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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility

[endless dividing an interval between numbers]

6 ideas
Lengths do not contain infinite parts; parts are created by acts of division [Aristotle, by Le Poidevin]
     Full Idea: Aristotle says that a length does not already contain, waiting to be discovered, an infinite number of parts; such parts only come into existence once they are defined by an act of division.
     From: report of Aristotle (Physics [c.337 BCE]) by Robin Le Poidevin - Travels in Four Dimensions 07 'Two'
     A reaction: If that is true of infinite parts then it must also be true of finite parts. So a cake has no parts at all until it is cut. That could play merry hell with discussions of mereology. Wholes are ontologically prior to parts.
A continuous line cannot be composed of indivisible points [Aristotle]
     Full Idea: No continuum can be composed of indivisibles: e.g. a line cannot be composed of points, the line being continuous and the points indivisibles.
     From: Aristotle (Physics [c.337 BCE], 231a23), quoted by Ian Rumfitt - The Boundary Stones of Thought 7.4
     A reaction: Rumfitt observes that ' the basic problem is to say what the ultimate parts of a continuum are, of they are not points'. Early modern philosophers had lots of proposals.
The continuum is not divided like sand, but folded like paper [Leibniz, by Arthur,R]
     Full Idea: Leibniz said the division of the continuum should not be conceived 'to be like the division of sand into grains, but like that of a tunic or a sheet of paper into folds'.
     From: report of Gottfried Leibniz (works [1690], A VI iii 555) by Richard T.W. Arthur - Leibniz
     A reaction: This from the man who invented calculus. This thought might apply well to the modern physicist's concept of a 'field'.
There is no continuum in reality to realise the infinitely small [Hilbert]
     Full Idea: A homogeneous continuum which admits of the sort of divisibility needed to realise the infinitely small is nowhere to be found in reality.
     From: David Hilbert (On the Infinite [1925], p.186)
     A reaction: He makes this remark as a response to Planck's new quantum theory (the year before the big works of Heisenberg and Schrödinger). Personally I don't see why infinities should depend on the physical world, since they are imaginary.
Between any two rational numbers there is an infinite number of rational numbers [Friend]
     Full Idea: Since between any two rational numbers there is an infinite number of rational numbers, we could consider that we have infinity in three dimensions: positive numbers, negative numbers, and the 'depth' of infinite numbers between any rational numbers.
     From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
     A reaction: This is before we even reach Cantor's staggering infinities (Ideas 8662 and 8663), which presumably reside at the outer reaches of all three of these dimensions of infinity. The 'deep' infinities come from fractions with huge denominators.
Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan]
     Full Idea: The problem with infinitesimals is that in some places they behaved like real numbers close to zero but in other places they behaved like zero.
     From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.2)
     A reaction: Colyvan gives an example, of differentiating a polynomial.