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Title: Mathematical continua & the intuitive idea of continuity
Author: Hudry, Jean-Louis
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2006
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How does philosophy understand the concept of continuity? The intuitive idea of continuity is about perceptual smoothness; but what looks smooth may be discontinuous, meaning that phenomenal continuity does not constitute a reliable definition. Metaphysics speaks of continuants with respect to temporal parts, but does not provide a definition of continuity. When properly defined, it is then associated with a minimal change divided into infinitesimal parts, which is an implicit reference to Leibniz's law of continuity such that a continuous change pertains to a geometric graph differentiable at arbitrary points. Yet, does it make sense to define continuity by means of discontinuous points? We must view Leibniz's definition as a transitory stage between two contradictory concepts, i.e. geometric and arithmetical continua. While Aristotle shows that a continuous line is infinitely divisible into lines, Dedekind defines an arithmetical continuum (or real line) as a complete domain of real numbers. This distinction opposes the intuitive idea of a smooth extension to a discontinuous and extensionless sequence of numbers, meaning that algebraic formalisms do not solve Zeno's geometric paradoxes but make them irrelevant. The consequences for physical continuity are such that an Aristotelian time is a smooth temporal interval devoid of indivisible parts; namely, instants of time are abstract limits and not physical durations. Arithmetical continuity defines a continuous time as isomorphic to a set of real numbers, but the measure of this extensionless structure is physically meaningless, and there is no physical argument to claim that a continuous time is a better model than a discrete time. Arithmetical continuity is omnipresent in modern mathematics; yet, it is fraught with difficulties in relation to the infinite. Cantor distinguishes an infinitely countable set of natural (or rational) numbers from an infinitely uncountable continuum. These infinite cardinalities imply the 'axiom' of choice, such that it is always possible to choose a unique element in a set over an infinite collection of disjoint, non-empty sets. Brouwer rejects this postulate because based on the unjustified idea that the infinite has a same ordering as the finite. He then claims that only infinitely incomplete sequences can be generated, since the nature of the infinite is to be merely potential. Others directly challenge arithmetic. C.S. Peirce suggests a topological geometry devoid of discrete numbers; however, it is clear that modern topology rests on an arithmetical ordering of real numbers and cannot be defined as pure geometry. More recently, J.L. Bell rejects the intuitive discontinuity of algebraic structures by defending an axiomatic system of smooth infinitesimals; yet, the identification of axiomatic smoothness with intuition neglects the necessity for any axiomatic property to belong to the axioms alone. Still, the construction of an axiomatic system can help us defend arithmetical continuity. Hilbert shows that a Euclidean model of geometry is isomorphic to an algebraic model, such that the axiom of continuity is satisfiable in either model. As for the absolute consistency of the axiomatic system, it requires a metamathematics, which aims to demonstrate the arithmetical infinite on finite logical grounds. First-order logic fails to define a continuum as a concrete object, since the uncountable set of all countable subsets is independent of any logic whose models have only countable domains (Lowenheim-Skolem theorem). By contrast, second-order logic makes sense of a continuum as an abstract set, which means that arithmetical continuity is nothing more than an ideal, hypothetical abstraction.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available