Changes of Setting and the History of Mathematics: A New Study of Frege

This thesis addresses an issue in the philosophy of Mathematics which is little discussed, and indeed little recognised. This issue is the phenomenon of a ‘change of setting’. Changes of setting are events which involve a change in a scientific framework which is fruitful for answering questions which were, under an old framework, intractable. The formulation of the new setting usually involves a conceptual re-orientation to the subject matter. In the natural sciences, such re-orientations are arguably unremarkable, inasmuch as it is possible that within the former setting for one’s thinking one was merely in error, and under the new orientation one is merely getting closer to the truth of the matter. However, when the subject matter is pure mathematics, a problem arises in that mathematical truth is (in appearance) timelessly immutable. The conceptions that had been settled upon previously seem not the sort of thing that could be vitiated. Yet within a change of setting that is just what seems to happen. Changes of setting, in particular in their effects on the truth of individual propositions, pose a problem for how to understand mathematical truth. Thus this thesis aims to give a philosophical analysis of the phenomenon of changes of setting, in the spirit of the investigations performed in Wilson (1992) and Manders (1987) and (1989). It does so in three stages, each of which occupies a chapter of the thesis: 1. An analysis of the relationship between mathematical truth and settingchanges, in terms of how the former must be viewed to allow for the latter. This results in a conception of truth in the mathematical sciences which gives a large role to the notion that a mathematical setting must ‘explain itself’ in terms of the problems it is intended to address. 2. In light of (1), I begin an analysis of the change of setting engendered in mathematical logic by Gottlob Frege. In particular, this chapter will address the question of whether Frege’s innovation constitutes a change of setting, by asking the question of whether he is seeking to answer questions which were present in the frameworks which preceded his innovations. I argue that the answer is yes, in that he is addressing the Kantian question of whether alternative systems of arithmetic are possible. This question arises because it had been shown earlier in the 19th century that Kant’s conclusion, that Euclid’s is the only possible description of space, was incorrect. 3. I conclude with an in-depth look at a specific aspect of the logical system constructed in Frege’s Grundgesetze der Arithmetik. The purpose of this chapter is to find evidence for the conclusions of chapter two in Frege’s technical work (as opposed to the philosophical). This is necessitated by chapter one’s conclusions regarding the epistemic interdependence of formal systems and informal views of those frameworks. The overall goal is to give a contemporary account of the possibility of setting-changes; it will turn out that an epistemic grasp of a mathematical system requires that one understand it within a broader, somewhat historical context

Frege\u27s Constraint and the Nature of Frege\u27s Foundational Program

Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ( ) or ‘Frege Constraint’ ( ), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how generalizes Frege’s views while comes closer to his original conceptions. Different authors diverge on the interpretation of and on whether it applies to definitions of both natural and real numbers. Our aim is to trace the origins of and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish from (§2). We discuss six rationales which may motivate the adoption of different instances of and (§3). We turn to the possible interpretations of (§4), and advance a Semantic (§4.1), arguing that while it suits Frege’s definition of natural numbers (4.1.1), it cannot reasonably be imposed on definitions of real numbers (§4.1.2), for reasons only partly similar to those offered by Crispin Wright (§4.1.3). We then rehearse a recent exchange between Bob Hale and Vadim Batitzky to shed light on Frege’s conception of real numbers and magnitudes (§4.2). We argue that an Architectonic version of is indeed faithful to Frege’s definition of real numbers, and compatible with his views on natural ones. Finally, we consider how attributing different instances of to Frege and appreciating the role of the Architectonic can provide a more perspicuous understanding of his foundational program, by questioning common pictures of his logicism (§5)

Finitism--an essay on Hilbert's programme

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1991.Includes bibliographical references (p. 213-219).by David Watson Galloway.Ph.D

Issues in commonsense set theory

The success of set theory as a foundation for mathematics inspires its use in artificial intelligence, particularly in commonsense reasoning. In this survey, we briefly review classical set theory from an AI perspective, and then consider alternative set theories. Desirable properties of a possible commonsense set theory are investigated, treating different aspects like cumulative hierarchy, self-reference, cardinality, etc. Assorted examples from the ground-breaking research on the subject are also given. © 1995 Kluwer Academic Publishers

Metaontology

The Ontological Question 'What exists?' dates back over two thousand five hundred years to the dawn of Western philosophy, and attempts to answer it define the province of ontology. The history of the Western philosophical tradition itself has been one of the differentiation and separation of the various sciences from the primordial stuff of ancient philosophy. Physics was first to break away from the tutelage of philosophy and established its independence in the seventeenth century. The other sciences followed suit fairly rapidly, with perhaps psychology being the last to separate. The results for modern philosophy - of this breakup of what was once a great empire over human reason - have been mixed. An inevitable result has been that questions considered in ancient times to belong to philosophy have fallen within the ambit of other disciplines. So speculations about the material composition and genesis of the universe that interested Thales, Heraclitus and Leucippus, are continued by contemporary cosmologists in well equipped research laboratories, and not by philosophers. However ontology, unlike cosmology, has not broken away from its parent discipline and the Ontological Question as to what exists is still argued by philosophers today. That ontology has failed to make the separation that cosmology has, is a reflection on the weakness of the methodology for settling ontological arguments. Unlike their great Rationalist predecessors, most modern philosophers do not believe that logic alone is sufficient to provide an answer as to what is. But neither do observation or experiment, in any direct way, seem to help us in deciding, for example, whether sets or intentions should be admitted to exist or not. In consequence, the status of ontology as an area of serious study has to depend on the devising of a methodology within which the Ontological Question can be tackled. The pursuit of such a methodology is the concern of metaontology and is also the concern of this thesis

Law's ontology and practical reason

The thesis is an attempt to reconcile law's dual nature, its factual dimension (its facticity) and its normative/evaluative dimension (its normativity), in a non-reductive manner. The tension between those two dimensions appears particularly acute when we try to discern some object of reference for our normative talk/discourse. Then the possibility of absence of such objects poses a high threat to the meaningfulness of the enterprise of law tout court. Faced with this danger lawyers usually end up reducing legal referents to physical, nonnormative entities. Palpable for our senses as those entities may be, they do not seem to eliminate the threat of meaninglessness posed to the legal enterprise, as they end up eliminating law's normativity. In contrast I argue that legal and broader practical norms can be reconstructed as abstract objects that are available to knowledge. The method employed, relies predominantly on a semantic explication of the 'objecthood' of norms along the lines of a neo-Fregean theory of mental content. Further, I employ an analysis of the meaning of legal expressions in order to show that a semantic account of legal 'objecthood' will be demarcated by the pragmatic-normative requirements that support the relevant practices in which legal meaning is generated (as is specified by some version of Wittgenstein's 'meaning as use' theory of meaning). I proceed to argue that those pragmatic requirements include some transcendental pragmatic norms which specify an ultimate practical or moral point of view against the background of which practical meaning is possible. Later, this point of view is specified as a Super-norm or Principle of Autonomy. This norm bestows an evaluative element upon the meaning of all practical expressions/sentences and, via the semantic explication of ontology, into the normative objects (rules, properties and so on) that correspond to them. Finally, it is claimed that legal norms are a species of practical norms, to the extent that both fall under the same criteria of validity that are specified by the point of view of the Norm ofAutonomy