Geoffrey Hellman

Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell’s insights. Problems of absolutism plague some versions, and, interestingly, Russell’s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modal-structuralism and a category theoretic approach as remaining non-absolutist options. It is suggested that these should be combined.

We develop a point-free construction of the classical one- dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence of "indecomposability" from a non-punctiform conception. It is surprising that such simple axioms as ours already imply the Archimedean property and that they determine an isomorphism with the Dedekind-Cantor structure of R as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own.

Standard proofs of generalized Bell theorems, aiming to restrict stochastic, local hidden-variable theories for quantum correlation phenomena, employ as a locality condition the requirement of conditional stochastic independence. The connection between this and the no-superluminary-action requirement of the special theory of relativity has been a topic of controversy. In this paper, we introduce an alternative locality condition for stochastic theories, framed in terms of the models of such a theory (§2). It is a natural generalization of a light-cone determination condition that is essentially equivalent to mathematical conditions that have been used to derive Bell inequalities in the deterministic case. Further, it is roughly equivalent to a condition proposed by Bell that, in one investigation, needed to be supplemented with a much stronger assumption in order to yield an inequality violated by some quantum mechanical predictions. It is shown here that this reflects a very general situation: from the proposed locality condition, even adding the strict anticorrelation condition and the auxiliary hypotheses needed to derive experimentally useful (and theoretically telling) inequalities, no Bell-type inequality is derivable. (These independence claims are the burden of §4.) A certain limitation on the scope of the proposed stochastic locality condition is exposed (§5), but it is found to be rather minor. The conclusion is thus supported that conditional stochastic independence, however reasonable on other grounds, is essentially stronger than what is required by the special theory.Our results stand in apparent contradiction with a class of derivations purporting to obtain generalized Bell inequalities from locality alone. It is shown in Appendix (B) that such proofs do not achieve their goal. This fits with our conclusion that generalized Bell theorems are not straightforward generalizations of theorems restricting deterministic hidden-variable theories, and that, in fact, such generalizations do not exist. This leaves open the possibility that a satisfactory, non-deterministic account of the quantum correlation phenomena can be given within the framework of the special theory.