Combining Texts

All the ideas for 'Science and Method', 'Rechnungsmethoden (dissertation)' and 'A Problem about Substitutional Quantification?'

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4 ideas

5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
10792 The substitutional quantifier is not in competition with the standard interpretation [Kripke, by Marcus (Barcan)]
     Full Idea: Kripke proposes that the substitutional quantifier is not a replacement for, or in competition with, the standard interpretation.
     From: report of Saul A. Kripke (A Problem about Substitutional Quantification? [1976]) by Ruth Barcan Marcus - Nominalism and Substitutional Quantifiers p.165
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.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
18256 Quantity is inconceivable without the idea of addition [Frege]
     Full Idea: There is so intimate a connection between the concepts of addition and of quantity that one cannot begin to grasp the latter without the former.
     From: Gottlob Frege (Rechnungsmethoden (dissertation) [1874], p.2), quoted by Michael Dummett - Frege philosophy of mathematics 22 'Quantit'
     A reaction: Frege offers good reasons for making cardinals prior to ordinals, though plenty of people disagree.
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
9831 Geometry appeals to intuition as the source of its axioms [Frege]
     Full Idea: The elements of all geometrical constructions are intuitions, and geometry appeals to intuition as the source of its axioms.
     From: Gottlob Frege (Rechnungsmethoden (dissertation) [1874], Ch.6), quoted by Michael Dummett - Frege philosophy of mathematics
     A reaction: Very early Frege, but he stuck to this view, while firmly rejecting intuition as a source of arithmetic. Frege would have known well that Euclid's assumption about parallels had been challenged.