PROF. J. K. ROBERTSON, professor of physics, Queen's University, Kingston, Ontario, in his presidential address to the Royal Society of Canada de-livered on May 21, 1945, argues that the notions of continuity and discontinuity as used in physics and other sciences are complementary and not just contradictory. When, in the eighteenth century, physical theory was occupied with the continuous aspect of things it was natural to suppose that the structure of the world must be either continuous or discontinuous, not both. Therefore, since continuity was apparent, there could be no discontinuity anywhere, as Leibniz argued. The chemical atomic theory and the kinetic theory of gases of the nineteenth century brought out the aspect of discontinuity for the first time, and recent sub-atomic physics has gone a stage further. At every stage, however, as Prof. Robertson emphasizes, theory really requires both continuity and discontinuity. This was first realized in the case of light, which in some aspects is continuous, and in others discontinuous ; and, later, electrons were found to play a similar dual part. Granted that the single electron is discrete, indefinite and indeterminate, it should not be inferred that the continuity, definiteness and strict causality found in the behaviour of large numbers of electrons are not equally genuine aspects of the physical world. Prof. Robertson refers briefly to the way in which the biological theory of evolution requires both a continuous long-term aspect and a discontinuous short-term one ; and he ends by asking the question, whether new discoveries in science which look like discontinuities in the historical development of thought are not also in other respects examples of the continuity of thought.
PROF. J. K. ROBERTSON, professor of physics, Queen's University, Kingston, Ontario, in his presidential address to the Royal Society of Canada de-livered on May 21, 1945, argues that the notions of continuity and discontinuity as used in physics and other sciences are complementary and not just contradictory. When, in the eighteenth century, physical theory was occupied with the continuous aspect of things it was natural to suppose that the structure of the world must be either continuous or discontinuous, not both. Therefore, since continuity was apparent, there could be no discontinuity anywhere, as Leibniz argued. The chemical atomic theory and the kinetic theory of gases of the nineteenth century brought out the aspect of discontinuity for the first time, and recent sub-atomic physics has gone a stage further. At every stage, however, as Prof. Robertson emphasizes, theory really requires both continuity and discontinuity. This was first realized in the case of light, which in some aspects is continuous, and in others discontinuous ; and, later, electrons were found to play a similar dual part. Granted that the single electron is discrete, indefinite and indeterminate, it should not be inferred that the continuity, definiteness and strict causality found in the behaviour of large numbers of electrons are not equally genuine aspects of the physical world. Prof. Robertson refers briefly to the way in which the biological theory of evolution requires both a continuous long-term aspect and a discontinuous short-term one ; and he ends by asking the question, whether new discoveries in science which look like discontinuities in the historical development of thought are not also in other respects examples of the continuity of thought.
The argument of this chapter is governed by the double meaning of its title: (a) the relationships between the concepts of discreteness and continuity in modern mathematics and physics; and (b) the relationships, both continuous and discontinuous, between mathematics and physics, as, from Galileo on, a mathematical-experimental science, with mathematics coming first in this conjunction. The project of modern physics was, thus, defined by an essential continuity between mathematics and physics, as a mathematical representation of nature, principally by means of continuous functions and then calculus, until the rise of quantum theory, specifically as quantum mechanics (QM) and quantum field theory (QFT). With QM, at least in certain interpretations, such as the one adopted in this chapter, quantum theory disconnected the ultimate constitution of nature from any mathematical representation, thus, making physics discontinuous with mathematics. As this chapter will argue, however, this new epistemological situation did not disconnect quantum physics from mathematics, but, on the contrary, led to relating abstract mathematics, such as that of Hilbert spaces over ℂ and operator algebra there, to discrete physical phenomena in terms of probabilities, thus reestablishing the connection, which was no longer representational, between mathematics and physics. This chapter will also argue that the development of the concepts of continuity and discontinuity, and the relationships between them, has acquired a great richness and complexity through the nineteenth and twentieth centuries in mathematics itself, richness and complexity that found their way into physics, in particular relativity and quantum theory. One of the intriguing aspects of this development is the idea, advanced by, among others, A. Grothendieck, following B. Riemann's comment on the subject, that the continuous may serve as an approximation of the discontinuous, rather than seeing, as is more common, the discontinuous as a mode of technical approach to the continuous. This chapter will discuss this idea, and the relationships between continuity and discontinuity in, and between, mathematics and physics in terms of two new concepts: reality without realism (RWR), applicable in both mathematics and physics, and the relationships between them, and ideality without idealism (IWI), the version of the concept of RWR, applicable specifically to mathematics.
Abstract Discrete features in the description of some natural phenomena date back to Greek atomists’ speculations on the existence of smallest parts of matter. The concept of continuity first entered science about a century later, with Aristotle’s definition: ‘Things are said to be continuous whenever there is one and the same limit of both wherein they overlap and which they possess in common.’ As examples of things continuous he mentioned lines, surfaces, solids, motions, time, and place. The most significant use tow hich he put his principle of continuity was in the area of biology, however.
Twentieth-Century Physics A century which has run less than a third of its course has already witnessed two great upheavals in physical science. These are associated with the words Relativity and Quanta, and have forced the physicist of to-day to view nature against a background of ideas which is very different from that of his nineteenth-century predecessor. The latter thought of nature as an assemblage of objects located in space and continually changing with the passage of time. It was something entirely detached from, and external to, himself; something which he could study and explore from a distance as the astronomer studies the surface of the sun through his telescope, or the explorer the desert from his aeroplane. He thought of the apparatus of his laboratory as the astronomer thinks of his telescope, or the explorer of his field-glass; it shewed him things which were there whether he looked at them or not, which had been there before the first man appeared on earth, and would still be there after the last man had been frozen to extinction. Finally he accepted a “common-sense” view of nature, believing that there was no great difference between appearance and reality; the possibility that things were not as they seemed might provide an admirable subject for a debating society of philosophers, but was of as little practical concern to the scientist as to the farm-labourer. Although he may not have realised it, this complex of beliefs constituted a philosophical creed in itself.
As has become evident from historical studies, science does not proceed in the coherent and predictable way that basic science texts would have us believe. I will argue that an excellent counter-example is an episode from the historical development of quantum mechanics in which the incompatibility of the particle and the wave representations of the electron and light were destined to be encompassed by two mathematically equivalent, but conceptually quite different theories. I shall argue that the appearance of two such different, yet equivalent, quantum theories was not surprising at all and I claim even predictable. As Einstein himself wrote in 1909: ‘It is my opinion that the next phase in the development of theoretical physics will bring us a theory of light that can be interpreted as a kind of fusion of the wave and the [particle] theory.’ Certainly, the interpretative content of Werner Heisenberg's and Erwin Schrödinger's theories could not have been more different. By mid-1926, the theoretical foundations had been laid for a scientific and emotional battleground between the particle and the wave. I suggest that an important element in the debate was not the incompatibility itself but actually coming to terms with ambiguity in science. For, in the end, ambiguous and vague interpretations of the same phenomena became part of science, where science was supposed to give a clear and unambiguous description of nature.
Newton thought that light was made up of particles, but then it was discovered, as we have seen here, that it behaves like a wave. Later, however (in the beginning of the twentieth century) it was found that light did indeed sometimes behave like a particle. Historically, the electron, for example, was thought to behave like a particle, and then it was found that in many respects it behaved like a wave. So it really behaves like neither. Now we have given up. We say: “it is like neither.” ere is one lucky break, however-electrons behave like light. e quantum behaviour of atomic objects (electrons, protons, neutron, photons, and so on) is the same for all, they are all ‘particle waves’ or whatever you want to call them.
The argument of this article is threefold. First, the article argues that from its rise in the sixteenth century to our own time, the advancement of modern physics as mathematical-experimental science has been defined by the invention of new mathematical structures. Second, the article argues that quantum theory, especially following quantum mechanics, gives this thesis a radically new meaning by virtue of the following two features: on the one hand, quantum phenomena are defined as essentially different from those found in all previous physics by purely physical features; and on the other, quantum mechanics and quantum field theory are defined by purely mathematical postulates, which connect them to quantum phenomena strictly in terms of probabilities, without, as in all previous physics, representing or otherwise relating to how these phenomena physically come about. While these two features may appear discordant, if not inconsistent, I argue that they are in accord with each other, at least in certain interpretations (including the one adopted here), designated as “reality without realism”, RWR, interpretations. This argument also allows this article to offer a new perspective on a thorny problem of the relationships between continuity and discontinuity in quantum physics. In particular, rather than being concerned only with the discreteness and continuity of quantum objects or phenomena, quantum mechanics and quantum field theory relate their continuous mathematics to the irreducibly discrete quantum phenomena in terms of probabilistic predictions while, at least in RWR interpretations, precluding a representation or even conception of how these phenomena come about. This subject is rarely, if ever, discussed apart from previous work by the present author. It is, however, given a new dimension in this article which introduces, as one of its main contributions, a new principle: the mathematical complexity principle.