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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. 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?

Richard Swinburne argues for simplicity on logical grounds:. According to Swinburne, since our choice of theory cannot be determined by data see Underdetermination and Duhem—Quine thesis , 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.

From the Tractatus Logico-Philosophicus :. In science , Occam's razor is used as a heuristic to guide scientists in developing theoretical models rather than as an arbiter between published models.

In chemistry , Occam's razor is often an important heuristic when developing a model of a reaction mechanism. An often-quoted version of this constraint which cannot be verified as posited by Einstein himself [50] says "Everything should be kept as simple as possible, but not simpler.

In the scientific method, parsimony is an epistemological , metaphysical or heuristic preference, not an irrefutable principle of logic or a scientific result.

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.

When scientists use the idea of parsimony, it has meaning only in a very specific context of inquiry. Several background assumptions are required for parsimony to connect with plausibility in a particular research problem.

The reasonableness of parsimony in one research context may have nothing to do with its reasonableness in another. It is a mistake to think that there is a single global principle that spans diverse subject matter.

It has been suggested that Occam's razor is a widely accepted example of extraevidential consideration, even though it is entirely a metaphysical assumption.

There is little empirical evidence that the world is actually simple or that simple accounts are more likely to be true than complex ones.

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.

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.

Appeals to simplicity were used to argue against the phenomena of meteorites, ball lightning , continental drift , and reverse transcriptase.

At the time, however, the atomic theory was considered more complex because it implied the existence of invisible particles that had not been directly detected.

Ernst Mach and the logical positivists rejected John Dalton 's atomic theory until the reality of atoms was more evident in Brownian motion , as shown by Albert Einstein.

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, 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. There are examples where Occam's razor would have favored 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 all consistent with available data.

A single instance of Occam's razor favoring a wrong theory falsifies the razor as a general principle. If multiple models of natural law make exactly the same testable predictions, they are equivalent and there is no need for parsimony to choose a preferred one.

For example, Newtonian, Hamiltonian and Lagrangian classical mechanics are equivalent. Physicists have no interest in using Occam's razor to say the other two are wrong.

Likewise, there is no demand for simplicity principles to arbitrate between wave and matrix formulations of quantum mechanics.

Science often does not demand arbitration or selection criteria between models that make the same testable predictions. 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 argues that the best way to explain altruism among animals is based on low-level i.

Altruism is defined by some evolutionary biologists e. Alexander, ; W. Hamilton, as behavior that is beneficial to others or to the group at a cost to the individual, and many posit individual selection as the mechanism that explains altruism solely in terms of the behaviors of individual organisms acting in their own self-interest or in the interest of their genes, via kin selection.

Williams was arguing against the perspective of others who propose selection at the level of the group as an evolutionary mechanism that selects for altruistic traits e.

Wilson, The basis for Williams' contention is that of the two, individual selection is the more parsimonious theory. In doing so he is invoking a variant of Occam's razor known as Morgan's Canon : "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.

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.

Zoology provides an example. Muskoxen , when threatened by wolves , form a circle with the males on the outside and the females and young on the inside.

This is an example of a behavior by the males that seems to be altruistic. The behavior is disadvantageous to them individually but beneficial to the group as a whole and was thus seen by some to support the group selection theory.

Another interpretation is kin selection: if the males are protecting their offspring, they are protecting copies of their own alleles.

Engaging in this behavior would be favored by individual selection if the cost to the male musk ox is less than half of the benefit received by his calf — which could easily be the case if wolves have an easier time killing calves than adult males.

It could also be the case that male musk oxen would be individually less likely to be killed by wolves if they stood in a circle with their horns pointing out, regardless of whether they were protecting the females and offspring.

That would be an example of regular natural selection — a phenomenon called "the selfish herd". Systematics is the branch of biology that attempts to establish patterns of genealogical relationship among biological taxa.

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, pheneticists contend that overall similarity 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 hypotheses of relative degree of relationship, based on shared, derived character states.

Cladistic parsimony is used to select as the preferred hypothesis of relationships the cladogram that requires the fewest implied character state transformations.

Critics of the cladistic approach often observe that for some types of tree, parsimony consistently produces the wrong results, regardless of how much data is collected this is called statistical inconsistency, or long branch attraction.

However, this criticism is also potentially true for any type of phylogenetic inference, unless the model used to estimate the tree reflects the way that evolution actually happened.

Because this information is not empirically accessible, the criticism of statistical inconsistency against parsimony holds no force.

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.

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 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. 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.

In the philosophy of religion , Occam's razor is sometimes applied to the existence of God. William of Ockham himself was a Christian.

He believed in God, and in the authority of Scripture; he writes that "nothing ought to be posited without a reason given, unless it is self-evident literally, known through itself or known by experience or proved by the authority of Sacred Scripture.

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.

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.

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: [60].

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 a god, some theists base their belief upon grounds independent of, or prior to, reason, making Occam's razor irrelevant.

Various arguments in favor of God establish God as a useful or even necessary assumption. Contrastingly some anti-theists hold firmly to the belief that assuming the existence of God introduces unnecessary complexity Schmitt , e.

Another application of the principle is to be found in the work of George Berkeley — 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 belief is that it's possible, given Berkeley's position, to find solipsism itself more in line with the razor than a God-mediated world beyond a single thinker.

Occam's razor may also be recognized in the apocryphal story about an exchange between Pierre-Simon Laplace and Napoleon.

It is said that in praising Laplace for one of his recent publications, the emperor asked how it was that the name of God, which featured so frequently in the writings of Lagrange , appeared nowhere in Laplace's.

At that, he is said to have replied, "It's because I had no need of that hypothesis. In his article "Sensations and Brain Processes" , J.

Smart invoked Occam's razor with the aim to justify his preference of the mind-brain identity theory over spirit-body dualism.

Dualists state that there are two kinds of substances in the universe: physical including the body and spiritual, which is non-physical. In contrast, identity theorists state that everything is physical, including consciousness, and that there is nothing nonphysical.

Though it is impossible to appreciate the spiritual when limiting oneself to the physical, Smart maintained that identity theory explains all phenomena by assuming only a physical reality.

Subsequently, Smart has been severely criticized for his use or misuse of Occam's razor and ultimately retracted his advocacy of it in this context.

Paul Churchland states that by itself Occam's razor is inconclusive regarding duality. In a similar way, Dale Jacquette stated that Occam's razor has been used in attempts to justify eliminativism and reductionism in the philosophy of mind.

Eliminativism is the thesis that the ontology of folk psychology including such entities as "pain", "joy", "desire", "fear", etc. 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.

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 [64] [65] have suggested a connection between Occam's razor and Kolmogorov complexity. One of the problems with the original formulation of the razor is that it only applies to models with the same explanatory power i.

A more general form of the razor can be derived from Bayesian model comparison, which is based on Bayes factors and can be used to compare models that don't fit the observations equally well.

These methods can sometimes optimally balance the complexity and power of a model. Generally, the exact Occam factor is intractable, but approximations such as Akaike information criterion , Bayesian information criterion , Variational Bayesian methods , false discovery rate , and Laplace's method are used.

Many artificial intelligence researchers are now employing such techniques, for instance through work on Occam Learning or more generally on the Free energy principle.

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. 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.

Some attempts have been made to re-derive known laws from considerations of simplicity or compressibility. According to Jürgen Schmidhuber , the appropriate mathematical theory of Occam's razor already exists, namely, Solomonoff's theory of optimal inductive inference [70] and its extensions.

Dowe's "Foreword re C. Wallace" [72] 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" [73] both for such discussions and for in section 4 discussions of MML and Occam's 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. 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.

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.

Occam's razor has met some opposition from people who have considered it too extreme or rash. Walter Chatton c.

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.

Leibniz's version took the form of a principle of plenitude , as Arthur Lovejoy has called it: the idea being that God created the most varied and populous of possible worlds.

Kant felt a need to moderate the effects of Occam's razor and thus created his own counter-razor: "The variety of beings should not rashly be diminished.

Karl Menger found mathematicians to be too parsimonious with regard to variables, so he formulated his Law Against Miserliness, which took one of two forms: "Entities must not be reduced to the point of inadequacy" and "It is vain to do with fewer what requires more.

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.

There is also Crabtree's Bludgeon , which cynically states that "[n]o set of mutually inconsistent observations can exist for which some human intellect cannot conceive a coherent explanation, however complicated.

From Wikipedia, the free encyclopedia. Philosophical principle of selecting the solution with the fewest assumptions.

For the aerial theatre company, see Ockham's Razor Theatre Company. See also: Pragmatism and Problem of Induction. Main article: Akaike information criterion.

Main article: Existence of God. Philosophy portal Psychology portal Science portal. 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.

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.

Australasian Journal of Philosophy. Retrieved 1 June International Journal for Philosophy of Chemistry. Retrieved 12 July Stanford Encyclopedia of Philosophy.

California: Stanford University. Physics in Canada. Bibcode : arXiv In Knowles, Dudley ed. Explanation and Its Limits.

Cambridge University Press. Ockam's Razor: A User's Manual. Luke Wadding, Louvain , reprinted Paris: Vives, p. See also Franklin, op cit. Philosophical Studies.

The Johns Hopkins University Press. Chap 9. Archived from the original on 28 April Retrieved 26 March Boing Boing.

Archived from the original on 31 March The linguistics Student's Handbook. Edinburgh: Edinburgh University Press.

A Dictionary of Philosophy. London: Pan Books. Archived from the original on 23 August Retrieved 12 June On the Shoulders of Giants.

Running Press. Retrieved 24 February Regula I. Regula II. Metaphysics Research Lab, Stanford University. Foundations of Physics Letters.

Bibcode : math. Bibcode : Entrp.. In Zalta, Edward N. Archived from the original on 6 July Vernon, VA: Mt. 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 are more testable.

The phrase Occam's razor did not appear until a few centuries after William of Ockham 's death in Libert Froidmont , in his On Christian Philosophy of the Soul , takes credit for the phrase, speaking of " novacula occami ".

AD stated, "We consider it a good principle to explain the phenomena by the simplest hypothesis possible. 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.

For if one thing were demonstrated from many and another thing from fewer equally known premises, clearly that is better which is from fewer because it makes us know quickly, just as a universal demonstration is better than particular because it produces knowledge from fewer premises.

Similarly in natural science, in moral science, and in metaphysics the best is that which needs no premises and the better that which needs the fewer, other circumstances being equal.

The Summa Theologica of Thomas Aquinas — states that "it is superfluous to suppose that what can be accounted for by a few principles has been produced by many.

William of Ockham circa — was an English Franciscan friar and theologian , an influential medieval philosopher and a nominalist. His popular fame as a great logician rests chiefly on the maxim attributed to him and known as Occam's razor.

The term razor refers to distinguishing between two hypotheses either by "shaving away" unnecessary assumptions or cutting apart two similar conclusions.

While it has been claimed that Occam's razor is not found in any of William's writings, [17] one can cite statements such as Numquam ponenda est pluralitas sine necessitate William of Ockham — Wikiquote "Plurality must never be posited without necessity" , which occurs in his theological work on the Sentences of Peter Lombard Quaestiones et decisiones in quattuor libros Sententiarum Petri Lombardi ; ed.

Nevertheless, the precise words sometimes attributed to William of Ockham, Entia non-sunt multiplicanda praeter necessitatem Entities must not be multiplied beyond necessity , [18] are absent in his extant works; [19] this particular phrasing comes from John Punch , [20] who described the principle as a "common axiom" axioma vulgare of the Scholastics.

This principle is sometimes phrased as Pluralitas non-est ponenda sine necessitate "Plurality should not be posited without necessity". 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. Bertrand Russell offers a particular version of Occam's razor: "Whenever possible, substitute constructions out of known entities for inferences to unknown entities.

Around , 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. Another technical approach to Occam's razor is ontological parsimony. This is considered a strong version of Occam's razor.

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.

This principle goes back at least as far as Aristotle, who wrote "Nature operates in the shortest way possible.

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.

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.

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

Occam's razor has gained strong empirical support in 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. 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.

Some increases in complexity are sometimes necessary, so there 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.

Even other empirical criteria, such as 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.

But if an alternative ad hoc hypothesis were indeed justifiable, its implicit conclusions would be empirically verifiable.

On a commonly accepted repeatability principle, these alternative theories have never been observed and continue to escape observation.

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.

This endless supply of elaborate competing explanations, called saving hypotheses, cannot be ruled out — except by using Occam's razor.

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.

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.

The probabilistic Bayesian basis for Occam's razor is elaborated by David J. MacKay in chapter 28 of his book Information Theory, Inference, and Learning Algorithms , [38] where he emphasizes that a prior bias in favour of simpler models is not required.

William H. Jefferys and James O. Berger generalize and quantify the original formulation's "assumptions" concept as the degree to which a proposition is unnecessarily accommodating to possible observable data.

This, again, reflects the mathematical relationship between key concepts in Bayesian inference namely marginal probability , conditional probability , and posterior probability.

The bias—variance tradeoff is a framework that incorporates the Occam's razor principle in its balance between overfitting i.

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".

This is again comparing a simple theory to a more complex theory where both explain the data equally well. The philosopher of science Elliott Sober once argued along the same lines as Popper, tying simplicity with "informativeness": The simplest theory is the more informative, in the sense that it requires less information to a question.

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. 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?

Richard Swinburne argues for simplicity on logical grounds:. According to Swinburne, since our choice of theory cannot be determined by data see Underdetermination and Duhem—Quine thesis , 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.

From the Tractatus Logico-Philosophicus :. In science , Occam's razor is used as a heuristic to guide scientists in developing theoretical models rather than as an arbiter between published models.

In chemistry , Occam's razor is often an important heuristic when developing a model of a reaction mechanism. An often-quoted version of this constraint which cannot be verified as posited by Einstein himself [50] says "Everything should be kept as simple as possible, but not simpler.

In the scientific method, parsimony is an epistemological , metaphysical or heuristic preference, not an irrefutable principle of logic or a scientific result.

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.

When scientists use the idea of parsimony, it has meaning only in a very specific context of inquiry. Several background assumptions are required for parsimony to connect with plausibility in a particular research problem.

The reasonableness of parsimony in one research context may have nothing to do with its reasonableness in another. It is a mistake to think that there is a single global principle that spans diverse subject matter.

It has been suggested that Occam's razor is a widely accepted example of extraevidential consideration, even though it is entirely a metaphysical assumption.

There is little empirical evidence that the world is actually simple or that simple accounts are more likely to be true than complex ones.

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.

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.

Appeals to simplicity were used to argue against the phenomena of meteorites, ball lightning , continental drift , and reverse transcriptase.

At the time, however, the atomic theory was considered more complex because it implied the existence of invisible particles that had not been directly detected.

Ernst Mach and the logical positivists rejected John Dalton 's atomic theory until the reality of atoms was more evident in Brownian motion , as shown by Albert Einstein.

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, 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. There are examples where Occam's razor would have favored 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 all consistent with available data.

A single instance of Occam's razor favoring a wrong theory falsifies the razor as a general principle. If multiple models of natural law make exactly the same testable predictions, they are equivalent and there is no need for parsimony to choose a preferred one.

For example, Newtonian, Hamiltonian and Lagrangian classical mechanics are equivalent. Physicists have no interest in using Occam's razor to say the other two are wrong.

Likewise, there is no demand for simplicity principles to arbitrate between wave and matrix formulations of quantum mechanics.

Science often does not demand arbitration or selection criteria between models that make the same testable predictions.

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 argues that the best way to explain altruism among animals is based on low-level i.

Altruism is defined by some evolutionary biologists e. Alexander, ; W. Hamilton, as behavior that is beneficial to others or to the group at a cost to the individual, and many posit individual selection as the mechanism that explains altruism solely in terms of the behaviors of individual organisms acting in their own self-interest or in the interest of their genes, via kin selection.

Williams was arguing against the perspective of others who propose selection at the level of the group as an evolutionary mechanism that selects for altruistic traits e.

Wilson, The basis for Williams' contention is that of the two, individual selection is the more parsimonious theory. In doing so he is invoking a variant of Occam's razor known as Morgan's Canon : "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.

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.

Zoology provides an example. Muskoxen , when threatened by wolves , form a circle with the males on the outside and the females and young on the inside.

This is an example of a behavior by the males that seems to be altruistic. The behavior is disadvantageous to them individually but beneficial to the group as a whole and was thus seen by some to support the group selection theory.

Another interpretation is kin selection: if the males are protecting their offspring, they are protecting copies of their own alleles.

Engaging in this behavior would be favored by individual selection if the cost to the male musk ox is less than half of the benefit received by his calf — which could easily be the case if wolves have an easier time killing calves than adult males.

It could also be the case that male musk oxen would be individually less likely to be killed by wolves if they stood in a circle with their horns pointing out, regardless of whether they were protecting the females and offspring.

That would be an example of regular natural selection — a phenomenon called "the selfish herd". Systematics is the branch of biology that attempts to establish patterns of genealogical relationship among biological taxa.

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, pheneticists contend that overall similarity 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 hypotheses of relative degree of relationship, based on shared, derived character states.

Cladistic parsimony is used to select as the preferred hypothesis of relationships the cladogram that requires the fewest implied character state transformations.

Critics of the cladistic approach often observe that for some types of tree, parsimony consistently produces the wrong results, regardless of how much data is collected this is called statistical inconsistency, or long branch attraction.

However, this criticism is also potentially true for any type of phylogenetic inference, unless the model used to estimate the tree reflects the way that evolution actually happened.

Because this information is not empirically accessible, the criticism of statistical inconsistency against parsimony holds no force.

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.

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 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. 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. In the philosophy of religion , Occam's razor is sometimes applied to the existence of God.

William of Ockham himself was a Christian. He believed in God, and in the authority of Scripture; he writes that "nothing ought to be posited without a reason given, unless it is self-evident literally, known through itself or known by experience or proved by the authority of Sacred Scripture.

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. 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.

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: [60].

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 a god, some theists base their belief upon grounds independent of, or prior to, reason, making Occam's razor irrelevant.

Various arguments in favor of God establish God as a useful or even necessary assumption. Contrastingly some anti-theists hold firmly to the belief that assuming the existence of God introduces unnecessary complexity Schmitt , e.

Another application of the principle is to be found in the work of George Berkeley — 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 belief is that it's possible, given Berkeley's position, to find solipsism itself more in line with the razor than a God-mediated world beyond a single thinker.

Occam's razor may also be recognized in the apocryphal story about an exchange between Pierre-Simon Laplace and Napoleon. It is said that in praising Laplace for one of his recent publications, the emperor asked how it was that the name of God, which featured so frequently in the writings of Lagrange , appeared nowhere in Laplace's.

At that, he is said to have replied, "It's because I had no need of that hypothesis. In his article "Sensations and Brain Processes" , J.

Smart invoked Occam's razor with the aim to justify his preference of the mind-brain identity theory over spirit-body dualism.

Dualists state that there are two kinds of substances in the universe: physical including the body and spiritual, which is non-physical.

In contrast, identity theorists state that everything is physical, including consciousness, and that there is nothing nonphysical.

Though it is impossible to appreciate the spiritual when limiting oneself to the physical, Smart maintained that identity theory explains all phenomena by assuming only a physical reality.

Subsequently, Smart has been severely criticized for his use or misuse of Occam's razor and ultimately retracted his advocacy of it in this context.

Paul Churchland states that by itself Occam's razor is inconclusive regarding duality. In a similar way, Dale Jacquette stated that Occam's razor has been used in attempts to justify eliminativism and reductionism in the philosophy of mind.

Eliminativism is the thesis that the ontology of folk psychology including such entities as "pain", "joy", "desire", "fear", etc. 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.

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 [64] [65] have suggested a connection between Occam's razor and Kolmogorov complexity. One of the problems with the original formulation of the razor is that it only applies to models with the same explanatory power i.

A more general form of the razor can be derived from Bayesian model comparison, which is based on Bayes factors and can be used to compare models that don't fit the observations equally well.

These methods can sometimes optimally balance the complexity and power of a model. Generally, the exact Occam factor is intractable, but approximations such as Akaike information criterion , Bayesian information criterion , Variational Bayesian methods , false discovery rate , and Laplace's method are used.

Many artificial intelligence researchers are now employing such techniques, for instance through work on Occam Learning or more generally on the Free energy principle.

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.

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.

Some attempts have been made to re-derive known laws from considerations of simplicity or compressibility. According to Jürgen Schmidhuber , the appropriate mathematical theory of Occam's razor already exists, namely, Solomonoff's theory of optimal inductive inference [70] and its extensions.

Dowe's "Foreword re C. Wallace" [72] 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" [73] both for such discussions and for in section 4 discussions of MML and Occam's razor.

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