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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 27 ideas from Peter Geach

10731 For abstractionists, concepts are capacities to recognise recurrent features of the world [Geach]
10732 If concepts are just recognitional, then general judgements would be impossible [Geach]
10733 The abstractionist cannot explain 'some' and 'not' [Geach]
10734 Only a judgement can distinguish 'striking' from 'being struck' [Geach]
10735 Abstraction from objects won't reveal an operation's being performed 'so many times' [Geach]
22489 'Good' is an attributive adjective like 'large', not predicative like 'red' [Geach, by Foot]
2567 You can't define real mental states in terms of behaviour that never happens [Geach]
2568 Beliefs aren't tied to particular behaviours [Geach]
8769 If someone has aphasia but can still play chess, they clearly have concepts [Geach]
8770 'Abstractionism' is acquiring a concept by picking out one experience amongst a group [Geach]
8771 'Or' and 'not' are not to be found in the sensible world, or even in the world of inner experience [Geach]
8772 We can't acquire number-concepts by extracting the number from the things being counted [Geach]
8773 Abstractionists can't explain counting, because it must precede experience of objects [Geach]
8774 The numbers don't exist in nature, so they cannot have been abstracted from there into our languages [Geach]
8775 A big flea is a small animal, so 'big' and 'small' cannot be acquired by abstraction [Geach]
8776 We cannot learn relations by abstraction, because their converse must be learned too [Geach]
8778 Blind people can use colour words like 'red' perfectly intelligently [Geach]
8777 If 'black' and 'cat' can be used in the absence of such objects, how can such usage be abstracted? [Geach]
8779 We can form two different abstract concepts that apply to a single unified experience [Geach]
8780 Attributes are functions, not objects; this distinguishes 'square of 2' from 'double of 2' [Geach]
8781 The mind does not lift concepts from experience; it creates them, and then applies them [Geach]
11910 Being 'the same' is meaningless, unless we specify 'the same X' [Geach]
8969 We should abandon absolute identity, confining it to within some category [Geach, by Hawthorne]
16075 Denial of absolute identity has drastic implications for logic, semantics and set theory [Wasserman on Geach]
12154 Are 'word token' and 'word type' different sorts of countable objects, or two ways of counting? [Geach, by Perry]
16073 Leibniz's Law is incomplete, since it includes a non-relativized identity predicate [Geach, by Wasserman]
12152 Identity is relative. One must not say things are 'the same', but 'the same A as' [Geach]