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Single Idea 18201

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

Full Idea

The existence of very general principles in mathematics are universally regarded as obvious, where on an empiricist view one would expect them to be bold hypotheses, about which a prudent scientist would maintain reserve.

Gist of Idea

General principles can be obvious in mathematics, but bold speculations in empirical science

Source

Charles Parsons (Mathematical Intuition [1980], p.152), quoted by Penelope Maddy - Naturalism in Mathematics

Book Ref

Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.106

A Reaction

This is mainly aimed at Quine's and Putnam's indispensability (to science) argument about mathematics.

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]