Online research in philosophy

We extend the topos-theoretic treatment given in previous papers of assigning values to quantities in quantum theory, and of related issues such as the Kochen-Specker theorem. This extension has two main parts: the use of von Neumann algebras as a base category (Section 2); and the relation of our generalized valuations to (i) the assignment to quantities of intervals of real numbers, and (ii) the idea of a subobject of the coarse-graining presheaf (Section 3).

A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework. The new methodology has allowed us to introduce the Infinity Computer working with such numbers (its simulator has already been realized). Examples dealing with divergent series, infinite sets, and limits are given.

Axiomatizations of measurement systems usually require an axiom--called an Archimedean axiom--that allows quantities to be compared. This type of axiom has a different form from the other measurement axioms, and cannot--except in the most trivial cases--be empirically verified. In this paper, representation theorems for extensive measurement structures without Archimedean axioms are given. Such structures are represented in measurement spaces that are generalizations of the real number system. Furthermore, a precise description of "Archimedean axioms" is given and it is shown that in all interesting cases "Archimedean axioms" are independent of other measurement axioms.

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.

Degree theories of vagueness build on the observation that vague predicates such as 'tall' and 'red' come in degrees. They employ an infinite-valued logic, where the truth values correspond to degrees of truth and are typically represented by the real numbers in the interval [0,1]. In this paper, the success with which the numerical assignments of such theories can capture the phenomenon of vagueness is assessed by drawing an analogy with the measurement of various physical quantities using real numbers. I argue that degree theories of vagueness are undermined by the failure of the necessary connectedness principle. Moreover, the semantics for the connectives entail that there must be a uniquely correct numerical assignment for the sentences, and this is implausible. Different senses of 'coming in degrees' are then distinguished; I argue that a confusion between them could be the source of the degree theorist's error, and the distinction illuminates the problem cases described earlier in the paper.

Our model of time is the classical continuum of real numbers, and our model of other measurable quantities that change over time is that of functions defined on real numbers with real numbers as values. This model is not derived from reality or from our experience of it, but imposed on reality; and the fit is very imperfect. In classical mathematics, the value of a function for any real number as argument is independent of its value for any other argument: the analogue is Hume's doctrine that events are loose and separate. This makes continuity in the change of any quantity a contingent law of physica, rather than a conceptual necessity. The article explores alternatives to this classical model.

On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to explain one way in which it may be extended to encompass the theory of real numbers, introducing the reals, by means of suitable further abstraction principles, as ratios of quantities.

The thesis that numbers are ratios of quantities has recently been advanced by a number of philosophers. While adequate as a definition of the natural numbers, it is not clear that this view suffices for our understanding of the reals. These require continuous quantity and relative to any such quantity an infinite number of additive relations exist. Hence, for any two magnitudes of a continuous quantity there exists no unique ratio. This problem is overcome by defining ratios, and hence real numbers, as binary relations between infinite standard sequences. This definition leads smoothly into a new definition of measurement consonant with the traditional view of measurement as the discovery or estimation of numerical relations. The traditional view is further strengthened by allowing that the additive relations internal to a quantity are distinct from relations observed in the behaviour of objects manifesting quantities. In this way the traditional theory can accommodate the theory of conjoint measurement. This is worth doing because the traditional theory has one great strength lacked by its rivals: measurement statements and quantitative laws are able to be understood literally. 1 This paper was completed in 1990-1. while the author was a visiting scholar at the Irvine Research Unit in Mathematical Behavioral Sciences. University of California. Irvine. The author wishes to thank the Director. Professor R. D. Luce, for the generous provision of space and facilities within the Unit and for his critical comments upon some of the ideas expressed herein: Professor L. Narens. for his trenchant criticisms: and the University of Sydney, for granting Special Study Leave and financial assistance to make the visit possible.

Quantities are naturally viewed as functions, whose arguments may be construed as situations, events, objects, etc. We explore the question of the range of these functions: should it be construed as the real numbers (or some subset thereof)? This is Carnap's view. It has attractive features, specifically, what Carnap views as ontological economy. Or should the range of a quantity be a set of magnitudes? This may have been Helmholtz's view, and it, too, has attractive features. It reveals the close connection between measurement and natural law, it makes dimensional analysis intelligible, and explains the concern of scientists and engineers with units in equations. It leaves the philosophical problem of the relation between the structure of magnitudes and the structure of the reals. What explains it? And is it always the same? We will argue that on the whole, construing the values of quantities as magnitudes has some advantages, and that (as Helmholtz seems to suggest in "Numbering and Measuring from an Epistemological Viewpoint") the relation between magnitudes and real numbers can be based on foundational similarities of structure.

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.