Combining Texts

All the ideas for 'Subjectivist's Guide to Objective Chance', 'On the Introduction of Transfinite Numbers' and 'An Outline of Empiricism'

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7 ideas

2. Reason / A. Nature of Reason / 7. Status of Reason
23248 Early empiricists said reason was just a useless concept introduced by philosophers [Galen, by Frede,M]
     Full Idea: The so-called Empiricists in Hellenistic times [as cited by Galen] denied the existence of reason, treating it as a useless theoretical postulate introduced by some philosophers
     From: report of Galen (An Outline of Empiricism [c.170], 87.4-9.28ff) by Michael Frede - Intro to 'Rationality in Greek Thought' p.3
     A reaction: I think 'be sensible' is understood by everyone, but 'use your reason' is far from obvious. The main role of reason seems to be as an identifier for human exceptionalism. Animals obviously make good judgements. Frede thinks the empiricists were right.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
13489 Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD]
     Full Idea: Von Neumann's decision was to start with the ordinals and to treat cardinals as a special sort of ordinal.
     From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by William D. Hart - The Evolution of Logic 3
     A reaction: [see Hart 73-74 for an explication of this]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
12336 A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou]
     Full Idea: In Von Neumann's definition an ordinal is a transitive set in which all of the elements are transitive.
     From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Alain Badiou - Briefings on Existence 11
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / g. Von Neumann numbers
18179 For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy]
     Full Idea: For Von Neumann the successor of n is n U {n} (rather than Zermelo's successor, which is {n}).
     From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8
18180 Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann]
     Full Idea: Von Neumann's version of the natural numbers is in fact preferred because it carries over directly to the transfinite ordinals.
     From: comment on John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9
15925 Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine]
     Full Idea: Each Von Neumann ordinal number is the set of its predecessors. ...He had shown how to introduce ordinal numbers as sets, making it possible to use them without leaving the domain of sets.
     From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Shaughan Lavine - Understanding the Infinite V.3
26. Natural Theory / D. Laws of Nature / 4. Regularities / b. Best system theory
9425 Lewis later proposed the axioms at the intersection of the best theories (which may be few) [Mumford on Lewis]
     Full Idea: Later Lewis said we must choose between the intersection of the axioms of the tied best systems. He chose for laws the axioms that are in all the tied systems (but then there may be few or no axioms in the intersection).
     From: comment on David Lewis (Subjectivist's Guide to Objective Chance [1980], p.124) by Stephen Mumford - Laws in Nature