11 ideas
23623 | Predicativism says only predicated sets exist [Hossack] |
Full Idea: Predicativists doubt the existence of sets with no predicative definition. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 02.3) | |
A reaction: This would imply that sets which encounter paradoxes when they try to be predicative do not therefore exist. Surely you can have a set of random objects which don't fall under a single predicate? |
23624 | The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack] |
Full Idea: The iterative conception justifies Power Set, but cannot justify a satisfactory theory of von Neumann ordinals, so ZFC appropriates Replacement from NBG set theory. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 09.9) | |
A reaction: The modern approach to axioms, where we want to prove something so we just add an axiom that does the job. |
23625 | Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack] |
Full Idea: The limitation of size conception of sets justifies the axiom of Replacement, but cannot justify Power Set, so NBG set theory appropriates the Power Set axiom from ZFC. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 09.9) | |
A reaction: Which suggests that the Power Set axiom is not as indispensable as it at first appears to be. |
23628 | The connective 'and' can have an order-sensitive meaning, as 'and then' [Hossack] |
Full Idea: The sentence connective 'and' also has an order-sensitive meaning, when it means something like 'and then'. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.4) | |
A reaction: This is support the idea that orders are a feature of reality, just as much as possible concatenation. Relational predicates, he says, refer to series rather than to individuals. Nice point. |
23627 | 'Before' and 'after' are not two relations, but one relation with two orders [Hossack] |
Full Idea: The reason the two predicates 'before' and 'after' are needed is not to express different relations, but to indicate its order. Since there can be difference of order without difference of relation, the nature of relations is not the source of order. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.3) | |
A reaction: This point is to refute Russell's 1903 claim that order arises from the nature of relations. Hossack claims that it is ordered series which are basic. I'm inclined to agree with him. |
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. |
21995 | Must production determine superstructure, or could it be the other way round? [Singer on Marx] |
Full Idea: Once the 'interaction' between the superstructure and the productive forces is admitted, is it still possible to maintain that production determines the superstructure, rather than the other way round? | |
From: comment on Karl Marx (Capital Vol. 1 [1867]) by Peter Singer - Marx 7 | |
A reaction: It is much harder to defend historical determinism if Singer is right about this. Modern capitalism won't admit of the sort of simple distinctions that mark was looking for. |
20577 | Even decently paid workers still have their produce bought with money stolen from them [Marx] |
Full Idea: Even if the workers are paid a fair wage, the whole thing still remains the age-old activity of the conqueror, who buys commodities from the conquered with the money has has stolen from them, | |
From: Karl Marx (Capital Vol. 1 [1867], p.728), quoted by Johanna Oksala - Political Philosophy: all that matters Ch.8 | |
A reaction: [Penguin edition cited] The word 'stolen' is obviously dubious here. 'Exploitation' is a much more accurate word. One might talk of 'blackmail' or 'extortion' rather than theft. |
7482 | Resurrection developed in Judaism as a response to martyrdoms, in about 160 BCE [Anon (Dan), by Watson] |
Full Idea: The idea of resurrection in Judaism seems to have first developed around 160 BCE, during the time of religious martyrdom, and as a response to it (the martyrs were surely not dying forever?). It is first mentioned in the book of Daniel. | |
From: report of Anon (Dan) (27: Book of Daniel [c.165 BCE], Ch.7) by Peter Watson - Ideas | |
A reaction: Idea 7473 suggests that Zoroaster beat them to it by 800 years. |