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Ideas for 'The Evolution of Logic', 'The Law of Peoples' and 'Reply to 'Rorarius' 2nd ed'

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8 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13463 There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD]
     Full Idea: Since we can map the transfinite ordinals one-one into the infinite cardinals, there are at least as many infinite cardinals as transfinite ordinals.
     From: William D. Hart (The Evolution of Logic [2010], 1)
13492 Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD]
     Full Idea: It is easier to generalize von Neumann's finite ordinals into the transfinite. All Zermelo's nonzero finite ordinals are singletons, but if ω were a singleton it is hard to see how if could fail to be the successor of its member and so not a limit.
     From: William D. Hart (The Evolution of Logic [2010], 3)
13459 The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD]
     Full Idea: We can show (using the axiom of choice) that the less-than relation, <, well-orders the ordinals, ...and that it partially orders the ordinals, ...and that it totally orders the ordinals.
     From: William D. Hart (The Evolution of Logic [2010], 1)
13491 The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD]
     Full Idea: The axiom of infinity with separation yields a least limit ordinal, which is called ω.
     From: William D. Hart (The Evolution of Logic [2010], 3)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
13446 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD]
     Full Idea: The real numbers were not isolated from geometry until the arithmetization of analysis during the nineteenth century.
     From: William D. Hart (The Evolution of Logic [2010], 1)
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
13509 We can establish truths about infinite numbers by means of induction [Hart,WD]
     Full Idea: Mathematical induction is a way to establish truths about the infinity of natural numbers by a finite proof.
     From: William D. Hart (The Evolution of Logic [2010], 5)
     A reaction: If there are truths about infinities, it is very tempting to infer that the infinities must therefore 'exist'. A nice, and large, question in philosophy is whether there can be truths without corresponding implications of existence.
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
13474 Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD]
     Full Idea: There is a familiar comparison between Euclid (unique parallel) and 'spherical' geometry (no parallel) and 'saddle' geometry (several parallels).
     From: William D. Hart (The Evolution of Logic [2010], 2)
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
13471 Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD]
     Full Idea: The thesis that no existence proposition is analytic is one of the few constants in philosophical consciences, but there are many existence claims in mathematics, such as the infinity of primes, five regular solids, and certain undecidable propositions.
     From: William D. Hart (The Evolution of Logic [2010], 2)