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Single Idea 9861

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

Full Idea

We see the same thing to be both one and an unlimited number at the same time.

Gist of Idea

The same thing is both one and an unlimited number at the same time

Source

Plato (The Republic [c.374 BCE], 525a)

Book Ref

Plato: 'Complete Works', ed/tr. Cooper,John M. [Hackett 1997], p.1141

A Reaction

Frege makes the same point, that a pair of boots is both two and one. The point is at its strongest in opposition to empirical accounts of arithmetic. However, Mill observes that pebbles can be both 5 and 3+2, without contradiction.

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]