Our ultimate purpose is to give an axiomatic treatment of recursion theory sufficient to develop the priority method. The direct or abstract approach is to keep in mind as clearly as possible the methods actually used in recursion theory, and then to formulate them explicitly. The indirect or experimental approach is to look first for other mathematical theories which seem similar to recursion theory, to formulate the analogies precisely, and then to search for an axiomatic treatment which covers not only (...) recursion theory but also the analogous theories as particular cases.The first approach is more general because it does not depend on the existence of analogues. A concrete mathematical theory, it seems, need have no such analogues and still be important, as e.g. classical number theory. In such a case, an axiomatic treatment may still be useful for exhibiting the mathematical structure of the theory considered and the assumptions on which it rests. However, it will lack one of the most heavily advertised advantages of the axiomatic method, namely, the “economy of thought” which results from an uniform theory for several different and interesting cases: we cannot hope for this if, by hypothesis, we know of only one particular case. In contrast, the second approach, if successful at all, is bound to achieve such “economy” because we start out with several interesting particular cases. Another possible virtue of the second approach is that of field work over insight: the abstract pattern that we are looking for and hoping to formalize in axioms, may not be evident in any one mathematical theory, but may spring to the eye if one happens to look simultaneously at several theories which happen to realize the pattern. (shrink)

This paper argues against the thesis of Professor Savan, that Spinoza's views about words and about the imagination are such that he could not consistently say, and indeed did not think, that philosophical truths can be expressed adequately in language. The evidence for this thesis is examined in detail, and it is argued that Spinoza should have distinguished between two types of imagination, corresponding roughly to Kant's transcendental and empirical imagination. Finally, it is suggested that the bulk of the argument (...) of the Ethics is conducted on the level of the ?second kind of knowledge?, reason, but that it also contains examples of the use of the first and third kinds of knowledge. (shrink)

Let ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let $M$ be a countable transitive model of ZF. The method of forcing extends $M$ to another model $M\lbrack G\rbrack$ of ZF (a "generic extension"). If the axiom of choice holds in $M$ it also holds in $M\lbrack G\rbrack$, that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, and we derive (...) an application to Boolean toposes. (shrink)

We propose partial measurements as a conceptual tool to understand how to operate with counterfactual claims in quantum physics. Indeed, unlike standard von Neumann measurements, partial measurements can be reversed probabilistically. We first analyze the consequences of this rather unusual feature for the principle of superposition, for the complementarity principle, and for the issue of hidden variables. Then we move on to exploring non-local contexts, by reformulating the EPR paradox, the quantum teleportation experiment, and the entanglement-swapping protocol for the situation (...) in which one uses partial measurements followed by their stochastic reversal. This leads to a number of counter-intuitive results, which are shown to be resolved if we give up the idea of attributing reality to the wavefunction of a single quantum system. (shrink)

We show that local realism applied to states characterized by a single quantum equally and coherently shared between a number of qubits (so-called W states) produces predictions incompatible with quantum theory. The origin of this incompatibility is shown to originate from the destructive interference of amplitude probabilities associated with nonlocal states, a phenomenon that has no classical analog.

There are some who would question the need to republish papers that have already appeared elsewhere. Walter Pauel once said that scholars should think in terms of books rather than research papers since the latter become lost in the literature. When he told me this year ago I was not entirely convinced. Surely the young scholar must publish papers to secure his academic position. In addition, throughout his career he attends conferences many of which will require the publication of his (...) papers in the resultant conference volumes. By their very nature such papers often discuss topics in greater detail than that scholar's subsequent books. In this case also the papers tend to become "lost" even when there exit extensive guides to the literature such as the Critical Bibliography published annually in Isis for historians of science. Many of my own papers over the past forty-five years have indeed appeared in such conference volumes as in journals. (shrink)