Aristotle claimed that the continuum, that is say a line, is not solely composed of points. Modern mathematics would agree, they additionally impose a topology to achieve cohesion.

Have their been any other philosophers who have discussed the continuum?

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    Modern mathematicians do not agree, Aristotle's idea is not related to topology. – Ron Maimon Oct 8 '12 at 4:17
  • To elaborate on Ron Maimon's comment: the topology is not associated to just the line, but to the space in which the line is defined. None of the ordinals with cardinality equal to the real numbers (assuming the axiom of choice, so that that collection of ordinals is well defined) are lines in any conventional sense, for instance, despite being "linearly ordered", because their elements are not elements of a topological space. And if you do define a topology, it is a property of the space, not the lines per se. A line worth the name is indeed a set of points — in an appropriate context. – Niel de Beaudrap Oct 16 '12 at 13:26

The theory of the continuum is central in Brouwer's intuitionism. See for example the excerpts in Mancosu (ed.), From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s. From my review of this book: A flavour of Brouwer's theory of the continuum is conveyed by its most conspicuous theorem: all functions are uniformly continuous (p. 39). The simplest classical counterexample is a function with y=0 if x<0 and y=1 otherwise. There are several good arguments for why the intuitionistic theorem is in better accord with intuition. First, the classical counterexample relies on "magical language" since, constructively, the function cannot be evaluated; for example, there are numbers that are "neither equal to 0 nor distinct from 0" (p. 51). Second, the theorem implies that the continuum cannot be split, in accordance with intuition (whereas it is split in the classical counterexample). "Indeed, Democritus argues with good reason that if I can break a stick, then it was from the outset not a whole. Strictest atomism is the inescapable conclusion of this." (Weyl, p. 135; see p. 124).

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A continuum is defined, for example, in Michael Potter's Set Theory and its Philosophy (on page 121). Note that he defines a continuum as a line (that is, a totally ordered set in which every open interval is non-empty) that has a countable dense subset and satisfies the least upper bound property. This is slightly stronger than Wikipedia's definition, which doesn't require the countable dense subset.

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