metaphysic and was thus eliminable. One potential problem with this view 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.

In the 20th century Philosophy of Mind, Occam's razor was supported by J. J. C. Smart, who in his article "Sensations and Brain Processes" (1959) stated that Occam's razor was the basis for his preference of the mind-brain identity theory over mind-body dualism. Dualists state that there are two kinds of substances in the universe: physical (including the body) and mental, which is nonphysical. In contrast identity theorists state that everything is physical, including consciousness, and that there is nothing nonphysical. The basis for the materialist position is that of the two competing theories, dualism and mind-brain identity, the identity theory is the simpler since it commits to fewer entities. Smart was criticized for his use of the razor and ultimately retracted his advocacy of it in this context.

Paul Churchland (1984) cites Occam's razor as the first line of attack against dualism, but states that by itself it is inconclusive. The deciding factor for Churchland is the greater explanatory prowess of a materialist position in the Philosophy of Mind as informed by findings in neurobiology.

Dale Jacquette (1994) states that Occam's razor is the rationale behind eliminativism and reductionism in the philosophy of mind. Eliminativism is the thesis that the ontology of folk psychology including such entities as "pain", "joy", "desire", "fear", etc., are eliminable in favor of an ontology of a completed neuroscience.

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.[52]

Probability theory and statistics

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 are various papers in scholarly journals deriving formal versions of Occam's razor from probability theory and applying it in statistical inference, and also of various criteria for penalizing complexity in statistical inference. Recent papers have suggested a connection between Occam's razor and Kolmogorov complexity.[53]

One of the problems with the original formulation of the principle is that it only applies to models with the same explanatory power (i.e. prefer the simplest of equally good models). A more general form of Occam's razor can be derived from Bayesian model comparison and Bayes factors, which can be used to compare models that don't fit the data equally well. These methods can sometimes optimally balance the complexity and power of a model. Generally the exact Occam factor is intractable but approximations such as Akaike Information Criterion, Bayesian Information Criterion, Variational Bayes, False discovery rate and Laplace approximation are used. Many artificial intelligence researchers are now employing such techniques.

William H. Jefferys and James O. Berger (1991) generalize and quantify the original formulation's "assumptions" concept as the degree to which a proposition is unnecessarily accommodating to possible observable data.[54] The model they propose balances the precision of a theory's predictions against their sharpness; theories which sharply made their correct predictions are preferred over theories which would have accommodated a wide range of other possible results. This, again, reflects the mathematical relationship between key concepts in Bayesian inference (namely marginal probability, conditional probability and posterior probability).

The statistical view leads to a more rigorous formulation of the razor than previous philosophical discussions. In particular, it shows that "simplicity" must first be defined in some way before the razor may be used, and that this definition will always be subjective. For example, in the Kolmogorov-Chaitin Minimum description length approach, the subject must pick a Turing machine whose operations describe the basic operations believed to represent "simplicity" by the subject. However one could always choose a Turing machine with a simple operation that happened to construct one's entire theory and would hence score highly under the razor. This has led to two opposing views of the objectivity of Occam's razor.

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[55] of small size as the proper basis for excluding arbitrarily complex instruction sets in the formulation of razors. Describing the program for the universal program as the "hypothesis", and the representation of the evidence as program data, it has been formally proven under ZF that "the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized."[56] 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[57][58]

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.[59][60]

According to Jürgen Schmidhuber, the appropriate mathematical theory of Occam's razor already exists, namely, Ray Solomonoff's theory of optimal inductive inference[61] and its extensions.[62] See discussions in[63] for the subtle distinctions between the algorithmic probability (ALP) work of Ray Solomonoff and the Minimum Message Length work of Chris Wallace, and see[64] both for such discussions and also (in sec. 4) discussions of MML and Ockham's razor. For a specific example of MML as Ockham's razor in the problem of decision tree induction, see.[65]

In literature and writing

Occam's razor has been recommended as a measure of how good the plot of a novel is. Simple and logical plots are easy to explain and this enhances the experience of the reader. The writer is also less likely to make an error while explaining the plot to the reader.[66]

In psychology and humor

Hanlon's razor – to explain human behavior, assume incompetence before malice.

Controversial aspects of the razor

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.[lower-alpha 3] Occam's razor is used to adjudicate between theories that have already passed "theoretical scrutiny" tests, and which are equally well-supported by the evidence.[lower-alpha 4] 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.

The other things in question are the evidential support for the theory.[lower-alpha 5] Therefore, a simpler but less correct theory should not be always preferred over a more complex but more correct one. It is this fact which gives the lie to the common misinterpretation of Occam's razor that "the simplest" one is usually the correct one. "Preferred" doesn't necessarily mean "correct". 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.[lower-alpha 6] Quine, in a discussion on definition, referred to these two perspectives as "economy of practical expression" and "economy in grammar and vocabulary", respectively.[69] The theory of relativity is often given as an example of the proliferation of complex words to describe a simple concept.

Galileo Galilei lampooned the misuse of Occam's razor in his Dialogue. The principle is represented in the dialogue by Simplicio. The telling point that Galileo presented ironically was that if you really wanted to start from a small number of entities, you could always consider the letters of the alphabet as the fundamental entities, since you could 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 of Chatton (c. 1290–1343) was a contemporary of William of Ockham (c. 1287–1347) who took exception to Occam's razor and Ockham's use of it. In response he devised his own anti-razor: "If three things are not enough to verify an affirmative proposition about things, a fourth must be added, and so on." Although there have been a number of philosophers who have formulated similar anti-razors since Chatton's time, no one anti-razor has perpetuated in as much notability 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 Gottfried Wilhelm Leibniz (1646–1716), Immanuel Kant (1724–1804), and Karl Menger (1902–1985). Leibniz's version took the form of a principle of plenitude, as Arthur Lovejoy has called it: The idea being that God created the 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."[70]

Karl Menger found mathematicians to be too parsimonious with regard to variables so he formulated his Law Against Miserliness which took one of two forms: "Entities must not be reduced to the point of inadequacy" and "It is vain to do with fewer what requires more." See "Ockham's Razor and Chatton's Anti-Razor" (1984) by Armand Maurer. A less serious, but (some might say) even more extremist anti-razor is 'Pataphysics, the "science of imaginary solutions" invented by Alfred Jarry (1873–1907). Perhaps the ultimate in anti-reductionism, "'Pataphysics seeks no less than to view each event in the universe as completely unique, subject to no laws but its own." Variations on this theme were subsequently explored by the Argentine writer Jorge Luis Borges in his story/mock-essay Tlön, Uqbar, Orbis Tertius. There is also Crabtree's Bludgeon, which takes a cynical view that "[n]o set of mutually inconsistent observations can exist for which some human intellect cannot conceive a coherent explanation, however complicated."

Finally, William Shakespeare's Hamlet asserts that "There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy."[68]