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 2004,0199228078]].
Click on the Idea Number for the full details 
back to texts

expand these ideas
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
9550

We only know relational facts about the empty set, but nothing intrinsic

9562

In simple type theory there is a hierarchy of null sets

9573

The null set is a structural position which has no other position in membership relation

9572

Realists about sets say there exists a null set in the real world, with no members

4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Set
9551

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
9549

The set theorist cannot tell us what 'membership' is

4. Formal Logic / F. Set Theory ST / 7. Natural Sets
9571

ZFU refers to the physical world, when it talks of 'urelements'

4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
9563

A pack of wolves doesn't cease when one member dies

5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
9561

The mathematics of relations is entirely covered by ordered pairs

5. Theory of Logic / K. Features of Logics / 2. Consistency
9552

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
9555

Continuum Hypothesis: no cardinal greater than alephnull but less than cardinality of the continuum

6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
9546

Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects

9553

Analytic geometry gave space a mathematical structure, which could then have axioms

6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / c. Nominalist structuralism
10192

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
9559

If a successful theory confirms mathematics, presumably a failed theory disconfirms it?

9566

No scientific explanation would collapse if mathematical objects were shown not to exist

18. Thought / E. Abstraction / 7. Abstracta by Equivalence
9568

I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes'

19. Language / B. Reference / 3. Direct Reference / b. Causal reference
9547

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
9574

'Gunk' is an individual possessing no parts that are atoms
