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.
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.
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.
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.
MORE than any other word of the language, the word atom is implicated with the history of human speculations concerning the nature uf things. It is introduced when people cease to content themselves with observing, and begin to philosophize. There are many of the fundamental and essential writings of the literature of physics in which it does not appear, or appears without warrant. These are the descriptions of things observed, the accounts of experiments, the records of measurements, on which the edifice of theoretical physics is founded. There are many articles of what is commonly called the “theoretical” sort in which it does not occur. Such are the papers on the motions of planets, on the vibrations of elastic solids, on the currents in electrical networks, on the courses of light-rays through optical systems-papers which are essentially descriptions, although they give the impression of being something greater and deeper because they relate to idealized cases, and are phrased in the laconic language of mathematics. When the word atom appears justifiably in a discourse, it means that the author has departed from the safe routine of describing observed and observable events, however selectively, however skilfully, however intelligently. It signifies that he has gone beyond the limits of observation, and has entered upon the audacious adventure of constructing by the side of the real universe an ideal one, which shall act as the real one does, and be intelligible through and through.
The genesis of classical science was accompanied by a transition from logical to mathematical analysis. This change did not mean a rejection of Aristotle's canons of logic; it was simply that these canons became inadequate. They underwent a certain generalization and, in the course of this, came closely to approximate mathematical analysis, the foundations of the calculus of the infinitesimal. Classical science no longer took as its point of departure the notion of motion from "something" into "something," as did Peripatetic physics and cosmology (for example, motion to "the natural place"). The initial concept is motion, considered from point to point and instant to instant. This kind of differential notion of motion was clearly expressed by Kepler when he wrote: "Where Aristotle sees direct oppositeness between two things, with no mediating links, there I, looking at geometry philosophically, find a mediated opposition; so that where Aristotle employs a single term, ‘other,’ I use two: ‘larger’ and ‘smaller.’" (1)
In recent years, the term quantum-like has been increasingly used in different disciplines, including neurosciences, psychological and socio-economical disciplines, claiming that some investigated phenomena show “something” in common with quantum processes and, therefore, they can be modeled using a sort of quantum formalism. Therefore, the increasing use of the term quantum-like calls for defining and sharing its meaning in order to properly adopt it and avoid possible misuse. In our opinion, the concept of quantum-like may be successfully applied to macroscopic phenomena and empirical sciences other than physics when at least two conditions are satisfied: a) the behavior of the investigated phenomena show logical analogies with quantum ones; b) it is possible to find a criterion of truth based on an experiential/scientific approach applied to a probabilistic model of description of the phenomena. This is only a first, small step in the approach to the concept of quantum-like, hopefully helpful to promote further discussion and achieve a better definition.
The present article is written by a theoretical physicist who, for some time, has endeavored to trace the origin of the concepts and principles of quantum theory to an empirical background broader than that afforded by delicate optical and mechanical experiments on a microphysical scale involving the microconstant h. In particular he tried to reduce the dominant role of probability in modern physics to irrefutable evidence of a quite general nature. As long as quantum probability is deduced from experience with seemingly uncontrollable micro-mechanisms—think of radioactive disintegration as the most typical example—there will always be the objection that one day we still may find a hidden causality behind those small scale gambling devices called atoms, although one example of an exactly predictable atomic event would make present-day quantum theory untenable. In order to dislodge the remaining classicists from their fortress of doubt one may try to show that quantum probability is not an alibi for a temporary impotence of predicting microphysical events, but is a positive conclusion drawn from experience of a most general type, namely, from the principle of continuity, the basic tenet of the philosophy of Leibniz.