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All the ideas for 'fragments/reports', 'On the Introduction of Transfinite Numbers' and 'Logic [1897]'

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8 ideas

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
13. Knowledge Criteria / C. External Justification / 2. Causal Justification
11052 Psychological logic can't distinguish justification from causes of a belief [Frege]
     Full Idea: With the psychological conception of logic we lose the distinction between the grounds that justify a conviction and the causes that actually produce it.
     From: Gottlob Frege (Logic [1897] [1897])
     A reaction: Thus Frege kicked the causal theory of justification well into touch long before it had even been properly formulated. That is not to say that there is no psychological aspect to logic, because there is.
26. Natural Theory / A. Speculations on Nature / 5. Infinite in Nature
1748 Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius]
     Full Idea: Archelaus was the first person to say that the universe is boundless.
     From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3
27. Natural Reality / G. Biology / 3. Evolution
5989 Archelaus said life began in a primeval slime [Archelaus, by Schofield]
     Full Idea: Archelaus wrote that life on Earth began in a primeval slime.
     From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus
     A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea.