Spacetime Visualisation and the Intelligibility of Physical Theories

Stud. Hist. Phil. Mod. Phys., Vol. 32, No. 2, pp. 243–265, 2001 # 2001 Elsevier Science Ltd. All rights reserved. Printed in Great Britain 1355-2198/01/$ - see front matter

Spacetime Visualisation and the Intelligibility of Physical Theories Henk W. de Regt* This paper argues that spacetime visualisability is not a necessary condition for the intelligibility of theories in physics. Visualisation can be an important tool for rendering a theory intelligible, but it is by no means a sine qua non. The paper examines the historical transition from classical to quantum physics, and analyses the role of visualisability (Anschaulichkeit) and its relation to intelligibility. On the basis of this historical analysis, an alternative conception of the intelligibility of scientific theories is proposed, based on Heisenberg’s reinterpretation of the notion of Anschaulichkeit. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Anschaulichkeit; Intelligibility; Quantum Theory; Spacetime; Understanding; Visualisation.

1. Introduction . How does science achieve understanding of nature? Erwin Schrodinger held a strong opinion about this issue: he claimed that the only way to obtain understanding of nature is to construct scientific theories which are visualisable in space and time. Arguing against the abstract, unvisualisable matrix mechanics of his rival Werner Heisenberg, he stated that we cannot really alter our manner of thinking in space and time, and what we cannot comprehend within it we cannot understand at all. There are such . things}but I do not believe that atomic structure is one of them (Schrodinger, 1928, p. 27). *Department of Philosophy, Vrije Universiteit, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands (e-mail: [email protected]). PII: S 1 3 5 5 - 2 1 9 8 ( 0 1 ) 0 0 0 0 7 - 7 243

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. Schrodinger (1996, p. 90) observed that modern science presupposes that nature is understandable, an assumption which goes back to the Greeks. Given these two premises, nature must be understandable in space and time. In other words: spatiotemporal visualisability is a necessary condition . for scientific understanding. Schrodinger’s conclusion was that an acceptable scientific theory must be visualisable, or in German: anschaulich. He employed this argument in defence of his theory of wave mechanics, the Anschaulichkeit of which supposedly made it superior to the rival theory of matrix mechanics. . On closer examination, one can distinguish two steps in Schrodinger’s defence of wave mechanics: first, he claims that in order to understand natural phenomena one needs an intelligible theory; second, he claims that only visualisable theories can be intelligible. The second step is the main theme of the present paper: what is the relation between spacetime visualisability and . intelligibility of physical theories? The first step in Schrodinger’s argument (the claim that only intelligible theories can provide an understanding of . phenomena) is an assumption which I share with Schrodinger; in Section 6 I will briefly defend it. This article is structured as follows. First, in Section 2, the situation in classical physics is briefly considered: what is the relation between visualisability and intelligibility? The three subsequent sections analyse the question of whether quantum theory is}or should be}visualisable. This is done by examining the historical development of Bohr’s ideas and the so-called old quantum theory (Section 3), the new quantum mechanics (Section 4), and the case of electron spin (Section 5). Finally, Section 6 presents a philosophical theory of intelligibility and understanding which accounts for the role of visualisation in achieving scientific understanding.

2. Visualisability and Intelligibility in Classical Physics . Schrodinger’s reflections on the possibility and desirability of visualisation in physics were induced by the evolution of quantum theory in the 1920s, which showed a development towards unvisualisability. How was the situation before the advent of quantum theory? Were the theories of classical physics always anschaulich, that is: were they visualisable, and thereby regarded as intelligible? In his book Zum Weltbild der Physik, Von Weizs.acker observes that physicists tend to identify anschaulich with ‘classical’ because ‘classical physics describes all physical phenomena as states of quantities in three-dimensional, Euclidean space and as changes of these states in one-dimensional, objective time’ (Von Weizs.acker, 1970, p. 51). Similarly, Dieks (this issue) emphasises that classical theories of physics can always be formulated as spacetime theories, in which physical quantities are represented as functions on a manifold of spacetime points. According to Dieks, ‘the spacetime points, forming the arena in which the game of theory is played, guarantee

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Anschaulichkeit and the associated kind of intelligibility’. But is this really true? It seems that, at least in practice, not all classical theories are immediately visualisable. Consider, for example, thermodynamics. Of course, this theory describes physical systems in three-dimensional space and their evolution in time. But it appears quite difficult to visualise thermodynamical properties of these systems (such as pressure, temperature and entropy), their evolution and the relations between them. Other cases in point are Maxwell’s theory of electromagnetism and the theory of statistical mechanics. To be sure, there does not seem to be an argument against the in principle possibility of visualisation in these cases (as would be the case with theories which cannot be formulated as spacetime theories). Nonetheless, visualisation seems practically impossible in some cases. The reason is of course that nineteenth-century physics had already become more abstract and had drifted away from the paradigm visualisable theory: classical mechanics. It follows that being a spacetime theory is a necessary condition for visualisability, but it does not appear to be a sufficient condition: cases like thermodynamics indicate that spacetime theories are sometimes hardly visualisable. The Anschaulichkeit of classical physics is therefore less trivial than is often suggested. What about the intelligibility of classical physics? Does the apparent unvisualisability of some classical theories inhibit their intelligibility? Did nineteenth-century physicists have problems with the lack of visualisability, . and did they doubt the intelligibility of those theories, just as Schrodinger questioned the intelligibility of quantum theory? William Thomson famously declared: ‘It seems to me that the test of ‘‘Do we or not understand a particular subject in physics?’’ is, ‘‘Can we make a mechanical model of it?’’ ’ (Thomson, 1884, pp. 131–132). And: ‘I never satisfy myself until I can make a mechanical model of a thing. If I can make a mechanical model I can understand it. As long as I cannot make a mechanical model all the way through I cannot understand. . .’ (ibid., pp. 270–271). Thomson thus equated intelligibility with the possibility of making mechanical models. While the emphasis is on mechanics, it is clear that there is an immediate connection with visualisability: a mechanical model is visualisable. Thomson’s vortex atoms exemplified this desire for mechanical models. Another example of such a visualisable mechanical model is James Clerk Maxwell’s well-known model of the electromagnetic ether. Indeed, Maxwell also regarded mechanical explanation of great importance because it promises understanding and intelligibility. Thus, in A Treatise on Electricity and Magnetism he states (after having discussed a mechanical analogy for the electromagnetic phenomenon of self-induction): It is difficult [. . .] for the mind which has once recognised the analogy between the phenomena of self-induction and those of the motion of material bodies, to abandon altogether the help of this analogy, or to admit that it is entirely superficial and misleading. The fundamental dynamical idea of matter, as capable by its motion of becoming the recipient of momentum and of energy, is so interwoven with our forms of thought that whenever we catch a glimpse of it in

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Maxwell was seriously concerned with the intelligibility of concepts and theories. Harman (1998, pp. 27–36) observes that Maxwell, influenced by William Hamilton and William Whewell, adopted a Kantian view of space and time, regarding them as necessary conditions of thinking and therefore of intelligibility, rather than as structures in reality. For example, Maxwell described geometric figures as ‘forms of thought and not of matter’ (quoted in Harman, 1998, p. 30). This Kantian strain is clearly present in the above quotation. Maxwell’s analogies, such as the hydrodynamical analogies in his early papers on electromagnetism and the mechanical ether-model, aimed at providing understanding, in the sense of rendering the phenomena intelligible. Harman (1998, pp. 87–88 and p. 90) emphasises that the geometrical nature of the analogies was essential in this respect. As Harman (1982, p. 9) observes, ‘mechanical models were not necessarily envisaged as representations of reality, but were seen as demonstrating that phenomena could in principle be represented by mechanisms; mechanical construction rendered phenomena intelligible’. It is evident that the leading nineteenth-century physicists Maxwell and Thomson regarded mechanical explanation as the best way to achieve understanding and intelligibility, and at least for Maxwell this was explicitly connected with the visualisability (geometrical nature) of mechanical models. Towards the end of the nineteenth century the programme of mechanical explanation ran into serious difficulties, and rival approaches burst onto the scene: phenomenalism, energetics, and the electromagnetic worldview. In the 1890s, in the context of the debates between these different viewpoints, Ludwig Boltzmann defended his so-called Bild-conception of scientific theories and discussed the role of visualisable models in science. Boltzmann’s conception of a scientific theory as a ‘picture’ (Bild) is quite general; it is first of all intended as a warning against taking a theory as a true representation of states of affairs and processes in nature: ‘no theory can be objective, actually coinciding with nature, but rather each theory is only a mental picture of phenomena, related to them as a sign is to designatum’ (Boltzmann, 1974, pp. 90–91).1 However, Boltzmann does not deny a role for pictures in the more specific sense of . und . visualisations. In his Vorlesungen uber Maxwells Theorie der Elektricitat des Lichtes (Boltzmann, 1891, 1893), which contains many mechanical models of electromagnetic phenomena, he argues that a mechanical analogy may help us to obtain a visualisation (Versinnlichung) of the consequences of a scientific theory. And an important function of visualisation is that it may lead to understanding. Indeed, Boltzmann argued that explanation by means of 1

See De Regt (1999a, pp. 114–118) for an extensive discussion of Boltzmann’s Bild-conception of theories.

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mechanical pictures is preferable because it leads to the most intelligible accounts of nature:2 If [. . .] all apparently qualitative changes were representable by the picture of motions or changes of arrangements of smallest parts, this would lead to an especially simple explanation of nature. In that case nature would appear to us at its most comprehensible [begreiflichsten], but we cannot compel her to this, we must leave open a possibility that this will not do and that we need in addition other pictures of other changes; understandably, it is precisely the more recent developments of physics that have made it prudent to allow for this possibility (Boltzmann, 1974, p. 143).

Boltzmann holds that it is a contingent fact that mechanical pictures appear most intelligible to us, and he admits that nature might ultimately resist this desired mode of understanding.3 In spite of this, however, we cannot easily change our intellectual capacities and therefore we must continue to look for mechanical pictures. That is what Boltzmann himself did and that is why he continued the programme of mechanical explanation even after 1900. The difficulties that mechanicism faced at that time prompted scientists and philosophers to weaken their commitment to realism (as exemplified by Boltzmann’s Bild-conception). But many held on to mechanical pictures as an ideal of intelligibility, as can be gleaned from the following passage from the entry ‘Model’ in the 1902 edition of the Encyclopaedia Britannica, written by Boltzmann: But while it was formerly believed that it was allowable to assume with a great show of probability the actual existence of such mechanisms in nature, yet nowadays philosophers postulate no more than a partial resemblance between the phenomena visible in such mechanisms and those which appear in nature. Here again it is perfectly clear that these models of wood, metal and cardboard are really a continuation and integration of our processes of thought; for, according to the view in question, physical theory is merely a mental construction of mechanical models, the working of which we make plain to ourselves by analogy of mechanisms we hold in our hands, and which have so much in common with natural phenomena as to help our comprehension of the latter (Boltzmann, 1974, p. 218).

In sum, reflection on the nature of classical physics leads to the conclusion that the spacetime framework, in which classical theories are formulated, is a necessary condition for visualisability, but not a sufficient condition. Many theories of nineteenth-century classical physics already had such a degree of abstraction that visualisation was not immediately possible. As visualisability (especially by means of mechanical pictures) was the ideal of intelligibility, nineteenth-century physicists did not take the intelligibility of classical theories 2

Incidentally, as I have shown elsewhere (De Regt, 1997), Boltzmann’s ideas strongly influenced . Schrodinger’s intellectual development and were the source of the latter’s commitment to Anschaulichkeit. 3 Boltzmann rejected an aprioristic, Kantian account of ‘laws of thought’ and opted for an evolutionary perspective instead; see De Regt (1999a, pp. 119–122).

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for granted. This is clear from their attempts to construct mechanical models in order to render electromagnetic and thermodynamic phenomena intelligible. Of course it was only in the twentieth century that the failure of these attempts became definitive. But the move away from visualisability and the associated intelligibility had already set in earlier, and therefore the transition from classical physics to quantum physics was (at least in this respect) less revolutionary than is often believed.

3. Quantum Theory and the Waning of Anschaulichkeit The theories of classical physics presuppose a spacetime framework, thereby meeting at least a necessary requirement for visualisability. We have seen that this does not yet guarantee visualisability: it is not a sufficient condition. But it was only in the twentieth century that a theory appeared on the scene that seemed to be unvisualisable in principle: quantum theory. Again, it should be emphasised that this was a gradual process: the idea that quantum theory is unvisualisable did not arrive at once. Only in 1925, with the advent of matrix mechanics, did the unvisualisability of quantum theory become complete. But intimations of unvisualisability were present in quantum theory from the very beginning. In the present section the first stage of the process towards unvisualisability will be described, focusing on the work of Niels Bohr, the main protagonist in the development of quantum theory until 1925. Bohr’s atomic model of 1913/1918, which formed the basis of the ‘old quantum theory’, was semi-mechanical, and thereby also semi-visualisable: the stationary states could be visualised in terms of electron orbits, but the transitions between states (the infamous ‘quantum jumps’) were unvisualisable. Another feature of the old quantum theory, which inhibited visualisation, was the dual nature of light (wave-particle). Bohr deemed spacetime description highly important; he plausibly regarded it as a prerequisite for understanding. However, around 1922 he began to contemplate the possibility that quantum theory failed in an essential way to provide visualisable pictures. His considerations, combined with a conviction that ‘every description of natural processes must be based on ideas which have been introduced and defined by the classical theory’,4 led him to an analysis of the way in which atomic theory departed from ‘the causal description in space and time that is characteristic of the classical mechanical description of nature’,5 finally resulting in the conception of complementarity. Although visualisability was not an explicit subject of discussion in Bohr’s work before 1922, it was implicitly contained in the conceptual problems that 4

Bohr (1976, Vol. 3, p. 458); from his 1923 article ‘On the Application of the Quantum Theory to Atomic Structure’. 5 Bohr (1976, Vol. 3, p. 571); from an unpublished manuscript ‘Problems of the Atomic Theory’, written in 1923 or 1924.

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had occupied him from 1911 on, namely those concerning the relation of the quantum theory to classical theories. As we saw in Section 2 above, classical spacetime description appears to be a necessary condition for visualisability. Therefore, the question whether the classical concepts ‘position’ and ‘momentum’ (of particles) can be retained in quantum theory needs to be answered in order to find an answer to the question whether atomic processes are visualisable. The relation between Bohr’s quantum theory and classical theory was expressed in the correspondence principle, which asserted that in the regime of large quantum numbers the predictions of quantum theory and classical theory should be the same. According to Radder (1991, pp. 203–208), one can distinguish three stages in Bohr’s attitude towards the correspondence principle. In the first stage, from 1913 to 1916, the relation between classical physics and quantum theory was seen as numerical correspondence: the correspondence only concerns the numerical results of calculations of the values of quantities. Bohr spoke of ‘agreement of calculations’, and in this way stressed the conceptual discontinuity between the two theories. From 1916 on, however, Bohr’s attitude towards the correspondence principle changed, issuing in an idea of conceptual correspondence: he now believed that there exists a far-reaching analogy between classical theory and quantum theory. Quantum theory is a ‘rational generalisation’ of classical theory: the transition probabilities are non-classical but the relation between radiation and motion is analogous to that in classical electrodynamics. This view nurtured expectations of being able to develop semi-classical accounts of the ‘mechanism’ of radiation. Unfortunately, these expectations were not fulfilled, as all semiclassical approaches failed. Consequently, the idea of conceptual correspondence was rejected, and a conception of numerical and formal correspondence was adopted (1923–1925). In this third stage, every hope of understanding the mechanism of transitions was given up. What are the implications of this development for the question of visualisability? In 1913, Bohr’s restricted view of a merely numerical correspondence reflected his awareness of the conceptual discontinuity between his model and classical theory, which had its source in the unvisualisable part of his model: the transitions (‘quantum jumps’). On the other hand, the stationary states were still described in a classical, visualisable manner. Thus, Bohr’s model was partly visualisable. From 1916 on, the successes of the application of the correspondence principle led Bohr to believe that the conceptual discontinuity between quantum and classical theory might not be as great as previously thought. This development, in which the role of classical concepts in the foundation of the theory increased, suggests that for Bohr there were in principle no objections to a possible visualisation of his atomic model. Nevertheless, Bohr was too well aware of the failure of classical physics to search for a complete return to classical representations, and it is unlikely that he would have supported a naively realistic interpretation of such a semiclassical representation (an orbital model in which electrons instantaneously

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jump from one orbit to another). The idea of conceptual correspondence was rejected again in the third stage (1923–1925). The reasons for this change were that Bohr’s orbital model was confronted with insoluble problems: it failed to account for the spectrum of helium and for the anomalous Zeeman effect. In this period, Bohr started to examine the role of pictures (Bilder) in physical theory. In 1922 he wrote, in a letter to Hffding: ‘One must in general and especially in new fields of work constantly be aware of the apparent or possible inadequacy of pictures’, and ‘my personal viewpoint is that these difficulties are of a kind that hardly leaves room for the hope of accomplishing a spatial and temporal description of them that corresponds to our usual sense-impressions’.6 A central factor in Bohr’s worries about the possibility of pictures was the wave-particle dilemma concerning light. Bohr did not accept Einstein’s concept of light-quanta as a description of reality. However, light-quanta appeared to be necessary to account for phenomena that involve interaction between radiation and matter. Consequently, it appeared ‘hardly possible to propose any picture which accounts, at the same time, for the interference phenomena and the photoelectric effect, without introducing profound changes in the viewpoints on the basis of which we have hitherto attempted to describe the natural phenomena’.7 Nevertheless, Bohr’s reluctance to accept light-quanta was so strong that even Compton’s experiments of 1923, which were generally regarded as a confirmation of the light-quantum hypothesis, did not change his opinion. Instead, he proposed, together with H. A. Kramers and J. C. Slater, an alternative theory of radiation}the BKS theory}in which strict energy conservation (and thereby causality) was renounced in order to describe interaction processes without the light-quantum (Bohr, Kramers and Slater, 1924).8 The central idea of the BKS theory is that atoms in stationary states are accompanied by a classical ‘virtual radiation field’, which is identical to the field of a set of harmonic oscillators of all possible frequencies of the atom (a set of so-called ‘virtual oscillators’). The field, which is called ‘virtual’ because it is not observable and does not carry momentum or energy, transmits probabilities for transitions in other atoms. Accordingly, any atom experiences a virtual field which is the sum of the virtual fields of all atoms present (including its own). Thus, through the field there is communication between distant atoms, which is non-causal because it merely changes the probability that a transition will occur (contrary to an emitted light-quantum, which, when absorbed, always induces a transition). Since the main purpose of the BKS theory was to show that a quantum theory of radiation without light-quanta remained possible, it can be regarded as an attempt to preserve the possibility of an unambiguous spacetime picture, 6

Bohr to Hffding, 22 September 1922; quoted in Favrholdt (1992, pp. 24–25). Bohr (1976, Vol. 3, p. 235); from the 1920 lecture ‘On the Interaction Between Light and Matter’. 8 My account of the BKS-theory is based mainly upon Dresden (1987, pp. 159–178). 7

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and thus visualisation, of atomic events.9 Whereas the wave-particle duality of radiation fundamentally excluded unambiguous visualisation, the BKS theory restored it by resolving the dichotomy in favour of the wave-picture. The price that had to be paid for this was, besides a renunciation of causality, an explicitly anti-realistic view of the postulated ‘virtual’ entities. Moreover, Bohr stressed that the BKS theory had a ‘formal’ character, analogously to his earlier emphasis on the formal nature of his 1913 atomic theory. As in 1913, he thus wanted to emphasise that the theory lacked explanatory force in the usual sense of the word (e.g. by providing a physical mechanism of the actual probability transmissions), a defect which perhaps could be repaired in the future.10 It appeared, however, that Bohr was willing to pay this price for retaining the possibility of unambiguous visualisable description. The BKS theory was experimentally disproved in April 1925 by Bothe and Geiger. Bohr’s response confirms the interpretation given above. He accepted that the experiment proved that there was a ‘coupling through radiation of the changes of state of widely-separated atoms’. This causal coupling refuted the virtual-radiation-field hypothesis, but it did not yet prove the reality of lightquanta. Instead, Bohr concluded that the result did ‘so thoroughly rule out the retention of the ordinary spacetime description of phenomena, that in spite of the existence of coupling, conclusions concerning a possible corpuscular nature of radiation lack a sufficient basis’.11 In other words, the existence of coupling ‘precludes the possibility of a simple description of the physical occurrences by means of visualisable pictures [anschaulicher Bilder]’.12 The refutation of the BKS theory marks the transition to a new phase in the development of quantum theory, a phase in which Bohr began to examine the limitations of a spacetime description and which finally led to his conception of complementarity. He was convinced that the refutation of the BKS theory pointed to ‘an essential failure of the pictures of space and time on which the description of natural phenomena has hitherto been based’.13 Above all, the fact that even the spatiotemporal picture of radiation (the wave theory of light) had its limitations worried Bohr. From 1925 on he had to accept the waveparticle duality of light. In the beginning he emphasised its formal character . This was one reason why Erwin Schrodinger appreciated the BKS theory; see De Regt (1997, pp. 472–473). 10 Bohr (1976, Vol. 5, p. 101) wrote: ‘At the present state of science it does not seem possible to avoid the formal character of the quantum theory which is shown by the fact that the interpretation of atomic phenomena does not involve a description of the mechanism of the discontinuous processes, which in the quantum theory of spectra are designated as transitions between stationary states of the atom’. Cf. Dresden (1987, p. 171), who defines Bohr’s conception of a ‘formal theory’ as ‘a theory which contains rules, relations, and connections between physical quantities, but the theory does not and sometimes cannot provide a physical mechanism or a physical justification for such relations’. 11 Bohr to Geiger, 21 April 1925, in Bohr (1976, Vol. 5, p. 354). Translation in Murdoch (1987, p. 29). 12 Bohr to Born, 1 May 1925, in Bohr (1976, Vol. 5, p. 497). Translation in Murdoch (1987, p. 29). 13 Bohr (1934, pp. 34–35); from the 1925 lecture ‘Atomic Theory and Mechanics’. 9

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(particularly of the particle-picture) (Bohr, 1934, p. 29). Thus, he admitted that there was a duality in the mathematical description, but refrained from giving a physical interpretation to it. In December 1925 he discussed the issue with Einstein, and afterwards stated: ‘In my opinion the possibility of obtaining a spacetime picture based on our usual concepts becomes ever more hopeless’.14 It appears that Bohr was still worried by the inadequacy of pictures ‘based on our usual concepts’, that is, visualisable pictures. Indeed, despite their apparent limitations, he became ever more convinced that these ‘usual’ (i.e. classical) concepts were indispensable for rendering atomic physics intelligible.

4. The Schism: Matrix Mechanics Versus Wave Mechanics In the early 1920s, Bohr’s students Wolfgang Pauli and Werner Heisenberg worked on the development of Bohr’s semi-mechanical atomic model. They invested all their energy in attempts to solve its problems, particularly the anomalous Zeeman effect, but to little avail. Heisenberg made ingenious adaptations to the model, but these violated basic principles of quantum theory and therefore proved unsatisfactory in the end (see De Regt, 1999b, pp. 412– 413). Pauli’s attempts were equally unsuccessful and even led him to a temporary retreat from atomic physics (ibid., pp. 414–416). When he took up the subject again, in the autumn of 1924, Pauli started a radically new programme for quantum theory in which all mechanical concepts, such as momenta and orbits, were relinquished completely. The first result of this approach was his ‘exclusion principle’. Pauli’s attack on mechanical models went hand in hand with a renunciation of visualisability: the rejection of the applicability of concepts of classical mechanics in the atomic domain entailed the impossibility of visualisation in the classical sense. Though Pauli did not explicitly regard this as an advantage, he certainly did not consider it a serious problem. Pauli’s letter to Bohr of 12 December 1924 reveals his attitude towards the aspect of unvisualisability (in fact, in this letter the notion of Anschaulichkeit is referred to for the first time in the Pauli correspondence). In it Pauli expressed the conviction ‘that not only the dynamic concept of force but also the kinematic concept of motion of the classical theory shall have to undergo fundamental changes’. In an appended note, he remarked with characteristic sarcasm: I consider this certain}despite our good friend Kramers and his colourful picture books}‘and the children, they love to listen.’ Even though the demand of these children for Anschaulichkeit is partly a legitimate and a healthy one, still this demand should never count in physics as an argument for the retention of fixed conceptual systems. Once the new conceptual systems are settled, then also these will be anschaulich.15 14

Bohr to Slater, 28 January 1926, in Bohr (1976, Vol. 5, p. 497). Pauli (1979, p. 188). Pauli referred to the popular book by Kramers and Holst: Bohr’s Atomtheori (Kramers and Holst, 1922), which contained drawings of atoms and orbiting electrons.

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Thus, Pauli rejected Anschaulichkeit if it was to be conceived as the visualisability of mechanical models. The concepts of classical mechanics had to be replaced by new ones, which would after some time become anschaulich as well, in the sense of ‘intelligible’. With respect to the way in which these new concepts had to be discovered, Pauli advocated an operationalist methodology. After giving credit to the energies and angular momenta of stationary states for being far more real than orbits, he stated: The (still unattained) goal must be to deduce these and all other physically real, observable characteristics of the stationary states from the (fixed) quantum numbers and quantum-theoretical laws. However, we should not want to clap the atoms into the chains of our preconceptions (to which in my opinion belongs the assumption of the existence of electron orbits in the sense of the usual kinematics) but we must on the contrary adjust our ideas to experience (Pauli, 1979, p. 189).

Pauli’s ideas significantly influenced Heisenberg’s work in the spring and summer of 1925, which laid the foundation for matrix mechanics, the first ‘quantum mechanics’ (coined by Max Born as a name for the long-sought foundation for quantum theory). Matrix mechanics, further developed by Heisenberg in collaboration with Born and Jordan, refrained from any visualisation of atomic structure, restricting itself to describing relations between observable quantities such as frequencies and intensities of spectral lines. This ‘theory of observables’ was framed in the then unusual and difficult mathematical language of matrices. It was recognised as successful and . promising by quantum physicists from the Copenhagen–Gottingen circle, among whom Bohr and Pauli. Outside this relatively small group, however, the theory was largely ignored. A main reason for this was obviously its difficult mathematics and its abstract, unvisualisable character. In addition, however, its successes were severely limited: only very simple cases could be solved by the matrix treatment, and initially the actual results did not supersede those of the old quantum theory (see Beller, 1999, Chapter 2). Half a year later, early in 1926, a rival quantum-mechanical theory appeared . on the scene: Schrodinger’s wave mechanics. Representing atomic structure in terms of wave functions, this theory appeared quite different from matrix mechanics. Most notably, it seemed to promise visualisation of atomic . processes as wave phenomena. In fact, for Schrodinger himself, this promise was the chief motivation for constructing wave mechanics. As was noted . already in Section 1, Schrodinger was strongly committed to the thesis that Anschaulichkeit is a necessary condition for intelligibility of theories.16 Because of the promise of visualisability, wave mechanics was happily received by many . physicists outside the Copenhagen–Gottingen community. This was not the only reason, however. The theory was easily applicable to concrete situations, and it solved problems that matrix theory was unable to handle. For example, . See De Regt (1997) for an analysis of Schrodinger’s commitment to Anschaulichkeit and a detailed description of his discovery of wave mechanics.

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it yielded the spectrum of hydrogen quite straightforwardly, in contrast to Pauli’s highly problematic treatment of the hydrogen atom with matrix methods (see Beller, 1999, p. 29). Nevertheless, wave mechanics had its own problems: describing free electrons by wave packets was troubled by dispersion. And, most importantly, visualisability of atoms was not as easy as it seemed, because n-particle systems are represented by wave functions in 3n-dimensional configuration space, not 3-dimensional real space. The ensuing competition between these two quantum-mechanical theories led to emotionally charged debates in which the issue of Anschaulichkeit . became the focus of attention. Schrodinger (1928, p. 46n) said of Heisenberg’s matrix mechanics: ‘I naturally knew about his theory, but was discouraged, if not repelled, by what appeared to me as very difficult methods of . transcendental algebra, and by the want of Anschaulichkeit’. Schrodinger emphasised the connection between visualisability and intelligibility, arguing that matrix mechanics is unintelligible because of its abstract, unvisualisable character. By contrast, wave mechanics is anschaulich and therefore, according . to Schrodinger, a theory that provides understanding of atomic phenomena. As noted above, the larger physics community was receptive to this forceful argument in favour of wave mechanics. And consequently, the proponents of matrix mechanics strongly felt the need to defend their theory against it. . Indeed, Heisenberg was well aware of Schrodinger’s advantage in this respect, and he was far from pleased. In a letter to Pauli, he angrily exclaimed: ‘The . more I think of the physical part of Schrodinger’s theory, the more abominable . I find it. What Schrodinger writes about Anschaulichkeit makes scarcely any sense’ (Pauli, 1979, p. 328). . Pauli had a more sophisticated reply available to Schrodinger’s plea for Anschaulichkeit. As we saw above, already in 1924}in the context of his criticism of mechanical models of the atom}he had argued that intelligibility did not necessarily have to be equated with visualisability. Pauli (1979, p. 188) reasoned that after a while quantum mechanics would automatically lead to a new conception of intelligibility: ‘Once the new conceptual systems are settled, then also these will be anschaulich’. In other words, Pauli wanted to replace the . ‘classical ideal of intelligibility’, endorsed by Schrodinger, by a new ‘revolutionary ideal of intelligibility’. Remarkably, it was Heisenberg}albeit in discussion with Pauli}who made an attempt to flesh out Pauli’s suggestion about a new conception of . Anschaulichkeit. Heisenberg’s 1927 paper ‘Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik’, as is clear from its title, intends to show that quantum theory (read: matrix mechanics) was not completely unanschaulich. As such, the paper was a direct response to . Schrodinger’s allegations.17 But how could Heisenberg maintain that matrix mechanics was anschaulich? This remarkable conclusion depended on two . See Beller (1999, pp. 67–78) for an analysis of the Heisenberg–Schrodinger ‘dialogue’ in this paper.

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feats: a redefinition of Anschaulichkeit and the derivation of the uncertainty relations. In the first sentence of his paper Heisenberg already presents his redefinition: ‘We believe we understand [anschaulich zu verstehen] the physical content of a theory when we can see its qualitative experimental consequences in all simple cases and when at the same time we have checked that the application of the theory never contains inner contradictions’.18 He proceeds by discussing to what extent the usual kinematic and mechanical concepts (position, path, velocity, energy) can be retained in quantum mechanics. He argues that under a suitable (operationalist) definition these concepts remain valid. However, discussion of a thought experiment in which the position of an electron is determined (the g-ray microscope) leads Heisenberg to the conclusion that for any pair of canonically conjugate quantities p and q the relation DpDq
h holds, where Dp is the precision with which the value of p can be determined and similarly for Dq. Heisenberg’s ‘uncertainty relation’, if applied to momentum and position, precisely specifies the limitations of a spacetime description in the atomic domain. So one might think that Heisenberg had definitively established the Unanschaulichkeit of quantum mechanics. Not so. He argues that the uncertainty relation allows one to analyse simple (thought) experiments, such as the g-ray microscope and the Stern–Gerlach experiment, and to arrive at qualitative predictions. And this amounts precisely to Anschaulichkeit, given his redefinition of the notion. Therefore, he concludes that ‘as we can think through qualitatively the experimental consequences of the theory in all simple cases, we will no longer have to look at quantum mechanics as unphysical [unanschaulich] and abstract’.19 Many authors have called Heisenberg’s redefinition of Anschaulichkeit a rhetorical trick (e.g. Cushing, 1994, p. 99; Beller, 1999, p. 70). Although it might indeed be a perverse definition of ‘visualisability’, I believe that it does offer a valuable new perspective on ‘intelligibility’ (and that is what Heisenberg intended: anschaulich zu verstehen). In Section 6, I will present a theory of scientific understanding based on Heisenberg’s suggested redefinition.

5. Visualisation: The Case of Electron Spin Before embarking upon a new theory of scientific understanding, however, I wish to conclude my historical analysis by focusing on the discovery of electron spin. This is an episode in the transition from the old quantum theory to the new quantum mechanics in which visualisability played a crucial role. 18

Quoted from the 1983 translation by Wheeler and Zurek, p. 62. Wheeler and Zurek (1983, p. 82). In a footnote appended to this statement, Heisenberg directly . replied to Schrodinger’s complaint about the unvisualisability of matrix mechanics. While . admitting that Schrodinger’s theory had provided additional insight into the quantum-mechanical . laws, he rejected the visualisable interpretation favoured by Schrodinger.

19

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Although matrix mechanics was a conceptual innovation, at the empirical level there was still a long way to go before it would prove its superiority to the old quantum theory. For example, the anomalous Zeeman effect, which played such an important role in the crisis, was still hardly understood. Pauli had suggested that it was caused by a peculiar property of the electron, namely a classically indescribable Zweideutigkeit (two-valuedness); the explanation of this Zweideutigkeit would in turn be an unmechanischer Zwang. In the autumn of 1925 Samuel Goudsmit and George Uhlenbeck advanced the hypothesis of electron spin as an alternative explanation. The idea of a spinning electron was completely in the spirit of the old quantum theory, since it fitted into the visualisable orbital model. According to the spin hypothesis, the electron is a rotating particle, which has an angular momentum ms (=  1/2) and a magnetic dipole moment 2mB ms (in which mB ¼ eh=4pme is the Bohr magneton and the g-factor 2 follows if one assumes that the electron is a sphere with surface charge). The spin thus served as a mechanical explanation of the Zweideutigkeit of the electron, replacing Pauli’s unmechanischer Zwang (see De Regt, 1999b, pp. 414–416). Ralph Kronig had been the first to propose the idea of spin in January 1925, when he was visiting Pauli and discussing the Zweideutigkeit of the electron. Pauli called it ‘a quite witty thought’, but rejected it (Kronig, 1960, p. 21). Obviously, the hypothesis was in flat contradiction with Pauli’s antimechanical programme. Moreover, it led to calculations of the doublet splitting in hydrogen that were wrong by a factor two, and was thus also empirically inadequate. Kronig was so discouraged by the negative response of Pauli, and later also of Heisenberg and Kramers, that he never published his idea. In October 1925 Goudsmit and Uhlenbeck independently of Kronig’s earlier proposal conceived the idea of spin again. Uhlenbeck later recalled how they got at it after reading Pauli’s paper on the exclusion principle: [In this paper] four quantum numbers were ascribed to the electron. This was done rather formally; no concrete picture was connected with it. To us, this was a mystery. We were so conversant with the proposition that every quantum number corresponds to a degree of freedom, and on the other hand with the idea of a point electron, which obviously had three degrees of freedom only, that we could not place the fourth quantum number. We could understand it only if the electron was assumed to be a small sphere that could rotate.20

Goudsmit and Uhlenbeck did not notice the problem of the wrong factor two in the doublet splitting (they did not calculate it), but they realised that there was another problem: the rotational velocity at the surface of the electron would far exceed the speed of light. However, their supervisor Ehrenfest considered the hypothesis valuable and sent their manuscript in for publication (Uhlenbeck and Goudsmit, 1925). Uhlenbeck’s account shows that their discovery was initiated by an attempt at visualisation. 20

Address delivered in April 1955. Quoted in Van der Waerden (1960, p. 213).

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In the reception of Goudsmit and Uhlenbeck’s proposal by Bohr, Heisenberg and Pauli, Anschaulichkeit also played an interesting role. It was Bohr who was the first to accept the spin hypothesis and this is plausibly connected with his attitude toward visualisability. As described in Section 3, after the refutation of the BKS theory Bohr was convinced that there were limits to the possibility of visualisation of atomic processes. Nonetheless, he was not willing to give up the idea of visualisation completely. Instead, he had come to believe that classical concepts were indispensable for understanding phenomena in the atomic domain. Thus, although he considered Heisenberg’s matrix mechanics a great step forward, he did not think that it was a satisfactory theory from the point of view of intelligibility. Of course, Bohr knew that a complete return to classically mechanical pictures was impossible, but he still wanted an intuitive, physical interpretation of the formalism, in which classical (e.g. mechanical) concepts had to play a role. This attitude might explain why Bohr was easily converted to the mechanical, visualisable spin hypothesis. Pais (1986) presents the amusing story of Bohr’s conversion, as Bohr himself told it to him. Initially, Bohr thought that the spin hypothesis was incapable of explaining doublet splitting, because this splitting can occur only in a magnetic field, whereas inside the atom there appears to be only the electric field of the nucleus. In December 1925 he travelled to Leiden to attend the Lorentz jubilee festivities. On the way he met Pauli in Hamburg, who asked him what he thought about spin, and Bohr expressed his doubts. In Leiden, he met Einstein and Ehrenfest, who immediately asked Bohr what he thought about spin. Bohr again expressed his doubts. Then Einstein pointed out that, according to relativity theory, in the rest frame of the electron the nucleus is a moving charge and thus produces a magnetic field. This field is parallel to the angular momentum of the orbit, and the effect is a spin-orbit coupling, giving rise to doublet splitting. Pais tells us: Bohr was at once convinced. When told of the factor two he expressed confidence that this problem would find a natural resolution. [. . .] After Leiden Bohr . travelled to Gottingen. There he was met at the station by Heisenberg and Jordan who asked what he thought about spin. Bohr replied that it was a great advance and explained about the spin-orbit coupling. [. . .] On his way home the train stopped at Berlin where Bohr was met at the station by Pauli, who had made the trip from Hamburg for the sole purpose of asking Bohr what he now thought about spin. Bohr said it was a great advance, to which Pauli replied: ‘eine neue Copenhagener Irrlehre’. After his return home Bohr wrote to Ehrenfest that he had become a prophet of the electron magnet gospel (Pais, 1986, pp. 278–279).

Uhlenbeck and Goudsmit’s second paper on the spinning electron (1926; dated December 1925) had an appended note by Bohr in which he wrote that spin ‘opens up a very hopeful prospect of our being able to account more extensively for the properties of elements by means of mechanical models, at

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least in the qualitative way characteristic of applications of the correspondence principle’.21 This statement confirms that Bohr still believed that mechanical models were valuable, although he knew that obviously such models could not be interpreted in a naively realistic manner. From that time on Bohr tried to convince Pauli and Heisenberg as well: ‘I have since [my trip to Leiden] had quite a difficult time trying to persuade Pauli and Heisenberg, who were so deep in the spell of the magical duality that they were most unwilling to greet any outway of the sort’.22 Heisenberg was indifferent to the aspect of visualisability. For him the wrong factor two in the doublet structure was the greatest obstacle. On 21 November he wrote to Goudsmit as well as Pauli about spin and expressed his concern about this problem. However, his attitude toward the idea was not altogether negative.23 Pauli apparently responded with fierce disapproval, as can be deduced from Heisenberg’s letter to Pauli of 24 November. There Heisenberg stated that he still had strong reservations about spin. Nevertheless, he was working on a matrix-mechanical treatment of the model, which he hoped would solve the problems. Heisenberg’s attitude toward spin changed under the influence of Bohr, and despite the unsolved empirical problems he gradually accepted the idea.24 Pauli, on the contrary, was persistent in his rejection of the visualisable spin model. In December 1925, in a letter to Bohr, he stated objections of a technical nature. He admitted that a definitive appraisal of the model could only be made from the perspective of the new quantum mechanics, but until this was accomplished he maintained his opinion: ‘I don’t like the thing!’.25 In February 1926 the difficulty of the factor two was resolved by L. Thomas, who had localised the problem in a wrong treatment of the relative motion of the electron. Bohr happily communicated this result to Pauli, but Pauli responded negatively: he expressed the conviction that Thomas’s calculations were not relevant to the problem. In March, Pauli presented an analysis of the disagreement about the spin hypothesis. Here the deeper motivation for his persistent rejection came again to the fore. He explained that his disparaging remark ‘neue Irrlehre’ (new heresy) had been motivated by his conviction that the more fundamental, general problems regarding spin (i.e. consistency with matrix mechanics) had to be solved before spin could be accepted as an explanation of the hydrogen spectrum and the Zeeman effect.26 One week later, however, Bohr had convinced Pauli that his objections to Thomas’s 21

Bohr quoted in Miller (1986, p. 138). Bohr to Kronig, 26 March 1926, in Bohr (1976, Vol. 5, p. 235). The ‘magical duality’ probably refers to Pauli’s Zweideutigkeit. 23 See Heisenberg to Pauli, 21 November 1925, in Pauli (1979, pp. 261–262); and Heisenberg to Goudsmit, 21 November 1925, quoted in Van der Waerden (1960, p. 214). 24 See Heisenberg’s letters to Pauli of 24 December 1925, 7 January 1926, 14 January 1926, and 27 January 1926, in Pauli (1979, p. 271, p. 280, p. 281, p. 282). 25 Pauli to Bohr, 30 December 1925, in Pauli (1979, p. 275). 26 Pauli to Bohr, 5 March 1926, in Pauli (1979, p. 302). 22

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calculations were unfounded. Pauli had made a mistake for which he now excused himself and concluded: ‘Now nothing else is left for me than to surrender completely!’.27 That even after this capitulation Pauli still had reservations becomes clear from a letter to Wentzel (8 May 1926). There he reported that the problems with respect to spin were solved, and that one now had to believe that there was much truth in it, but he added: ‘I believe it only with a heavy heart, because the difficulties which it causes for the theory of the structure of the electron are very substantial’.28 The editors of the Pauli correspondence note that he probably referred to the fact that the rotational velocity of the electron, when conceived as a spinning particle, exceeds the speed of light. Pauli’s reluctant surrender and his objection indicate that he was still convinced that a mechanical, visualisable interpretation of spin was misguided, even though he could not deny that the spin hypothesis successfully accounted for the empirical data. His dissatisfaction was, however, removed, when in 1927 he succeeded in fitting the concept into the formalism of quantum mechanics, thereby abandoning its classically visualisable interpretation. The resulting theory of spin is still used, and contains what are now known as the ‘Pauli spin matrices’. In 1955 Pauli looked back on this episode and remarked: After a brief period of spiritual and human confusion, caused by a provisional restriction to Anschaulichkeit, a general agreement was reached following the substitution of abstract mathematical symbols, as for instance psi, for concrete pictures. Especially the concrete picture of rotation has been replaced by mathematical characteristics of the representations of the group of rotations in three-dimensional space (Pauli, 1964, Vol. 2, p. 1239).

While formally this is true, the fact remains that the ‘concrete picture of rotation’ is still a useful guide for physical understanding of electron spin. To mention but one example, the experimental determination of magnetic moments of free electrons and muons in a so-called Penning trap, consisting of electric and magnetic fields, is described as follows in Perkins (1982, p. 323): ‘The effect of the field is to confine the electron or muon ion in a potential well in which they make very small oscillations. Due to the magnetic field, the particle performs circular cyclotron orbits in the horizontal plane, and the particle spin also precesses about the field direction’. One may agree with Pauli that no concrete picture should be attached to concepts such as precession, but I doubt that many physicists could do completely without the visual imagery (which invites us to connect the concept of precession with the behaviour of spinning electrons in particular circumstances).29 27

Pauli to Bohr, 12 March 1926, in Pauli (1979, p. 310). Pauli to Wentzel, 8 May 1926, in Pauli (1979, p. 322). 29 Cf. Feynman (1965, pp. 7–10), who puts inverted commas around the term ‘precession’ but whose description of the phenomenon invites a similar visualisation. 28

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6. Visualisation as a Tool for Scientific Understanding The historical analysis in the preceding sections has made it clear that visualisation often plays a role in rendering scientific theories intelligible. In the present section I will present a theory of scientific understanding that accounts . for this role of visualisation.30 Contrary to Schrodinger, I do not want to maintain that visualisability is a necessary prerequisite for intelligibility. On my view, it is merely a ‘tool’ for obtaining understanding. But though there may be other tools for the same purpose, visualisation is surely a quite effective tool, which has proved its mettle many times in the past. One might ask why one should want intelligible theories in the first place? As . mentioned in Section 1, Schrodinger claimed that having an intelligible theory is a condition for scientific understanding of the phenomena. The historical examples presented above support this idea: physicists aim at understanding of the phenomena by constructing theories which can be rendered intelligible (e.g. by making mechanical models). Another argument for it is given by John Ziman, who observes that ‘scientific communications must not only be comprehensible, they also typically enable comprehension. [. . .] They help us to understand the world’ (Ziman, 2000, p. 289; italics in the original). Ziman (ibid., p. 290) subsequently argues that ‘[w]hen [. . .] we say that an individual ‘‘understands’’ a non-cognitive entity, we imply that she has established an internal mental structure representing that entity}in particular an internal mental structure that might well be held by another individual with whom she desires to communicate on the same subject’. Ziman rightly emphasises the importance of communication in the enterprise of achieving scientific understanding. As the search for understanding cannot be accomplished directly by one individual, the ability to communicate results is essential. Accordingly, the intelligibility of ‘mental representations’ (e.g. theories) is crucial for communication and for achieving understanding. This conclusion can be formulated more precisely as a Criterion for Understanding Phenomena: CUP: A phenomenon P is understood iff a theory T of P exists which is intelligible (and meets the usual logical and empirical requirements). Now the question arises: when is a theory intelligible? The problem is to find a general criterion for the intelligibility of theories, which allows for variation in intelligibility standards: it should incorporate understanding by means of . visualisation (as defended by Schrodinger) as well as understanding by means of abstract thinking (as defended by Pauli). As argued elsewhere (De Regt and Dieks, to appear), intelligibility consists in being able to ‘grasp’ how the predictions are brought about by the theory. This admittedly vague demand for an intuitive feeling for how the theory ‘works’ can be given more substance by invoking Heisenberg’s redefinition of Anschaulichkeit, discussed 30

See Dieks and De Regt (1998), and in particular De Regt and Dieks (to appear), for a more detailed presentation of this theory.

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in Section 4.31 A modified version of this redefinition furnishes a general Criterion for the Intelligibility of Theories: CIT: A scientific theory T is intelligible to scientist S iff S can recognise qualitatively characteristic consequences of T without performing exact calculations. A feature of CIT is that it captures the contextual nature of intelligibility: a scientific theory is intelligible to a particular scientist if she is able to recognise its consequences. CIT explicitly refers to the users of the theory (the scientists), and intelligibility thereby comes to depend on such contextual factors as the capacities, background knowledge and background beliefs of these users. Accordingly, the criterion allows for the possibility that a scientific theory considered unintelligible by some (at one time) will be regarded as intelligible by others (possibly at a later time). How precisely do contextual factors contribute to attaining scientific understanding? In order to recognise consequences of a theory intuitively, and to be able to argue about them, one needs ‘conceptual tools’. With the help of these tools one can circumvent the calculating stage and ‘jump to the conclusion’. Obviously, the availability and applicability of tools is a contextual matter. There is no universal tool for understanding, but a variety of ‘toolkits’, containing particular tools for particular people in particular situations. Which tools scientists have at their disposal depends partly on the (historical, social, or disciplinary) context in which they find themselves. As has become clear in the previous sections, visualisation is a prime example of a conceptual tool which can be used to obtain insight into the consequences of scientific theories. Indeed, it is regarded by many scientists as . an almost indispensable help in doing science. Not only Schrodinger, but physicists such as Faraday and Feynman emphasised the importance of visualisation in scientific practice. Consider Feynman’s (1965, Vol. 2, p. 4-8) discussion of the idea of field lines in electrostatics: although intuitive application of this idea is possible only in very simple situations, it is quite useful to get a feeling of how electrostatic systems behave. And this, according to Feynman (ibid., p. 2-1), is precisely what it means to have a physical understanding of the situation in question: ‘if we have a way of knowing what should happen in given circumstances without actually solving the equations, then we ‘‘understand’’ the equation, as applied to these circumstances’. The examples cited above, from Maxwell’s ether model to Heisenberg’s g-ray microscope, provide further evidence for the usefulness of visualisation in achieving this kind of understanding. But while visualisation can be an important tool, it would be erroneous to maintain that it is essential for understanding. For example, many theoretical physicists have developed a familiarity with, and intuition for, the behaviour of 31

Heisenberg (1927, p. 172). Incidentally, Feynman (1965, Vol. 2, p. 2-1) attributes a similar view to Dirac.

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the solutions of the equations they use, which enables them to have a feeling for the qualitative behaviour of the described systems without invoking picturable physical mechanisms. Physicists like Pauli, Heisenberg and Dirac clearly possessed this ability, and Pauli could persistently reject visualisable models because his capacity for abstract reasoning was so well developed. The claim such physicists usually make, namely that they really understand the theory they are working with, is perfectly legitimate. Criterion CIT implies that such different tools as visualisable physical mechanisms on the one hand and rather abstract reasoning on the other, can fit in with the same central aim of science, namely achieving understanding. An implication of CIT is that theories may become intelligible when we get accustomed to them and develop the ability to use them in an intuitive way. In this sense, the present view rehabilitates an intuition which has often been discredited: the idea that understanding is related to familiarity. Friedman (1974, p. 9), for example, deems the view that ‘scientific explanations give us understanding of the world by relating (or reducing) unfamiliar phenomena to familiar ones’}the so-called ‘familiarity view’ of scientific understanding}‘rather obviously inadequate’. Indeed, there is a well-known argument against it: science often explains phenomena that are directly known and quite familiar to us by means of theories and concepts of a strange and unfamiliar (sometimes even counter-intuitive) character. Nonetheless, the intuition behind the familiarity view should not, and need not, be relinquished completely. The ‘conceptual toolkit thesis’ associated with CIT implies that only if tools are familiar to their users can they be used for their purpose, namely making predictions without entering into explicit calculations. However, this familiarity does not necessarily involve the idea that the user should be acquainted with the concepts in question from everyday experience. One can get accustomed to the mathematical concepts used in a theory or to an initially weird physical interpretation of a theory, to such an extent that it becomes possible to reason in the associated terms in an intuitive manner. Many theoretical physicists will share Pauli’s view that visualisation is superfluous (contrary to experimental physicists who will favour the approach taken in e.g. Perkins (1982)). CIT incorporates the intuition on which the familiarity view is based; but it does not lead to the consequence that we should understand phenomena in terms of things which are more familiar from everyday experience than the phenomena themselves. While this article has been concerned mainly with quantum mechanics before 1930, more recent developments confirm the analysis presented here. Quantum field theory is mathematically far more intricate and abstract than standard quantum mechanics, but its history shows that there is still a role for visualisation and that this role fits the analysis given above. The first formulations of quantum electrodynamics, developed by Julian Schwinger and Sin-Itoro Tomonaga in 1947–1948, were field-theoretical formalisms which were mathematically rigorous but unamenable to visualisation. In 1949 Richard Feynman proposed his

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diagrammatic approach which provides a visualisation of the interaction processes in terms of spacetime trajectories, albeit one that cannot be taken as a one-to-one representation of actual occurrences in nature. As Feynman’s presentation of his approach was not based on mathematical proof, it was regarded with suspicion (Miller, 1986, p. 170). However, as soon as Freeman Dyson proved that Feynman’s theory was equivalent with that of Schwinger and Tomonaga, most physicists were converted to Feynman’s ‘intuitive method’. One notices a striking similarity with the 1926 competition between matrix and wave mechanics. In an interview with Silvan Schweber (quoted in Schweber, 1994, pp. 465– 466), Feynman stated that ‘visualisation in some form or other is a vital part of my thinking’. His description of how precisely his thinking worked fits in with our analysis of scientific understanding, and led him to conclude: ‘I see the character of the answer before me}that’s what the picturing is’. On the relation between the visualising and the calculating stage Feynman had this to say: ‘Ordinarily I try to get the pictures clearer but in the end, the mathematics can take over and can be more efficient in communicating the idea than the picture’. As an example, he described how he solved the liquid helium problem by visualising the situation: I didn’t know how to write a damn thing and there was nothing I could do but keep on picturing [. . .], and I couldn’t get anything down mathematically [. . .]. So the whole thing was worked out first, in fact was published first as a descriptive thing [. . .] which doesn’t carry much weight, but to me was the real answer. I really understood it and I was trying to explain it (quoted in Schweber, 1994, p. 466).

Note the distinction between understanding and explanation in the last sentence: the former is provided by visualisation, the latter by mathematical derivation. Obviously, Feynman was an exceptional genius, but the success of his diagrammatic method in scientific practice indicates that the minds of most physicists work in a similar way and that visualisation is still a useful tool for achieving understanding. Incidentally, even Schwinger maintained that he employed visual imagery. Interviewed by Schweber (1994, p. 365), he said: ‘I suppose you can’t avoid having mental pictures. If somebody talks about an . electron I don’t think I see Schrodinger’s equation. Everybody pictures something; it’s a point moving. There has to be some sort of picture’. But, Schwinger continues, after the initial ‘physical impetus}which to some extent is visual, but certainly not in the elaborate way of Feynman’, . mathematical analysis takes over (ibid., p. 366). Like Schrodinger and Pauli a quarter of a century before, the difference between Feynman and Schwinger nicely illustrates the variation in conceptual tools that may be employed for obtaining understanding. Visualisation can be an effective road to intelligibility but it is not the only one.

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Acknowledgements}I thank Dennis Dieks for helpful comments. This research was supported in part by the Netherlands Organisation for Scientific Research (NWO).

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