The role of a posteriori mathematics in physics

Studies in History and Philosophy of Modern Physics xxx (2017) 1e10

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The role of a posteriori mathematics in physics Edward MacKinnon California State University, East Bay, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 6 May 2017 Received in revised form 27 August 2017 Accepted 5 September 2017 Available online xxx

The calculus that co-evolved with classical mechanics relied on definitions of functions and differentials that accommodated physical intuitions. In the early nineteenth century mathematicians began the rigorous reformulation of calculus and eventually succeeded in putting almost all of mathematics on a set-theoretic foundation. Physicists traditionally ignore this rigorous mathematics. Physicists often rely on a posteriori math, a practice of using physical considerations to determine mathematical formulations. This is illustrated by examples from classical and quantum physics. A justification of such practice stems from a consideration of the role of phenomenological theories in classical physics and effective theories in contemporary physics. This relates to the larger question of how physical theories should be interpreted. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Mathematics Physics Maxwell Quantum field theory The standard model

The a priori/a posteriori distinction, used by Kant, plays no role in mathematics. It does, however, play a role in philosophical interpretations of mathematically formulated theories. Philosophers of science have developed competing accounts of what scientific theories should be. In a syntactic account a scientific theory is presented, or reconstructed, as a deductive system based on axioms and employing both logical and mathematical rules of deduction. In a semantic account a theory is presented as a mathematical structure, such as phase space or Hilbert space, interpreted through a family of models. In both programs interpreting a theory is a matter of imposing a physical interpretation on a mathematical formalism. The math plays an a priori role. Both approaches insist that the mathematics used must have a validity independent of any physical interpretation imposed on it. The math, accordingly, must conform to rigorous standards. The practice of many physicists often runs contrary to this interpretative methodology. This happens in two basic ways. The first is by beginning with physical assumptions and letting the physics determine the type of math used in the theory formulation. The second concerns justification, rather than selection. Physicists often justify mathematical arguments on physical rather than mathematical grounds. In both cases the math plays a methodologically a posteriori role. The criticism that such math is not rigorous is effectively countered by the claim: Too much rigor leads to rigor mortis. We will consider some examples of this practice in

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both classical and quantum physics and the reflect on their significance. We begin by considering the conceptual matrix from which calculus emerged. 1. The mathematics of classical physics Pythagoras, Plato, and their disciples speculated on mathematical forms having some kind of existence independent of physical reality, or a pure a priori math. Aristotle's account of subalternation assigned mathematics a more a posteriori role in physical explanations. Arithmetic and geometry were regarded as idealizations derived from physical reality by abstraction and idealization: of numbers from units and of geometrical forms from physical shapes. Neither the Greeks, nor the Alexandrians, nor their Arabic successors ever developed a quantitative science of qualities. At the start of the Scientific Revolution, in the early Seventeenth century, a quantitative treatment of qualities was a common practice. A sketchy history can bring out the conceptual factors involved in the transition.1 In de Interpretatione (chap. 1) Aristotle developed the idea that things causally determine our concepts of them. The most basic concepts fit in his ordered list of categories: substance, quantity, quality, relation, place, time, situation, state, action, and passion. The first three categories have a conceptual ordering that determines the way quantities are treated. A quality, such as color

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See MacKinnon 2012 for a more detailed history.

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presupposes extension, which in turn presupposes a substance that is extended and colored. A discussion of the quantity of a quality perverts the proper conceptual ordering. Aristotle's doctrine of categories was transmitted to medieval Scholastics, before the revival of Aristotelianism, through translations of Porphry's Isagoge, which summarized the doctrine. Theological problems required a discussion of the quantity of qualities. The accepted doctrine, vividly illustrated in Dante's Divine Comedy that one's post-death assignment to a particular level in heaven, purgatory, or hell, depended on the degree (or lack) of sanctifying grace at the moment of death. Grace was regarded as a property of the soul, albeit a supernatural one. This property required quantification. Thomas Aquinas seems to have been the first to explicitly treat the quantification of qualities. (Summa Theologiae, 1,Q. 42, a. 1, ad 1) Under the later Nominalists this matured into a doctrine of the intensification and remission of qualities. This supplies the pivot transforming Aristotelian natural philosophy into mathematical physics. Newton's intermingling of physical and mathematical concepts presents a distinctive problem. As he explained it (Principia, p.38, Newton 1952a) the demonstrations are shorter by the method of indivisibles, his version of differential calculus, but since he deemed this harsh he followed the general method of ratios. Since his geometrical method now seems harsher we will rely on his theory of indivisibles. Quantities that vary continuously are called ’fluents', while their rates of change are called ’fluxions'. The basic problem of differentiation is: given the ratio of two fluents, to find the ratio of their fluxions.2 This relied on physical concepts: “And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of their quantities not before they vanish, nor afterward, but with which they vanish.” (Principia, p. 39) This physical justification led both Newton and Euler to express the ratios with which quantities vanish as 0/0 ¼ n. Euler tried to put calculus on an analytic, rather than a geometric basis. Here ’analytic’ loosely means a function that can be expressed by a simple formula, y ¼ f(x), or by a Taylor expansion. Yet, he too relied on expressions of the form, 0/0 ¼ n for the ratio of vanishing quantities. As Kline (Chap 13) has shown, 17th century scientists realized that the new calculus lacked an adequate foundation. Their reliance on calculus as a tool was effectively justified by a mixture of physical and theological considerations. Using calculus to solve physical problems led to correct results. Kepler and Galileo developed the idea that God created the world in accord with math-ematical forms. When Maupertuis introduced his Least Action Principle in 1744, he relied on a theological justification. The laws of matter must possess the perfection worthy of God's creation. Later physicists retained the a priori idea of a world fashioned in accord with mathematical forms even when they discounted the theological justification. The co-evolution of physics and mathematics reached a branching point in the early nineteenth century. Two men led the differing developments. Laplace pioneered a new style of mathematical physics. In place of Lagrange's analytic mechanics Laplace developed a style that Poisson later dubbed ‘physical mechanics’. Even by the standards of his time his math was not rigorous. He used approximations and power series in which he regularly dropped terms that were deemed insignificant on physical grounds. He treated math as a tool, not a system. His younger contemporary, Cauchy, instituted a program of developing calculus with no explicit reliance on physical notions. Like Gauss and Fourier Cauchy continued to work on physics problems and this work inspired new

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This is developed in Newton's ”Tractatus de Quadruture Curvarum”, first published in 1704. It is translated in Whiteside, pp.

developments in mathematics. However, mathematicians, following Cauchy's lead, abandoned physicalistic reasoning and geometric foundations in calculus in favor of analysis, and eventually set-theoretic foundations. (See Grattan-Guinness, 1980). These two trends epitomize what we have been referring to as a posteriori and a priori mathematics. A simplified comparison of the divergent formulations of calculus can bring out the difference by focusing on three terms, ‘function’, ‘infinitesimal’ and ‘continuous’. In the old formulation3 calculus is concerned with quantities that vary continuously and so can take on all possible values within boundary limits. This may be extended to quantities that have a limited number of discontinuities. A function expresses the relation between one quantity, the dependent variable, and another, the independent variable. It can usually be expressed through a simple analytic formula. y ¼ f(x), The basic unit of change for a continuous variable is an infinitesimal, or a differential. Infinitesimals are treated like quantities in the sense that x þ dx is a legitimate addition. Similarly, a derivative may be interpreted as the ratio of two differentials at the opposite extreme a set-theoretical formulation a function f: S / T, is a mapping that assigns to each element, s, of the domain, S, an element f(s)of the range, T. Abstract algebra is a broad topic. We will simply consider a set-theoretic formulation of operations. For this purpose, we can consider an algebra an ordered set, (A, o), with one 0 or more operations. A function f: (A, o)/ (A , o') maps elements of 0 A onto A and also carries the operation o into the operation o’. The functions of special concern here are homeomorphisms, in which the image of A is a proper subset of A‘. Finally, if mathematical continuity is not based on any notion of continuous quantities, then the fact that an interval is treated as a non-denumerable infinity of elements does not determine its length, differentiability, or integrability. Borel sets replace intuitive notions of continuity by beginning with point sets and then constructing aggregates that cover the interval. A Borel space is a set M with a s algebra. It elements are Borel sets. This is the smallest family of subsets of < that includes the open sets and is closed under complementation and countable intersections. A measure is a function whose domain is some class of sets and whose range is an aggregate of nonnegative real numbers. Borel sets supply a constructive method of assigning measures to sets so that the measure of an interval is the same as its length; congruent sets have equal measures; and the measure of a countable union of non-overlapping sets is equal to the sum of the measures of the individual sets. Borel sets do not presuppose physical continuity. Infinitesimals play no role in a settheoretical formulation. The concept of an infinitesimal as a number greater than 0, but smaller than any assignable number, does not accord with the Archimedean axiom: For any numbers a, b, where a and b are positive numbers and a < b, there exists an n, a natural number, such that na > b. Set theory provides a foundation for mathematics, one with well-known difficulties. Other foundations have been proposed, such as category theory, in which maps are basic elements. We will briefly consider two proposed foundations that allow infinitesimals. The first is the non-standard analysis developed primarily by Abraham Robinson, a severe critic of established set theory. (Robinson 1966) In non-standard theory set theory the natural numbers, N, are embedded in a larger set *N, which includes infinitely large numbers. This process is extended from the natural to the reals by embedding < in *<. The inverse of the infinitely large numbers in *< are infinitesimals. The second method of

3 This is presented in old calculus books such as the highly popular texts by Granville, later Granville and Smith, and finally Granville, Smith, and Longley, published between 1929 and 1962.

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reviving infinitesimals is smooth analysis (Bell, 1998). The fundamental object in a smooth world, S, is an indefinitely extensible homogeneous straight line, R. This incorporates a non-standard logic, effectively intuitionist logic. This does not have a principle of excluded middle, as in either (ε > 0) or (ε ¼ 0). This allows nilsquare infinitesimals, (ε s 0, but ε2 ¼ 0), effectively reproducing the idea of an infinitesimal as a number greater than 0, but smaller than any assignable number. These new foundations played no role in the developments we are considering. However, category theory has been proposed as a basis for reconstruction scientific theories that overcomes the syntactic/semantic polarity (Halvorson, 2016).

1.1. Nineteenth century physics Thanks to an army of historians of science the development of physics in the nineteenth and twentieth centuries has received detailed expositions. We will exploit this work by siphoning off examples of the way some leading physicists related math to physics in the three main branches of nineteenth century physics, mechanics, thermodynamics, and electrodynamics. A common problem they faced was one of discovering, or hypothesizing, relations between quantities including quantities characterizing theoretical entities. These relations were then given a mathematical formulation. The idealization of quantities as continuous justified the use of differentials Physicists had a critical awareness of the difference Fourier had introduced between a depth and a phe-nomenological account. Rankine attempted a depth account of thermodynamics by basing it on kinetic theory. Clausius, Thomson, Maxwell, and Helmholtz insisted on a phenomenological account, though they had made significant contributions to kinetic theory. A phenomenological account chiefly deals with empirical relations and low level theories. This introduces a new dimension in the physics/mathematics distinction, one of relating mathematical formalisms to physical reality as described, or as embodied in a model. Fortunately, Maxwell gave a detailed account of how a phenomenological account should be developed. In his presidential address to the Mathematical Physical Section of the British Association he considered different ways of relating mathematics to physics. (Niven, Vol. II, 215e227). At one extreme are the ‘mathematicians’ (i.e.,French physicists) who find satisfaction with pure quantities represented by symbols. At the other extreme are those who project their whole physical energy on scenes they conjure up. Maxwell advocated a mean between these extremes, geometrical reasoning about physical quantities. In his Theory of Heat (Maxwell 1972) developed the study of heat as a phenomenological science. Here he carefully develops a consistent phenomenological conceptualization. States of bodies are interpreted as varying continuously. Changes of state are described in terms of gross properties, rather than through molecular motions. The solid state can sustain a longitudinal pressure, while a fluid cannot. A gas is distinguished from a liquid by its ability to expand until it fills the available boundaries. Without introducing any molecular hypotheses, he expanded this phenomenological account to include the quantitative concepts characterizing thermal phenomena: temperature, heat, specific heat, latent heat, heat of fusion, heat of vaporization, and methods of measuring them. (Maxwell 1972, 16e31). To develop thermodynamics, which for Maxwell was the explanation of thermal phenomena through mechanical principles, it was first necessary to introduce mechanical concepts. These, as Maxwell saw it, have a different status. In mechanics, length, mass, and time are taken as fundamental. They supply the units in terms of which one can define other mechanical concepts. On this basis, Maxwell defines: density, specific gravity, uniform velocity,

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momentum, force, work, kinetic energy, and potential energy. (Maxwell 1972, 76e91). Thermal phenomena are related to mechanics on two distinct levels, that of empirical generalizations, and that of a causal account. On the first, or phenomenological level, the crucial law is energy conservation. This Maxwell interpreted as a mechanical, rather than a thermal, law, because it could be strictly established only for mechanical systems. On a functional level, thermodynamics relates to mechanics by the assumption that heat is a form of energy, and that energy can be expended to do work. Ideally, kinetic theory should supply a causal basis for thermodynamics, but this cannot be done because the forms that energy takes in molecular motion and structure is not adequately understood. (Maxwell 1972. chap. XXII). However, kinetic theory can be used to give a plausible account of changes of state and could ideally put entropy on a non-statistical basis. “If we conceive of a being, finite like us, but with faculties so sharpened that he can follow every molecule in its course, he would be able to do what is at present impossible to us.” (Maxwell 1972, 308. It was P. G. Tait who dubbed this being ‘Maxwell's demon’). Thus, Maxwell developed two distinct conceptualizations of reality. The mechanical conceptualization was considered basic, while the thermal conceptualization was regarded as phenomenological. Each supported distinctive mathematical formulations. The principle of energy conservation supplied a bridge be-tween the two. The thermal account supported treatments of temperature, heat, and energy as continuous. Though the mechanics of kinetic theory had not yet been adequately developed, it did not support treating these quantities as continuous. The need for mathematical rigor is more acute when we are dealing with high-level mathematically formulated theory. The supreme example of such a theory in nineteenth century physics is electrodynamics. The basic problem confronting Maxwell, his competitors, and successors was one of deciding which electromagnetic theory should be accepted. Since the competing theories were eventually given equivalent formulations the disputes centered on the physical significance accorded terms and functions. Maxwell accepted Faraday's experimental research and initially attempted to give it a mathematical formulation by relying on the method of physical analogies. This method, which Maxwell adapted from Thomson, was presented as a middle ground between an explanatory reliance on physical hypotheses, which might not be true, and a purely analytic approach focusing on the mathematical formalism and often slighting its physical interpretation. A reliance on physical analogies was, in Maxwell's view, particularly appropriate in electrical studies because of the lack of an adequate causal account. Hence, he attempted to adapt mathematical methods from fields with a structural similarity. (Niven, Vol. I, p. 156)4 In his 1856 paper, he interpreted Faraday's lines of force as fine tubes of variable cross-section carrying an incompressible fluid. (p. 155) This provided a basis for adapting the theory of an ideal incompressible fluid, with the attendant ideas of continuity, sources, sinks, variable flow, and resistance. Maxwell's hope was that the mathematical relations he established would suggest physical relations, which could then be tested experimentally. In his 1861 paper, he introduced the hypothesis of rotating molecular vortices and distortable idle wheels which transmit rotation from one vortex to another. (p. 451) This model supported a mathematical

4 More detailed accounts may be found in Heimann, 1970, Buchwald (1977a, 1977b) and Nersessian (1984, 2008). I will refer to Maxwell's papers by their page number in Niven and to his Treatise by paragraph numbers. (Niven, 1965, Maxwell, 1954 [1873].

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treatment of electrical current, displacement current, magnetic forces, electromotive forces, potential, and the hypothesis that light consists of transverse undulations of the ether. Yet, in Maxwell's opinion, this model was simply provisional. Maxwell's definitive 1864 paper (p. 526) changed this reliance on physical analogies in two distinct but related ways. First, instead of molecular vortices, he now relied on the general assumption of an ethereal medium filling space, permeating bodies, capable of being set in motion and of transmitting motion from one part to another, and also of communicating that motion to gross matter so as to heat it and affect it in various ways. (Niven, Vol. I. p. 258). The second novel feature is a reliance on dynamics, rather than mechanics. Since about 1838 British physicists had been using ‘dynamics’ for an explanation based on Lagrange's analytic mechanics, rather than on any particular mechanical model. (See Harman, 1982, pp. 25e27). Maxwell was acutely concerned with the question of which aspects of his work could be interpreted as true of reality and which merely reflected his reliance on models. He was strongly committed to field theory, rather than action at a distance, as the means by which force is communicated. This commitment entailed attaching a realistic physical significance to three basic concepts: ‘displacement’, ‘energy’, and ‘electromotive force’. All three, however, presented problems. Maxwell had inherited from Faraday, not merely the electromagnetic field, but also an underlying tension between explaining this field through lines of force and explaining it through polarization of a dielectric medium. In either case a field can transmit force only if something in the field moves or, in Maxwell's terminology, if there is displacement. In Maxwell's molecular vortex model the displacement is in a direction opposite to the electrical field; in the dynamical model it is in the same direction. Such difficulties notwithstanding, Maxwell was committed to displacement as the underlying physical reality, with charge and current as phenomenological manifestations of displacement. Maxwell always insisted that ’energy' be interpreted literally, since all energy is equivalent to mechanical energy which can be precisely specified. To defend field theory over distance theory it seemed necessary to insist that energy is stored in the field, rather than merely on the surface of conductors. However, as Maxwell admitted, he did not know how energy is stored in the field or how electromagnetic force is transmitted through the field. This is the background to the Treatise. This long confusing work is concerned with two basic difficulties. The first, and most perplexing, was that Maxwell did not have a causal account of electrical phenomena. From his mentor, William Whewell, he accepted the difference between a phenomenological account and a depth account in terms of true causes. In the Treatise he candidly admits that he does not know what electricity is (35), or the direction and velocity of current (570). His earlier account involved stress in the medium, but he does not know how stress originates (644), or what light really is (821). He had a phenomenological, rather than a depth account (574). The second difficulty concerned the competition between his medium account and the competing distance accounts, which had been updated to include the derivation of an electromagnetic theory of light. Could the empirical data decide which is correct? In two separate sections, he cites possible experiments that might support the medium over the distance account. With painstaking honesty Maxwell notes that there was no unambiguous empirical support for his predictions on the medium account. He reluctantly concluded that the two mutually incompatible theories were empirically equivalent, though he still preferred medium theory on intuitive grounds. (846e866). Further developments came from a fusion of elements in both traditions.

The Maxwellians eventually dropped the primacy Maxwell had accorded displacement and accepted Helmholtz's idea of atoms of electricity, aka electrons, as sources. (See Buchwald 1977a) When Helmholtz returned to the study of electrodynamics he presented a general formula, which he thought might supply a basis for experimental tests of the competing theories. Franz Neumann had derived all the electrodynamic effects for a closed electric circuit from a formula for the potential. Helmholtz generalized this potential formula to express the potential which a current element ds carrying current i exerts on a current element ds' carrying current j, where the two are separated by a distance r:

P ¼ A2ij ds$dst þ ð1  kÞðr$dsÞðr$ds Þ t

(1)

A is a constant with the value 1/c, where c is the velocity of light. The term, k, is a trial constant. If k ¼ 1 this formula reduces to Weber's potential. If k ¼ þ1 the formula reproduces Franz Neumann's potential; while k ¼ 0 reproduces Maxwell's potential. The parts of this expression multiplied by k become zero when integrated over a closed circuit. If experiments were to decide between the competing theories, Helmholtz concluded, it would be necessary to use open circuits as the testing ground, an extremely difficult process. The Maxwell limit, k ¼ 0, is unique in two respects. First, it allowed only transverse vibrations. Second, only the Maxwell limit yielded electromagnetic vibrations with a velocity equal to the velocity of light. The experiments that Helmholtz and his aides conducted in the 1870's generally had negative results. Helmholtz himself gradually became convinced that the Maxwell potential formula was correct. However, interpreting Maxwell's theory as a limiting case of Helmholtz's potential formula, as most Continental electricians did, led to serious conceptual difficulties. (Buchwald 1977a, 1977b, 177e193).
ge
, Heinrich Hertz, conducted a series of exHelmholtz's prote periments that were interpreted as supporting Maxwell's account of a dielectric medium supporting transverse vibrations. In 1885 he became a professor at Karlsruhe and in 1888 began his epochal researches. In experiments on the propagation of electromagnetic radiation in air he showed that these electromagnetic waves have a finite velocity and that they can be refracted, polarized, diffracted, reflected, and produce interference effects. This undercut distance theories and precipitated a consensus within the European physics community of the correctness of Maxwell's idea of a dielectric medium transmitting electromagnetic vibrations. However, this did not entail accepting Maxwell's theory. Hertz's diligent study of Maxwell's Treatise led him to conclude that he did not know what Maxwell's theory was. (Hertz, 1962 [1892], 20). He concluded “Maxwell's theory is Maxwell's system of equations”. (Hertz, 1962, p. 21). The transformation of Maxwell's mixture of a depth level account, largely unsuccessful, and a phenomenological account coupled to a mathematical formulation into a hypotheticaldeductive system (Hertz, 1962, 138) undercut most of the proposed ontological props. The electrical charge, e, was determined by conservation laws and measurements, rather than particle assumptions. The displacement lost its mechanical significance. The specific dielectric capacity, ε, and the specific magnetic capacity, m, were reinterpreted as characterizing properties of material bodies, rather than the medium. From beginning to end the development of electrodynamics was conditioned by a dialectic between physical concepts and the mathematical expressions. The distinctive new concepts, charge, current, displacement, electrical fields, magnetic fields, vibrations in the ether, and the functional relations introduced determined the mathematical expressions that were used or developed. The

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novel factor was the phenomenological/depth distinction. The development of thermodynamics as a phenomenological science and the grudging recognition of electrodynamics as a phenomenological science implied that mathematical expressions related to physical reality as described or modeled. This left open the question of the relation between these descriptions and models and the reality described. This problem took on increased significance with the development of quantum physics. 2. Quantum physics Quantum physics from its inception was embroiled in a struggle to relate mathematical formulations to physical concepts. 5 Planck had derived the Wien black-body radiation law by developing a model of black bodies as a collection of oscillators and deriving a formula for the entropy of an individual oscillator, v2 S=vU 2 ¼ a=U, where S is entropy and U energy. When measurements of black body radiation did not fit Wien's law Planck tried modifications of his derivation. In October 1900 he proposed the formula. v2 S=vU 2 ¼ a=Uðb þ UÞ where a and b are constants. This led to the Planck radiation formula that fit all the available experimental measurements. Planck then began what he called “an intensive investigation of the true meaning” of his new formula. He did not advocate the idea that radiation is quantized. Einstein's explanation of the photoelectric effect introduced the idea of radiation quantized in bundles, ε ¼ hn, but based his derivation of Wien's, rather than Planck's, radiation law. €dinger Inspired guesses by de Broglie, Heisenberg, and Schro launched quantum mechanics in the mid-1920s. They involved three mutually incompatible physical accounts: de Broglie's matter €dinger's waves; Heisenberg's reliance on observables; and Schro €dinger demonstrated the mathetentative wave hypothesis. Schro matical equivalence of the wave and matrix formulations. He rejected Born's probability interpretation of his wave function. Dirac's transformation theory provided a more general formalism with wave and matrix mechanics as special cases. John von Neumann's celebrated treatise put the formulation of QM on a rigorous basis through a Hilbert-space formulation. He thought this was necessary because “The methods of Dirac · · · in no way satisfies the requirements of mathematical rigor.” (von Neumann 1955 [1932], p. viii). The new quantum mechanics (QM) proved successful in resolving the chief difficulties encountered in the final stages of the older semi-classical quantum theory and in solving new problems. There was a widespread realization that the new QM rested on shaky foundations. In keeping with the theme of this article we will focus on relatively a posteriori approaches to interpreting quantum mechanics.6 Here Bohr has a unique position, one that is routinely misinterpreted by associating Bohr with the Copenhagen interpretation of quantum mechanics.7 We will present Bohr's position in its starkest form without supporting arguments. Quantum physics, he insisted, should be interpreted as a rational generalization of classical physics. The mathematical formalism is an inferential tool, not a theory to be interpreted. “Its physical content is exhausted by its power to formulate statistical laws governing observations obtained under conditions specified in plain language.” (Bohr 1963, p.

5 More detailed accounts of these developments may be found in: Jammer, 1966; MacKinnon 1982a; and in Mehra & Rechenberg, 1982. 6 An historical survey of interpretations of quantum mechanics is given in MacKinnon 2016. A more systematic development is given in Ruetsche, 2011. 7 Interpretations that distinguish Bohr from Copenhagen are given in Honner, 1987 and Gomatam, 2007.

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12) An assistant (Petersen 1963) cites Bohr's claim: “We are suspended in language. Apart from language we do not even know up and down.” His ‘plain language’ is ordinary language extended to include the basic concepts and laws of classical physics.8 Developments in the mid-1920s had led to a crisis. The common view of the crisis was that different experiments presupposed contradictory properties in the objects treated: electrons as particles and waves; radiation as continuous and discrete; atomic electrons as traveling, and not traveling, in elliptical orbits. Bohr concluded that these difficulties showed the essential failure of the pictures in space and time on which the description of natural phenomena have hitherto been based. What he introduced was something akin to a Gestalt shift from the objects studied to the linguistic and conceptual system used to describe experiments, whether real or thought, and report results. Each such account was regarded as an epistemology irreducible unit. Thus, in the idealized double-slit experiment one cannot properly ask which slit the electron went through. The only meaningful way to ask such a question is to close one slit. Then one has a different experiment. In an experimental setup in which the ‘particle’ cluster of concepts is used one can rely on the standard material inferences these concepts support, e.g., retrodicting an electron's trajectory. The guidelines for extending classical concepts and restricting their usage in quantum contexts carried over to the formalism. One replaced classical concepts, like ‘position’, ‘momentum’ and ‘energy’ by the now familiar mathematical operators, solved the resulting equation, and interpreted the conclusion as possible experimental results. Bohr's closest associates at this time shared the view that quantum mechanics should be interpreted as a rational generalization of classical physics. Heisenberg claimed “… The Copenhagen interpretation regards things and processes which are describable in terms of classical concepts, i.e., the actual, as the foundation of any physical interpretation.” (Heisenberg, 1958, p. 145). Pauli contrasted Reichenbach's attempt to formulate quantum mechanics as an axiomatic theory with his own position: “Quantum mechanics is a much less radical procedure. It can be considered the minimal generalization of the classical theory which is necessary to reach self-consistent description of microphenomena, in which the finiteness of the quantum of action is essential.” (Pauli, 1964, p. 1404) Subsequently, philosophers interpreted Bohr's scattered epistemological pronouncements as comments on the interpretation of quantum mechanics as a mathematically formulated theory. In this context, Bohr's measurement-based position seemed outdated, incoherent, and even bizarre. For present purposes, it is important to see how far this physics-first approach can be extended in the development of quantum physics. This is important not merely because it exemplifies an a posteriori approach to the mathematics used but also for Ockhamist reasons. If an interpretation of the mathematical formalism as an operational tool suffices then it is difficult to justify any more realistic interpretation. We will do this by focusing on two physicists who implemented Bohr's position that the physics generates the math. The Dirac formalism expressing states of quantum systems by the directions of vectors in an abstract space came into general use in physics only after Messiah's well known textbook put the Dirac formalism on an acceptable mathematical formulation. (Messiah, 1964). This reformulation omitted Dirac's argument justifying the formalism. A summary of Dirac's argument highlights the difference. Dirac justifies the representation of states by vectors through

8 A detailed account of this extended ordinary language is given in MacKinnon 2011, chap. 5.

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an analysis of measurements. A simplified recasting of his argument highlights the problematic features. Consider a beam of light consisting of a single photon plane-polarized at an oblique angle relative to the optic axis of a tourmaline crystal. Either the whole photon passes, in which case it is observed to be polarized perpendicular to the optic axis, or nothing goes through. The initial oblique polarization, accordingly, must be considered a superposition of states of parallel and perpendicular polarization. Again, consider another single-photon beam passed through an interferometer so that it gets split into two components that subsequently interfere. Prior to the interference, the photon must be considered to be in a translational state, which is a superposition of the translational states associated with the two components. Since particle states obey a superposition principle, they should be represented by mathematical quantities that also obey a superposition principle, vectors. The physics generates the mathematics (Dirac, 1958, pp. 4e14). The experiments cited had never been performed. What Dirac actually did was to consider familiar phenomena of polarization and interference, translate them from wave to particle language, and then extrapolate from large-scale statistics involving many photons back to the behavior of an individual photon. This is an experimental basis in a Bohrian sense. It highlights the conceptual inadequacy of classical concepts in describing paired experimental results. Here the pairing comes from a state preparation followed by a measurement. Polarization or interference of individual photons had never been observed. Like Bohr, Dirac treats the mathematical formulation as a tool, and often justifies mathematical formulations by physical considerations. The first edition of his Principles relied on his transformation theory. In the second edition, he introduced vectors “ in a suitable vector space with a sufficiently large number of dimensions.”. In the third edition, he introduced his bra-ket notation and simply postulated a conjugate imaginary space with the needed properties. He assumed that the vector space he postulated must be more general than a Hilbert space, because it includes continuous vectors that cannot be normalized (Dirac, 1958, pp. 40, 48). He only spoke of the Hilbert-space formulation of quantum mechanics when he became convinced that it should be abandoned (Dirac, 1964). Messiah, like von Neumann, developed a statistical interpretation of QM and did not apply the superposition principle to individual systems. In this context, the Dirac argument from physical superposition of states of an individual system to a mathematical representation that also obeys a superposition principle has no foundation. Physicists generally learned the Dirac formulation through Messiah's elegant mathematical presentation and then failed to realize that Dirac's presentation represented better physics. The application of the superposition principle to individual states proved indispensable in particle physics and supplies a foundational principle for quantum computers and quantum entanglement. Julian Schwinger developed QM on a strict measurement basis and extended his strict measurement methodology to quantum electrodynamics and QFT. (Schwinger, 1959) Though his contributions are well known, his methodology of developing QFT has never become the mainstream method. It is far easier to develop QFT by begin-ning with classical or pseudo-classical Lagrangians and quantization procedures than to rely on a systematic following of Schwinger's variational methods. However, a systematic following of this strict methodology supplies an apt tool for clarifying the limits of valid applicability of a measurement-based interpretation of QM. Schwinger described his early student years as “unknown to him, a student of Dirac's” (Schweber, 1994, p. 278). Before beginning his freshman year at C$C$N.Y. he had studied Dirac's Principles

and, at age 16, wrote his first paper, never published, “On the Interaction of Several Electrons”, generalizing the Dirac- FockPodolsky many-time formulation of quantum electrodynamics. Schwinger repeatedly claimed that he was a self-taught physicist. That he could write such a paper before beginning college coupled to his notorious practice of skipping classes in college bears out this appraisal. The primary influence on his early education as a quantum physicist was Dirac’s Principles. He also seems to have been influenced by Bohr's accounts of the significance of measurement and the doctrine of complementarity. Schwinger's systematic redevelopment of the Dirac formulation was presented in a series of lectures, some of which were privately circulated.9 Schwinger explicitly puts QM on an epistemological basis: “Quantum mechanics is a symbolic expression of the laws of microscopic measurement.” (Schwinger 1970, 1). Accordingly, he begins with the distinctive features capturing these measurements. This, for Schwinger, is the fact that successive measurements can yield incompatible results. Since state preparations also capture this feature Schwinger actually uses state preparations, rather than complete measurements as his starting point. (See MacKinnon, 2007) He begins by symbolizing a measurement, M, of a quantity, A, as an operation that sorts an ensemble into sub-ensembles characterized by their A values, M(ai) The paradigm case is a Stern-Gerlach filter sorting a beam of atoms into two or more beams. This is a type one measurement. An immediate repetition would yield the same results. There is no reduction of the wave packet or recording of numerical results. An idealization of successive measurements is used to characterize the distinguishing feature of these microscopic measurements. Symbolically


0 00 
0 
0 
00  M a M a ¼ d a ;a M a This 0

Mða Þ ¼

can Yk

be

expanded

into a

(2) complete

measurement,

0

Mðai Þ where ai stands for a complete set of compati¼1

ible physical quantities. Using A, B, C and D for complete sets of compatible quantities, a more general compound measurement is one in which systems are accepted only in the state B ¼ bi and emerge in the state, A ¼ ai, e.g., an S-G filter that only accepts atoms with sz ¼ þ1 and only emits atoms with sx ¼ þ1. This is symbolized M(ai, bi). If this is followed by another compound measurement M(ci, di), the net result is equivalent to an overall measurement that only accepts systems in state di and emits systems in state ai. Symbolically,

Mðai ; bi ÞMðci ; di Þ ¼ < bi jci > Mðai ; di Þ:

(3)

For this to be interpreted as a measurement < bijci > must be a number characterizing systems with C ¼ ci that are accepted as having B ¼ bi. The totality of such numbers, < atjbt > , is called the transformation function, relating a description of a system in terms of the complete set of compatible physical quantities, B, to a description in terms of the complete compatible set, A. In the edition of Dirac's Principles that Schwinger studied, the transformation function was basic. A little manipulation reveals that N, the total number of states in a complete measurement, is independent of the choice of complete physical quantities. For N states the measurement symbols form an algebra of dimensionality N.2 These measurement operators form a set that is linear, associative, and noncommutative under multiplication.

9 His 1970 is a slightly revised version of his 1952 and 1959 lectures. See Gottfried (1966, pp. 192e213) for a summary and Mehra et al., 2000, pp. 344-355.

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E. MacKinnon / Studies in History and Philosophy of Modern Physics xxx (2017) 1e10

To get a physical interpretation of this algebra consider the sequence of selective measurements M(b')M(a')M(b'). This differs from a simple or repeated measurement M(bt) in virtue of the disturbance produced by the intermediate M(at) measurement. This suggests M(b')M(a')M(b') ¼ p(a‘, b')M(b), where. p(a', b') ¼ < a'jbt’> < b'ja' > . Since this is invariant under the transformation, < a'jb' > / l (a') < a'jbt’ > l (b1), where l (a'), l (b') are arbitrary numbers, Schwinger argues that only the product, p(a', P b') should be accorded physical significance. Using a' p(a', b')¼1 Schwinger interprets this as a probability and imposes the restriction,

 0  0 0 0 < b at > ¼ < a bt > *

(4)

The use of complex numbers in the measurement algebra implies the existence of a dual algebra in which all numbers are replaced by complex conjugate numbers. This algebra of measurement operators can be expanded into a geometry of states. Introduce the fictional null (or vacuum) state, 0, and then expand M(a', b') as a product, M(a' 0)M(0, b'). Let M(0, b') ¼ F(b'), the annihilation of a system in state b', and M(a', 0) ¼ J(a'), the creation of a system in state a'. These play the role of the state vectors, F(b') ¼ < bt\'j and J(at) ¼ jat > . With the convenient fiction that every Hermitian operator symbolizes a property and every unit vector a state one can calculate standard expectation values. Schwinger follows Dirac's precedent of not assuming Hilbert space, but simply constructing a functional space on the basis of physical arguments. This practice can generate the same type of criticisms concerning rigor that the Dirac formulation encountered. Thus, Eq. (4) introduces the complex numbers needed for a complex vector space. Dirac had introduced a complex vector space on physical grounds. The direction of a vector represents the state of a system. The state of a photon includes polarization, best represented by a phase in a complex space. In Schwinger's reformulation, measurements and transformations are basic, rather than states and vectors. In this context, Eq. (4) seems to lack physical justification, since measurements only yield real numbers. Gottfried (1966, p. 201) justifies the use of a complex space by appealing to the wisdom of hindsight. This difficulty supplied the point of departure for a series of papers in which Accardi sought to put this algebra on a more secure basis (Accardi, 1995). The work of Messiah and Accardi illustrates a familiar pattern. Physicists make a breakthrough, often using physical arguments to justify mathematical formulations. A more rigorous reformulation is part of a subsequent mopping up operation. In such a reformulation, mathematical formulations are taken as basic and physical concepts and models are adjusted to fit. This dialectic between physical concepts and mathematical formulations plays a stronger role in the development of quantum mechanics than in other fields. This is primarily because the basic physical concepts used stem from classical physics and no classical model can fit all quantum phenomena. (MacKinnon, 2016) The development of quantum electrodynamics (QED) fits this general scenario. QED relies on series expansions in powers of a, the fine structure constant, where a z 1/137. As the expansion is carried out to order n, the number of terms (or Feynman diagrams) is of order n!. This is not a convergent series. Physicists tame this mathematical difficulty by physical considerations. The higher order terms are identified as physical processes, usually virtual processes such as an electron emitting and then absorbing a photon or a photon emitting and then absorbing an electron-positron pair. These processes involve higher energies, or smaller distances. QED is an Effective Field Theory (EFT) valid only for a limited range of energies. The higher order terms should be handled by a new higher energy EFT. Their contribution to the

7

observed results at QED energies can be accommodated by a suitable redefinition of coupling constants. The weak infinities are illustrated by expression of the form a5log(b/a). This diverges in the limit a / 0. This limit fits the assumption of point particles. Excluding higher energy terms effectively excludes a reliance on the point mass particles that produce the weak infinities. The limits of Schwinger's a posteriori methodology became manifest in the development of the standard model of particle physics. Schwinger rejected Gell- Mann's quark hypothesis (Schwinger, 1964). When this hypothesis proved successful Schwinger effectively dropped out of quantum field theory. The standard model of particle physics is, arguably, the most successful theory in the history of physics. It covers all basic interactions: hadron-hadron; hadron-lepton; and lepton-lepton. These ground atomic and nuclear physics. It has been well confirmed by extensive experimental results. Yet, there is a widespread agreement that this theory has serious defects and should eventually be replaced. The standard model is not a properly unified theory. It patches together strong, electromagnetic, and weak interactions. It relies on non-convergent series. It also relies on 19 parameters, like coupling constants, whose values are set by experimental data rather than by the theory. Diverse attempts to improve this situation reflect the polarity we have been considering. The pole emphasizing the need for rigorous mathematics includes the developers of algebraic, axiomatic, and local quantum field theory. On the assumption that algebraic is the most basic we will label this AQFT (Ruetsche, 2002).10 The available formulations of AQFT have sharp physical limitations. They treat single particle states, interactions by perturbation theory, and collisions by Smatrix theory. They do not include the particle interactions basic to the standard model. The hope is that further developments of AQFT will include, and even go beyond, this physics. The notion of Effective Field Theories (EFT) lies closer to the opposite, or a posteriori, pole. An EFT is a functioning theory presumed to be valid only within a limited energy range. The key idea of an EFT is to separate low-energy, relatively long-range interactions, from high-energy, relatively short-range interactions. EFT calculations eliminate high-energy contributions and then compensate for the effect of the elimination.11 Renormalization methods clarify the nature of this compensation. The high-energy interactions have the effect of modifying the coupling constants. The low-energy calculations effectively include this by relying on experimental values for these constants. This general methodology has been extended to Effective Theories (ETs) that are not field theories (Rohrlich, 2001; Kane, 2000; chap. 3). In Rohrlich's account, which we will adapt later, mature theories have complex structures involving: central terms, a semantics, a mathematical component, and an ontology specific to the domain of a theory's valid applicability. Altmanspacher and Primas (2002) develop a related notion of ’relative onticity’. In both cases the relative ontology proper to an energy level is not presented as an account of what reality is objectively. The goal or reducing relative ontologies to an ultimate ontology is either ignored or postponed. The more immediate problem is one of finding a bridge between two relative ontologies, e.g., gravitational

10 Ruetsche, 2011 presents an excellent survey. Fraser, 2009 argues that a rigorous axiomatic formulation should supply the basis for an interpretation. MacKinnon, 2008a, 2008b disputes these claims. 11 Georgi, 1993, Manohar, 1996, and Kaplan, 2005 present general accounts of EFTs. The philosophical significance of EFTs has been treated in Hartmann 2001, and Castellani, 2002.

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force in Newtonian mechanics and curved space in General Relativity. This problematic can be illustrated by the history of theories of light. A Kuhnian scenario suggests a sequence of paradigm replacements: Newton's corpuscular theory; a wave theory; the semi-classical treatment of radiation in non-relativistic quantum mechanics; and quantum electrodynamics. One difficulty this scenario presents is that the ‘replaced’ theories are still functioning. Geometric optics, treating light as rays that travel in straight lines, is used in the design of cameras, microscopes, telescopes, and eyeglasses. The wave theory is used in diffraction gratings, and interferometers, as well as in explaining chromatic aberration. The concept of light as photons is used in explaining laser beams and masers. In each case the physical conceptualization of light as rays, waves, or photons grounds the mathematics used.

3. Conclusion We can move the problematic we have been treating into a larger perspective. The most basic question philosophers of physics treat is: What is the relation between physics, mathematics and reality? We can simplify this by beginning with polar extremes and working towards a middle position. Adapting Ruetsche's terminology we will label the polar extremes: Mathematical Imperialism and Calculational Pragmatism. We will divide the Mathematical Imperialists into two groups, the purists and the theorists. The leading purists are Plato, in ancient times, and Max Tegmark in modern times. Plato uses the analogy of a line divided into two major unequal segments, the sensible and the intelligible, with each of these divided into two sections. (Republic 509). The ascent to pure truth begins in the bottom half of the sensible segment with shadows and images, proceeds to the things imaged, people, animals, man-made objects. The lower part of the intelligible segment is concerned with mathematical forms and assumptions that lead to conclusions. The uppermost section passes from assumptions to the basic principles, the unchanging eternal forms of the Good, the Beautiful, and the True, and does so with no reliance on images. The objects of the lower levels receive from the Good not only their knowability, but also their being and essence. Max Tegmark, a theoretical cosmologist, argues that all science can be considered a mixture of mathematics and baggage, where the baggage is the concepts we impose on mathematical forms (Tegmark, 2014). The lower sciences, concerned with people, animals and man-made objects, are chiefly baggage with only a functional use of mathematics. As we ascend through the sciences to theoretical physics the amount of baggage decreases and the explanatory role of mathematics increases. An extrapolation suggests that the ultimate Theory of Everything will have only mathematics with no baggage. This, in turn, suggests the basis thesis of his book. External physical reality is a mathematical structure, an abstract immutable entity existing outside of space and time. The theorists, in our simple systematization, focus on individual mathematically formulated theories. A theory is regarded as a mathematical formulation that can support one or more physical interpretations. This requires a mathematical formalism that is consistent independent of a physical interpretation. Functioning physical theories do not meet this requirement. So, what are interpreted are idealized reconstructions. These reconstructed theories are interpreted by holding them up to reality. What must the world be like if this theory is true of it? A positive answer yields an ontology of reality. Skeptical evaluations of such realist answers suggest a retreat to an epistemological position. A theory should be evaluated for its empirical adequacy, not its truth.

Calculational pragmatists can also be divided into purists and theorists. A pure calculational pragmatist has a standard response to metaphysical and epistemological questions about the significance of scientific theories. Shut up and calculate! A theoretical calculational pragmatist is concerned with the interpretation of theories. However, she insists, theories are not interpreted by holding them up to reality. They are evaluated by checking theoretical predictions against experimental results as reported. This supports a piecemeal progressive interpretation of individual theories, but not an interpretation of the relation between physics, mathematics and reality. The discussion of Effective Theories suggests a way of exploring the territory between these two polar extremes. A functional theory includes a conceptualization of some domain of reality supporting a mathematical formulation. We are using ‘conceptualization’ rather than ‘relative ontology’ or ‘model’ to highlight a key feature. A conceptualization supports material inferences. A formal inference, e.g. All A is B and all B is C. So, all A is C, does not depend on the meaning of the terms. A material inference, which should not be confused with material implication (p I q), does depend on the meaning of concepts and is often expressed as an enthymeme. This dog is a pit bull. So, it is dangerous. A conceptualization of some domain relies relies on a categorial system classifying the members of the domain and specifying their basic properties and relations. When such a conceptualization supports a mathematical formulation, then the introduction or justification of a mathematical formula may depend on physical reasons. Light travels in rays. So, when it is reflected the angle of incidence equals the angle of reflection. Light travels in waves. So, interference patterns should be explained by a mathematical formulation of superposition. In a pure mathematical perspective such mixed reasoning may look like sloppy math. In teaching and using physics this represents a normal practice. Maxwell's treatment of heat, previously considered, explicitly makes use of two different conceptualizations of the same domain. The phenomenological conceptualization relies on gross properties of bodies that vary continuously. All the basic concepts are defined in terms of continuous properties of bodies. The kinetic conceptualization is based on the mechanics of molecular motion. The mechanical conceptualization was considered basic, while the thermal conceptualization was regarded as phenomenological. Each supported distinctive mathematical formulations. The principle of energy conservation supplied a bridge between the two. The thermal account supported treatments of temperature, heat, and energy as continuous. Though the mechanics of kinetic theory had not yet been adequately developed, it did not support treating these quantities as continuous. Our earlier references to Maxwell's Treatise pointed out that he was developing electromagnetism as a phenomenological science because he lacked a knowledge of what electricity and light really are. The Treatise is concerned with a detailed comparison of two different conceptualizations, the distance theory developed by Continental physicists, and Maxwell's field theory. These competing conceptualizations support different mathematical treatments.12 The method appropriate to distance theory is to begin with the part and work towards the whole. This method is grounded in the concept of a particle. Integrating over a volume is equivalent to adding the effect of the charges contained in the volume. But, Maxwell notes, “To conceive of a particle, however, requires a process of abstraction, since all our perceptions are related to extended bodies, so that the idea of the all that is in our

12 This is developed in each of the four sections of the Treatise. A summary is given on pp. 528-530.

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consciousness at a given instant is perhaps as primitive an idea as that of an individual thing.”(529) The medium theorist conceptually begins with the whole and works towards the parts, which are treated as continuous, rather than discrete. The appropriate mathematical method is one of partial differential equations coupled to integrations over all of space, rather than just the space containing charged bodies. Thus, the two methods are based on idealizations of different fundamental features of the macroscopic concept of matter: particle vs. continuous distribution. The later electromagnetism that merged aspects of both approaches undercut the significance of Maxwell's contrasting analyses. This, coupled to the difficulties involved in reading the Treatise led to a systematic neglect of Maxwell's analysis of the relation between mathematics and physics. The analogy between Maxwell's analysis and the interpretation of QFT and the Standard Model is striking. Instead of relating a phenomenological and a to-be-developed depth model of the same phenomena, quantum physicists relate an effective theory to a to-bedeveloped grand unifying theory. The experimental practice of QFT relies almost exclusively on the concept of localized particles. This is inescapable, because particle accelerators and detectors supply the localization. Idealized formulations of algebraic and axiomatic field theory do not support the idea of localized particles. This results in two competing conceptualizations of the same domain, supporting different mathematical formulations. The Standard Model relies on a conceptualization that is not regarded as ultimate. Can AQFT go beyond this in supplying an ontology of reality? When a functioning physical theory is interpreted as a systematization on a phenomenological level, or as an effective theory, then there is no sharp separation between physical and mathematical reasoning. A physical conceptualization supports the mathematical formulation and often supplies justifications for particular mathematical inferences. Such a mixture is especially operative at the formative stage when a physicist is seeking a mathematical form that fits his or her intuition. Inferences from physical considerations to mathematical formulations, that play no role in reconstructed theories, often looks like a reasonable inferences in a practice-centered perspective. Such reflections raise the further question. Which perspective is basic? An answer that reflects a received tradition in philosophy is that the practice-centered perspective is basis when one is doing physics. But if one is making physical theories objects of study then the math-centered perspective supplies a basis for answering ontological and epsistemological questions about the theory. The material presented here does not support this division of labor. Effective theories are grounded in a conceptualization of some particular domain or of some energy range. The mixture of conceptual and mathematical inferences it supports is tested and revised to fit reality as reported in experimental data. What this yields is not an ontology of basic reality, but a revised conceptualization that is functionally adequate. The mathematical formalism employed may be abstracted and adapted to other purposes, Thomson and Maxwell adapted the mathematical formalism of fluid dynamics to represent electromagnetic fields. Dirac adapted the mathematical formulation of simple harmonic oscillators to treat the emission and absorption of photons. The mathematical formalism may also be made into an object of study. If this formalism is given a rigorous formulation, as an axiomatic system, then the object of study is clearly delineated. It is a set of axioms and whatever may be deduced from the axioms by the allowed rules of inference. Even without an axiomatic formulation a formalism, regarded as an abstract structure, may be interpreted through models. Such a rigorous reformulation can play an important role in interpreting physical theories. It prunes the functioning theory by revealing which features are essential and

9

which are contingent or accidental. Can an interpretation of a rigorously reformulated theory go beyond the conceptualization of the functioning theory and yield an ontology of reality? The answer to this question depends on the theory treated. If a theory is accepted as fundamental and true, then it is reasonable to ask what kind of reality it is true of. The Standard Model is the most fundamental theory presently available. Its empirical support is extensive and precise. Yet, this is widely regarded as an effective theory valid only for a limited energy range. The ongoing attempts to go beyond the standard model might eventually lead to a more fundamental theory. String theory is the leading candidate for such a successor theory. This theory, however, lacks empirical support. It may well be that the distinguishing features of quantum mechanics: superposition, interference, distributed probability, and entanglement, may also be features of an ultimate theory. Anticipations of distinguishing features, however, do not a theory make. Absent a grand unifying theory, it is reasonable to assume that physics will continue to advance through effective theories. These will undoubtedly include a posteriori mathematics.

References Accardi, L. (1995). Can mathematics help solving the interpretational problems of quantum mechanics? Il Nuovo Cimento, 110B, 685e721. Altmanspacher, H., & Primas, H. (2002). Epistemic and ontic quantum realities. Bell, J. L. (1998). A primer of infinitesimal analysis. Cambridge: Cambridge University Press. Bohr, N. (1963). Essays 1958-1962 on atomic physics and human knowledge. New York: Wiley. Buchwald, J. Z. (1977a). William Thomson and the mathematization of Faraday's electrostatics. His- Torical Studies in the Physical Sciences, 8, 101e136. Buchwald, J. Z. (1977b). William Thomson and the mathematization of Faraday's electrostatics. His- Torical Studies in the Physical Sciences, 8, 101e136. Castellani, E. (2002). Reductionism, emergence, and effective field theories. Studies in History and Philosophy of Modern Physics B, 33, 251e267. Dirac, P. (1958). The principles of quantum mechanics, 4th ed. Oxford: Clarendon Press. Dirac, P. (1964). Foundations of quantum theory. Lecture at Yeshiva University. Fraser, D. (2009). 32766. The problem of theory choice for the interpretation of quantum field theory. In C. Bicchieri, & J. McK. Alexander (Eds.), Proceedings of the 2006 philosophy of science associatiossn meeting. East Lansing, Michigan: Philosophy of Science Association. ProCite field[8]: eds. Georgi, H. (1993). Effective field theory. Ann. Rev. Nuclear Physics, 43, 209e252. Gomatam, R. (2007). Bohr's interpretation and the copenhagen interpretationeare the two incom- patible? (pp. 736e748) Chicago: The University of Chicago Press. Gottfried, K. (1966). Quantum mechanics. Volume I: Fundamentals. New York: W. A. Benjamin. Granville, W., Smith, P., & Longley, W. (1962). Elements of the differential and integral calculus. New York: John Wiley. Grattan-Guinness, I. (1980). From the calculus to set theory, 1630-1910. London: Duckworth. Halvorson, H. (2016). Categories of scientific theories. PhilSciArchives. #11923. Hartmann, S. (2001). Effective field theories, reductionism and scientific explanation. Studies in History and Philosophy of Modern Physics, 32B, 267e304. Harman, P. M. (1982). Energy, force, and Matter: The conceptual development of nineteenth-. Cam- bridge: Cambridge University Press. Heimann, P. M. (1970). Maxwell and the modes of consistent representation. Archive for History of Exact Sciences, 6, 170e213. Heisenberg, W. (1958). Physics and Philosophy: The revolution in modern science. New York: Harpers. Hertz, H. (1962). [1892]. Electric waves. New York: Dover Reprint. Honner, J. (1987). The description of Nature: Niels Bohr and the philosophy of quantum physics. Oxford: Clarendon Press. Jammer, M. (1966). The conceptual development of quantum mechanics. New York: McGraw-Hill. Kane, G. (2000). Supersymmetry. Cambridge, Mass: Perseus Publishing. Kaplan, D. (2005). Five lectures on effective field theory. arXiv:nucl-th/0510023. MacKinnon, E. (1982a). Scientific explanation and atomic physics. Chicago: University of Chicago Press. MacKinnon, E. (2007). Schwinger and the ontology of quantum field theory. Foundations of Science, 12, 295e323. MacKinnon, E. (2008a). The standard model as a philosophical challenge. Philosophy of Science, 75, 447e457. MacKinnon, E. (2008b). Why interpret quantum physics? Open Journal of Philosophy, 6. MacKinnon, E. (2012). Interpreting Physics: Language and the classical/quantum divide. Dordrecht: Springer.

Please cite this article in press as: MacKinnon, E., The role of a posteriori mathematics in physics, Studies in History and Philosophy of Modern Physics (2017), http://dx.doi.org/10.1016/j.shpsb.2017.09.002

10

E. MacKinnon / Studies in History and Philosophy of Modern Physics xxx (2017) 1e10

MacKinnon, E. (2016). Why interpret quantum physics? Open Journal of Philosophy, 6. Maxwell, J. C. (1972). Theory of heat. London: Longmans, Green and Co. Maxwell, J. C. (1954). A treatise on electricity and magnetism. New York: Dover reprint. Mehra, J., & Rechenberg, H. (1982). The quantum theory of Planck, Einstein, Bohr, and Sommerfeld: Its foundation and the rise of its difficulties, 1900-1925. New York: Springer. Mehra, J., Rechenberg, H., Milton, K. A., & Schwinger, J. (2000). Climbing the Mountain: The scientific biography of Julian Schwinger. New York: Oxford University Press. Messiah, A. (1964). Quantum mechanics (Vol. I). Amsterdam: North Holland. Nersessian, N. (1984). Faraday to Einstein: Constructing meaning in scientific theories. Dor- drecht: Martinus Nijhoff. Nersessian, N. (2008). Creating scientific concepts. Cambridge: The MIT Press. von Neumann, J. (1955). Mathematical foundations of quantum mechanics, translated by robert T. Beyer. Princeton: Princeton University Press. Newton, I. (1952a). Mathematical principles of natural philosophy. Chicago: Encyclopedia Britannica, Inc. Niven, W. D. (1965). The scientific papers of James clerk Maxwell. New York: Dover.

Petersen, A. (1963). The philosophy of Niels Bohr. Bulletin of the Atomic Scientists, 19, 8e14. Pauli, W. (1964). Collected scientific papers. New York: Interscience. Rohrlich, F. (2001). Cognitive scientific realism. Philosophy of Science, 68, 185e202. Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland Publishing Company. Ruetsche, L. (2002). Interpreting quantum field theory. Philosophy of Science, 69, 348e378. Ruetsche, L. (2011). Interpreting quantum theories. Oxford: Oxford University Press. Schweber, S. S. (1994). QED and the men who made it. Princeton: Princeton University Press. Schwinger, J. (1959). The algebra of microscopic measurement. Proc. Natl. Acad. Sci. U.S.A., 45, 1542. Schwinger, J. (1964). Field theory of matter. II. Physical Review, 136, B1821eB1824. Schwinger, J. (1970). Particles, sources, and fields. Reading, Mass: Addison-Wesley. Tegmark, M. (2014). Our mathematical universe :My quest for the ultimate nature of reality. New York: Alfred A. Knopf. Whiteside, Derek T., The Mathematical Works of Isaac Newton, Vol. 1. New York: Johnson Reprint Co.

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