Combining Texts

All the ideas for 'Dthat', 'Paradoxes of the Infinite' and 'Elements of Law and Justice'

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4 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory

9987	An aggregate in which order does not matter I call a 'set' [Bolzano]
	Full Idea: An aggregate whose basic conception renders the arrangement of its members a matter of indifference, and whose permutation therefore produces no essential difference, I call a 'set'.
	From: Bernard Bolzano (Paradoxes of the Infinite [1846], §4), quoted by William W. Tait - Frege versus Cantor and Dedekind IX
	A reaction: The idea of 'sets' was emerging before Cantor formalised it, and clarified it by thinking about infinite sets. Nowadays we also have 'ordered' sets, which rather contradicts Bolzano, and we also expect the cardinality to be determinate.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite

10856	A truly infinite quantity does not need to be a variable [Bolzano]
	Full Idea: A truly infinite quantity (for example, the length of a straight line, unbounded in either direction) does not by any means need to be a variable.
	From: Bernard Bolzano (Paradoxes of the Infinite [1846]), quoted by Brian Clegg - Infinity: Quest to Think the Unthinkable §10
	A reaction: This is an important idea, followed up by Cantor, which relegated to the sidelines the view of infinity as simply something that could increase without limit. Personally I like the old view, but there is something mathematically stable about infinity.

14. Science / C. Induction / 2. Aims of Induction

19387	Hypotheses come from induction, which is comparison of experiences [Leibniz]
	Full Idea: We construct a hypothesis on the basis of an induction, that is on the basis of a comparison of experiences.
	From: Gottfried Leibniz (Elements of Law and Justice [1669], p.2)
	A reaction: This fits the traditional positivist picture of science (observe-hypothesise-predict-observe). I like the definition of induction as 'comparison of experiences', because it doesn't reduce it to sequences of objects, and points to coherence.

19. Language / B. Reference / 3. Direct Reference / b. Causal reference

14080	Are causal descriptions part of the causal theory of reference, or are they just metasemantic? [Kaplan, by Schaffer,J]
	Full Idea: Kaplan notes that the causal theory of reference can be understood in two quite different ways, as part of the semantics (involving descriptions of causal processes), or as metasemantics, explaining why a term has the referent it does.
	From: report of David Kaplan (Dthat [1970]) by Jonathan Schaffer - Deflationary Metaontology of Thomasson 1
	A reaction: [Kaplan 'Afterthought' 1989] The theory tends to be labelled as 'direct' rather than as 'causal' these days, but causal chains are still at the heart of the story (even if more diffused socially). Nice question. Kaplan takes the meta- version as orthodox.