Combining Texts

Ideas for 'The Boundary Stones of Thought', 'Foucault: a very short introduction' and 'Varieties of Ontological Dependence'

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4 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
18842 Maybe an ordinal is a property of isomorphic well-ordered sets, and not itself a set [Rumfitt]
     Full Idea: Menzel proposes that an ordinal is something isomorphic well-ordered sets have in common, so while an ordinal can be represented as a set, it is not itself a set, but a 'property' of well-ordered sets.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.2)
     A reaction: [C.Menzel 1986] This is one of many manoeuvres available if you want to distance mathematics from set theory.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
18834 Infinitesimals do not stand in a determinate order relation to zero [Rumfitt]
     Full Idea: Infinitesimals do not stand in a determinate order relation to zero: we cannot say an infinitesimal is either less than zero, identical to zero, or greater than zero. ….Infinitesimals are so close to zero as to be theoretically indiscriminable from it.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 7.4)
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
18846 Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry) [Rumfitt]
     Full Idea: One of the motivations behind Cantor's and Dedekind's pioneering explorations in the field was the ambition to give real analysis a new foundation in set theory - and hence a foundation independent of geometry.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 9.6)
     A reaction: Rumfitt is inclined to think that the project has failed, although a weaker set theory than ZF might do the job (within limits).
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
17312 It is more explanatory if you show how a number is constructed from basic entities and relations [Koslicki]
     Full Idea: Being the successor of the successor of 0 is more explanatory than being predecessor of 3 of the nature of 2, since it mirrors more closely the method by which 2 is constructed from a basic entity, 0, and a relation (successor) taken as primitive.
     From: Kathrin Koslicki (Varieties of Ontological Dependence [2012], 7.4)
     A reaction: This assumes numbers are 'constructed', which they are in the axiomatised system of Peano Arithmetic, but presumably the numbers were given in ordinary experience before 'construction' occurred to anyone. Nevertheless, I really like this.