Combining Texts

All the ideas for 'Science and Method', 'Mathematics: Form and Function' and 'Summa'

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4 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets

18189	ZFC could contain a contradiction, and it can never prove its own consistency [MacLane]
	Full Idea: We have at hand no proof that the axioms of ZFC for set theory will never yield a contradiction, while Gödel's second theorem tells us that such a consistency proof cannot be conducted within ZFC.
	From: Saunders MacLane (Mathematics: Form and Function [1986], p.406), quoted by Penelope Maddy - Naturalism in Mathematics
	A reaction: Maddy quotes this, while defending set theory as the foundation of mathematics, but it clearly isn't the most secure foundation that could be devised. She says the benefits of set theory do not need guaranteed consistency (p.30).

6. Mathematics / A. Nature of Mathematics / 2. Geometry

10245	One geometry cannot be more true than another [Poincaré]
	Full Idea: One geometry cannot be more true than another; it can only be more convenient.
	From: Henri Poincaré (Science and Method [1908], p.65), quoted by Stewart Shapiro - Philosophy of Mathematics
	A reaction: This is the culminating view after new geometries were developed by tinkering with Euclid's parallels postulate.

9. Objects / B. Unity of Objects / 2. Substance / c. Types of substance

16616	Substances 'substand' (beneath accidents), or 'subsist' (independently) [Eustachius]
	Full Idea: It is proper to substance both to stretch out or exist beneath accidents, which is to substand, and to exist per se and not in another, which is to subsist.
	From: Eustachius a Sancto Paulo (Summa [1609], I.1.3b.1.2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 06.2
	A reaction: This reflects Aristotle wavering between 'ousia' being the whole of a thing, or the substrate of a thing. In current discussion, 'substance' still wavers between a thing which 'is' a substance, and substance being the essence.

9. Objects / C. Structure of Objects / 2. Hylomorphism / a. Hylomorphism

16585	Prime matter is free of all forms, but has the potential for all forms [Eustachius]
	Full Idea: Everyone says that prime matter, considered in itself, is free of all forms and at the same time is open to all forms - or, that matter is in potentiality to all forms.
	From: Eustachius a Sancto Paulo (Summa [1609], III.1.1.2.3), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 03.1
	A reaction: This is the notorious doctrine developed to support the hylomorphic picture derived from Aristotle. No one could quite figure out what prime matter was, so it faded away.