Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

On the relation between the normative and the empirical in the philosophy of science.

The relation between the normative and the empirical in the philosophy of science is examined by investigating apriori and aposteriori approaches to methodology. The apriori is usually equated with the prescriptive, and the aposteriori with the descriptive. It is argued that this equation is mistaken, and that neither a purely apriori nor a purely aposteriori approach to methodology can succeed. Methodologies based on probability are used as illustrations. Purely apriori and purely aposteriori approaches are examined in Parts I and II respectively. The former are investigated through the intuitionism of J.M. Keynes and the analytic method of Carnap. Dutch Book arguments are also considered as apriori arguments. I conclude that an apriori approach is irredeemably flawed, in that it can never meet the goal it sets for itself of producing a self-evidently justified set of rules for science. Purely aposteriori approaches are investigated in the second Part by focussing on R. Giere's and W.V.O. Quine's proposals for a naturalised epistemology. It is argued that a purely empirical approach is caught on the horns of a dilemma: if it is defended on aposteriori grounds then the argument is circular, and if on apriori grounds it is self-refuting. Thus it is shown that the aposteriori approach too cannot serve as the foundations for methodology. However, I shall argue that Quine's project has been misunderstood, and that in fact Quine argues for aposteriori methodology from conventionalist grounds. The possibility of a conventionalist approach to the philosophy of science which avoids the problems of the purely empirical and of the purely apriori approach is explored in the third Part of this thesis. Karl Popper's early advocacy of such a conventionalist approach is discussed. The final chapter is devoted to showing how certain flaws in Popper's and Quine's conventionalist approaches may be mended. It is concluded that the conventionalist approach to methodology provides an adequate framework for the relation between the normative and the empirical in the philosophy of science

Probability and nonclassical logic

Classical tautologies have probability. Classical contradictions have probability. These familiar features reflect a connection between standard probability theory and classical logic. In contexts in which classical logic is questioned—to deal with the paradoxes of self-reference, or vague propositions, for the purposes of scientific theory or metaphysical anti-realism—we must equally question standard probability theory. Section 1 covers the intended interpretation of ‘nonclassical logic’ and ‘probability’. Section 2 reviews the connection between classical logic and classical probability. Section 3 briefly reviews salient aspects of nonclassical logic, laying out a couple of simple examples to fix ideas. Section 4 explores modifications of probability theory. The variations laid down will be motivated initially by formal analogies to the classical setting. In section 5, however, we look at two foundational justifications for the presentations of ‘nonclassical probabilities’ that are arrived at. Sections 6-7 describe extensions of the nonclassical framework: to conditionalization and decision theory in particular. Section 8 will consider some alternative approaches, and section 9 evaluates progress

Approaches to analysis with infinitesimals following Robinson, Nelson, and others

This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis. We highlight some applications including (1) Loeb's approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson's and related frameworks to the multiverse view as developed by Hamkins. Keywords: axiomatisations, infinitesimal, nonstandard analysis, ultraproducts, superstructure, set-theoretic foundations, multiverse, naive integers, intuitionism, soritical properties, ideal elements, protozoa

How does probability theory generalize logic?

Nearly half a century ago Popper [1959], appendices *iv and *v, presented a number of related axiomatizations of the theory of probability in each of which p(x | z) is defined for all x and z, even z = yy' (where concatenation turns out to represent meet, and the accent complementation). These systems are too little known amongst mathematicians. Popper went on to claim that his systems provide a context within which it is possible to give fully correct definitions of the relation of derivability between sentences (in the sense of the classical sentential calculus), and of the property of sentential demonstrability, by means of the formulas z |- x =Df p(x | zx') = 1 |- x =Df p(x | x') = 1. This claim has been challenged by Stalnaker [1970], Harper [1975], and by Leblanc & van Fraassen [1979]. The challenge has never been properly answered (though a start was made in Popper & Miller [1994], §4). The aim of this talk is to answer it, and to contrast Popper’s enterprise with what is known as probabilistic semantics

The Logical Writings of Karl Popper

This open access book is the first ever collection of Karl Popper's writings on deductive logic. Karl R. Popper (1902-1994) was one of the most influential philosophers of the 20th century. His philosophy of science ("falsificationism") and his social and political philosophy ("open society") have been widely discussed way beyond academic philosophy. What is not so well known is that Popper also produced a considerable work on the foundations of deductive logic, most of it published at the end of the 1940s as articles at scattered places. This little-known work deserves to be known better, as it is highly significant for modern proof-theoretic semantics. This collection assembles Popper's published writings on deductive logic in a single volume, together with all reviews of these papers. It also contains a large amount of unpublished material from the Popper Archives, including Popper's correspondence related to deductive logic and manuscripts that were (almost) finished, but did not reach the publication stage. All of these items are critically edited with additional comments by the editors. A general introduction puts Popper's work into the context of current discussions on the foundations of logic. This book should be of interest to logicians, philosophers, and anybody concerned with Popper's work