|
15927
|
Definition just needs negation, known variables, conjunction, disjunction, substitution and quantification [Weyl, by Lavine]
|
|
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.
|
|
From:
report of Hermann Weyl (works [1917]) by Shaughan Lavine - Understanding the Infinite V.3
|
|
A reaction:
Lavine summarises this as 'first-order logic with parameters'.
|
|
9143
|
Implicit definitions must be satisfiable, creative definitions introduce things, contextual definitions build on things [Fine,K, by Cook/Ebert]
|
|
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.
|
|
17304
|
As causation links across time, grounding links the world across levels [Schaffer,J]
|
|
Full Idea:
Grounding is something like metaphysical causation. Just as causation links the world across time, grounding links the world across levels. Grounding connects the more fundamental to the less fundamental, and thereby backs a certain form of explanation.
|
|
From:
Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], Intro)
|
|
A reaction:
Obviously you need 'levels' for this, which we should take to be structural levels.
|
|
17308
|
Explaining 'Adam ate the apple' depends on emphasis, and thus implies a contrast [Schaffer,J]
|
|
Full Idea:
Explaining why ADAM ate the apple is a different matter from explaining why he ATE the apple, and from why he ate THE APPLE. ...In my view the best explanations incorporate ....contrastive information.
|
|
From:
Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], 4.3.1)
|
|
A reaction:
But why are the contrasts Eve, or throwing it, or a pear? It occurs to me that this is wrong! The contrast is with anything else which could have gone in subject, verb or object position. It is a matter of categories, not of contrasts.
|
|
9144
|
Fine's 'procedural postulationism' uses creative definitions, but avoids abstract ontology [Fine,K, by Cook/Ebert]
|
|
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 [Fine,K]
|
|
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.
|
|
9142
|
Fine considers abstraction as reconceptualization, to produce new senses by analysing given senses [Fine,K, by Cook/Ebert]
|
|
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.
|
|
17305
|
I take what is fundamental to be the whole spatiotemporal manifold and its fields [Schaffer,J]
|
|
Full Idea:
I myself would prefer to speak of what is fundamental in terms of the whole spatiotemporal manifold and the fields that permeate it, with parts counting as derivative of the whole.
|
|
From:
Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], 4.1.1)
|
|
A reaction:
Not quite the Parmenidean One, since it has parts, but a nice try at updating the great man. Note the reference to 'fields', suggesting that this view is grounded in the physics rather than metaphysics. How many fields has it got?
|
|
17307
|
Nowadays causation is usually understood in terms of equations and variable ranges [Schaffer,J]
|
|
Full Idea:
The leading treatments of causation work within 'structural equation models', with events represented via variables each of which is allotted a range of permitted values, which constitute a 'contrast space'.
|
|
From:
Jonathan Schaffer (Grounding, Transitivity and Contrastivity [2012], 4.3.1)
|
|
A reaction:
Like Woodward's idea that causation is a graph, this seems to be a matter of plotting or formalising correlations between activities, which is a very Humean approach to causation.
|