13412
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Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf]
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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.
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From:
Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.165)
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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.
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13413
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We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf]
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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.
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From:
Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.166)
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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.
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13411
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If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [Benacerraf]
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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.
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From:
Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.164)
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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.
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16689
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The schools said spirits lack extension, and wonder how many could dance on a needle's point [More,H]
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Full Idea:
Many, not without reason, laugh at those ridiculous fancies of the schools, that rashly take away all extensions from spirits, whether souls or angels, and then dispute how many of them booted and spurred may dance on a needle's point at once.
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From:
Henry More (Immortality of the Soul [1659], III.2.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 17.3
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A reaction:
This famous idea originated with William Chillingworth. More's version is the classic one. Pasnau cites Aquinas Summa 1a 52.3 as discussing the actual question (and says this couldn't happen).
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