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)
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)
Why should anyone think that what is so deeply puzzling about the quantum world is really a matter of logic? Why should anyone think that studying the logic of quantum mechanics should be a route to a proper understanding of quantum mechanics? There is a tradition in the philosophy of quantum mechanics, a tradition stronger among philosophical commentators than among physicists, which sees the subject as the business of understanding quantum mechanical language and the way that language relates to the world. It is an idea more strongly held by philosophers – it comes much more naturally to them – because the analytic tradition, in which most Anglo-American philosophers of science are raised, is generally much exercised about words and how they hook on to things. The analytic tradition in the philosophy of quantum mechanics is not unreasonable. Everyone admits that we cannot picture the microphysical world, that a graphic or iconic representation of quantum systems is impossible. Therefore understanding quantum mechanics must be a matter of understanding the logic of the words and the mathematics of quantum mechanics. If this seems implausible, it is because of an ambiguity in our use of the verb ‘to understand’. We understand the physical world, and we understand physics. But physics is not the physical world. It is something of an entirely different sort, a human product, a way of representing the world.
The growth of modern science resulted in a radically new cognitive approach to many problems of classical philosophy. In the context of contemporary theoretical physics, if only one avoids easy apologetic, it is no longer possible to defend this version of epistemological empiricism that was defended half a century ago. Whereas in the 1930’s the traditional distinction between theoretical and observational terms was called into question, due to its arbitrary nature, in the 1980’s the basic philosophical opposition between material and immaterial (resp. between physical and mathematical) objects seems arbitrary since the very notion of matter appears nothing but a terminological relic of the bygone epoch.
The non-classical nature of quantum theory has led to a number of efforts to formalize the logic of statements whose universe of discourse is the subatomic (see, for example, [3], [6], [8], and [10]).It might be hoped that ideally the fruit of such efforts would be a formalism not totally divorced from the classical logic, but incorporating or incorporated in it in such a manner as to unify the logic of the micro-and the macro-worlds.Indeed, this has been the general direction taken by these efforts, including that which we principally examine and expand upon here.*Josef Jauch, [6], pp.67-89, developed a "propositional calculus of quantum mechanics".He objected to calling a system like his a "logic of quantum mechanics"; however, our use of 'logic' here is not, I think, the use to which Jauch objected.A logic, for our present purposes, is a mathematical formalism having a reasonable interpretation as a theory of deduction; its actual application as normative of argumentation in reality is a matter of empirical judgment.This seems close to the way Jauch wished to use "propositional calculus" but more general, allowing for extension to a quantification theory of quantum mechanics, for example.Although Jauch's presentation of a propositional calculus of quantum mechanics is mathematically interesting and suggestive, there are a number of problems connected with it.It is our purpose to engage some of these problems and to present a rigorously defined formalism intended to come close to doing the job intended by Jauch for his logic.The success or lack of success of this effort as related to actual application of this formalism in the foundations of quantum theory is, I think, a matter for the physicists to decide; the empirical judgment involved in this application is not in the scope of this paper.If a formalism is to have a reasonable interpretation as a theory of deduction, it seems clear that it must include a relation which, when it holds between two elements of the formalism, may be reasonably interpreted as expressing that one of the elements 'follows from' the other, in •The author wishes to thank Robert Piziak for advice and comments in the course of research for this paper.
The Copenhagen Interpretation of quantum mechanics has become the standard way of explaining the physical philosophical meaning of the mathematics of microscopic behavior. It developed from the thought of people who attended the Institute for Theoretical Physics of the University of Copenhagen during the 40 years that Niels Bohr directed it, from 1920 to 1962. Bohr took a large part in developing the Copenhagen Interpretation, he didn't escape all controversy over it (although the quote above is a bit of an exaggeration). Bohr did die before the experimental theoretical developments that have transposed the debate over the appropriateness adequacy of the Copenhagen Interpretation into a new key, he may well have been lucky to have escaped when he did. There is no authoritative codification of CI [the Copenhagen Interpretation], write T Gornitz C. E von Weizsacker of the Max Planck Institute in Starnberg, West Germany in' their formal presentation to the recent Loyola Conference on Mathematical Interpretational Problems in Relativistic Quantum Theory, and under the conditions of its origin there probably could not be one.... All prominent authors of the time who wrote on these questions, like Bohr, [Werner] Heisenberg, [Wolfgang] Pauli, J[ohn] v[on] Neumann on one side, [Albert] Einstein [Erwin] Schrodinger on the other, fell, so we feel, into some stammering when they tried to express their own positions. Nevertheless there is what might be called a Copenhagen attitude. It concentrates on observable phenomena, makes statistics probability central to its understanding, considers quantum mechanics primarily a theory of knowledge or information becomes a kind of physica negativa (analogous to a theologia negativa) in trying not to say too much about things that can't be seen or felt. For example, it doesn't much care whether the orbits of electrons in atoms are real or not and, in fact, tends to regard them as unreal. It appears in most textbooks as the standard way of looking at things.
The relationship between physical notions and the structures expressing them is an issue of primary importance the history and philosophy of pre-modern natural philosophy and of modern science. Historical evidence suggests that, when investigating the relationship between physics and mathematics, one should pay attention to the material, sensually perceivable forms mediating it: spoken or written words, drawings, artefacts, descriptions of experimental set-ups or, as the case discussed the present paper, symbolic notation. The reason for this is that, historically, physical notions and their expressions have very often been developed as one, and only distinguished from each other at a later stage. This was for example the case with what we today regard as mathematical and classical mechanics, which were developed the course of the eighteenth century as the discipline of the analysis of motion. When considered from a later point of view, both the physical notions and the concepts involved historical developments appear as being in flux, vague, and, as far as the side is concerned, non-rigorous. In such cases, verbal description, mechanical artefacts or symbols do not simply mediate between physical notions and structures, as might be the case at a later stage, but represent both of them one, allowing the historical actors to construct and manipulate physical-mathematical meaning by following, at times, their physical reasoning and, at times, their criteria of rigour. The history of quantum mechanics offers many examples of how new physics and new mathematics can emerge as one, with symbolic notation playing a central role shaping both. Among the many creators of quantum theory, Paul Dirac was particularly sensitive to questions of notation:
When we consider and analyze physical events with the purpose of creating corresponding mathematical models we often assume that the mathematical apparatus used in modeling, at least the simplest mathematical apparatus, is infallible. In particular, this relates to the use of infinitely small and infinitely large numbers in arithmetic and the use of Newton's definition of a limit in analysis. We believe that is where the main problem lies in contemporary study of nature (physics, genetics). We have introduced a new concept of “Observer's Mathematics—Mathematics of Relativity” (see references). Observer's mathematics creates new arithmetic, algebra, geometry, topology, analysis and logic which do not contain the concept of continuum, but locally coincide with the standard fields. The concept of a quantum arises automatically from the human thought process, rather than from physics. From this point of view classical mechanics is brought closer to quantum mechanics.