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Single Idea 9976

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

Full Idea

Frege's own conception of abstraction (although he disapproves of the term) is in agreement with the view that abstracting from the particular nature of the elements of M would yield the concept under which fall all sets equipollent to M.

Clarification

'Equipollent' means they map one-to-one onto each other

Gist of Idea

Frege accepts abstraction to the concept of all sets equipollent to a given one

Source

comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by William W. Tait - Frege versus Cantor and Dedekind III

Book Ref

'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.44

A Reaction

Nice! This shows how difficult it is to slough off the concept of abstractionism and live with purely logical concepts of it. If we 'construct' a set, then there is a process of creation to be explained; we can't just think of platonic givens.

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]