Set-Theoretic Constructions

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...) of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

Constructive and intuitionistic Zermelo-Fraenkel set theories are axiomatic theories of sets in the style of Zermelo-Fraenkel set theory (ZF) which are based on intuitionistic logic. They were introduced in the 1970's and they represent a formal context within which to codify mathematics based on intuitionistic logic. They are formulated on the basis of the standard first order language of Zermelo-Fraenkel set theory and make no direct use of inherently constructive ideas. In working in constructive and intuitionistic ZF we can thus (...) to some extent rely on our familiarity with ZF and its heuristics. -/- Notwithstanding the similarities with classical set theory, the concepts of set defined by constructive and intuitionistic set theories differ considerably from that of the classical tradition; they also differ from each other. The techniques utilised to work within them, as well as to obtain metamathematical results about them, also diverge in some respects from the classical tradition because of their commitment to intuitionistic logic. In fact, as is common in intuitionistic settings, a plethora of semantic and proof-theoretic methods are available for the study of constructive and intuitionistic set theories. -/- The entry introduces the main features of Constructive and Intuitionistic ZF and offers links to the relevant bibliography. (shrink)

This paper briefly traces the evolution of the function concept until its modern set theoretic definition, and then investigates its relationship to the pre-formal notion of variable dependence. I shall argue that the common association of pre-formal dependence with the modern function concept is misconceived, and that two different notions of dependence are actually involved in the classic and the modern viewpoints, namely effective and functional dependence. The former contains the latter, and seems to conform more to our pre-formal conception (...) of dependence. The idea of effective dependence is further investigated in connection with the notions of function content and intensionality. Finally, the relevance of the distinction between the two kinds of dependence to mathematical practice is considered. (shrink)

We show that a paraconsistent set theory proposed in Weber (2010) is strong enough to provide a quite classical nonprimitive notion of identity, so that the relation is an equivalence relation and also obeys full substitutivity: a = b -> F(b)). With this as background it is shown that the proposed theory also proves the negation of x=x. While not by itself showing that the proposed system is trivial in the sense of proving all statements, it is argued that this (...) outcome makes the system inadequate. (shrink)

Philosophy of science in the 20th century is to be considered as mostly characterized by a fundamentally systematic heuristic attitude, which looks to mathematics, and more generally to the philosophy of mathematics, for a genuinely and epistemologically legitimate form of knowledge. Rooted in this assumption, the book provides a formal reconsidering of the dynamics of scientific theories, especially in the field of the physical sciences, and offers a significant contribution to current epistemological investigations regarding the validity of using formal (especially: (...) model-theoretic) methods of analysis, as developed principally by Stegmüller, Sneed, Suppes, Moulines, “to bring the airy flights of analytical philosophy back down to earth”, to borrow Stephan Hartmann’s provocative statement. At the same time, the volume represents a comprehensive account of the epistemic content of physical theories, the logic of theory change in science, and specific (inter-)theoretical core aspects of scientific progress, particularly in the form suggested informally by Thomas Kuhn. As C. Ulises Moulines writes in the preface, “there is no other example in present-day literature (in any language) on this topic, i.e. the formal analysis of the ideographic characterization of the dynamics of theories between Kuhn’s theory of science and structural epistemology, that is as systematic and complete as Perrone’s work”. (shrink)

In his ontological works Kurt Grelling tries to give a rigorous analysis of the foundations of the so-called Gestalt-psychology. Gestalten are peculiar emergent qualities, ontologically dependent on their foundations, but nonetheless non reducible to them. Grelling shows that this concept, as used in psychology and ontology, is often ambiguous. He distinguishes two important meanings in which the word “Gestalt” is used: Gestalten as structural aspects available to transposition and Gestalten as causally self-regulating wholes. Gestalten in the first meaning are, according (...) to Grelling, “equivalence classes of correspondences”, while Gestalten as self-regulating wholes have more to do with relations of ontological dependence. Grelling’s clarification of the concept of Gestalt is doubtless an excellent piece of philosophical analysis, but at the end of the day it turns out that his analysis captures at best only a part of intuitions traditionally connected with the notion of Gestalt. (shrink)