15927
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Definition just needs negation, known variables, conjunction, disjunction, substitution and quantification [Weyl, by Lavine]
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Full Idea:
For mathematics, Weyl arrived (by 1917) at a satisfactory list of definition principles: negation, identification of variables, conjunction, disjunction, substitution of constants, and existential quantification over the domain.
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From:
report of Hermann Weyl (works [1917]) by Shaughan Lavine - Understanding the Infinite V.3
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A reaction:
Lavine summarises this as 'first-order logic with parameters'.
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9143
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Implicit definitions must be satisfiable, creative definitions introduce things, contextual definitions build on things [Fine,K, by Cook/Ebert]
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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.
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From:
report of Kit Fine (The Limits of Abstraction [2002], 060) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 3
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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.
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6019
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If someone squashed a horse to make a dog, something new would now exist [Mnesarchus]
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Full Idea:
If, for the sake of argument, someone were to mould a horse, squash it, then make a dog, it would be reasonable for us on seeing this to say that this previously did not exist but now does exist.
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From:
Mnesarchus (fragments/reports [c.120 BCE]), quoted by John Stobaeus - Anthology 179.11
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A reaction:
Locke would say it is new, because the substance is the same, but a new life now exists. A sword could cease to exist and become a new ploughshare, I would think. Apply this to the Ship of Theseus. Is form more important than substance?
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9144
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Fine's 'procedural postulationism' uses creative definitions, but avoids abstract ontology [Fine,K, by Cook/Ebert]
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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.
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From:
report of Kit Fine (The Limits of Abstraction [2002], 100) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 4
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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.
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10141
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Many different kinds of mathematical objects can be regarded as forms of abstraction [Fine,K]
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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.
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From:
Kit Fine (The Limits of Abstraction [2002], I.4)
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A reaction:
This result, if persuasive, would be just the sort of unified account which the whole problem of abstact ideas requires.
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9142
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Fine considers abstraction as reconceptualization, to produce new senses by analysing given senses [Fine,K, by Cook/Ebert]
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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.
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From:
report of Kit Fine (The Limits of Abstraction [2002], 035) by R Cook / P Ebert - Notice of Fine's 'Limits of Abstraction' 2
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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.
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