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Full Idea
Many different kinds of mathematical objects (natural numbers, the reals, points, lines, figures, groups) can be regarded as forms of abstraction, with special theories having their basis in a general theory of abstraction.
Gist of Idea
Many different kinds of mathematical objects can be regarded as forms of abstraction
Source
Kit Fine (The Limits of Abstraction [2002], I.4)
Book Ref
Fine,Kit: 'The Limits of Abstraction' [OUP 2008], p.46
A Reaction
This result, if persuasive, would be just the sort of unified account which the whole problem of abstact ideas requires.
| 9142 | Fine considers abstraction as reconceptualization, to produce new senses by analysing given senses [Fine,K, by Cook/Ebert] |
| 9143 | Implicit definitions must be satisfiable, creative definitions introduce things, contextual definitions build on things [Fine,K, by Cook/Ebert] |
| 9144 | Fine's 'procedural postulationism' uses creative definitions, but avoids abstract ontology [Fine,K, by Cook/Ebert] |
| 10135 | We can abstract from concepts (e.g. to number) and from objects (e.g. to direction) [Fine,K] |
| 10136 | Points in Euclidean space are abstract objects, but not introduced by abstraction [Fine,K] |
| 10137 | Abstractionism can be regarded as an alternative to set theory [Fine,K] |
| 10138 | An object is the abstract of a concept with respect to a relation on concepts [Fine,K] |
| 10141 | Many different kinds of mathematical objects can be regarded as forms of abstraction [Fine,K] |
| 10143 | 'Creative definitions' do not presuppose the existence of the objects defined [Fine,K] |
| 10144 | Postulationism says avoid abstract objects by giving procedures that produce truth [Fine,K] |
| 10145 | Abstracts cannot be identified with sets [Fine,K] |