Ideas from 'Replies on 'Limits of Abstraction'' by Kit Fine [2005], by Theme Structure

[found in 'Philosophical Studies' (ed/tr -) [- ,]].

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1. Philosophy / F. Analytic Philosophy / 7. Limitations of Analysis
Concern for rigour can get in the way of understanding phenomena
                        Full Idea: It is often the case that the concern for rigor gets in the way of a true understanding of the phenomena to be explained.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
                        A reaction: This is a counter to Timothy Williamson's love affair with rigour in philosophy. It strikes me as the big current question for analytical philosophy - of whether the intense pursuit of 'rigour' will actually deliver the wisdom we all seek.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
There is no stage at which we can take all the sets to have been generated
                        Full Idea: There is no stage at which we can take all the sets to have been generated, since the set of all those sets which have been generated at a given stage will itself give us something new.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1)
4. Formal Logic / G. Formal Mereology / 3. Axioms of Mereology
We might combine the axioms of set theory with the axioms of mereology
                        Full Idea: We might combine the standard axioms of set theory with the standard axioms of mereology.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1)
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
If you ask what F the second-order quantifier quantifies over, you treat it as first-order
                        Full Idea: We are tempted to ask of second-order quantifiers 'what are you quantifying over?', or 'when you say "for some F" then what is the F?', but these questions already presuppose that the quantifiers are first-order.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005])
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Assigning an entity to each predicate in semantics is largely a technical convenience
                        Full Idea: In doing semantics we normally assign some appropriate entity to each predicate, but this is largely for technical convenience.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Dedekind cuts lead to the bizarre idea that there are many different number 1's
                        Full Idea: Because of Dedekind's definition of reals by cuts, there is a bizarre modern doctrine that there are many 1's - the natural number 1, the rational number 1, the real number 1, and even the complex number 1.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
                        A reaction: See Idea 10572.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
Why should a Dedekind cut correspond to a number?
                        Full Idea: By what right can Dedekind suppose that there is a number corresponding to any pair of irrationals that constitute an irrational cut?
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
Unless we know whether 0 is identical with the null set, we create confusions
                        Full Idea: What is the union of the singleton {0}, of zero, and the singleton {φ}, of the null set? Is it the one-element set {0}, or the two-element set {0, φ}? Unless the question of identity between 0 and φ is resolved, we cannot say.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
Set-theoretic imperialists think sets can represent every mathematical object
                        Full Idea: Set-theoretic imperialists think that it must be possible to represent every mathematical object as a set.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1)
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Logicists say mathematics can be derived from definitions, and can be known that way
                        Full Idea: Logicists traditionally claim that the theorems of mathematics can be derived by logical means from the relevant definitions of the terms, and that these theorems are epistemically innocent (knowable without Kantian intuition or empirical confirmation).
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 2)
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / b. Levels of abstraction
A generative conception of abstracts proposes stages, based on concepts of previous objects
                        Full Idea: It is natural to have a generative conception of abstracts (like the iterative conception of sets). The abstracts are formed at stages, with the abstracts formed at any given stage being the abstracts of those concepts of objects formed at prior stages.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1)
                        A reaction: See 10567 for Fine's later modification. This may not guarantee 'levels', but it implies some sort of conceptual priority between abstract entities.
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Abstraction-theoretic imperialists think Fregean abstracts can represent every mathematical object
                        Full Idea: Abstraction-theoretic imperialists think that it must be possible to represent every mathematical object as a Fregean abstract.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1)
We can combine ZF sets with abstracts as urelements
                        Full Idea: I propose a unified theory which is a version of ZF or ZFC with urelements, where the urelements are taken to be the abstracts.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1)
We can create objects from conditions, rather than from concepts
                        Full Idea: Instead of viewing the abstracts (or sums) as being generated from objects, via the concepts from which they are defined, we can take them to be generated from conditions. The number of the universe ∞ is the number of self-identical objects.
                        From: Kit Fine (Replies on 'Limits of Abstraction' [2005], 1)
                        A reaction: The point is that no particular object is now required to make the abstraction.