Hale proposes a neo-logicist definition of real numbers by abstraction as ratios defined on a complete ordered domain of quantities (magnitudes). I argue that Hale's definition faces insuperable epistemological and ontological difficulties. On the epistemological side, Hale is committed to an explanation of measurement applications of reals which conflicts with several theorems in measurement theory. On the ontological side, Hale commits himself to the necessary and a priori existence of at least one complete ordered domain of quantities, which is extremely (...) implausible because science treats the logical structure of quantities as subject to experimentally and theoretically motivated refinements and revisions. (shrink)

Chaos-related obstructions to predictability have been used to challenge accounts of theory validation based on the agreement between theoretical predictions and experimental data. These challenges are incomplete in two respects: they do not show that chaotic regimes are unpredictable in principle and, as a result, that there is something conceptually wrong with idealized expectations of correct predictions from acceptable theories, and they do not explore whether chaos-induced predictive failures of deterministic models can be remedied by stochastic modeling. In this paper (...) we appeal to an asymptotic analysis of state space trajectories and their numerical approximations to show that chaotic regimes are deterministically unpredictable even with unbounded resources. Additionally, we explain why stochastic models of chaotic systems, while predictively successful in some cases, are in general predictively as limited as deterministic ones. We conclude by suggesting that the way in which scientists deal with such principled obstructions to predictability calls for a more comprehensive approach to theory validation, on which experimental testing is augmented by a multifaceted mathematical analysis of theoretical models, capable of identifying chaos-related predictive failures as due to principled limitations which the world itself imposes on any less-than-omniscient epistemic access to some natural systems. (shrink)

Although Hume's analysis of geometry continues to serve as a reference point for many contemporary discussions in the philosophy of science, the fact that the first Enquiry presents a radical revision of Hume's conception of geometry in the Treatise has never been explained. The present essay closely examines Hume's early and late discussions of geometry and proposes a reconstruction of the reasons behind the change in his views on the subject. Hume's early conception of geometry as an inexact non-demonstrative science (...) is argued to be a consequence of his attempt to discredit geometrical proofs of infinite divisibility of extension by anchoring the meaning of geometrical concepts in inherently inexact qualitative measurement procedures. This measurement-based attack on the exactness and certainty of geometry is analyzed and shown to be both self-refuting and inconsistent with the general epistemological framework of the Treatise. The revised conception of geometry as a demonstrative science in the first Enquiry is then interpreted as Hume's response to the failure of his earlier attempt to discredit geometrical proofs of infinite divisibility of extension. (shrink)

In this paper we motivate and develop the analytic theory of measurement, in which autonomously specified algebras of quantities (together with the resources of mathematical analysis) are used as a unified mathematical framework for modeling (a) the time-dependent behavior of natural systems, (b) interactions between natural systems and measuring instruments, (c) error and uncertainty in measurement, and (d) the formal propositional language for describing and reasoning about measurement results. We also discuss how a celebrated theorem in analysis, known as Gelfand (...) representation, guarantees that autonomously specified algebras of quantities can be interpreted as algebras of observables on a suitable state space. Such an interpretation is then used to support (i) a realist conception of quantities as objective characteristics of natural systems, and (ii) a realist conception of measurement results (evaluations of quantities) as determined by and descriptive of the states of a target natural system. As a way of motivating the analytic approach to measurement, we begin with a discussion of some serious philosophical and theoretical problems facing the well-known representational theory of measurement. We then explain why we consider the analytic approach, which avoids all these problems, to be far more attractive on both philosophical and theoretical grounds. (shrink)

The central argument for functionalism is the so-called argument from multiple realizations. According to this argument, because a functionally characterized system admits a potential infinity of structurally diverse physical realizations, the functional organization of such systems cannot be captured in a law-like manner at the level of physical description (and, thus, must be treated as a principally autonomous domain of inquiry). I offer a rebuttal of this argument based on formal modeling of its premises in the framework of automata theory. (...) In this formal model I exploit the so-called minimal (universal) realizations of automata behaviors to show that the argument from multiple realizations is not just invalid but is refutable, in the sense that its premises (when made formally precise) entail the very opposite of the functionalist's conclusion. (shrink)

Putnam's ``model-theoretic'' argument against metaphysical realism presupposes that an ideal scientific theory is expressible in a first order language. The central aim of this paper is to show that Putnam's ``first orderization'' of science, although unchallenged by numerous critics, makes his argument unsound even for adequate theories, never mind an ideal one. To this end, I will argue that quantitative theories, which dominate the natural sciences, can be adequately interpreted and evaluated only with the help of so-called theories of measurement (...) whose epistemological and methodological purpose is to justify systematic assignments of quantitative values to objects in the world. And, in order to fulfill this purpose, theories of measurement must have an essentially higher order logical structure. As a result, Putnam's argument fails because much of science turns out to rest on essentially higher order theoretical assumptions about the world. (shrink)

In today's philosophy of science, scientific theories are construed as abstract mathematical objects: formal axiomatic systems or classes of set-theoretic models. By focusing exclusively on the logico-mathematical structure of theories, however, this approach ignores their essentially cognitive nature: that theories are conceptualizations of the world produced by some cognitive agents. As a result, traditional philosophical analyses of scientific theories are incapable of coherently accounting for the relevant relations between highly abstract and idealized models in science and concrete empirical phenomena in (...) the world. ;As a more accurate and comprehensive alternative to traditional analyses of scientific theorizing, the dissertation proposes and develops a conception of scientific theories which makes explicit the essential role of cognitive agents--theorizers--in generating abstract scientific models and systematically relating such models to experimentally controlled parts of the world. ;Specifically, by utilizing the conceptually powerful formal framework of the Theory of Categories, the new approach clarifies and makes formally precise the cognitive status of several important aspects of scientific theorizing, including the notion of a scientific model as theorizers' mathematical conceptualization of the world jointly determined by theorizers' physical interactions with the world and their cognitive access to mathematical objects, the distinction between models in science and models in pure mathematics, the distinction between provisional and successful scientific models, the notion of a systematic correspondence between successful scientific models and the world which does not impose upon the world the mathematical structure of scientific models, and the way in which theorizers' modeling strategy determines the general structure of scientific knowledge. (shrink)

The main aspect of the intuitionist ontology of mathematicsis the conception of mathematical objects as products of the human mind. This paper argues that so long as the existence of mathematical objects is made dependent on thehuman mind , the intuitionist ontology is refutable in that it is inconsistent with our well-confirmed beliefs about what is physically possible. At the same time, it is also argued that the intuitionistś attempt to remove this inconsistency by endowing the mind with various highly (...) idealized features and capacities will erase any significant ontological difference between Intuitionism and Platonism. (shrink)