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12215
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The existence of numbers is not a matter of identities, but of constituents of the world [Fine,K]
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Full Idea:
On saying that a particular number exists, we are not saying that there is something identical to it, but saying something about its status as a genuine constituent of the world.
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From:
Kit Fine (The Question of Ontology [2009], p.168)
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A reaction:
This is aimed at Frege's criterion of identity, which is to be an element in an identity relation, such as x = y. Fine suggests that this only gives a 'trivial' notion of existence, when he is interested in a 'thick' sense of 'exists'.
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10529
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If Hume's Principle can define numbers, we needn't worry about its truth [Fine,K]
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Full Idea:
Neo-Fregeans have thought that Hume's Principle, and the like, might be definitive of number and therefore not subject to the usual epistemological worries over its truth.
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From:
Kit Fine (Precis of 'Limits of Abstraction' [2005], p.310)
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A reaction:
This seems to be the underlying dream of logicism - that arithmetic is actually brought into existence by definitions, rather than by truths derived from elsewhere. But we must be able to count physical objects, as well as just counting numbers.
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10530
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Hume's Principle is either adequate for number but fails to define properly, or vice versa [Fine,K]
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Full Idea:
The fundamental difficulty facing the neo-Fregean is to either adopt the predicative reading of Hume's Principle, defining numbers, but inadequate, or the impredicative reading, which is adequate, but not really a definition.
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From:
Kit Fine (Precis of 'Limits of Abstraction' [2005], p.312)
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A reaction:
I'm not sure I understand this, but the general drift is the difficulty of building a system which has been brought into existence just by definition.
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12209
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The indispensability argument shows that nature is non-numerical, not the denial of numbers [Fine,K]
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Full Idea:
Arguments such as the dispensability argument are attempting to show something about the essentially non-numerical character of physical reality, rather than something about the nature or non-existence of the numbers themselves.
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From:
Kit Fine (The Question of Ontology [2009], p.160)
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A reaction:
This is aimed at Hartry Field. If Quine was right, and we only believe in numbers because of our science, and then Field shows our science doesn't need it, then Fine would be wrong. Quine must be wrong, as well as Field.
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9224
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Proceduralism offers a version of logicism with no axioms, or objects, or ontological commitment [Fine,K]
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Full Idea:
My Proceduralism offers axiom-free foundations for mathematics. Axioms give way to the stipulation of procedures. We obtain a form of logicism, but with a procedural twist, and with a logic which is ontologically neutral, and no assumption of objects.
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From:
Kit Fine (Our Knowledge of Mathematical Objects [2005], 1)
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A reaction:
[See Ideas 9222 and 9223 for his Proceduralism] Sounds like philosophical heaven. We get to take charge of mathematics, without the embarrassment of declaring ourselves to be platonists. Someone, not me, should evaluate this.
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9223
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My Proceduralism has one simple rule, and four complex rules [Fine,K]
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Full Idea:
My Proceduralism has one simple rule (introduce an object), and four complex rules: Composition (combining two procedures), Conditionality (if A, do B), Universality (do a procedure for every x), and Iteration (rule to keep doing B).
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From:
Kit Fine (Our Knowledge of Mathematical Objects [2005], 1)
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A reaction:
It sounds like a highly artificial and private game which Fine has invented, but he claims that this is the sort of thing that practising mathematicians have always done.
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