12 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. |
17462 | A single object must not be counted twice, which needs knowledge of distinctness (negative identity) [Rumfitt] |
Full Idea: One requirement for a successful count is that double counting should be avoided: a single object should not be counted twice. ...but that is to make a knowledgeable judgement of distinctness - to resolve a question of identity in the negative. | |
From: Ian Rumfitt (Concepts and Counting [2002], III) | |
A reaction: He also notes later (p.65) that you must count all and only the right things. |
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. |
17461 | Some 'how many?' answers are not predications of a concept, like 'how many gallons?' [Rumfitt] |
Full Idea: We hit trouble if we hear answers to some 'How many?' questions as predications about concepts. The correct answer to 'how many gallons of water are in the tank?' may be 'ten', but that doesn''t mean ten things instantiate 'gallon of water in the tank'. | |
From: Ian Rumfitt (Concepts and Counting [2002], I) | |
A reaction: Rumfitt makes the point that a huge number of things instantiate that concept in a ten gallon tank of water. No problem, says Rumfitt, because Frege wouldn't have counted that as a statement of number. |
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. |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |