Ideas from 'A Structural Account of Mathematics' by Charles Chihara [2004], by Theme Structure

[found in 'A Structural Account of Mathematics' by Chihara,Charles [OUP 2004,0-19-922807-8]].

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4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
Realists about sets say there exists a null set in the real world, with no members
We only know relational facts about the empty set, but nothing intrinsic
In simple type theory there is a hierarchy of null sets
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
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
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 / D. Concepts / 6. Abstract Concepts / g. Abstracta by equivalence
I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes'
19. Language / D. Theories of 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