Combining Texts

All the ideas for 'Identity and Spatio-Temporal Continuity', 'Building Blocks of Mathematical Logic' and 'On the Foundations of Logic and Arithmetic'

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4 ideas

5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic

17699	Variables are auxiliary notions, and not part of the 'eternal' essence of logic [Schönfinkel]
	Full Idea: A variable in a proposition of logic ....has the status of a mere auxiliary notion that is really inappropriate to the constant, 'eternal' essence of the propositions of logic.
	From: Moses Schönfinkel (Building Blocks of Mathematical Logic [1924], §1)
	A reaction: He presumably thinks that what the variables stand for (and he mentions 'argument places' and 'operators') will be included in the essence. My attention was caught by the thought that he takes logic to have an essence.

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism

17697	The existence of an arbitrarily large number refutes the idea that numbers come from experience [Hilbert]
	Full Idea: The standpoint of pure experience seems to me to be refuted by the objection that the existence, possible or actual, of an arbitrarily large number can never be derived through experience, that is, through experiment.
	From: David Hilbert (On the Foundations of Logic and Arithmetic [1904], p.130)
	A reaction: Alternatively, empiricism refutes infinite numbers! No modern mathematician will accept that, but you wonder in what sense the proposed entities qualify as 'numbers'.

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

17698	Logic already contains some arithmetic, so the two must be developed together [Hilbert]
	Full Idea: In the traditional exposition of the laws of logic certain fundamental arithmetic notions are already used, for example in the notion of set, and to some extent also of number. Thus we turn in a circle, and a partly simultaneous development is required.
	From: David Hilbert (On the Foundations of Logic and Arithmetic [1904], p.131)
	A reaction: If the Axiom of Infinity is meant, it may be possible to purge the arithmetic from the logic. Then the challenge to derive arithmetic from it becomes rather tougher.

9. Objects / A. Existence of Objects / 5. Individuation / e. Individuation by kind

13128	'Ultimate sortals' cannot explain ontological categories [Westerhoff on Wiggins]
	Full Idea: 'Ultimate sortals' are said to be non-subordinated, disjoint from one another, and uniquely paired with each object. Because of this, the ultimate sortal cannot be a satisfactory explication of the notion of an ontological category.
	From: comment on David Wiggins (Identity and Spatio-Temporal Continuity [1971], p.75) by Jan Westerhoff - Ontological Categories §26
	A reaction: My strong intuitions are that Wiggins is plain wrong, and Westerhoff gives the most promising reasons for my intuition. The simplest point is that objects can obviously belong to more than one category.