8) in the cases of surfaces and bodies, the common boundaries are lines, and lines or surfaces, respectively, and in the case of time they are moments. Lines, in particular, are continuous because "it is possible to find a common boundary at which its parts join together, a point" ( Categories 6, 5 a1 –2, in Aristotle 1984, p. For Aristotle, number -by which he meant the positive integers greater than or equal to two -is discrete, whereas measurable magnitude -lines, surfaces, bodies, time, and place -are continuous. 371)." However, while ingredients of the standard ancient conceptionare already found in the writings of some of the pre-Socratics, it was Aristotle (384 –322 BCE), inspired by the writings of the geometers of his day, who provided its earliest systematic philosophical treatment.Ĭentral to Aristotle's analysis is the distinction between discrete and continuous quantity whereas the former lack, the latter have, a common boundary at which the parts join to form a unity. Thus, not only is the continuum infinitely divisible, but through the process of division it cannot be reduced to discrete indivisible elements that are, as Anaxagoras picturesquely put it, "separated from one another as if cut off with an axe (ibid. 500 –428 BCE) when he observed that "Neither is there a smallest part of what is small, but there is always a smaller (for it is impossible that what is should cease be )" (Kirk, Raven, and Schofield 1983, p. Some of the essential characteristics of what emerged as the standard ancient conception were already described by Anaxagoras of Clazomenae (c. The Aristotelian Conceptionīefore the Cantor-Dedekind philosophy the idea of the continuum stood in direct opposition to the discrete and was generally thought to be grounded in our intuition of extensive magnitude, in particular of spatial or temporal magnitude, and of the motion of bodies through space. These three periods are considered in this entry in historical turn. Speculation regarding the nature and structure of continua and of continuous phenomena more generally therefore naturally falls into three periods: the period of the emergence and eventual domination of the Cantor-Dedekind philosophy, and the periods before and after. Whereas the constructivist and predicativist theories have their roots in the early twentieth-century debates on the foundations of mathematics and were born from critiques of the Cantor-Dedekind theory, the infinitesimalist theories were intended to either provide intuitively satisfying (and,in some cases, historically rooted) alternatives to the Cantor-Dedekind conception that have the power to meet the needs of analysis or differential geometry, or to situate the Cantor-Dedekind system of real numbers in a grander conception of an arithmetic continuum. The period that has transpired since its emergence as the standard philosophy has been especially fruitful in this regard, having witnessed the rise of a variety of constructivist and predicativist theories of real numbers and corresponding theories of analysis as well as the emergence of a number of alternative theories that make use of infinitesimals. Since its inception, however, there has never been a time at which the Cantor-Dedekind philosophy has either met with universal acceptance or has been without competitors. During this period the Cantor-Dedekind philosophy of the continuum also emerged as a pillar of standard mathematical philosophy that underlies the standard formulation of analysis, the standard analytic and synthetic theories of the geometrical linear continuum, and the standard axiomatic theories of continuous magnitude more generally. Since their appearance, the late nineteenth-century constructions of the real numbers have undergone set-theoretical and logical refinement, and the systems of rational and integer numbers on which they are based have themselves been given a set-theoretic foundation. In honor of Georg Cantor (1845 –1918) and Richard Dedekind (1831 –1916), who first proposed this mathematico-philosophical thesis, the presumed correspondence between the two structures is sometimes called the Cantor-Dedekind axiom. In accordance with this view, the geometric linear continuum is assumed to be isomorphic with the arithmetic continuum, the axioms of geometry being so selected to ensure this would be the case. In the decades bracketing the turn of the twentieth century, the real number system was dubbed the arithmetic continuum because it was held that this number system is completely adequate for the analytic representation of all types of continuous phenomena.
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