Combining Texts

All the ideas for 'fragments/reports', 'On the Reality of Accidents' and 'Paradoxes of the Infinite'

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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.

8. Modes of Existence / E. Nominalism / 4. Concept Nominalism

19382	Abstracta are abbreviated ways of talking; there are just substances, and truths about them [Leibniz]
	Full Idea: I consider abstracta not as real things but as abbreviated ways of talking ...and to that extent I am a nominalist, at least provisionally ...It suffices to posit only substances as real things, and, to assert truths about these.
	From: Gottfried Leibniz (On the Reality of Accidents [1688]), quoted by Richard T.W. Arthur - Leibniz
	A reaction: I am a modern nominalist, in my hostility to a serious ontological commitment to abstracta. You get into trouble, though, if you say there are only objects or substances. Physics says reality may all be 'fields', or something.... 'Truths' is good.

21. Aesthetics / C. Artistic Issues / 7. Art and Morality

468	Musical performance can reveal a range of virtues [Damon of Ath.]
	Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice.
	From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where?