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Single Idea 17697

[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism ]

Full Idea

The standpoint of pure experience seems to me to be refuted by the objection that the existence, possible or actual, of an arbitrarily large number can never be derived through experience, that is, through experiment.

Gist of Idea

The existence of an arbitrarily large number refutes the idea that numbers come from experience

Source

David Hilbert (On the Foundations of Logic and Arithmetic [1904], p.130)

Book Ref

'From Frege to Gödel 1879-1931', ed/tr. Heijenoort,Jean van [Harvard 1967], p.130

A Reaction

Alternatively, empiricism refutes infinite numbers! No modern mathematician will accept that, but you wonder in what sense the proposed entities qualify as 'numbers'.

The 19 ideas with the same theme [denials that mathematics is rooted in experience]:

9861 The same thing is both one and an unlimited number at the same time [Plato]
21963 It is possible that an omnipotent God might make one and two fail to equal three [Descartes]
16928 Mathematics cannot be empirical because it is necessary, and that has to be a priori [Kant]
12411 Mill is too imprecise, and is restricted to simple arithmetic [Kitcher on Mill]
5656 Empirical theories of arithmetic ignore zero, limit our maths, and need probability to get started [Frege on Mill]
14776 That two two-eyed people must have four eyes is a statement about numbers, not a fact [Peirce]
8633 There is no physical difference between two boots and one pair of boots [Frege]
9577 The naïve view of number is that it is like a heap of things, or maybe a property of a heap [Frege]
17697 The existence of an arbitrarily large number refutes the idea that numbers come from experience [Hilbert]
5399 Maths is not known by induction, because further instances are not needed to support it [Russell]
17806 It is untenable that mathematics is general physical truths, because it needs infinity [Curry]
10735 Abstraction from objects won't reveal an operation's being performed 'so many times' [Geach]
8884 The phenomenal concept of an eleven-dot pattern does not include the concept of eleven [Sosa]
18201 General principles can be obvious in mathematics, but bold speculations in empirical science [Parsons,C]
17614 The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
9610 Numbers are not abstracted from particulars, because each number is a particular [Brown,JR]
9612 There is an infinity of mathematical objects, so they can't be physical [Brown,JR]
10131 If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman]
10005 Arithmetic doesn’t simply depend on objects, since it is true of fictional objects [Hofweber]