The Mathematical Formalism and the Standard Way of Thinking about It

David Z Albert

Satoru Yamamoto

Hilary Putnam

The thing that strikes everyone who looks at quantum mechanics is ‘superposition of states’. For example, one can have a hydrogen atom in such a condition that the probability that it is at one energy level is 25% and the probability that it is at the next higher energy level is 75%. Now, the problem is that one should not think of this as meaning that the atom is either at the first energy level or the second but we don't know which. What then does it mean ? That is the question! That is what ‘interpretations’ of quantum mechanics are all about. I shall not review the argument to show that one cannot think of it in a classical way, that one cannot think of it as meaning that the energy level is one or the other (nor can one think of it as meaning that the hydrogen atom is at an in-between energy level). Physicists gave up that way of viewing it (which is, unfortunately, the only way of viewing it that one can ‘explain to a barmaid’, in Rutherford's phrase) long before there were more-or-less formal proofs that one cannot view it that way. Formal proofs that there are no hidden variables are not, I think, what has played a role in the thinking of physicists; what physicists are more impressed by is the fact that if one tries to think of it that way then it doesn't square with any intelligible physical picture at all.

Arkady Plotnitsky

Beginning with its introduction by W. Heisenberg, quantum mechanics was often seen as an overly abstract theory, mathematically and physically, vis‐à‐vis classical physics or relativity. This perception was amplified by the fact that, while the quantum‐mechanical formalism provided effective predictive algorithms for the probabilistic predictions concerning quantum experiments, it appeared unable to describe, even by way idealization, quantum processes themselves in space and time, in the way classical mechanics or relativity did. The aim of the present paper is to reconsider the nature of mathematical and physical abstraction in modern physics by offering an analysis of the concept of “physical fact” and of the concept of “physical concept,” in part by following G. W. F. Hegel’s and G. Deleuze’s arguments concerning the nature of conceptual thinking. In classical physics, relativity, and quantum physics alike, I argue, physical concepts are defined by the following main features—1) their multi‐component multiplicity; 2) their essential relations to problems; 3) and the interactions between physical, mathematical, and philosophical components within each concept. It is the particular character of these interactions in quantum mechanics, as defined by its essentially predictive (rather than descriptive) nature, that distinguishes it from classical physics and relativity.

Edward Nelson

Arkady Plotnitsky, Andrei Yu. Khrennikov, Gregg S. Jaeger +2 more

Beginning with its introduction by W. Heisenberg, quantum mechanics was often seen as an overly abstract theory, mathematically and physically, vis‐à‐vis classical physics or relativity. This perception was amplified by the fact that, while the quantum‐mechanical formalism provided effective predictive algorithms for the probabilistic predictions concerning quantum experiments, it appeared unable to describe, even by way idealization, quantum processes themselves in space and time, in the way classical mechanics or relativity did. The aim of the present paper is to reconsider the nature of mathematical and physical abstraction in modern physics by offering an analysis of the concept of “physical fact” and of the concept of “physical concept,” in part by following G. W. F. Hegel’s and G. Deleuze’s arguments concerning the nature of conceptual thinking. In classical physics, relativity, and quantum physics alike, I argue, physical concepts are defined by the following main features—1) their multi‐component multiplicity; 2) their essential relations to problems; 3) and the interactions between physical, mathematical, and philosophical components within each concept. It is the particular character of these interactions in quantum mechanics, as defined by its essentially predictive (rather than descriptive) nature, that distinguishes it from classical physics and relativity.