Combining Texts

All the ideas for 'fragments/reports', 'Set Theory and the Continuum Hypothesis' and 'Commentary on Euclid's 'Elements''

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4 ideas

1. Philosophy / D. Nature of Philosophy / 3. Philosophy Defined

23367	Even pointing a finger should only be done for a reason [Epictetus]
	Full Idea: Philosophy says it is not right even to stretch out a finger without some reason.
	From: Epictetus (fragments/reports [c.57], 15)
	A reaction: The key point here is that philosophy concerns action, an idea on which Epictetus is very keen. He rather despise theory. This idea perfectly sums up the concept of the wholly rational life (which no rational person would actually want to live!).

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism

14248	We could accept the integers as primitive, then use sets to construct the rest [Cohen]
	Full Idea: A very reasonable position would be to accept the integers as primitive entities and then use sets to form higher entities.
	From: Paul J. Cohen (Set Theory and the Continuum Hypothesis [1966], 5.4), quoted by Oliver,A/Smiley,T - What are Sets and What are they For?
	A reaction: I find this very appealing, and the authority of this major mathematician adds support. I would say, though, that the integers are not 'primitive', but pick out (in abstraction) consistent features of the natural world.

14. Science / D. Explanation / 2. Types of Explanation / g. Causal explanations

13165	Geometrical proofs do not show causes, as when we prove a triangle contains two right angles [Proclus]
	Full Idea: Geometry does not ask 'why?' ..When from the exterior angle equalling two opposite interior angles it is shown that the interior angles make two right angles, this is not a causal demonstration. With no exterior angle they still equal two right angles.
	From: Proclus (Commentary on Euclid's 'Elements' [c.452], p.161-2), quoted by Paolo Mancosu - Explanation in Mathematics §5
	A reaction: A very nice example. It is hard to imagine how one might demonstrate the cause of the angles making two right angles. If you walk, turn left x°, then turn left y°, then turn left z°, and x+y+z=180°, you end up going in the original direction.

18. Thought / E. Abstraction / 1. Abstract Thought

9569	The origin of geometry started in sensation, then moved to calculation, and then to reason [Proclus]
	Full Idea: It is unsurprising that geometry was discovered in the necessity of Nile land measurement, since everything in the world of generation goes from imperfection to perfection. They would naturally pass from sense-perception to calculation, and so to reason.
	From: Proclus (Commentary on Euclid's 'Elements' [c.452]), quoted by Charles Chihara - A Structural Account of Mathematics 9.12 n55
	A reaction: The last sentence is the core of my view on abstraction, that it proceeds by moving through levels of abstraction, approaching more and more general truths.