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All the ideas for 'Cours d'Analyse', 'Expositio super viii libros' and 'works'

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6 ideas

5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
The values of variables can't determine existence, because they are just expressions [Ryle, by Quine]
     Full Idea: Ryle objected somewhere to my dictum that 'to be is to be the value of a variable', arguing that the values of variables are expressions, and hence that my dictum repudiates all things except expressions.
     From: report of Gilbert Ryle (works [1950]) by Willard Quine - Reply to Professor Marcus p.183
     A reaction: I have a lot of sympathy with Ryle's view, and I associate it with the peculiar Millian view that we can somehow replace a name in a sentence with the actual physical object. Objects can't be parts of sentences - and maybe they can't be 'values'.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
Values that approach zero, becoming less than any quantity, are 'infinitesimals' [Cauchy]
     Full Idea: When the successive absolute values of a variable decrease indefinitely in such a way as to become less than any given quantity, that variable becomes what is called an 'infinitesimal'. Such a variable has zero as its limit.
     From: Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4
     A reaction: The creator of the important idea of the limit still talked in terms of infinitesimals. In the next generation the limit took over completely.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
When successive variable values approach a fixed value, that is its 'limit' [Cauchy]
     Full Idea: When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the 'limit' of all the others.
     From: Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4
     A reaction: This seems to be a highly significan proposal, because you can now treat that limit as a number, and adds things to it. It opens the door to Cantor's infinities. Is the 'limit' just a fiction?
11. Knowledge Aims / A. Knowledge / 1. Knowledge
Knowledge is a quality existing subjectively in the soul [William of Ockham]
     Full Idea: Knowledge is a certain quality which exists in the soul as its subject ('existens subiective in anima').
     From: William of Ockham (Expositio super viii libros [1340], Prologue)
     A reaction: One might say here that knowledge is a property, and so it might not be susceptible to further analysis. It invites the question of how you could know by introspection that you have got it, which would be an extreme internalist view.
Sometimes 'knowledge' just concerns the conclusion, sometimes the whole demonstration [William of Ockham]
     Full Idea: Sometimes 'knowledge' means evident cognition of the conclusion alone, sometimes of the demonstration as a whole.
     From: William of Ockham (Expositio super viii libros [1340], Prologue)
     A reaction: 'Demonstration' will be something like Greek 'logos' - full understanding, ability to explain and give reasons. William is certainly right about normal usage. I know the answer in a quiz, without any requirement for justifications.
11. Knowledge Aims / B. Certain Knowledge / 1. Certainty
Knowledge is certain cognition of something that is true [William of Ockham]
     Full Idea: Knowledge is certain cognition of something that is true.
     From: William of Ockham (Expositio super viii libros [1340], Prologue)
     A reaction: This view has problems. William is not facing up to the sceptical questions which can shake any degree of certainty, and also that someone who lacked self-confidence might know many things while always feeling uncertain about them. 'Cognition' must go!