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
                        Full Idea: In the Gödelian realistic view of set theory the statement that there is a null set as the assertion of the existence in the real world of a set that has no members.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6)
                        A reaction: It seems to me obvious that such a claim is nonsense on stilts. 'In the beginning there was the null set'?
We only know relational facts about the empty set, but nothing intrinsic
                        Full Idea: Everything we know about the empty set is relational; we know that nothing is the membership relation to it. But what do we know about its 'intrinsic properties'?
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5)
                        A reaction: Set theory seems to depend on the concept of the empty set. Modern theorists seem over-influenced by the Quine-Putnam view, that if science needs it, we must commit ourselves to its existence.
In simple type theory there is a hierarchy of null sets
                        Full Idea: In simple type theory, there is a null set of type 1, a null set of type 2, a null set of type 3..... (Quine has expressed his distaste for this).
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 07.4)
                        A reaction: It is bad enough trying to individuate the unique null set, without whole gangs of them drifting indistinguishably through the logical fog. All rational beings should share Quine's distaste, even if Quine is wrong.
The null set is a structural position which has no other position in membership relation
                        Full Idea: In the structuralist view of sets, in structures of a certain sort the null set is taken to be a position (or point) that will be such that no other position (or point) will be in the membership relation to it.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6)
                        A reaction: It would be hard to conceive of something having a place in a structure if nothing had a relation to it, so is the null set related to singeton sets but not there members. It will be hard to avoid Platonism here. Set theory needs the null set.
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?
                        Full Idea: What is it about the intrinsic properties of just that one unit set in virtue of which Bill Clinton is related to just it and not to any other unit sets in the set-theoretical universe?
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5)
                        A reaction: If we all kept pet woodlice, we had better not hold a wood louse rally, or we might go home with the wrong one. My singleton seems seems remarkably like yours. Could we, perhaps, swap, just for a change?
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
The set theorist cannot tell us what 'membership' is
                        Full Idea: The set theorist cannot tell us anything about the true relationship of membership.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5)
                        A reaction: If three unrelated objects suddenly became members of a set, it is hard to see how the world would have changed, except in the minds of those thinking about it.
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
ZFU refers to the physical world, when it talks of 'urelements'
                        Full Idea: ZFU set theory talks about physical objects (the urelements), and hence is in some way about the physical world.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 11.5)
                        A reaction: This sounds a bit surprising, given that the whole theory would appear to be quite unaffected if God announced that idealism is true and there are no physical objects.
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
A pack of wolves doesn't cease when one member dies
                        Full Idea: A pack of wolves is not thought to go out of existence just because some member of the pack is killed.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 07.5)
                        A reaction: The point is that the formal extensional notion of a set doesn't correspond to our common sense notion of a group or class. Even a highly scientific theory about wolves needs a loose notion of a wolf pack.
5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
The mathematics of relations is entirely covered by ordered pairs
                        Full Idea: Everything one needs to do with relations in mathematics can be done by taking a relation to be a set of ordered pairs. (Ordered triples etc. can be defined as order pairs, so that <x,y,z> is <x,<y,z>>).
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 07.2)
                        A reaction: How do we distinguish 'I own my cat' from 'I love my cat'? Or 'I quite like my cat' from 'I adore my cat'? Nevertheless, this is an interesting starting point for a discussion of relations.
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
                        Full Idea: In first-order logic a set of sentences is 'consistent' iff there is an interpretation (or structure) in which the set of sentences is true. ..For Frege, though, a set of sentences is consistent if it is not possible to deduce a contradiction from it.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 02.1)
                        A reaction: The first approach seems positive, the second negative. Frege seems to have a higher standard, which is appealing, but the first one seems intuitively right. There is a possible world where this could work.
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
                        Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 05.1)
                        A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers.
6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects
                        Full Idea: Hilbert's geometrical axioms were existential in character, asserting the existence of certain geometrical objects (points and lines). Euclid's postulates do not assert the existence of anything; they assert the possibility of certain constructions.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 01.1)
                        A reaction: Chihara says geometry was originally understood modally, but came to be understood existentially. It seems extraordinary to me that philosophers of mathematics can have become more platonist over the centuries.
Analytic geometry gave space a mathematical structure, which could then have axioms
                        Full Idea: With the invention of analytic geometry (by Fermat and then Descartes) physical space could be represented as having a mathematical structure, which could eventually lead to its axiomatization (by Hilbert).
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 02.3)
                        A reaction: The idea that space might have axioms seems to be pythagoreanism run riot. I wonder if there is some flaw at the heart of Einstein's General Theory because of this?
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / c. Nominalist structuralism
We can replace existence of sets with possibility of constructing token sentences
                        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.
                        From: report of Charles Chihara (A Structural Account of Mathematics [2004]) by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' 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?
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?
                        Full Idea: If mathematics shares whatever confirmation accrues to the theories using it, would it not be reasonable to suppose that mathematics shares whatever disconfirmation accrues to the theories using it?
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 05.8)
                        A reaction: Presumably Quine would bite the bullet here, although maths is much closer to the centre of his web of belief, and so far less likely to require adjustment. In practice, though, mathematics is not challenged whenever an experiment fails.
No scientific explanation would collapse if mathematical objects were shown not to exist
                        Full Idea: Evidently, no scientific explanations of specific phenomena would collapse as a result of any hypothetical discovery that no mathematical objects exist.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 09.1)
                        A reaction: It is inconceivable that anyone would challenge this claim. A good model seems to be drama; a play needs commitment from actors and audience, even when we know it is fiction. The point is that mathematics doesn't collapse either.
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes'
                        Full Idea: What I refer to as an 'equivalence class' (of line segments of a particular length) is an open sentence in my Constructibility Theory. I just use this terminology of the Platonist for didactic purposes.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 09.10)
                        A reaction: This is because 'equivalence classes' is committed to the existence of classes, which is Quinean Platonism. I am with Chihara in wanting a story that avoids such things. Kit Fine is investigating similar notions of rules of construction.
19. Language / B. Reference / 3. Direct Reference / b. Causal reference
Mathematical entities are causally inert, so the causal theory of reference won't work for them
                        Full Idea: Causal theories of reference seem doomed to failure for the case of reference to mathematical entities, since such entities are evidently causally inert.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], 01.3)
                        A reaction: Presumably you could baptise a fictional entity such as 'Polonius', and initiate a social causal chain, with a tradition of reference. You could baptise a baby in absentia.
27. Natural Theory / B. Modern Physics / 4. Standard Model / a. Concept of matter
'Gunk' is an individual possessing no parts that are atoms
                        Full Idea: An 'atomless gunk' is defined to be an individual possessing no parts that are atoms.
                        From: Charles Chihara (A Structural Account of Mathematics [2004], App A)
                        A reaction: [Lewis coined it] If you ask what are a-toms made of and what are ideas made of, the only answer we can offer is that the a-toms are made of gunk, and the ideas aren't made of anything, which is still bad news for the existence of ideas.