Combining Philosophers

All the ideas for Oswald Veblen, Hugh MacColl and Proclus

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4 ideas

4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set

21548	The null class is the class with all the non-existents as its members [MacColl, by Lackey]
	Full Idea: In 1905 the Scottish logician Hugh MacColl published a paper in which he argued that the null class in logic should be taken as the class with all the non-existents as its members.
	From: report of Hugh MacColl (Symbolic Reasoning [1905]) by Douglas Lackey - Intros to Russell's 'Essays in Analysis' p.95
	A reaction: For the null object (zero) Frege just chose one sample concept with an empty extension. MacColl's set seems to have a lot of members, given that it is 'null'. How many, I wonder? Russell responded to this paper.

5. Theory of Logic / A. Overview of Logic / 2. History of Logic

10247	We have no adequate logic at the moment, so mathematicians must create one [Veblen]
	Full Idea: Formal logic has to be taken over by mathematicians. The fact is that there does not exist an adequate logic at the present time, and unless the mathematicians create one, no one else is likely to do so.
	From: Oswald Veblen (Presidential Address of Am. Math. Soc [1924], 141), quoted by Stewart Shapiro - Philosophy of Mathematics
	A reaction: This remark was made well after Frege, but before the advent of Gödel and Tarski. That implies that he was really thinking of meta-logic.

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.