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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism

[denials that mathematics is rooted in experience]

19 ideas
The same thing is both one and an unlimited number at the same time [Plato]
It is possible that an omnipotent God might make one and two fail to equal three [Descartes]
Mathematics cannot be empirical because it is necessary, and that has to be a priori [Kant]
Mill is too imprecise, and is restricted to simple arithmetic [Kitcher on Mill]
Empirical theories of arithmetic ignore zero, limit our maths, and need probability to get started [Frege on Mill]
That two two-eyed people must have four eyes is a statement about numbers, not a fact [Peirce]
There is no physical difference between two boots and one pair of boots [Frege]
The naÔve view of number is that it is like a heap of things, or maybe a property of a heap [Frege]
The existence of an arbitrarily large number refutes the idea that numbers come from experience [Hilbert]
Maths is not known by induction, because further instances are not needed to support it [Russell]
It is untenable that mathematics is general physical truths, because it needs infinity [Curry]
Abstraction from objects won't reveal an operation's being performed 'so many times' [Geach]
The phenomenal concept of an eleven-dot pattern does not include the concept of eleven [Sosa]
General principles can be obvious in mathematics, but bold speculations in empirical science [Parsons,C]
The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
There is an infinity of mathematical objects, so they can't be physical [Brown,JR]
Numbers are not abstracted from particulars, because each number is a particular [Brown,JR]
If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman]
Arithmetic doesnít simply depend on objects, since it is true of fictional objects [Hofweber]