Combining Philosophers

All the ideas for Augustin-Louis Cauchy, Alistair Mitchell and Cian Dorr

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6 ideas

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals

18085	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

18084	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?

8. Modes of Existence / E. Nominalism / 1. Nominalism / c. Nominalism about abstracta

14596	Call 'nominalism' the denial of numbers, properties, relations and sets [Dorr]
	Full Idea: Just as there are no numbers or properties, there are no relations (like 'being heavier than' or 'betweenness'), or sets. I will provisionally use 'nominalism' for the conjunction of these four claims.
	From: Cian Dorr (There Are No Abstract Objects [2008], 1)
	A reaction: If you are going to be a nominalist, do it properly! My starting point in metaphysics is strong sympathy with this view. Right now [Tues 22nd Nov 2011, 10:57 am GMT] I think it is correct.

8. Modes of Existence / E. Nominalism / 5. Class Nominalism

14597	Natural Class Nominalism says there are primitive classes of things resembling in one respect [Dorr]
	Full Idea: Natural Class Nominalists take as primitive the notion of a 'natural' class - a class of things that all resemble one another in some one respect and resemble nothing else in that respect.
	From: Cian Dorr (There Are No Abstract Objects [2008], 4)
	A reaction: Dorr rejects this view because he doesn't believe in 'classes'. How committed to classes do you have to be before you are permitted to talk about them? All vocabulary (such as 'resemble') seems metaphysically tainted in this area.

10. Modality / A. Necessity / 11. Denial of Necessity

14598	Abstracta imply non-logical brute necessities, so only nominalists can deny such things [Dorr]
	Full Idea: If there are abstract objects, there are necessary truths about these things that cannot be reduced to truths of logic. So only the nominalist, who denies that there are any such things, can adequately respect the idea that there are no brute necessities.
	From: Cian Dorr (There Are No Abstract Objects [2008], 4)
	A reaction: This is where two plates of my personal philosophy grind horribly against one another. I love nominalism, and I love natural necessities. They meet like a ring-species in evolution. I'll just call it a 'paradox', and move on (swiftly).

14. Science / C. Induction / 3. Limits of Induction

7295	Maybe induction is only reliable IF reality is stable [Mitchell,A]
	Full Idea: Maybe we should say that IF regularities are stable, only then is induction a reliable procedure.
	From: Alistair Mitchell (talk [2006]), quoted by PG - Db (ideas)
	A reaction: This seems to me a very good proposal. In a wildly unpredictable reality, it is hard to see how anyone could learn from experience, or do any reasoning about the future. Natural stability is the axiom on which induction is built.