14 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. |
| 16052 | 'Superdupervenience' is supervenience that has a robustly materialistic explanation [Horgan,T] |
| Full Idea: The idea of a ontological supervenience that is robustly explainable in a materialistically explainable way I hereby dub 'superdupervenience'. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §4) | |
| A reaction: [He credits William Lycan with the actual word] His assumption prior to this introduction is that mere supervenience just adds a new mystery. I take supervenience to be an observation of 'tracking', which presumably needs to be explained. |
| 16053 | 'Global' supervenience is facts tracking varying physical facts in every possible world [Horgan,T] |
| Full Idea: The idea of 'global supervenience' is standardly expressed as 'there are no two physically possible worlds which are exactly alike in all physical respects but different in some other respect'. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §5) | |
| A reaction: [Jaegwon Kim is the source of this concept] The 'local' view will be that they do indeed track, but they could, in principle, come apart. A zombie might be a case of them possibly coming apart. Zombies are silly. |
| 16056 | Don't just observe supervenience - explain it! [Horgan,T] |
| Full Idea: Although the task of explaining supervenience has been little appreciated and little discussed in the philosophical literature, it is time for that to change. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §8) | |
| A reaction: I would offer a strong addition to this: be absolutely sure that you are dealing with two distinct things in the supervenience relationship, before you waste time trying to explain how they relate to one another. |
| 16054 | Physicalism needs more than global supervenience on the physical [Horgan,T] |
| Full Idea: Global supervenience seems too weak to capture the physical facts determining all the facts. …There could be two spatio-temporal regions alike in all physical respects, but different in some intrinsic non-physical respect. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §5) | |
| A reaction: I.e. there might be two physically identical regions, but one contains angels and the other doesn't (so the extra fact isn't tracking the physical facts). Physicalism I take to be the simple denial of the angels. Supervenience is an explanandum. |
| 16055 | Materialism requires that physics be causally complete [Horgan,T] |
| Full Idea: Any broadly materialistic metaphysical position needs to claim that physics is causally complete. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §6) | |
| A reaction: Since 'physics' is a human creation, I presume he means that physical reality is causally complete. The interaction problem that faced Descartes seems crucial - how could something utterly non-physical effect a physical change? |
| 16057 | Instrumentalism normally says some discourse is useful, but not genuinely true [Horgan,T] |
| Full Idea: Instrumentalist views typically attribute utility to the given body of discourse, but deny that it expresses genuine truths. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §8) | |
| A reaction: To me it is obvious to ask why anything could have a high level of utility (especially in accounts of the external physical world) without being true. Falsehoods may sometimes (though I doubt it) be handy in human life, but useful in chemistry…? |