13 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. |
| 21597 | Logical connectives have the highest precision, yet are infected by the vagueness of true and false [Russell, by Williamson] |
| Full Idea: Russell says the best chance of avoiding vagueness are the logical connectives. ...But the vagueness of 'true' and 'false' infects the logical connectives too. All words are vague. Russell concludes that all language is vague. | |
| From: report of Bertrand Russell (Vagueness [1923]) by Timothy Williamson - Vagueness 2.4 | |
| A reaction: This relies on the logical connectives being defined semantically, in terms of T and F, but that is standard. Presumably the formal uninterpreted syntax is not vague. |
| 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. |
| 9051 | Since natural language is not precise it cannot be in the province of logic [Russell, by Keefe/Smith] |
| Full Idea: Russell takes it that logic assumes precision, and since natural language is not precise it cannot be in the province of logic at all. | |
| From: report of Bertrand Russell (Vagueness [1923]) by R Keefe / P Smith - Intro: Theories of Vagueness §1 | |
| A reaction: I find this view congenial. It seems to me that the necessary prelude to logic is to do everything you can to eliminate ambiguity and vagueness from the sentences at issue. We want the proposition, or logical form. If there isn't one, forget it? |
| 9054 | Vagueness is only a characteristic of representations, such as language [Russell] |
| Full Idea: Vagueness and precision alike are characteristics which can only belong to a representation, of which language is an example. | |
| From: Bertrand Russell (Vagueness [1923], p.62) | |
| A reaction: Russell was the first to tackle the question of vagueness, and he may have got it right. If we are unable to decide which set an object belongs in (red or orange) that is a problem for our conceptual/linguistic scheme. The object still has a colour! |
| 7458 | The reliability of witnesses depends on whether they benefit from their observations [Laplace, by Hacking] |
| Full Idea: The credibility of a witness is in part a function of the story being reported. When the story claims to have infinite value, the temptation to lie for personal benefit is asymptotically infinite. | |
| From: report of Pierre Simon de Laplace (Philosophical Essay on Probability [1820], Ch.XI) by Ian Hacking - The Emergence of Probability Ch.8 | |
| A reaction: Laplace seems to especially have reports of miracles in mind. This observation certainly dashes any dreams one might have of producing a statistical measure of the reliability of testimony. |
| 3441 | If a supreme intellect knew all atoms and movements, it could know all of the past and the future [Laplace] |
| Full Idea: An intelligence knowing at an instant the whole universe could know the movement of the largest bodies and atoms in one formula, provided his intellect were powerful enough to subject all data to analysis. Past and future would be present to his eyes. | |
| From: Pierre Simon de Laplace (Philosophical Essay on Probability [1820]), quoted by Mark Thornton - Do we have free will? p.70 |