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12556
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Mathematics is just about ideas, so whether circles exist is irrelevant [Locke]
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Full Idea:
All the discourses of mathematicians concerning conic sections etc. concern not the existence of any of those figures, but their demonstrations, which depend on their ideas, are the same, whether there be any square or circle existing in the world or no.
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From:
John Locke (Essay Conc Human Understanding (2nd Ed) [1694], 4.04.08)
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A reaction:
If the full-blown platonic circle really existed, we would have the epistemic problem not only of getting in causal contact with it, but also of knowing whether our idea of it was the correct idea. We can't know that, so we just work with our idea.
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7782
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Every simple idea we ever have brings the idea of unity along with it [Locke]
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Full Idea:
Amongst all the ideas we have… there is none more simple, than that of unity, or one… every idea in our understanding, every thought in our minds, brings this idea along with it.
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From:
John Locke (Essay Conc Human Understanding (2nd Ed) [1694], 2.16.01)
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A reaction:
If every idea we think of necessarily brings another idea along with it, that makes you suspect that the accompanying idea is innate. If I derive the concept of the sun from experience, do I also derive the idea that my concept is a unity?
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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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