Combining Texts

All the ideas for 'What is so bad about Contradictions?', 'Continuity and Irrational Numbers' and 'The Poetics'

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7 ideas

2. Reason / B. Laws of Thought / 3. Non-Contradiction

9123	Someone standing in a doorway seems to be both in and not-in the room [Priest,G, by Sorensen]
	Full Idea: Priest says there is room for contradictions. He gives the example of someone in a doorway; is he in or out of the room. Given that in and out are mutually exclusive and exhaustive, and neither is the default, he seems to be both in and not in.
	From: report of Graham Priest (What is so bad about Contradictions? [1998]) by Roy Sorensen - Vagueness and Contradiction 4.3
	A reaction: Priest is a clever lad, but I don't think I can go with this. It just seems to be an equivocation on the word 'in' when applied to rooms. First tell me the criteria for being 'in' a room. What is the proposition expressed in 'he is in the room'?

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers

17611	We want the essence of continuity, by showing its origin in arithmetic [Dedekind]
	Full Idea: It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity.
	From: Richard Dedekind (Continuity and Irrational Numbers [1872], Intro)
	A reaction: [He seeks the origin of the theorem that differential calculus deals with continuous magnitude, and he wants an arithmetical rather than geometrical demonstration; the result is his famous 'cut'].

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts

10572	A cut between rational numbers creates and defines an irrational number [Dedekind]
	Full Idea: Whenever we have to do a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut.
	From: Richard Dedekind (Continuity and Irrational Numbers [1872], §4)
	A reaction: Fine quotes this to show that the Dedekind Cut creates the irrational numbers, rather than hitting them. A consequence is that the irrational numbers depend on the rational numbers, and so can never be identical with any of them. See Idea 10573.

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic

17612	Arithmetic is just the consequence of counting, which is the successor operation [Dedekind]
	Full Idea: I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself is nothing else than the successive creation of the infinite series of positive integers.
	From: Richard Dedekind (Continuity and Irrational Numbers [1872], §1)
	A reaction: Thus counting roots arithmetic in the world, the successor operation is the essence of counting, and the Dedekind-Peano axioms are built around successors, and give the essence of arithmetic. Unfashionable now, but I love it. Intransitive counting?

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits

18087	If x changes by less and less, it must approach a limit [Dedekind]
	Full Idea: If in the variation of a magnitude x we can for every positive magnitude δ assign a corresponding position from and after which x changes by less than δ then x approaches a limiting value.
	From: Richard Dedekind (Continuity and Irrational Numbers [1872], p.27), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.7
	A reaction: [Kitcher says he 'showed' this, rather than just stating it]

10. Modality / B. Possibility / 1. Possibility

22518	The actual must be possible, because it occurred [Aristotle]
	Full Idea: Actual events are evidently possible, otherwise they would not have occurred.
	From: Aristotle (The Poetics [c.347 BCE], 1451b18)
	A reaction: [quoted online by Peter Adamson] Seems like common sense, but it's important to have Aristotle assert it.

21. Aesthetics / B. Nature of Art / 8. The Arts / b. Literature

16566	Poetry is more philosophic than history, as it concerns universals, not particulars [Aristotle]
	Full Idea: Poetry is something more philosophic and of graver import than history, since its statements are rather of universals, whereas those of history are singulars.
	From: Aristotle (The Poetics [c.347 BCE], 1451b05)
	A reaction: Hm. Characters in great novels achieve universality by being representated very particularly. Great depth of mind seems required to be a poet, but less so for a historian (though there is, I presume, no upward limit on the possible level of thought).