9612
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There is an infinity of mathematical objects, so they can't be physical [Brown,JR]
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Full Idea:
A simple argument makes it clear that all mathematical arguments are abstract: there are infinitely many numbers, but only a finite number of physical entities, so most mathematical objects are non-physical. The best assumption is that they all are.
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From:
James Robert Brown (Philosophy of Mathematics [1999], Ch. 2)
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A reaction:
This, it seems to me, is where constructivists score well (cf. Idea 9608). I don't have an infinity of bricks to build an infinity of houses, but I can imagine that the bricks just keep coming if I need them. Imagination is what is unbounded.
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9610
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Numbers are not abstracted from particulars, because each number is a particular [Brown,JR]
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Full Idea:
Numbers are not 'abstract' (in the old sense, of universals abstracted from particulars), since each of the integers is a unique individual, a particular, not a universal.
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From:
James Robert Brown (Philosophy of Mathematics [1999], Ch. 2)
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A reaction:
An interesting observation which I have not seen directly stated before. Compare Idea 645. I suspect that numbers should be thought of as higher-order abstractions, which don't behave like normal universals (i.e. they're not distributed).
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9620
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Empiricists base numbers on objects, Platonists base them on properties [Brown,JR]
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Full Idea:
Perhaps, instead of objects, numbers are associated with properties of objects. Basing them on objects is strongly empiricist and uses first-order logic, whereas the latter view is somewhat Platonistic, and uses second-order logic.
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From:
James Robert Brown (Philosophy of Mathematics [1999], Ch. 4)
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A reaction:
I don't seem to have a view on this. You can count tomatoes, or you can count red objects, or even 'instances of red'. Numbers refer to whatever can be individuated. No individuation, no arithmetic. (It's also Hume v Armstrong on laws on nature).
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9645
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Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR]
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Full Idea:
If we define p as '3 if Goldbach's Conjecture is true' and '5 if Goldbach's Conjecture is false', it seems that p must be a prime number, but, amazingly, constructivists would not accept this without a proof of Goldbach's Conjecture.
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From:
James Robert Brown (Philosophy of Mathematics [1999], Ch. 8)
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A reaction:
A very similar argument structure to Schrödinger's Cat. This seems (as Brown implies) to be a devastating knock-down argument, but I'll keep an open mind for now.
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