Combining Texts

All the ideas for 'Mahaprajnaparamitashastra', 'Causes and Counterfactuals' and 'Continuity and Irrational Numbers'

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10 ideas

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]

23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues

7903	The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]
	Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom.
	From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88)
	A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate').

26. Natural Theory / C. Causation / 1. Causation

8430	Causal statements are used to explain, to predict, to control, to attribute responsibility, and in theories [Kim]
	Full Idea: The function of causal statements is 1) to explain events, 2) for predictive usefulness, 3) to help control events, 4) with agents, to attribute moral responsibility, 5) in physical theory. We should judge causal theories by how they account for these.
	From: Jaegwon Kim (Causes and Counterfactuals [1973], p.207)
	A reaction: He suggests that Lewis's counterfactual theory won't do well on this test. I think the first one is what matters. Philosophy aims to understand, and that is achieved through explanation. Regularity and counterfactual theories explain very little.

26. Natural Theory / C. Causation / 9. General Causation / c. Counterfactual causation

8396	Many counterfactuals have nothing to do with causation [Kim, by Tooley]
	Full Idea: Kim has pointed out that there are a number of counterfactuals that have nothing to do with causation. If John marries Mary, then if John had not existed he would not have married Mary, but that is not the cause of their union.
	From: report of Jaegwon Kim (Causes and Counterfactuals [1973], 5.2) by Michael Tooley - Causation and Supervenience
	A reaction: One might not think that this mattered, but it leaves the problem of distinguishing between the causal counterfactuals and the rest (and you mustn't mention causation when you are doing it!).

8429	Counterfactuals can express four other relations between events, apart from causation [Kim]
	Full Idea: Counterfactuals can express 'analytical' dependency, or the fact that one event is part of another, or an action done by doing another, or (most interestingly) an event can determine another without causally determining it.
	From: Jaegwon Kim (Causes and Counterfactuals [1973], p.205)
	A reaction: [Kim gives example of each case] Counterfactuals can even express a relation that involves no dependency. Or they might just involve redescription, as in 'If Scott were still alive, then the author of "Waverley" would be too'.

8428	Causation is not the only dependency relation expressed by counterfactuals [Kim]
	Full Idea: The sort of dependency expressed by counterfactual relations is considerably broader than strictly causal dependency, and causal dependency is only one among the heterogeneous group of dependency relationships counterfactuals can express.
	From: Jaegwon Kim (Causes and Counterfactuals [1973], p.205)
	A reaction: In 'If pigs could fly, one and one still wouldn't make three' there isn't even a dependency. Kim has opened up lines of criticism which make the counterfactual analysis of causation look very implausible to me.

26. Natural Theory / D. Laws of Nature / 9. Counterfactual Claims

4781	Many counterfactual truths do not imply causation ('if yesterday wasn't Monday, it isn't Tuesday') [Kim, by Psillos]
	Full Idea: Kim gives a range of examples of counterfactual dependence without causation, as: 'if yesterday wasn't Monday, today wouldn't be Tuesday', and 'if my sister had not given birth, I would not be an uncle'.
	From: report of Jaegwon Kim (Causes and Counterfactuals [1973]) by Stathis Psillos - Causation and Explanation §3.3
	A reaction: This is aimed at David Lewis. The objection seems like commonsense. "If you blink, the cat gets it". Causal claims involve counterfactuals, but they are not definitive of what causation is.