16150
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One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
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Full Idea:
If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it.
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From:
Plato (Parmenides [c.364 BCE], 144a)
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A reaction:
This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed.
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8681
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The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend]
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Full Idea:
The main philosophical problem with the position of platonism or realism is the epistemic problem: of explaining what perception or intuition consists in; how it is possible that we should accurately detect whatever it is we are realists about.
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From:
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.5)
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A reaction:
The best bet, I suppose, is that the mind directly perceives concepts just as eyes perceive the physical (see Idea 8679), but it strikes me as implausible. If we have to come up with a special mental faculty for an area of knowledge, we are in trouble.
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8716
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Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend]
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Full Idea:
There are not enough constraints in the Formalist view of mathematics, so there is no way to select a direction for trying to develop mathematics. There is no part of mathematics that is more important than another.
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From:
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.6)
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A reaction:
One might reply that an area of maths could be 'important' if lots of other areas depended on it, and big developments would ripple big changes through the interior of the subject. Formalism does, though, seem to reduce maths to a game.
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