Combining Texts

All the ideas for 'talk', 'Grundlagen (Foundations of Theory of Manifolds)' and 'Thoughts without Laws'

unexpand these ideas     |    start again     |     specify just one area for these texts

4 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
15946 Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine]
     Full Idea: Cantor's development of set theory began with his discovery of the progression 0, 1, ....∞, ∞+1, ∞+2, ..∞x2, ∞x3, ...∞^2, ..∞^3, ...∞^∞, ...∞^∞^∞.....
     From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite VIII.2
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
15911 Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine]
     Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence).
     From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4
11. Knowledge Aims / A. Knowledge / 4. Belief / a. Beliefs
12582 The function of beliefs is to produce beliefs-that-p when p [Millikan]
     Full Idea: Presumably it is a proper function of the belief-manufacturing mechanisms in John to produce beliefs-that-p only if and when p.
     From: Ruth Garrett Millikan (Thoughts without Laws [1986], p.69), quoted by Christopher Peacocke - A Study of Concepts 5.2
     A reaction: This is the 'teleological' account of belief, which is trying to fit belief into an evolutionary view of humans. It is doubtful whether you can say mental states are just their 'proper' function, because then piano-playing becomes a puzzle.
14. Science / C. Induction / 3. Limits of Induction
7295 Maybe induction is only reliable IF reality is stable [Mitchell,A]
     Full Idea: Maybe we should say that IF regularities are stable, only then is induction a reliable procedure.
     From: Alistair Mitchell (talk [2006]), quoted by PG - Db (ideas)
     A reaction: This seems to me a very good proposal. In a wildly unpredictable reality, it is hard to see how anyone could learn from experience, or do any reasoning about the future. Natural stability is the axiom on which induction is built.