more from this thinker
|
more from this text
Single Idea 10735
[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
]
Full Idea
For an understanding of arithmetic the grasp of an operation's being performed 'so many times' is quite indispensable; and abstraction of a feature from groups of nuts cannot give us this grasp.
Gist of Idea
Abstraction from objects won't reveal an operation's being performed 'so many times'
Source
Peter Geach (Abstraction Reconsidered [1983], p.170)
Book Ref
'Knowledge and Mind', ed/tr. Ginet,C/Shoemaker,S [OUP 1983], p.170
A Reaction
I end up defending the empirical approach to arithmetic because remarks like this are so patently false. Geach seems to think we arrive ready-made in the world, just raring to get on with some counting. He lacks the evolutionary perspective.
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]
|