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. |
| 3102 | Why don't we experience or remember going to sleep at night? [Magee] |
| Full Idea: As a child it was incomprehensible to me that I did not experience going to sleep, and never remembered it. When my sister said 'Nobody remembers that', I just thought 'How does she know?' | |
| From: Bryan Magee (Confessions of a Philosopher [1997], Ch.I) | |
| A reaction: This is actually evidence for something - that we do not have some sort of personal identity which is separate from consciousness, so that "I am conscious" would literally mean that an item has a property, which it can lose. |
| 23857 | People in power always try to increase their power [Weil] |
| Full Idea: Every human group that exercises power does so …in such a way as to increase that power. | |
| From: Simone Weil (Prospects: Proletarian Revolution? [1933], p.15) | |
| A reaction: Not so true in smaller institutions, but at the centre of power you can control how power is distributed, so the temptation is too much. |
| 23856 | Spontaneous movements are powerless against organised repression [Weil] |
| Full Idea: A spontaneous movement is fundamentally impotent when it comes to fighting against organised forces of repression. | |
| From: Simone Weil (Prospects: Proletarian Revolution? [1933], p.2) | |
| A reaction: Her example is the Paris Commune of 1870. Hence revolution requires prior penetration of the corridors of power. Hence the phenomenon of 'entryism' of more radical people into reformist parties. |
| 23859 | True democracy is the subordination of society to the individual [Weil] |
| Full Idea: In the subordination of society to the individual lies the definition of true democracy, and that of socialism as well. | |
| From: Simone Weil (Prospects: Proletarian Revolution? [1933], p.19) | |
| A reaction: This is the simplest definition of the liberal view. The big difference is whether this subordination is the starting point of political thinking, or the end result at which it aims. |
| 23858 | War is perpetuated by its continual preparations [Weil] |
| Full Idea: War perpetuates itself under the form of preparation for war. | |
| From: Simone Weil (Prospects: Proletarian Revolution? [1933], p.16) | |
| A reaction: There are periods when military preparations are scaled down, but a reason is always found to scale them back up again. |
| 23860 | Even if a drowning man is doomed, he should keep swimming to the last [Weil] |
| Full Idea: A man who is thrown overboard in the middle of the ocean ought not to let himself drown, even though there is very litte chance of his reaching safety, but to go on swimming till exhausted. | |
| From: Simone Weil (Prospects: Proletarian Revolution? [1933], p.21) | |
| A reaction: You might survive a little longer if you don't exhaust yourself! Not clear where her authority for 'ought' comes from, but it expresses an interesting attitude. |