21382 | 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). |
18081 | 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. |
18080 | 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] |
18091 | 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] |
18085 | 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. |
18086 | 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 |
18110 | 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) |
18083 | 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) |
18834 | 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) |