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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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7485
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For Pythagoreans 'one' is not a number, but the foundation of numbers [Pythagoras, by Watson]
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Full Idea:
For Pythagoreans, one, 1, is not a true number but the 'essence' of number, out of which the number system emerges.
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From:
report of Pythagoras (reports [c.530 BCE], Ch.8) by Peter Watson - Ideas Ch.8
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A reaction:
I think this is right! Counting and numbers only arise once the concept of individuality and identity have arisen. Counting to one is no more than observing the law of identity. 'Two' is the big adventure.
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15896
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Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine]
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Full Idea:
Cantor grafted the Power Set axiom onto his theory when he needed it to incorporate the real numbers, ...but his theory was supposed to be theory of collections that can be counted, but he didn't know how to count the new collections.
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From:
report of George Cantor (The Theory of Transfinite Numbers [1897]) by Shaughan Lavine - Understanding the Infinite I
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A reaction:
I take this to refer to the countability of the sets, rather than the members of the sets. Lavine notes that counting was Cantor's key principle, but he now had to abandon it. Zermelo came to the rescue.
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3053
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Pythagoras taught that virtue is harmony, and health, and universal good, and God [Pythagoras, by Diog. Laertius]
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Full Idea:
Pythagoras taught that virtue is harmony, and health, and universal good, and God.
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From:
report of Pythagoras (reports [c.530 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 08.1.19
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A reaction:
I like the link with health, because I consider that a bridge over the supposed fact-value gap. Very Pythagorean to think that virtue is harmony. Plato liked that thought.
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5244
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For Pythagoreans, justice is simply treating all people the same [Pythagoras, by Aristotle]
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Full Idea:
Some even think that what is just is simple reciprocity, as the Pythagoreans maintained, because they defined justice simply as having done to one what one has done to another.
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From:
report of Pythagoras (reports [c.530 BCE], 28) by Aristotle - Nicomachean Ethics 1132b22
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A reaction:
One wonders what Pythagoreans made of slavery. Aristotle argues that officials, for example, have superior rights. The Pythagorean idea makes fairness the central aspect of justice, and that must at least be partly right.
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