The role of probability arguments in the history of science

Studies in History and Philosophy of Science 41 (2010) 95–104

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The role of probability arguments in the history of science Friedel Weinert Department of Social Sciences and Humanities, University of Bradford, Richmond Road, Bradford BD7 1DP, UK

a r t i c l e

i n f o

Article history: Received 29 May 2008 Received in revised form 20 April 2009

Keywords: Copernicanism Darwinism Atom models Bayesianism Likelihoods Probability arguments

a b s t r a c t The paper examines Wesley Salmon’s claim that the primary role of plausibility arguments in the history of science is to impose constraints on the prior probability of hypotheses (in the language of Bayesian confirmation theory). A detailed look at Copernicanism and Darwinism and, more briefly, Rutherford’s discovery of the atomic nucleus reveals a further and arguably more important role of plausibility arguments. It resides in the consideration of likelihoods, which state how likely a given hypothesis makes a given piece of evidence. In each case the likelihoods raise the probability of one of the competing hypotheses and diminish the credibility of its rival, and this may happen either on the basis of ‘old’ or ‘new’ evidence. Ó 2009 Elsevier Ltd. All rights reserved.

When citing this paper, please use the full journal title Studies in History and Philosophy of Science

1. Introduction Wesley Salmon has discussed the importance of plausibility arguments in the practice of science. He has related the role of plausibility arguments to the Bayesian apparatus of theory confirmation. Salmon argues that plausibility arguments can be of several kinds: formal (i.e. a scientific theory must be internally consistent), pragmatic (i.e. it matters from which tradition a hypothesis originated) and most importantly material (i.e. concerning questions of simplicity and symmetry but also the plausibility of causal mechanisms) (Salmon, 2005, p. 76). Salmon argues that plausibility considerations play a crucial role in the context of justification, rather than the context of discovery. In particular he sees their primary role as constituting constraints on the prior probabilities of hypotheses. The purpose of this paper is to examine the role of plausibility arguments in some striking cases in the history of science and to consider Salmon’s views regarding their role in the context of justification. In particular the paper will concentrate on ‘material’ plausibility arguments, which will be considered more specifically as probability arguments regarding physical processes and parameters. The result is that probability arguments of this kind do not simply impose constraints on prior probabilities but are important in a consideration of likelihoods—that is, the question of how likely a given hypothesis renders the evidence. Such material probability E-mail address: [email protected] 0039-3681/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.shpsa.2009.12.003

arguments were employed by the Copernicans and Darwinians, and had their role cut out in modern physics. Such probability arguments are important in the assessment of competing hypotheses when the evidence is well-known but insufficient to constitute a crucial experiment. Even when the evidence is insufficiently strong to decide between rival hypotheses, scientists will advance probability arguments regarding physical mechanisms and processes in order to give a differential weighting to the credibility of the competing hypotheses. This use of likelihoods, on the basis of established evidence, happens in periods of pre-revolutionary science (Copernicus) and in periods of scientific revolutions (Darwin). But even when there is no disagreement about a given model, new evidence may generate probability arguments, which, as Rutherford showed with respect to atom models, will discredit one model at the expense of its rival model. Although Rutherford and his co-workers started their experiments on the basis of their belief in Thomson’s ‘plum-pudding-model’, the results of large-angle scattering experiments led to likelihood considerations, which considerably diminished the plausibility of Thomson’s model. 2. Copernicanism In 1543, Nicolas Copernicus introduced his heliocentric hypothesis, according to which the mean sun is the centre of the (then

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known) universe and the earth becomes a planet, orbiting the sun like the other planets. The introduction of a ‘mathematical’ model of heliocentrism has generally not been hailed as a scientific revolution. The reasons advanced by historians of science for this negative assessment relate to the apparent lack of progress of the heliocentric model over the geocentric model. Historians of science typically point out that at the time of publication of Copernicus’s book, De revolutionibus, the heliocentric model was observationally equivalent to the geocentric model. Although Copernicus reports his own observations at Frauenburg, they do not amount to new discoveries, unlike, for example, Galileo’s later discovery of the Jupiter moons, which could threaten the geocentric model. Both the heliocentric and the geocentric models can account for the apparent planetary motions. Although Copernicus switches the central earth of geocentrism for the central sun of heliocentrism, he uses Greek geometric devices (such as epicycles and eccentric circles) to account for the motion of the planets (Fig. 1). Copernicus still adheres to the Greek postulate of circular motion for all celestial objects and criticizes Ptolemy for his employment of the equant (an off-centre geometric device to make a planet’s motion appear uniform around an epicycle) (Fig. 2). Copernicus follows Ptolemy in regarding the various Greek geometric devices as mathematically equivalent. Thus Copernicus accepts the ‘equipollence of hypotheses’: all mathematical hypotheses are regarded as equivalent if they can correctly account for the ‘appearances’ and no further question is asked as to the correctness of these hypotheses in terms of their physical nature. Yet the Copernican system went beyond mathematical astronomy. It also contained physical arguments about the location and the motion of the earth and the arrangement of the planets in the solar system (cf. Kokowki, 2004). Finally, Copernicus lacks any convincing physical mechanism by which the motion of the planets could be understood. Although Kepler was interested in ‘physical causes’, his speculations about magnetic rays from the sun keeping the planets in orbit fail to provide a convincing explanatory account of planetary motion. It was only when Newton combined gravitational forces with the first law of mechanics that a physical explanation of planetary motion within classical mechanics became available. These are weighty arguments in favour of withholding the honorific title of ‘scientific revolutionary’ from Copernicus. Yet it would be a mistake, in the author’s opinion, to deny any progress to the Copernican model, as some historians of science are wont

Fig. 2. The equant. Explanation of retrograde motion with a new geometric device, the equant. This representation is supposed to be a closer fit of the model to the data than the elementary model. From the point of view of the equant, the motion of the planet on the epicycle would appear uniform. Further flexibility is introduced by letting the Earth either sit at the Centre of the deferent or off-centre, as indicated in the diagram. (Adapted from Weinert, 2009, Fig. 1.5b.)

to do. According to de Price, for example, the mathematical part of the Copernican treatise is ‘little more than a re-shuffled version of the Almagest’ (Price, 1962, p. 215). The work of Copernicus can at least be described as a Copernican turn, understood here as a shift in perspective with respect to the location and motion of the earth in the solar system. This turn contributed to the Copernican revolution, which was completed in the work of Kepler, Galileo and Newton. The Copernican shift of perspective gives the heliocentric model several advantages (Weinert, 2009, Ch. 1, Sect. 4). In order to appreciate this turn it is important to go beyond a consideration of observational data and their mathematical analysis in terms of geometric devices and consider other criteria. It has not been sufficiently appreciated in the literature that the proponents of Copernicanism employed probability arguments in favour of the heliocentric model. 3. Probability arguments The Copernicans—the defenders of the heliocentric model from Copernicus to Galileo—employ probability arguments in order to support their claim that the Copernican model is cognitively more adequate than the geocentric model (despite the drawbacks mentioned above). The Copernicans employ these probability arguments in order to claim that the Copernican model of the solar system is more plausible than the geocentric model and that it leads to a more coherent view of the solar system. It is important to note that these probability arguments refer to physical events and processes, which are assumed to operate in the solar system, and which are modelled in the Copernican model. These probabilities are not prima facie subjective degrees of beliefs on the part of individual scientists, as they are understood in some versions of Bayesianism. Rather, the probability arguments regarding physical processes were meant to support the belief in heliocentrism. The Copernican probability arguments can be divided into two types: 3.1. The position of the earth

Fig. 1. To explain retrograde motion, a planet is not carried around the major sphere, the deferent, but on a epicycle, moving in the same direction as the deferent. This device was used by the Greeks to create a better fit of the observations with their geocentric models. (Adapted from Weinert, 2009, Fig. 1.5a.)

The Copernicans claim, firstly, that a shift in the location of the earth from a stationary ‘central’ position to a planetary position leads to a better model of the spatial (topologic) arrangement of

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the planets in the solar system. Note that this topologic arrangement of the planets—their order and relative distance from the sun—is achieved in the Copernican model on the basis of Greek geometric devices, which became discredited through the work of Kepler and Newton. Nevertheless, various figures in the history of Copernicanism—Copernicus himself, and his pupil Rheticus— argued that the location of the earth amongst the planets, orbiting the sun, results in a greater coherence of the planetary model. The Copernican hypothesis ‘harmonizes the observations’ (Rheticus, 1959 [1540], p. 163) in a way that the geocentric model fails to do. The geocentric view is unable to explain the ‘remarkable symmetry and interconnection’ of planetary motions (ibid., p. 145). In order to understand the importance of the emphasis on coherence on the part of the Copernicans it is helpful to appreciate that Ptolemy treated each planet’s motion separately and not in relation to other planets. This procedure had the unwelcome consequence that, at first, Ptolemy was unable, in his Algamest, to determine the order of the planets on the basis of evidence. In his later work, he adopted a ‘nesting hypothesis’, which ‘allowed to fix the order and the distances of the planets from the Earth’ (Goldstein, 2002, p. 220). Copernicus is the first modern astronomer who models the then known planets as a system, such that all the planets and their correlations are taken into account. Copernicus stresses in his book that the ‘mobility of the earth binds together the order and magnitude of the orbital circles of wandering stars’ (Copernicus, 1995 [1543], Bk V, Introduction). Copernicus and Rheticus argue that a ‘planetary earth’ model is more plausible because of the explanatory advantage it accrues: it leads to a correct ordering of the planets in the solar system and a new estimate of the relative distances of the planets from the sun. Copernicus rejects the ‘nesting hypothesis’ and adopts the period-distance relationship, known from antiquity, as a technique to determine the topologic arrangement of the planets in his system. Copernicus concludes ‘that all these arguments make it more likely that the earth moves than that it is at rest’ (Copernicus, 1995 [1543], Bk. I, Ch. 8).

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The plausibility of a model of the planetary earth with dual ‘mobility’ (annual and daily rotation) was justified by appeal to the greater probability of the location of the earth amongst the orbiting planets. This shift in the location of the earth greatly increases the coherence of the planetary model. The dual ‘mobility’ of the earth also provides a natural explanation of one of the great problems in Greek astronomy, the so-called ‘retrograde motion of planets’ (i.e. the apparent temporary suspension of the normal west-to-east motion, such that a planet seemed to move from east to west for a certain period of time, as seen from the earth) (see Fig. 3). The Copernicans argue that the observational ‘retrograde’ motion of the planets is more plausibly explained as an appearance due to the location of the moving earth in the planetary system rather than as a consequence of epicyclic motion of planets around a central earth on the geocentric model. The Copernican model also partly explains the different orbital periods of the planets by placing them at the correct relative distances from the sun, even though Copernicus still assumed uniform circular motion, so that he needed minor epicycles. In the case of the Copernican model, we do well to emphasize a distinction between the topologic and algebraic structure of the model. In Copernicus’s case it is only the topologic structure—the spatial arrangement of the planets—which is in good agreement with the reality of the solar system. The underlying algebraic structure, which accounts for the observable phenomena, had to undergo many refinements before it could be regarded as a physical explanation of planetary motion. In fact, Copernicus himself entertained doubts about the underlying geometric devices: he remained agnostic when he asked himself which of the various geometric devices correspond to physical features of the universe. Hence his adoption of the aforementioned ‘equipollence of hypotheses’. Copernicus was a realist with respect to the topologic structure of his model but an instrumentalist with respect to its algebraic structure.

Fig. 3. A simplified scheme of the appearance of retrograde motion of Venus as seen from by an Earth-bound observer. The observer ‘marks’ the position of Venus against the background stars as the planet prepares to overtake Earth in its orbit (position 1). When Venus has overtaken Earth, the observer makes a second observation: as expected, Venus has moved from west to east (position 2). But at a later stage, a third observation reveals an apparent and abnormal retracing of the orbit of Venus towards the west. In a heliocentric view this is due to the relative position of the Earth with respect to Venus around the Sun. (Adapted from Weinert, 2009, Fig. 1.4.)

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It should be noted at this point that the order of the planets was contested and that as late as 1651 Giovanni Riccioli’s textbook of astronomy presented six different astronomical models (Barrow, 2008, p. 23). Each of these models attempted to account for the appearances. Only the Copernican model is sun-centred and it may claim an explanatory advantage due to its coherence, which naturally explains both retrograde motion and the different orbital periods of the planets around the sun. From today’s perspective the Copernican system also accounts more naturally for the phases of Venus because its topologic order is more realistic than that of its earth-centred rivals. The problem was that the topologic order of the planets was not firmly established at that time. At least the Tychonic system—in which the moon and the sun revolve around the central earth, whilst the other planets revolve around the sun—can equally account for the phases of Venus. But to account for the variations in the seasonal motion of the sun, the Tychonic model has to introduce a deferent-epicycle system, which makes it less economic than the Copernican system. Furthermore, the Copernicans could appeal to a second plausibility argument, which increased the explanatory power of the heliocentric model. 3.2. The rotation of the ‘fixed’ stars The second probability argument exploits the dual mobility of the earth. It is favoured by Kepler. It is briefly mentioned in Copernicus (1995 [1543], Bk. I, §6) and appears in Galileo’s Dialogue (1953 [1632]). More precisely, the argument is concerned with the apparent rotational velocity of the outer sphere of ‘fixed’ stars. On both the geocentric and the heliocentric model the rotation of the sphere of ‘fixed’ stars requires explanation, but the physical consequences of the explanation are strikingly different, depending on whether a stationary or rotating earth is assumed. Kepler’s argument states that we should attach more plausibility to the heliocentric view because the evidence—the apparent motion of the ‘fixed’ stars in a twenty-four-hour rhythm about the earth—is more plausible on the view that the earth rotates on its own axis. It is physically more probable that the earth turns once on its own axis in twenty-four hours than that the sphere of the fixed stars moves ‘at incalculable speed’, in the same period, around a stationary earth (Kepler, 1995 [1618-1621], Bk. IV, Pt. I, §3). Kepler continues, it is more probable that the sphere of the fixed stars should be 2,000 or 1,000 times wider than the ancients said than that it should be 24,000 times faster than Copernicus said. (ibid., §4, p. 43) Similar considerations are advanced by Galileo’s spokesman Salviati in the Dialogue (1953 [1632], Second Day). Salviati complains that geocentrism attributes ‘incredible speeds’ to the sphere of fixed stars, which violates the simplicity of Nature’s ways. The same observation of rotation can be explained by the ‘moderate’ speed of the earth on its own axis. Salviati also points out that the natural relationship between the distance of the planets from the sun and their orbital periods is disturbed on geocentric assumptions (Galilei, 1962 [1632], p. 116–120). Saturn takes thirty years for the completion of its orbit but the sphere of the fixed stars, much further away, improbably requires a diurnal rotation. But it was Michael Maestlin, Kepler’s teacher at Tübingen, who carried out calculations to demonstrate the ‘improbability’ of a rotation of the celestial sphere. Consider the divergent probabilities

that follow from a modern reconsideration of the angular velocities involved under the two scenarios.1 Under some simplifying assumptions, the angular velocity of the earth for an observer at the equator is 464 m=s ¼ 1670 km=h. The geocentric view, by contrast, has to assume an angular velocity of the ‘fixed’ stars about the stationary earth. A calculation produces a value of 4:62  105 m=s ¼ 1:66  106 km=h. It is such an enormous rotational velocity of the stars—1.66 million km/h, compared to 1670 km/h for the earth at the equator—which the Copernicans considered improbable on mechanical grounds. By comparison, the orbital velocity of the earth around the sun is 30 km/s and the velocity of the solar system around the galactic centre is 225 km/s. Thus Kepler and Maestlin base their plausibility arguments on a physical feature of the heliocentric model. The daily rotation of the earth is the physical cause of the apparent rotation of the ‘fixed’ stars. This rotation is more probable, given the speeds involved, than the rotation attributed to the fixed stars in the geocentric model. 3.3. Bayesian considerations We can now turn to Salmon’s considerations to throw some further light on the epistemological status of the Copernican hypothesis in relation to the geocentric view. For the purpose of this exercise we shall reconstruct the historical Copernicans as ‘Bayesian’ Copernicans. Salmon relates plausibility arguments in the history of science to Bayesian considerations. In Salmon’s view plausibility arguments help scientists to decide which of competing hypotheses is to be regarded as more plausible, prior to subjecting them to particular evidence. That is, plausibility arguments help scientists to decide on the prior probability— P(h)—of a hypothesis before they consider it for testing. Generally, a hypothesis is regarded as more plausible if it fits into the established corpus of beliefs. For instance, Riccioli’s astronomy textbook (1651) presents the majority of astronomical models as favouring an earth-centred model of the solar system. In such a situation, the Copernicans did well to concern themselves, in terms of the Bayesian apparatus, with the likelihoods—PðejhÞ—of the rival models and not with their prior probabilities. Likelihoods measure how likely various hypotheses make the given evidence, here the apparent rotation of the stars. The Copernicans asked how probable the rival hypotheses (geocentrism, heliocentrism) made the evidence. They considered the problem of physical processes, which could account for the apparent rotation of the stars. As they found a rotation of the fixed stars ‘improbable’, they advanced probability arguments against the geocentric view. For the purpose of this exercise we can treat both theories as deterministic. The Copernican hypothesis, h1, is deterministic, since the evidence in question, for example the motion of the earth with respect to the ‘fixed’ stars, follows deductively from the heliocentric but not from the geocentric hypothesis. The diurnal motion of the stars, from east to west, follows deductively from the geocentric hypothesis, h2, which of course assumes that the earth is stationary. As the evidence, e, is well known—it is the rotational period of the ‘fixed’ stars—and entailed by both hypotheses, the likelihoods of the two hypotheses equal 1: Pðejh1 Þ ¼ 1 & Pðejh2 Þ ¼ 1. In terms of observational evidence, both geocentrism and heliocentrism faced a stalemate. Nevertheless, the Copernicans maintained that the heliocentric view is more plausible. In Bayesian terms the Copernicans argue that the posterior probability of h1 is greater than the posterior probability of h2. As there was no striking evidence for the diurnal rotation of

1 To arrive at these figures we assume a circular motion of the earth on its own axis at the equator and a circular motion of the sphere of the fixed stars in a twenty-four-hour period around the earth. The equation for the angular velocity in both cases is mEarth=Stars ¼ xr ¼ dpf ; f ¼ 1=T ¼ 8:644 s. The radius of the earth is 6.37  105 m and the radius of the earth-star distance is taken to be 1.27  1010 m in line with Ptolemy’s views. Note that the angular velocity of the earth 45° to the north of the equator is only 1180 km/h. If we adopt Maestlin’s smaller earth-star distance estimate—1.5  109 m—we get a rotational velocity of 1,132 German miles per ‘pulse’. If we count 4,000 pulses per hour and take the traditional measure of 1 German mile = 7,532 m, the figure increases to 3.4  107 km/h, which, in Maestlin’s words, ‘truly exceeds all belief’ (Tredwell, 2004, p. 318).

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the earth until Bradley’s observation of the aberration of star light in 1728 and, later, Foucault’s pendulum experiment in 1850, we can regard the heliocentric and geocentric views regarding the motion of the earth as observationally equivalent but structurally different. The only arguments available were the physical probability arguments about rotational velocities. In order to argue this point about plausibility the Copernicans took into account some relevant physical process, b, which we take to be physical regularities about rotational velocities. (A separate issue is that from Kepler to Newton new observational evidence—the orbit of comets, the topology of the moon and the discovery of the Jupiter moons—began to favour the heliocentric model over the geocentric model, since it seemed to enjoy a better fit with the available evidence.) In Bayesian terms the question is not simply whether the observational evidence, the rotation of the planets and the stars supports one hypothesis more than its rival. Rather, the question is whether one hypothesis, enhanced by arguments about a physical process regarding rotational velocities of celestial objects, b, receives more support from the evidence than does its rival. Bayes’s Theorem should therefore be written as:

Pðhje & bÞ ¼

Pðejh & bÞPðhÞ PðeÞ

ð1Þ

Kepler’s plausibility argument, in Bayesian terms, states that we should attach more plausibility to the heliocentric view, h1, because one particular piece of evidence— the apparent motion of the ‘fixed’ stars in a twenty-four-hour rhythm about the earth—is more plausible on the assumption that the earth rotates on its own axis. According to equation (1), the calculation of the posterior probability of a hypothesis—PðhjeÞ—requires the calculation of the prior probability—P(h)—of the hypothesis under consideration, and the likelihood. But, to start with prior probability, the Copernicans may have found it difficult to say what the prior probability of heliocentrism was (that is, irrespective of the observational consequences to be tested) because heliocentrism was formulated as a rival model to geocentrism in the face of accepted and established observations, going back to the Greeks. The Copernicans faced different attitudes regarding the prior probability of the heliocentric hypothesis. Some representatives of the Church, such as Andreas Osiander (who oversaw the publication of the Copernican Treatise and wrote an anonymous Preface for it) and Cardinals Barberini and Bellarmine, were willing to grant that the prior probability was equal to that of geocentrism and both were low, at any rate. Other opponents of heliocentrism, like Luther and his principal lieutenant Philip Melanchton, believed on religious grounds that the prior probability of a moving earth was much less probable than that of a stationary earth of geocentrism. Copernicans often conceded that the prior probability of the heliocentric hypothesis made heliocentrism less plausible than the geocentric model. But they proceeded to employ their probability arguments to show that heliocentrism was more plausible than geocentrism. In order to do so, the Copernicans considered, in Bayesian terms, the likelihoods—Pðejh1 Þ & Pðeh2 Þ—of the rival hypotheses with respect to some particular evidence. The evidence—the observable rotation of the sphere of fixed stars—is more likely on account of h1 than on account of h2, where both models are augmented by considerations of physical processes. According to the ‘Bayesian’ Copernicans hypothesis h1 makes the evidence more likely than hypothesis h2 : Pðejh1 & b1 Þ > Pðejh2 & b2 Þ. As the determination of prior probabilities as coherent personal degrees of beliefs has caused controversy, Bayesians often point out that the relative likelihoods of the evidence with respect to alternative hypotheses does all the evaluative work. (Hawthorne

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1993, §2.2, p. 115; original emphasis; cf. Salmon, 2005, p. 107; Dose, 2003, 2005; Sober, 2008, §1.3) This seems to be the case in the problem situation faced by the Copernicans and other problem situations considered in this paper. As we shall argue in Section 5, the role of probability arguments in the context of likelihoods, rather than priors, is of considerable importance in the history of science. Although the diurnal rotation of the fixed stars seems implausible, this implausibility does not necessarily imply that the geocentric view is mistaken on all other evidence. In the history of science many claims seemed implausible at first and later turned out to be correct. Plausibility considerations illustrate the differential nature of confirmation on the Bayesian view. The weight of evidence affects the credibility distributions of the rival hypotheses. The angular velocity of the ‘fixed’ stars is one piece of evidence; it diminishes the credibility of the geocentric model. By symmetry of argument, the more plausible explanation—the rotation of the earth on its own axis—does not necessarily imply that the heliocentric model has to be accepted as ‘true’. The early Copernican models were certainly not true. But the plausibility considerations bestow more credibility on the heliocentric view. Over time the cumulative evidence began to speak in favour of one explanatory model and against some rival account. The evidence shifts the credibility towards one model and away from its rival. 4. Darwinism In this section we begin by considering Darwin’s own reflections on plausibility, which we will then reconstruct in Bayesian terms for further clarification. 4.1. Darwinian probability arguments Although there is generally much less doubt that Darwin’s Origin of species (1968 [1859]) constituted a genuine revolution in science, similar probability arguments are to be found in the defence by the Darwinians against the arguments of natural theology in favour of design. In terms of the Bayesian apparatus the Darwinians were also concerned with likelihoods: PðejhÞ, although the evidence was more varied than in the case of Copernicus. The Darwinians too asked how likely the evidence was in the face of the theory of natural selection as opposed to the theory of special creations. The Darwinians needed to show that to account for the diversity of species, separate acts of creation were less likely than the principle of natural selection. At Darwin’s time there was a more general consensus about the fact of evolution than on the underlying mechanism involved. Lamarck, for instance, had proposed his theory of use inheritance (1809), which Darwin rejected in favour of natural selection. Darwin needed to convince his readers that natural selection was a plausible mechanism, which could elegantly account for many observations. In the Origin of species Darwin repeatedly appeals to probability considerations to argue against the theory of special creations and in favour of natural selection. As with the Copernicans the appeal is to a material process, not to a coherent belief structure, although the former is meant to support the latter. Darwin argues that a naturalistic process like natural selection is more probable, renders the evidence more coherent and the evolutionary theory more plausible: If then we have under nature variability and a powerful agent always ready to act and select, why should we doubt that variation in any way useful to beings, under their excessively complex relations of life, would be preserved, accumulated, and inherited? . . . What limit can we put to this power, acting during long ages and rigidly scrutinising the whole constitution,

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structure and habits of each creature,—favouring the good and rejecting the bad? I can see no limit to this power, in slowly and beautifully adapting each form to the most complex relations of life. The theory of natural selection, even if we looked no further than this, seems to me to be in itself probable. (Darwin, 1968 [1859], p. 443) Still, as there was at Darwin’s time no direct evidence for natural selection, he employs a battery of facts, which on the one hand tend to ‘corroborate’ the probability of natural selection as a force in nature (ibid., p. 263), whilst on the other hand they tend to discredit a process like design as improbable. These facts can conveniently be summarized under the aspects of the biodiversity of species, the extinction of old and the emergence of new species and the affinities between species (homologies and analogies). While the theory of natural selection provides a coherent explanation of these facts, the same facts remain inexplicable ‘on the ordinary view of creation’ (ibid., p. 417). Darwin contrasts natural selection with the ‘agency of a miracle’ (ibid., p. 349). In the sixth edition of the Origin Darwin claims: It can hardly be supposed that a false theory would explain, in so satisfactory a manner as does the theory of natural selection, the several large classes of facts above specified. It has recently been objected that this is an unsafe method of arguing: but it is a method used in judging common events of life, and has often been used by the greatest natural philosophers. The undulatory theory of light has thus been arrived at; and the belief of the revolution of the Earth on its own axis was until lately supported by hardly any direct evidence. (Quoted in Nola and Sankley, 2007, p. 120) Darwin’s appeal to the Copernican case is somewhat misleading, because the geocentric model seemed to explain ‘several large classes of facts’, although it was entirely false, both in its topologic and algebraic structure. So the fact that the evolutionary theory seemed to explain coherently so many facts should hardly be a recommendation for its ‘truth’. Strictly speaking, the argument is about the likelihood of the rival hypotheses in the face of the known evidence. It does require, however, that the evidence assigns differential probability weights to the contrasting explanations. This can be seen very clearly if we consider the modern descendants of the old rivalry, namely evolutionary biology and modern design scenarios, from a Bayesian point of view. 4.2. Bayesian considerations The role of priors and likelihoods can also be appreciated in a case where the underlying theory is statistical in nature and two rival models, h1 and h2, face each other over the evidence. The battleground is typically the ‘perfection’ of a particular organ, such as the eye, whose emergence Darwin explains on the theory of descent with modifications (Darwin, 1968 [1859], pp. 217–218). Taking the emergence of the eye as a particular piece of evidence, e, the modern design theorists claim that the probability of the evidence is much greater on the hypothesis of design, h2, than on the rival hypothesis of cumulative evolution, h1.

PðejdesignÞ  Pðejcumulativ e ev olutionÞ: In this case the assessment of the priors, Pðh1 Þ, Pðh2 Þ causes greater concern than in the previous case. While on an instrumentalist reading of the geometric devices employed in heliocentrism the prior probabilities of geocentrism and heliocentrism, on given observational data, could be regarded as approximately equal, the assessment of the priors of the rival hypotheses has been wildly divergent throughout the history of Darwinism. The evolutionary biologist will give a relatively high value to Pðh1 Þ, whilst the de-

sign theorist will only grant it a low value. For the other prior, Pðh2 Þ the situation is exactly reversed. The design theorist will give it a high probability and the evolutionary biologist a low probability. Bayes’s theorem tells us that this disagreement about the priors, for a given likelihood ratio, will affect the posterior probability of h. The standard answer in the Bayesian literature to this problem is to point out that large disagreement about the priors will be ‘washed out’ with the accumulation of evidence. Even large discrepancies in the numerical values of the priors can be ignored because a sufficient amount of evidence will eventually converge towards the correct theory. This convergence seems to reflect the fact that even scientists who routinely disagree in their views towards hypotheses, and their probability, will eventually agree on one model as more plausible than its rivals. However, in the case of Darwinism no such ‘swamping of priors’ has taken place. Today intelligent design theory attacks the theory of natural selection on the ground of its ‘implausibility’ in the face of evidence. According to strict Bayesianism, prior probabilities reflect personal degrees of beliefs of the scientists in their hypotheses. This emphasis on the subjective nature of the Bayesian probabilities should not be exaggerated. Yesterday’s prior is today’s posterior distribution and today’s posterior distribution is tomorrow’s prior (Howson & Urbach, 1993, pp. 118, 411; cf. Shimony, 1970). Scientific problems are embedded in a certain historical problem situation, in which certain solutions, for example the relative confirmation of certain hypotheses, already possess a certain credibility. In the words of physicists, who use the Bayesian method, there is a difference between ‘improper’ and ‘informative’ priors (Dose, 2003, pp. 1428, 1441, 1447; cf. Sober, 2008). As we have seen in the case of Copernicanism, the heliocentric model arose as a rival to existing solutions and claimed to be more plausible on account of the probability of its proposed physical processes. The assessment of priors should therefore be affected by antecedent successes and failures of a body of hypotheses. In such a situation, Salmon assigns a particular role to plausibility considerations. They help to assess the prior probability of hypotheses such as natural selection and design scenarios. The Darwinians had no direct evidence for the hypothesis of natural selection but it received support from other evidence, coming from areas like anatomy, embryology and palaeontology, whilst the design hypothesis relied on what Darwin dubbed ‘miracle agency’. A lack of agreement on priors is not an obstacle, since it is still possible to advance the argument via a consideration of the respective likelihoods. Bayesianism expresses the comparative analysis of rival conceptions in the following version of Bayes’s theorem:

Pðh1 jeÞ Pðejh1 Þ Pðh1 Þ ¼  Pðh2 jeÞ Pðejh2 Þ Pðh2 Þ

ð2Þ

If the ratio of priors is fixed, the arguments shift to the level of the likelihoods, Pðejh1 Þ and Pðejh2 Þ. Dembski, a proponent of intelligent design theory, agrees that ‘only the likelihoods are relevant’ (Dembski, 1998, Ch. 3). In the case at hand, the rival hypotheses, h1 and h2, are not deterministic so that Pðejh1;2 Þ < 1, and the evidence is taken to be the complex structure of the eye. But what value should we give to these likelihoods? The design theorist has an answer with respect to h1. The likelihood of the evidence—complex organs such as the eye or the Krebs cycle—on hypothesis h1 is extremely low, say as low as 1039. If Pðejh1 Þ is so extremely unlikely, it follows from Bayes’s Theorem that the evidence cannot confirm h1 since the prior probability of h1 will be higher than its posterior probability (op. cit., Ch. 3.1). The evolutionary biologist will disagree with the evaluation of the likelihood Pðejh1 Þ, pointing out that computer simulations show that complex organs such as the eye can be built in forty

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generations and that evolutionary theory stipulates cumulative rather than random selection. Hence the likelihood Pðejh1 Þ ¼ 1039 is far too low. Modern evolutionary biologists have been interested in finding a ‘plausible sequence of alterations from a light sensitive spot all the way to a fully developed lens eye’ (Nilsson and Pelger, 1997, p. 294). According to their computer simulations it would take ‘less than 364.000 years for a camera eye to evolve from a light sensitive patch’ (ibid., p. 301). The fact that there is evidence that animals had eyes as early as 550 million years ago and that eyes evolved independently at least 40 times in the evolutionary history of species considerably increases the likelihood of the evidence, e, under the hypothesis of natural selection. One of the advantages of the Bayesian apparatus is that it allows a comparison of rival models in terms of the likelihood ratio:

LR ¼

Pðejh1 Þ Pðejh2 Þ

ð3Þ

The evolutionary biologist will therefore want to assess the likelihood of the evidence under the hypothesis of design. As Darwin realized, it is not just the hypotheses, h1 and h2, which are under consideration. Each of the embedding theories proposes a mechanism under which the evidence is produced. In the case of cumulative evolution it is natural selection, b1. In the case of design it is an act of deliberate design, b2, on the part of an otherwise unspecified intelligent agency. Bayesians generally agree that the posterior probability PðhjeÞ of a hypothesis is crucially dependent on three basic quantities: PðhÞ; PðejhÞ; Pðej:hÞ because P(e) can be written as:

PðeÞ ¼ PðejhÞPðhÞ þ Pðej:hÞPð:hÞ

ð4aÞ

or more generally:

PðeÞ ¼ PðejhÞPðhÞ þ

X

Pðejhi ÞPðhi Þ

ð4bÞ

i

ðHowson & Urbach; 1993; p: 119; Dembski; 1998; Ch: 3Þ So equation (4a) should be extended to include terms for these mechanisms (b1, b2, where h and –h have been replaced by h1 and h2):

PðeÞ ¼ Pðejh1 & b1 ÞPðh1 Þ þ Pðejh2 & b2 ÞPðh2 Þ

ð5Þ

But there is a difference between b1 and b2: b1 is testable in principle and has been tested under laboratory conditions and observed in nature (as in the case of industrial melanism). But b2 (acts of design) is untestable in principle and will command little credibility in the scientific community (see Sober, 1999, 2002). It is of course logically possible to infer design from observations of complex organs, such as the eye. But this logical possibility does not mean that this inference is admissible on scientific grounds. The problem is that the permissible inference to design requires a leap of faith, which cannot be demanded by testable evidence. In the absence of direct tests, Darwin employed his probability argument to argue effectively that Pðejh1 & b1 Þ > Pðejh2 & b2 Þ: Even if the hypothesis of cumulative evolution has a relatively low prior probability—as the Darwinians conceded—the evidence still increases the posterior probability of the hypothesis on the strength of the likelihood ratio. W. Salmon sees plausibility arguments as constraints on prior probabilities, in the context of justification. They serve to ensure ‘informative’ priors. Salmon would probably argue that plausibility considerations favour an explanation in terms of natural selection, simply because such a natural process fits better into our established mechanical world view than appeal to supernatural agency. While this argument makes sense in some cases, both in ordinary life and the history of science, it faces difficulties in specific contexts:

101

a) In revolutionary periods of science, ‘improbable’ hypotheses (such as a rotating earth or natural selection) are often proposed as a replacement for established views but the majority view dismisses them as ‘implausible’. b) In the history of science we often see rival models aiming to explain the accepted evidence; an analysis of particular cases shows that the competition happens on the level of likelihood claims. When the prior probabilities are either approximately equivalent—as in the case of geocentrism and heliocentrism— or wildly divergent—as in the case of design theory and natural selection—the likelihoods often become the vehicle to carry the argument forward. Both in the case of Copernicanism and Darwinism probability considerations were of considerable importance. The case of Rutherford’s nucleus model will add new aspects to this view and show that likelihood considerations are widespread in the history of science.

5. Priors and likelihoods This analysis of the use of probability arguments in the Copernican and Darwinian models respectively harbours some interesting philosophical lessons. Copernicus and Rheticus had no new observational data to offer in support of the heliocentric view. They had no observational evidence in support of a rotating earth. Darwin had no direct evidence in support of natural selection, although evolutionary evidence from anatomy, embryology and homologies called for a naturalistic explanation. Yet the Copernicans and Darwinians appeal to theoretical values: coherence and probability. These values are used as eliminative measures when faced with the choice of two rival hypotheses in the face of particular evidence and not in order to determine the prior probability of the rival hypotheses. In 1543 the geocentric and heliocentric models were observationally equivalent although different in their topologic structures. But the probability arguments persuaded the early Copernicans that their model had cognitive advantages over their rival model, especially in terms of coherence. After the observations of Tycho Brahe of what is now known as a supernova (1572), the trajectory of comets through the solar system (1577–1596) and Galileo’s discovery of the Jupiter moons (1610), the evidence shifted in favour of the Copernican model. But Kepler was interested in the physical causes of planetary motion. He argues in terms of physical parameters such as the rotational velocities of celestial objects. Kepler could rely on Maestlin’s calculations (1596) to argue his case that it is physically more plausible to widen the then known cosmos by the power of 3 than to attribute ‘incalculable speeds’ to the outer sphere of ‘fixed’ stars. The argument is not, as readers of Popper and Kuhn would expect, that the heliocentric model explains more data, explains them better, in greater depth and makes more and better predictions than the geocentric model. The argument is that even in the absence of such cognitive superiority of one model over its rival, the plausibility of physical parameters has to be taken into consideration. On account of such physical considerations the heliocentric model is to be preferred because the heliocentric model makes more probable physical assumptions than the geocentric model. In the history of astronomy these physical assumptions became testable. Similar considerations apply to the Darwinian case. The range of phenomena can be rendered more coherent on the theory of descent with modifications. Therefore the theory of evolution is more plausible than its rival, the theory of special creations. In order to support these conclusions, Darwin argues in favour of the probability of a physical mechanism—the operation of natural selection—for which he had no direct evidence. With his emphasis on plausibility arguments as constraints on the determination of the probability of priors, Salmon has

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highlighted an important function of such arguments, especially with respect to the problem of priors in subjective Bayesianism. Recall that priors are supposed to express the probability of a hypothesis, in the absence of any particular evidence for it. Salmon would probably accept that they should be ‘informative’ priors, in the sense that there always exists a body of accepted beliefs, in one area, which may guide the assessment of priors in another. For instance, in Darwin’s time, it was generally accepted that the earth was no more than ninety-eight million years old, because this age had been calculated by Lord Kelvin (1863) on the basis of cooling rates of the earth. Kelvin took this estimate to throw doubt on the prior probability of natural selection. Whilst the Darwinians acknowledged that this estimate posed a problem for the theory of natural selection and its emphasis on gradualism, they nevertheless considered the hypothesis of natural selection to be a better explanation of the facts. This shows that such general plausibility arguments, especially of the material kind, have little bite in periods of scientific rivalry between several competing models. In such revolutionary periods, as we observe them in the Copernican and Darwinian traditions respectively, the power of plausibility considerations shifts towards probability arguments involving physical processes. Our analysis of the use of probability arguments in Copernicanism and Darwinism regarding physical processes confirms Salmon’s view that plausibility arguments are important in the context of justification. However, a consideration of the use of such arguments in the case of Copernicanism and Darwinism has shown that Kepler and Darwin put the emphasis, in Bayesian terms, on the likelihoods, because they proposed to consider the likelihood of a particular piece of evidence in the presence of probable physical processes. In terms of the prior probabilities of their respective hypotheses, both Copernicus and Darwin admitted that the prior probability of their respective hypotheses may be regarded as low compared to that of the predominant views. The reason there is so little emphasis on the priors is that both the Copernicans and the Darwinians entered a conceptual space in which rival models already existed with a similar claim to plausibility in the face of accepted evidence. On a synchronic view these respective rival theories are observationally equivalent but structurally different. Yet both the Copernicans and the Darwinians claim cognitive superiority for their respective models because in both cases they appeal to the probability of the operation of a physical process in nature, as against some improbable process (be it acts of creation or incalculable speeds). These probability arguments belong to Salmon’s category of material plausibility arguments. We must conclude that constraints on priors are only one function of plausibility arguments. There is a second important role in the use of probability arguments in the likelihood ratios. In fact, Salmon himself hints at the role of ‘plausible scenarios’ in the determination of likelihoods, but says little about their role (Salmon, 2005, pp. 108–111). Salmon argues that in the comparison between two hypotheses their likelihood ratio is often important, especially when the other probabilities involved in the calculation of posterior probabilities are not easily available. But as we have seen, in the light of above considerations, equation (3) should be enhanced by the respective physical mechanisms and processes to which both the Copernicans and the Darwinians appealed. Expression (3)—the likelihood ratio—thus becomes:

LR ¼

Pðejh1 & b1 Þ Pðejh2 & b2 Þ

ð6Þ

A number of writers have observed that in Bayesianism the shifting fate of hypotheses can be gleaned from the work of the likelihood ratios:

In Bayesian induction the relative likelihoods of the evidence with respect to alternative hypotheses does all the evaluative work . . . The whole point of using Bayes’s theorem to evaluate posterior probabilities of hypotheses (relative to evidence) is that the likelihoods in Bayes’s theorem are supposed to provide a fairly objective way of assessing the impact of evidence on the plausibilities of hypotheses. (Hawthorne 1993, §2.2, pp. 115– 116; original emphasis; cf. Howson & Urbach, 1993, p. 29; Salmon, 2005, p. 107; Dose, 2003, 2005; Sober, 2008, Chs. 1, 2) According to equation (6) ‘the numerical value of a likelihood ratio represents the number of times more (or less) likely the occurrence of outcome e would be if h1 were true than it would be if h2 were true’ (Hawthorne, 1993, p. 117; original emphasis). The likelihood ratios are important when, as in the case of Copernicanism, the rival models can be regarded as observationally equivalent such that the prior probabilities of hypotheses 1 and 2 cancel out. They are equally important when the priors are divergent, but their ratio is fixed, since it is still possible to make progress by considering the credibility of the respective models in terms of likelihoods. Clearly, then, Bayesianism is a way of illustrating why hypotheses perform differentially in the light of evidence. And likelihood considerations are quite widespread in science, as the following case will show. 6. Further role for probability considerations It should be observed that the Copernican and Darwinian cases touch on the problem of old evidence, which according to Bayesianism fails to increase the posterior probability of a hypothesis under test (see Nola and Sankley, 2007, §9. 3 for discussion). If, for instance, the hypothesis entails the evidence, such that Pðejh1 Þ ¼ 1, and we set the value of old evidence P(e) equal to 1, then the posterior probability of the hypothesis turns out to be equal to its prior probability. Note, however, that the Copernicans and Darwinians introduced a new hypothesis regarding the probability of a physical process in the defence of their respective theories. They argue that a central feature of their respective theories— namely the location and dual motion of the earth and natural selection—make the evidence more likely than the rival hypotheses. In this way the old evidence does bestow confirmatory value differentially because it bears on a contrast in the central structural features of the rival models. As neither geocentrism nor design scenarios offer plausible causal explanations of the known evidence, the physical process proposed in the respective rival models shifts the credibility distributions over the space of rival models. The Copernicans and Darwinians were engaged in eliminative induction because they argued that their probability arguments shifted the weight of credibility towards their models, which may at first have had lower prior probabilities. Plausibility considerations thus play a dual role in the context of justification. First, Salmon’s plausibility considerations may help to constrain the ‘informative’ priors but may also be shelved, as in Darwin’s problem with the age of the earth; second, more specific probability arguments, appealing to physical processes, help to constrain the likelihoods. So far we have discussed the role of probability arguments in the support of likelihoods during exceptional periods of science, like the Copernican and Darwinian revolutions. Known evidence inspired new explanatory hypotheses. But plausibility considerations in the specific sense are not restricted to revolutionary periods of science. It is worth considering a further episode in the history of science: the scattering experiments, which established Rutherford’s nucleus model of the atom over the plum-pudding model. This episode differs from the previous ones in two respects: first, there was no disagreement about the prior—Rutherford and

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his co-workers worked on the basis of the accepted Thomson model; second, however, new unexpected evidence precipitated the established model into a crisis and inspired, on the strength of numerical likelihood considerations, a new model of the atom. This indicates that considerations based on likelihoods—PðejhÞ—are quite common in the history of science. 6.1. Rutherford’s plausibility considerations In his analysis of the famous scattering experiments, Rutherford employs probability arguments in order to determine the likelihood of rival models concerning the nature of atomic structure. Unlike the two cases considered above, Rutherford could rely on new evidence. The scattering experiments were carried out by Rutherford’s collaborators, Marsden and Geiger, between 1909 and 1911. They involved the scattering of a-particles off thin gold foils. They yielded a surprising result—surprising on account of the accepted model of atomic structure, the so-called plum pudding model. This model had been proposed by E. Thomson, the discoverer of the electron (1897), at the beginning of the twentieth century. According to the plum pudding model, an atom consisted of a sphere of positive electricity, within which the negatively charged electrons were arranged in concentric rings. Importantly, the atom had no nucleus (Weinert, 2000). But when Rutherford’s collaborators shot doubly ionized helium atoms (a-particles) at a thin foil made up of gold atoms, they noticed that about 1 in 8000 helium nuclei were scattered at angles greater than 90 degrees. In order to account for these results Rutherford scrutinized the plum pudding model, which he and his collaborators had formerly accepted, and found that it could not account for the large-angle scattering. First, there is agreement on the experimental results: It is well known that the a and b particles suffer deflexions from their rectilinear paths by encounters with atoms of matter. This scattering is far more marked for the b [electrons] than for the a particle on account of the much smaller momentum and energy of the former particle. (Rutherford, 1911, p. 669) The generally accepted view, the plum-pudding model, explains the observations as a result of many small scatterings of the projectile from the target. But the large-angle scatterings cannot be explained on this model: It has generally been supposed that the scattering of a pencil of

a or b rays in passing through a thin plate of matter is the result of a multitude of small scatterings by the atoms of matter traversed. The observations, however, of Geiger and Marsden on the scattering of a rays indicate that some of the a particles, about 1 in 20,000, were turned through an average angle of 90° in passing through a layer of gold-foil about 0.00004 cm thick. (Ibid.) Rutherford then turns to probability considerations to weigh the relative credibility of the established atom model. Geiger showed later that the most probable angle of deflexion for a pencil of a particles traversing a gold-foil of this thickness was about 0°.87. A simple calculation based on the theory of probability shows that the chance of an a particle being deflected through 90° is vanishingly small. In addition, it will be seen later that the distribution of the a particles for various angles of large deflexion does not follow the probability law to be expected if such large deflexions are made up of a large number of small deviations. (Ibid.) From these probability considerations Rutherford finally infers the adequacy of the new nucleus model of the atom.

Fig. 4. Rutherford’s scattering diagram (1911).

It seems reasonable to suppose that the deflexions through a large angle is due to a single atomic encounter, for the chance of a second encounter of a kind to produce a large deflexion must in most cases be exceedingly small. A simple calculation shows that the atom must be a seat of an intense electric field in order to produce such a large deflexion at a single encounter. (Ibid.) Rutherford’s reasoning consists of two steps. First, he advances energy considerations, taken from classical physics. The heavy helium nuclei (a-particles) could not be deflected by multiple scatterings through the light electrons embedded in a sea of positive electricity, as the plum pudding models had to assume. But if one postulated a model of the atom, which resembled a planetary system, the large-angle scattering found a natural explanation. Secondly, Rutherford therefore postulates a heavy nucleus inside the atomic structure, around which the electrons orbited. The scattering at various angles of the projectiles could then be explained as a function of the closeness of the incoming nuclei to the nucleus of the gold atoms. (See Fig. 4) The electrons played no part in these considerations. In other words, Rutherford concluded that a nucleus model of the atom was more plausible because the postulation of a physical nucleus made the observed and measured scattering processes much more probable than the rival plum pudding model. 6.2. Bayesian considerations As before, we can throw further light on this case by considering Rutherford’s arguments in terms of Bayesian likelihood terms. We consider two likelihoods:

Pðejh1 & b1 Þ & Pðejh2 & b2 Þ; where e is the wide-angle scattering result of the experiments, h1 is Rutherford’s nucleus hypothesis, h2 is Thomson’s plum-pudding hypothesis and b1 and b2 are the respective mechanisms, for example large-angle scattering (b1) and multiple small-angle scatterings (b2). Then Rutherford’s calculations show: In these calculations, it is assumed that the a particles scattered through a large angle suffer only one large deflexion. For this to

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hold, it is essential that the thickness of the scattering material should be so small that the chance of a second encounter involving another large deflexion is very small. If, for example, the probability of a single deflexion / in passing through a thickness t is 1/1000, the probability of two successive deflexions each of value / is 1/106, and is negligibly small. (Rutherford, 1911, p. 675) Clearly, then, the plum-pudding hypothesis, h2, with its assumption of a ‘multitude of small scatterings’, makes the observation of large-angle scattering exceedingly unlikely. And so Thomson’s hypothesis, h2, considerably lowers the likelihood of the evidence, possibly below the level of experimental observation at that time and in contradiction to the actual observation. In comparison, the postulation of a nucleus—as the physical seat of the deflection— makes the large-angle scattering much more likely and explains the surprising result of deflection through angles of more than 90°. Again, the Bayesian considerations have to include the different mechanisms, under which the phenomena are to be explained. 7. Conclusion Salmon writes that ‘in the history of science we typically have several hypotheses or theories, T1, T2, T3, . . . that have been advanced to explain a phenomenon’ (Salmon, 2005, p. 119). The task is then either to eliminate a defective hypothesis or, if that is not possible, to lower its credibility. But as there is a rivalry between the competing hypotheses, which can be expressed in the Bayesian framework, the loss of credibility of one hypothesis constitutes a gain of credibility for its rival. This shift in credibility is not necessarily due to new evidence. In two cases that we have considered, probability arguments are employed to affect a differential distribution of credibility over the space of competing hypotheses. The probability arguments concern physical processes underlying some specific piece of evidence and are used in the likelihoods rather than to constrain the priors. The point is to show that the assumption of some physical parameter in one hypothesis makes the established evidence more likely than its rival. Rutherford, however, was prompted to use probability arguments on account of new evidence about large-angle scattering: he employed them against the plum pudding model and in favour of his nucleus model, which stipulates a new mechanism behind the phenomena. On occasion of the 300th anniversary of the death of Johannes Kepler (1630), Albert Einstein praised Kepler’s achievement as ‘a particularly fine example of the truth that knowledge cannot spring from experience alone but from the comparison of the inventions of the intellect with observed fact’ (Einstein, 1954, p. 266). If we include in the ‘invention of the intellect’ plausibility considerations and probability arguments, then this statement reflects an important fact about the context of justification in the history of science.

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