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8784 | 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. |
8787 | 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. |
8788 | 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. |
8783 | 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. |
8786 | 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. |