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Single Idea 9577

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

Full Idea

The most naïve opinion of number is that it is something like a heap in which things are contained. The next most naïve view is the conception of number as the property of a heap, cleansing the objects of their particulars.

Gist of Idea

The naïve view of number is that it is like a heap of things, or maybe a property of a heap

Source

Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.323)

Book Ref

-: 'Mind July 1972' [-], p.323

A Reaction

A hundred toothbrushes and a hundred sponges can be seen to contain the same number (by one-to-one mapping), without actually knowing what that number is. There is something numerical in the heap, even if the number is absent.

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]
9612 There is an infinity of mathematical objects, so they can't be physical [Brown,JR]
9610 Numbers are not abstracted from particulars, because each number is a particular [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]