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5 ideas
23626 | Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack] |
Full Idea: The transfinite ordinal numbers are important in the theory of proofs, and essential in the theory of recursive functions and computability. Mathematics would be incomplete without them. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.1) | |
A reaction: Hossack offers this as proof that the numbers are not human conceptual creations, but must exist beyond the range of our intellects. Hm. |
23621 | Numbers are properties, not sets (because numbers are magnitudes) [Hossack] |
Full Idea: I propose that numbers are properties, not sets. Magnitudes are a kind of property, and numbers are magnitudes. …Natural numbers are properties of pluralities, positive reals of continua, and ordinals of series. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], Intro) | |
A reaction: Interesting! Since time can have a magnitude (three weeks) just as liquids can (three litres), it is not clear that there is a single natural property we can label 'magnitude'. Anything we can manage to measure has a magnitude. |
23622 | We can only mentally construct potential infinities, but maths needs actual infinities [Hossack] |
Full Idea: Numbers cannot be mental objects constructed by our own minds: there exists at most a potential infinity of mental constructions, whereas the axioms of mathematics require an actual infinity of numbers. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], Intro 2) | |
A reaction: Doubt this, but don't know enough to refute it. Actual infinities were a fairly late addition to maths, I think. I would think treating fictional complete infinities as real would be sufficient for the job. Like journeys which include imagined roads. |
21559 | We need rules for deciding which norms are predicative (unless none of them are) [Russell] |
Full Idea: We need rules for deciding what norms are predicative and what are not, unless we adopt the view (which has much to recommend it) that no norms are predicative. ...[146] A predative propositional function is one which determines a class. | |
From: Bertrand Russell (Difficulties of Transfinite Numbers and Types [1905], p.141) | |
A reaction: He is referring to his 'no class' theory, which he favoured at that time. |
21558 | 'Predicative' norms are those which define a class [Russell] |
Full Idea: Norms (containing one variable) which do not define classes I propose to call 'non-predicative'; those which do define classes I shall call 'predicative'. | |
From: Bertrand Russell (Difficulties of Transfinite Numbers and Types [1905], p.141) |