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Single Idea 10192
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
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Full Idea
Chihara's 'constructability theory' is nominalist - mathematics is reducible to a simple theory of types. Instead of talk of sets {x:x is F}, we talk of open sentences Fx defining them. Existence claims become constructability of sentence tokens.
Gist of Idea
We can replace existence of sets with possibility of constructing token sentences
Source
report of Charles Chihara (A Structural Account of Mathematics [2004]) by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' p.81
Book Ref
-: 'Bulletin of Symbolic Logic' [-], p.81
A Reaction
This seems to be approaching the problem in a Fregean way, by giving an account of the semantics. Chihara is trying to evade the Quinean idea that assertion is ontological commitment. But has Chihara retreated too far? How does he assert existence?
The
22 ideas
from Charles Chihara
8758
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We could talk of open sentences, instead of sets
[Chihara, by Shapiro]
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10265
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Chihara's system is a variant of type theory, from which he can translate sentences
[Chihara, by Shapiro]
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8759
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We can replace type theory with open sentences and a constructibility quantifier
[Chihara, by Shapiro]
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10264
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Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ'
[Chihara, by Shapiro]
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18151
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Could we replace sets by the open sentences that define them?
[Chihara, by Bostock]
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10192
|
We can replace existence of sets with possibility of constructing token sentences
[Chihara, by MacBride]
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9547
|
Mathematical entities are causally inert, so the causal theory of reference won't work for them
[Chihara]
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9549
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The set theorist cannot tell us what 'membership' is
[Chihara]
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9550
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We only know relational facts about the empty set, but nothing intrinsic
[Chihara]
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9551
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What is special about Bill Clinton's unit set, in comparison with all the others?
[Chihara]
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9552
|
Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced
[Chihara]
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9553
|
Analytic geometry gave space a mathematical structure, which could then have axioms
[Chihara]
|
9559
|
If a successful theory confirms mathematics, presumably a failed theory disconfirms it?
[Chihara]
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9561
|
The mathematics of relations is entirely covered by ordered pairs
[Chihara]
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9562
|
In simple type theory there is a hierarchy of null sets
[Chihara]
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9563
|
A pack of wolves doesn't cease when one member dies
[Chihara]
|
9566
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No scientific explanation would collapse if mathematical objects were shown not to exist
[Chihara]
|
9568
|
I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes'
[Chihara]
|
9571
|
ZFU refers to the physical world, when it talks of 'urelements'
[Chihara]
|
9573
|
The null set is a structural position which has no other position in membership relation
[Chihara]
|
9572
|
Realists about sets say there exists a null set in the real world, with no members
[Chihara]
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9574
|
'Gunk' is an individual possessing no parts that are atoms
[Chihara]
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