Combining Texts

All the ideas for 'Letters to a German Princess', 'Abduction and Induction' and 'Logicism, Some Considerations (PhD)'

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7 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers

13412	Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf]
	Full Idea: Not all numbers could possibly have been learned à la Frege-Russell, because we could not have performed that many distinct acts of abstraction. Somewhere along the line a rule had to come in to enable us to obtain more numbers, in the natural order.
	From: Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.165)
	A reaction: Follows on from Idea 13411. I'm not sure how Russell would deal with this, though I am sure his account cannot be swept aside this easily. Nevertheless this seems powerful and convincing, approaching the problem through the epistemology.

13413	We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf]
	Full Idea: Both ordinalists and cardinalists, to account for our number words, have to account for the fact that we know so many of them, and that we can 'recognize' numbers which we've neither seen nor heard.
	From: Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.166)
	A reaction: This seems an important contraint on any attempt to explain numbers. Benacerraf is an incipient structuralist, and here presses the importance of rules in our grasp of number. Faced with 42,578,645, we perform an act of deconstruction to grasp it.

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers

13411	If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [Benacerraf]
	Full Idea: If we accept the Frege-Russell analysis of number (the natural numbers are the cardinals) as basic and correct, one thing which seems to follow is that one could know, say, three, seventeen, and eight, but no other numbers.
	From: Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.164)
	A reaction: It seems possible that someone might only know those numbers, as the patterns of members of three neighbouring families (the only place where they apply number). That said, this is good support for the priority of ordinals. See Idea 13412.

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

13415	An adequate account of a number must relate it to its series [Benacerraf]
	Full Idea: No account of an individual number is adequate unless it relates that number to the series of which it is a member.
	From: Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.169)
	A reaction: Thus it is not totally implausible to say that 2 is several different numbers or concepts, depending on whether you see it as a natural number, an integer, a rational, or a real. This idea is the beginning of modern structuralism.

14. Science / D. Explanation / 3. Best Explanation / a. Best explanation

14790	'Abduction' is beginning a hypothesis, particularly if it includes preference of one explanation over others [Peirce]
	Full Idea: The first starting of a hypothesis and the entertaining of it …is an inferential step which I propose to call 'abduction'. This will include a preference for any one hypothesis over others which would equally explain the facts.
	From: Charles Sanders Peirce (Abduction and Induction [1901], I)
	A reaction: I take there to be no more important function within human thought than the procedure by which we give preference to one particular explanation. It only makes sense, I think, if we take it as part of a coherence theory of justification.

14791	Abduction involves original suggestions, and not just the testing involved in induction [Peirce]
	Full Idea: It is of the nature of abduction to involve an original suggestion; while typical induction has no originality in it, but only tests a suggestion already made.
	From: Charles Sanders Peirce (Abduction and Induction [1901], I)
	A reaction: Peirce's 'abduction' is not, then, just the choice of a best explanation. He came up with the idea because he was keen to capture the creative and imaginative character of rational thought.

26. Natural Theory / D. Laws of Nature / 1. Laws of Nature

5467	Euler said nature is instrinsically passive, and minds cause change [Euler, by Ellis]
	Full Idea: Euler thought the powers necessary for the maintenance of the changing universe would turn out to be just the passive ones of inertia and impenetrability. There are no active powers, he urged, other than those of God and living beings.
	From: report of Leonhard Euler (Letters to a German Princess [1765]) by Brian Ellis - The Philosophy of Nature: new essentialism Ch.4
	A reaction: Very significant, I think, for revealing the religious framework behind early theories of natural laws. If there is nothing external to impose powers and movements on nature, the source must be sought within - hence essentialism.