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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals

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9 ideas
Things get smaller without end [Anaxagoras]
     Full Idea: Of the small there is no smallest, but always a smaller.
     From: Anaxagoras (fragments/reports [c.460 BCE], B03), quoted by Gregory Vlastos - The Physical Theory of Anaxagoras II
     A reaction: Anaxagoras seems to be speaking of the physical world (and probably writing prior to the emergence of atomism, which could have been a rebellion against he current idea).
Nature uses the infinite everywhere [Leibniz]
     Full Idea: Nature uses the infinite in everything it does.
     From: Gottfried Leibniz (works [1690]), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.1
     A reaction: [The quote can be tracked through Kitcher's footnote] He seems to have had in mind the infinitely small.
A tangent is a line connecting two points on a curve that are infinitely close together [Leibniz]
     Full Idea: We have only to keep in mind that to find a tangent means to draw a line that connects two points of a curve at an infinitely small distance.
     From: Gottfried Leibniz (works [1690]), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.1
     A reaction: [The quote can be tracked through Kitcher's footnote]
Infinitesimals are ghosts of departed quantities [Berkeley]
     Full Idea: The infinitesimals are the ghosts of departed quantities.
     From: George Berkeley (The Analyst [1734]), quoted by David Bostock - Philosophy of Mathematics 4.3
     A reaction: [A famous phrase, but as yet no context for it]
Values that approach zero, becoming less than any quantity, are 'infinitesimals' [Cauchy]
     Full Idea: When the successive absolute values of a variable decrease indefinitely in such a way as to become less than any given quantity, that variable becomes what is called an 'infinitesimal'. Such a variable has zero as its limit.
     From: Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4
     A reaction: The creator of the important idea of the limit still talked in terms of infinitesimals. In the next generation the limit took over completely.
Weierstrass eliminated talk of infinitesimals [Weierstrass, by Kitcher]
     Full Idea: Weierstrass effectively eliminated the infinitesimalist language of his predecessors.
     From: report of Karl Weierstrass (works [1855]) by Philip Kitcher - The Nature of Mathematical Knowledge 10.6
Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock]
     Full Idea: Non-standard natural numbers will yield non-standard rational and real numbers. These will include reciprocals which will be closer to 0 than any standard real number. These are like 'infinitesimals', so that notion is not actually a contradiction.
     From: David Bostock (Philosophy of Mathematics [2009], 5.5)
With infinitesimals, you divide by the time, then set the time to zero [Kitcher]
     Full Idea: The method of infinitesimals is that you divide by the time, and then set the time to zero.
     From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 10.2)
Infinitesimals do not stand in a determinate order relation to zero [Rumfitt]
     Full Idea: Infinitesimals do not stand in a determinate order relation to zero: we cannot say an infinitesimal is either less than zero, identical to zero, or greater than zero. ….Infinitesimals are so close to zero as to be theoretically indiscriminable from it.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.4)