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Single Idea 16928
[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
]
Full Idea
Mathematical propositions are always judgements a priori, and not empirical, because they carry with them necessity, which cannot be taken from experience.
Gist of Idea
Mathematics cannot be empirical because it is necessary, and that has to be a priori
Source
Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 268)
Book Ref
Kant,Immanuel: 'Prolegomena to Any Future Metaphysic', ed/tr. Lucas,Peter G. [Manchester UP 1971], p.18
A Reaction
Presumably there are necessities in the physical world, and we might discern them by generalising about that world, so that mathematics is (by a tortuous abstract route) a posteriori necessary? Just a thought…
The
19 ideas
with the same theme
[denials that mathematics is rooted in experience]:
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9861
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The same thing is both one and an unlimited number at the same time
[Plato]
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21963
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It is possible that an omnipotent God might make one and two fail to equal three
[Descartes]
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16928
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Mathematics cannot be empirical because it is necessary, and that has to be a priori
[Kant]
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12411
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Mill is too imprecise, and is restricted to simple arithmetic
[Kitcher on Mill]
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5656
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Empirical theories of arithmetic ignore zero, limit our maths, and need probability to get started
[Frege on Mill]
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14776
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That two two-eyed people must have four eyes is a statement about numbers, not a fact
[Peirce]
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8633
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There is no physical difference between two boots and one pair of boots
[Frege]
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9577
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The naïve view of number is that it is like a heap of things, or maybe a property of a heap
[Frege]
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17697
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The existence of an arbitrarily large number refutes the idea that numbers come from experience
[Hilbert]
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5399
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Maths is not known by induction, because further instances are not needed to support it
[Russell]
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17806
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It is untenable that mathematics is general physical truths, because it needs infinity
[Curry]
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10735
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Abstraction from objects won't reveal an operation's being performed 'so many times'
[Geach]
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8884
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The phenomenal concept of an eleven-dot pattern does not include the concept of eleven
[Sosa]
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18201
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General principles can be obvious in mathematics, but bold speculations in empirical science
[Parsons,C]
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17614
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The connection of arithmetic to perception has been idealised away in modern infinitary mathematics
[Maddy]
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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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9610
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Numbers are not abstracted from particulars, because each number is a particular
[Brown,JR]
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10131
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If mathematics is not about particulars, observing particulars must be irrelevant
[George/Velleman]
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10005
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Arithmetic doesn’t simply depend on objects, since it is true of fictional objects
[Hofweber]
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