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Single Idea 9149

[filed under theme 18. Thought / E. Abstraction / 2. Abstracta by Selection ]

Full Idea

In abstracting from the elements of a doubleton to obtain 2, we do not wish to abstract away from all features of the objects. We wish to take account of the fact that the two objects are distinct; this alone should be preserved under abstraction.

Clarification

A 'doubleton' is a set with two members

Gist of Idea

To obtain the number 2 by abstraction, we only want to abstract the distinctness of a pair of objects

Source

Kit Fine (Cantorian Abstraction: Recon. and Defence [1998], §3)

Book Ref

-: 'Journal of Philosophy' [-], p.10

A Reaction

This is Fine's strategy for meeting Frege's objection to abstraction, summarised in Idea 9146. It seems to use the common sense idea that abstraction is not all-or-nothing. Abstraction has degrees (and levels).

Related Idea

Idea 9146 After abstraction all numbers seem identical, so only 0 and 1 will exist! [Fine,K]

The 24 ideas with the same theme [mental acts which create abstract concepts]:

9789 You can't abstract natural properties to make Forms - objects and attributes are defined together [Aristotle]
9070 We learn primitives and universals by induction from perceptions [Aristotle]
10506 Mathematics can be abstracted from sensible matter, and from individual intelligible matter [Aquinas]
9104 A universal is the result of abstraction, which is only a kind of mental picturing [William of Ockham]
23640 Only mature minds can distinguish the qualities of a body [Reid]
23653 If you can't distinguish the features of a complex object, your notion of it would be a muddle [Reid]
19577 Everything is a chaotic unity, then we abstract, then we reunify the world into a free alliance [Novalis]
9145 We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor]
13454 Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor]
9976 Frege accepts abstraction to the concept of all sets equipollent to a given one [Tait on Frege]
9361 We have to separate the mathematical from physical phenomena by abstraction [Lewis,CI]
10644 A 'felt familiarity' with universals is more primitive than abstraction [Price,HH]
10646 Our understanding of 'dog' or 'house' arises from a repeated experience of concomitances [Price,HH]
9031 The basic concepts of conceptual cognition are acquired by direct abstraction from instances [Price,HH]
10557 Abstract objects are captured by second-order modal logic, plus 'encoding' formulas [Zalta]
12657 Abstractionism claims that instances provide criteria for what is shared [Fodor]
9149 To obtain the number 2 by abstraction, we only want to abstract the distinctness of a pair of objects [Fine,K]
9150 We should define abstraction in general, with number abstraction taken as a special case [Fine,K]
9981 Abstraction is 'logical' if the sense and truth of the abstraction depend on the concrete [Tait]
9982 Cantor and Dedekind use abstraction to fix grammar and objects, not to carry out proofs [Tait]
10141 Many different kinds of mathematical objects can be regarded as forms of abstraction [Fine,K]
10229 Simple types can be apprehended through their tokens, via abstraction [Shapiro]
9919 The old debate classified representations as abstract, not entities [Burgess/Rosen]
8917 The Way of Abstraction used to say an abstraction is an idea that was formed by abstracting [Rosen]