Challenging the myth that mathematical objects can be defined into existence, Bigelow here employs Armstrong's metaphysical materialism to cast new light on mathematics. He identifies natural, real, and imaginary numbers and sets with specified physical properties and relations and, by so doing, draws mathematics back from its sterile, abstract exile into the midst of the physical world.

A formal theory of quantity T Q is presented which is realist, Platonist, and syntactically second-order (while logically elementary), in contrast with the existing formal theories of quantity developed within the theory of measurement, which are empiricist, nominalist, and syntactically first-order (while logically non-elementary). T Q is shown to be formally and empirically adequate as a theory of quantity, and is argued to be scientifically superior to the existing first-order theories of quantity in that it does not depend upon empirically (...) unsupported assumptions concerning existence of physical objects (e.g. that any two actual objects have an actual sum). The theory T Q supports and illustrates a form of naturalistic Platonism, for which claims concerning the existence and properties of universals form part of natural science, and the distinction between accidental generalizations and laws of nature has a basis in the second-order structure of the world. (shrink)

The nature of measurement is a topic of central concern in the philosophy of science and, indeed, measurement is the essential link between science and mathematics. Professor Ellis's book, originally published in 1966, is the first general exposition of the philosophical and logical principles involved in measurement since N. R. Campbell's Principles of Measurement and Calculation, and P. W. Bridgman's Dimensional Analysis. Professor Ellis writes from an empiricist standpoint. His object is to distinguish and define the basic concepts in measurement, (...) for example: scale, quantity, unit. dimension, number and probability. He discusses the problem of classifying scales of measurement and the special logical problems associated with each kind of scale. A translation of mach's Critique on the Concept of Temperature, which gives his views on the nature of measurement more fully than in any of his other works, is given as an appendix. (shrink)

Developing some suggestions of Ramsey (1925), elementary logic is formulated with respect to an arbitrary categorial system rather than the categorial system of Logical Atomism which is retained in standard elementary logic. Among the many types of non-standard categorial systems allowed by this formalism, it is argued that elementary logic with predicates of variable degree occupies a distinguished position, both for formal reasons and because of its potential value for application of formal logic to natural language and natural science. This (...) is illustrated by use of such a logic to construct a theory of quantity which is argued to be scientifically superior to existing theories of quantity based on standard categorial systems, since it yields real-valued scales without the need for unrealistic existence assumptions. This provides empirical evidence for the hypothesis that the categorial structure of the physical world itself is non-standard in this sense. (shrink)