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| 9143 | Implicit definitions must be satisfiable, creative definitions introduce things, contextual definitions build on things |
| Full Idea: Fine distinguishes 'implicit definitions', where we must know it is satisfiable before it is deployed, 'creative definitions', where objects are introduced in virtue of the definition, ..and 'contextual definitions', based on established vocabulary. | |||
| From: report of Kit Fine (The Limits of Abstraction [2002], 060) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 3 | |||
| A reaction: Fine is a fan of creative definition. This sounds something like the distinction between cutting nature at the perceived joints, and speculating about where new joints might be inserted. Quite a helpful thought. |
| 10143 | 'Creative definitions' do not presuppose the existence of the objects defined |
| Full Idea: What I call 'creative definitions' are made from a standpoint in which the existence of the objects that are to be assigned to the terms is not presupposed. | |||
| From: Kit Fine (The Limits of Abstraction [2002], II.1) |
| 10145 | Abstracts cannot be identified with sets |
| Full Idea: It is impossible for a proponent of both sets and abstracts to identify the abstracts, in any reasonable manner, with the sets. | |||
| From: Kit Fine (The Limits of Abstraction [2002], IV.1) | |||
| A reaction: [This observation emerges from a proof Fine has just completed] Cf Idea 10137. The implication is that there is no compromise view available, and one must choose between abstraction or sets as one's account of numbers and groups of concepts. |
| 10136 | Points in Euclidean space are abstract objects, but not introduced by abstraction |
| Full Idea: Points in abstract Euclidean space are abstract objects, and yet are not objects of abstraction, since they are not introduced through a principle of abstraction of the sort envisaged by Frege. | |||
| From: Kit Fine (The Limits of Abstraction [2002], I.1) | |||
| A reaction: The point seems to be that they are not abstracted 'from' anything, but are simpy posited as basic constituents. I suggest that points are idealisations (of smallness) rather than abstractions. They are idealised 'from' substances. |
| 10144 | Postulationism says avoid abstract objects by giving procedures that produce truth |
| Full Idea: A procedural form of postulationism says that instead of stipulating that certain statements are true, one specifies certain procedures for extending the domain to one in which the statement will in fact be true, without invoking an abstract ontology. | |||
| From: Kit Fine (The Limits of Abstraction [2002], II.5) | |||
| A reaction: The whole of philosophy might go better if it was founded on procedures and processes, rather than on objects. The Hopi Indians were right. |
| 9144 | Fine's 'procedural postulationism' uses creative definitions, but avoids abstract ontology |
| Full Idea: Fine says creative definitions can found mathematics. His 'procedural postulationism' says one stipulates not truths, but certain procedures for extending a domain. The procedures can be stated without invoking an abstract ontology. | |||
| From: report of Kit Fine (The Limits of Abstraction [2002], 100) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 4 | |||
| A reaction: (For creative definitions, see Idea 9143) This sounds close in spirit to fictionalism, but with the emphasis on the procedure (which can presumably be formalized) rather than a pure act of imaginative creation. |
| 10141 | Many different kinds of mathematical objects can be regarded as forms of abstraction |
| 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. | |||
| From: Kit Fine (The Limits of Abstraction [2002], I.4) | |||
| A reaction: This result, if persuasive, would be just the sort of unified account which the whole problem of abstact ideas requires. |
| 10135 | We can abstract from concepts (e.g. to number) and from objects (e.g. to direction) |
| Full Idea: A principle of abstraction is 'conceptual' when the items upon which it abstracts are concepts (e.g. a one-one correspondence associated with a number), and 'objectual' if they are objects (parallel lines associated with a direction). | |||
| From: Kit Fine (The Limits of Abstraction [2002], I) |
| 9142 | Fine considers abstraction as reconceptualization, to produce new senses by analysing given senses |
| Full Idea: Fine considers abstraction principles as instances of reconceptualization (rather than implicit definition, or using the Context Principle). This centres not on reference, but on new senses emerging from analysis of a given sense. | |||
| From: report of Kit Fine (The Limits of Abstraction [2002], 035) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 2 | |||
| A reaction: Fine develops an argument against this view, because (roughly) the procedure does not end in a unique result. Intuitively, the idea that abstraction is 'reconceptualization' sounds quite promising to me. |
| 10137 | Abstractionism can be regarded as an alternative to set theory |
| Full Idea: The uncompromising abstractionist rejects set theory, seeing the theory of abstractions as an alternative, rather than as a supplement, to the standard theory of sets. | |||
| From: Kit Fine (The Limits of Abstraction [2002], I.1) | |||
| A reaction: There is also a 'compromising' version. Presumably you still have equivalence classes to categorise the objects, which are defined by their origin rather than by what they are members of... Cf. Idea 10145. |
| 10138 | An object is the abstract of a concept with respect to a relation on concepts |
| Full Idea: We can see an object as being the abstract of a concept with respect to a relation on concepts. For example, we may say that 0 is the abstract of the empty concept with respect to the relation of one-one correspondence. | |||
| From: Kit Fine (The Limits of Abstraction [2002], I.2) | |||
| A reaction: This is Fine's attempt to give a modified account of the Fregean approach to abstraction. He says that the reference to a relation will solve the problem of identity between abstractions. |