Ideas of Charles Chihara, by Theme

[American, fl. 1998, Professor at the University of California, at Berkeley.]

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4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
We only know relational facts about the empty set, but nothing intrinsic
In simple type theory there is a hierarchy of null sets
Realists about sets say there exists a null set in the real world, with no members
The null set is a structural position which has no other position in membership relation
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Set
What is special about Bill Clinton's unit set, in comparison with all the others?
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
The set theorist cannot tell us what 'membership' is
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
ZFU refers to the physical world, when it talks of 'urelements'
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
A pack of wolves doesn't cease when one member dies
Could we replace sets by the open sentences that define them?
We could talk of open sentences, instead of sets
5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
The mathematics of relations is entirely covered by ordered pairs
5. Theory of Logic / K. Features of Logics / 2. Consistency
Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum
6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects
Analytic geometry gave space a mathematical structure, which could then have axioms
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / c. Nominalist structuralism
We can replace existence of sets with possibility of constructing token sentences
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
Chihara's system is a variant of type theory, from which he can translate sentences
We can replace type theory with open sentences and a constructibility quantifier
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ'
7. Existence / D. Theories of Reality / 10. Ontological Commitment / e. Ontological commitment problems
If a successful theory confirms mathematics, presumably a failed theory disconfirms it?
No scientific explanation would collapse if mathematical objects were shown not to exist
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes'
19. Language / C. Reference / 3. Direct Reference / b. Causal reference
Mathematical entities are causally inert, so the causal theory of reference won't work for them
27. Natural Reality / A. Physics / 1. Matter / i. Modern matter
'Gunk' is an individual possessing no parts that are atoms