Combining Texts

All the ideas for 'Philosophical Fragments', 'Comments on a Certain Broadsheet' and 'Logicism, Some Considerations (PhD)'

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9 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.

7. Existence / A. Nature of Existence / 5. Reason for Existence

16007	I assume existence, rather than reasoning towards it [Kierkegaard]
	Full Idea: I always reason from existence, not towards existence.
	From: Søren Kierkegaard (Philosophical Fragments [1844], p.40)
	A reaction: Kierkegaard's important premise to help show that theistic proofs for God's existence don't actually prove existence, but develop the content of a conception. [SY]

10. Modality / A. Necessity / 2. Nature of Necessity

16013	Nothing necessary can come into existence, since it already 'is' [Kierkegaard]
	Full Idea: Can the necessary come into existence? That is a change, and everything that comes into existence demonstrates that it is not necessary. The necessary already 'is'.
	From: Søren Kierkegaard (Philosophical Fragments [1844], p.74)
	A reaction: [SY]

12. Knowledge Sources / A. A Priori Knowledge / 4. A Priori as Necessities

2602	What experience could prove 'If a=c and b=c then a=b'? [Descartes]
	Full Idea: Please tell me what the corporeal motion is that is capable of forming some common notion to the effect that 'things which are equal to a third thing are equal to each other'.
	From: René Descartes (Comments on a Certain Broadsheet [1644], p.366)

18. Thought / D. Concepts / 2. Origin of Concepts / c. Nativist concepts

2600	The mind's innate ideas are part of its capacity for thought [Descartes]
	Full Idea: I have never written or taken the view that the mind requires innate ideas which are something distinct from its own faculty of thinking.
	From: René Descartes (Comments on a Certain Broadsheet [1644], p.365)

2601	Qualia must be innate, because physical motions do not contain them [Descartes]
	Full Idea: The ideas of pains, colours, sounds etc. must be all the more innate if, on the occasion of certain corporeal motions, our mind is to be capable of representing them to itself, for there is no similarity between these ideas and the corporeal motions.
	From: René Descartes (Comments on a Certain Broadsheet [1644], p.365)
	A reaction: Simple and brilliant! We know perfectly well that there is no redness zooming through the air from a tomato (or the air would be pink!). Redness occurs when the light arrives, so we add the redness, so it is innate.