Ideas from 'Logicism in the 21st Century' by B Hale / C Wright [2007], by Theme Structure

[found in 'Oxf Handbk of Philosophy of Maths and Logic' (ed/tr Shapiro,Stewart) [OUP 2007,978-0-19-532592-8]].

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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Various strategies try to deal with the ontological commitments of second-order logic
                        Full Idea: Quine said higher-order logic is 'set theory in sheep's clothing', and there is concern about the ontology that is involved. One approach is to deny quantificational ontological commitments, or say that the entities involved are first-order objects.
                        From: B Hale / C Wright (Logicism in the 21st Century [2007], 8)
                        A reaction: [compressed] The second strategy is from Boolos. This question seems to be right at the heart of the strategy of exploring our ontology through the study of our logic.
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / d. Hume's Principle
Neo-logicism founds arithmetic on Hume's Principle along with second-order logic
                        Full Idea: The result of joining Hume's Principle to second-order logic is a consistent system which is a foundation for arithmetic, in the sense that all the fundamental laws of arithmetic are derivable within it as theorems. This seems a vindication of logicism.
                        From: B Hale / C Wright (Logicism in the 21st Century [2007], 1)
                        A reaction: The controversial part seems to be second-order logic, which Quine (for example) vigorously challenged. The contention against most attempts to improve Frege's logicism is that they thereby cease to be properly logical.
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / e. Caesar problem
The Julius Caesar problem asks for a criterion for the concept of a 'number'
                        Full Idea: The Julius Caesar problem is the problem of supplying a criterion of application for 'number', and thereby setting it up as the concept of a genuine sort of object. (Why is Julius Caesar not a number?)
                        From: B Hale / C Wright (Logicism in the 21st Century [2007], 3)
                        A reaction: One response would be to deny that numbers are objects. Another would be to derive numbers from their application in counting objects, rather than the other way round. I suspect that the problem only real bothers platonists. Serves them right.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology
                        Full Idea: It is only if logic is metaphysically and epistemologically privileged that a reduction of mathematical theories to logical ones can be philosophically any more noteworthy than a reduction of any mathematical theory to any other.
                        From: B Hale / C Wright (Logicism in the 21st Century [2007], 8)
                        A reaction: It would be hard to demonstrate this privileged position, though intuitively there is nothing more basic in human rationality. That may be a fact about us, but it doesn't make logic basic to nature, which is where proper reduction should be heading.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
Logicism might also be revived with a quantificational approach, or an abstraction-free approach
                        Full Idea: Two modern approaches to logicism are the quantificational approach of David Bostock, and the abstraction-free approach of Neil Tennant.
                        From: B Hale / C Wright (Logicism in the 21st Century [2007], 1 n2)
                        A reaction: Hale and Wright mention these as alternatives to their own view. I merely catalogue them for further examination. My immediate reaction is that Bostock sounds hopeless and Tennant sounds interesting.
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
One first-order abstraction principle is Frege's definition of 'direction' in terms of parallel lines
                        Full Idea: An example of a first-order abstraction principle is Frege's definition of 'direction' in terms of parallel lines; a higher-order example (which refers to first-order predicates) defines 'equinumeral' in terms of one-to-one correlation (Hume's Principle).
                        From: B Hale / C Wright (Logicism in the 21st Century [2007], 1)
                        A reaction: [compressed] This is the way modern logicians now treat abstraction, but abstraction principles include the elusive concept of 'equivalence' of entities, which may be no more than that the same adjective ('parallel') can be applied to them.