Talks

The way we combine operators in quantum theory depends on the causal relationship involved. For spacelike separated spacetime regions we use the tensor product. For immediately sequential regions of spacetime we use the direct product. In the latter case we lose information – that is we cannot go from the direct product of two operators to the two original operators. This is a kind of compression. We will see that such compression is associated with causal adjacency. We will situate this in the context of a much broader framework potentially suitable for developing a theory of quantum gravity.

Once again the problem of indistinguishability has been recently tackled. The question is why indistinguishability, in quantum mechanics but not in classical one, forces a changes in statistics. Or, what is able to explain the difference between classical and quantum statistics? The answer given regards the structure of their state spaces: in the quantum case the measure is discrete whilst in the classical case it is continuous. Thus the equilibrium measure on classical phase space is continuous, whilst on Hilbert space it is discrete. Put in other words, this difference goes along the way followed for a long time, it refers to the different nature of elementary particles. Answer of this type completely obscure the probabilistic side of the question. We are able to give in fully probability terms a deduction of the equilibrium probability distribution for the elements of a finite abstract system. Specializing this distribution we reach equilibrium distributions for classical particles, bosons and fermions. Moreover we are able to deduce Gentile's parastatistics too.

Taking for granted that the mathematical apparatus for describing probabilities in quantum mechanics is well-understood via work of von Neumann, Lüders, Mackey, and Gleason, we present an overview of different interpretations of “probability” in quantum mechanics bearing on physics and experiment, with the aim of clarifying the meaning and place of so-called “objective” interpretations of quantum probability. The dichotomy “objective/subjective” is unfortunate, we argue, as we should distinguish two different dimensions integral to the concept of probability. The first concerns the values of probability functions, viz. what the real numbers measure, e.g. relative frequencies of experimental outcomes, or strengths of physical dispositions (objective-1), vs. degrees of belief of idealized agents (subjective-1), etc. But a second dimension is also important, concerning the domain of definition, the “events” or “bearers” of probability, what the probabilities are probabilities of. Relative frequencies of what, described how, or strengths of dispositions to do what, described how, degrees of belief in what, etc. Reminding ourselves of the quantum mechanical phenomenon of incompatible observables, we recall that contradictions are standardly avoided by describing probabilities as pertaining to “measurement outcomes” rather than “possessed properties”: thereby, subjective elements are introduced into the very description of the “events”. (Interpretations qualify as objective-2 if they avoid “bad words” like “measurement” as primitive, in favor of possessed properties or physical interactions; as subjective-2 if such terms are employed in an essential way.) This leads to a two-by-two matrix of interpretative possibilities. The remainder of our talk consists in filling in the blanks (which the reader is invited to try for him/herself) and providing commentary on the relative advantages and disadvantages, which go to the heart of the problem of interpreting quantum theory. Given our scheme, it turns out that “objective version of Copenhagen” makes good sense; this is one locus of “propensities”, which can be made sense of, we claim, along the lines of pre-hidden-variables Bohm (his 1951 text), not to be confused with Popper. We close by noting a serious deficiency in recent Bayesian approaches to quantum probability (lying in the subj-1, subj.-2 quadrant), viz. its explanatory impoverishment. But I’ve already given too many hints.

It is often suggested that the special theory of relativity is incompatible with any notion of the passage of time. I shall try to show, following in the footsteps of Abner Shimony, that there is transience to be found in Minkowski spacetime, but this transience is local rather than global.

After having been a Whiteheadian for decades, Abner, under the influence of Lovejoy’s book, "The Revolt against Dualism,” no longer accepts Whitehead’s philosophy. In this paper I try to challenge this change of heart, as well as suggest a modification of Whitehead’s philosophy that allows for an elegant interpretation of the EPR/Bell correlations.

Abner Shimony mentions that his undergraduate years at Yale in the forties provided an introduction to three profound philosophers that influenced his thought – Alfred North Whitehead, Charles Sanders Peirce and Kurt Gödel. For all three, mathematics played a central role in the unfolding of their lives and thought. This paper will focus on the earliest of this trio, and focus on Peirce’s complex and rich views on the nature and practice of mathematics, attending first to the necessary and foundational nature Peirce ascribes to mathematics and to the place of hypothesis, diagrams, and observation in its development. For Peirce the foundational nature of mathematics arises from the absence of a need to ground it in any further discipline, even a discipline such as logic. The contextualization of a number of these issues in the work of the British mathematicians George Boole and J.J. Sylvester will be explored as well as the powerful yet complex influence of his mathematician father, Benjamin Peirce. Then the role of mathematics in the empirical sciences will be considered, and the meeting point traced in Peirce between the abstract, general and necessary, qualities that figure in mathematics, and the particular and empirical that constitute the description of nature. As various commentators have noted, in the epistemological concerns that dominate Peirce's thinking, intriguing and interesting tensions appear between, on the one hand, his understanding of mathematics and, on the other, his view of an evolving nature which is governed by chance and our knowledge of which is always fallible and thus open to revision. The meeting place, the theme of Wigner’s famous essay reflecting on the effectiveness of mathematics in the natural sciences, is at the heart of the enterprise of mathematical physics. And arguably Peirce’s writings here serve to bring out a number of deep and persistent issues that attend exploring this topic.

In the context of Bell-type experiments, two related notions of "separability" are offered, one of which is logically stronger than the other. It is shown that the weaker of these is logically equivalent to the statistical independence condition widely taken to have been refuted by the results of experiments testing the Bell inequalities. Some consequences of the analysis are discussed.

I will show Abner how to construct Minkowski's space-time diagrams directly from Einstein's two postulates and some very elementary plane geometry. This geometric route into special relativity was developed while teaching the subject to nonscientists, but some of its features may be unfamiliar to physicists and philosophers.