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. |
| 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. |
| 15990 | Every individual thing which exists has an essence, which is its internal constitution [Locke] |
| Full Idea: I take essences to be in everything that internal constitution or frame for the modification of substance, which God in his wisdom gives to every particular creature, when he gives it a being; and such essences I grant there are in all things that exist. | |
| From: John Locke (Letters to Edward Stillingfleet [1695], Letter 1), quoted by Simon Blackburn - Quasi-Realism no Fictionalism | |
| A reaction: This is the clearest statement I have found of Locke's commitment to essences, for all his doubts about whether we can know such things. Alexander says (ch.13) Locke was reacting against scholastic essence, as pertaining to species. |
| 15994 | If it is knowledge, it is certain; if it isn't certain, it isn't knowledge [Locke] |
| Full Idea: What reaches to knowledge, I think may be called certainty; and what comes short of certainty, I think cannot be knowledge. | |
| From: John Locke (Letters to Edward Stillingfleet [1695], Letter 2), quoted by Simon Blackburn - Quasi-Realism no Fictionalism | |
| A reaction: I much prefer that fallibilist approach offered by the pragmatists. Knowledge is well-supported belief which seems (and is agreed) to be true, but there is a small shadow of doubt hanging over all of it. |
| 22381 | Being a good father seems to depend on intentions, rather than actual abilities [Foot] |
| Full Idea: Being a good father, or daughter, or friend seems to depend on one's intentions, rather than on such things as cleverness and strength. | |
| From: Philippa Foot (Goodness and Choice [1961], p.138) | |
| A reaction: Not sure about that. In wartime a good father might need to be actually brave, and in times of hardship be actually economically successful. 'He meant well, but he was a hopeless father'? |
| 22380 | Some words, such as 'knife', have a meaning which involves its function [Foot] |
| Full Idea: The word 'knife' names an object in respect of its function. That is not to say (simply) that it names an object which has a function, but also that the function is involved in the meaning of the word. | |
| From: Philippa Foot (Goodness and Choice [1961], p.134) | |
| A reaction: It seems faintly possible that someone (a child, perhaps) could know the word and recognise the object, but not know what the object is for. Ditto with other things which have functional names. |