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. |
| 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. |
| 7082 | Nature requires causal explanations, but society requires clarification by reasons and motives [Weber, by Critchley] |
| Full Idea: Weber coined the distinction between explanation and clarification, saying that natural phenomena require causal explanation, while social phenomena require clarification by giving reasons or offering possible motives for how things are. | |
| From: report of Max Weber (works [1905]) by Simon Critchley - Continental Philosophy - V. Short Intro Ch.7 | |
| A reaction: This is music to the ears of property dualists and other non-reductivists, but if you go midway in the hierarchy of animals (a mouse, say) the distinction blurs. Weber probably hadn't digested Darwin, whose big impact came around 1905. |
| 6000 | The goal is rationality in the selection of things according to nature [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined the goal to be rationality in the selection and rejection of the things according to nature. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: This captures the central Stoic idea quite nicely. 'Live according to nature', but this always meant 'live according to reason', because that is (as Aristotle had taught) the essence of our nature. This only makes sense if reason and nature coincide. |
| 22155 | We are disenchanted because we rely on science, which ignores values [Weber, by Boulter] |
| Full Idea: Weber contends that modern western civilisation is 'disenchanted' because our society's method of arriving at beliefs about the world, that is, the sciences, is unable to address questions of value. | |
| From: report of Max Weber (works [1905]) by Stephen Boulter - Why Medieval Philosophy Matters 6 | |
| A reaction: This idea, made explicit by Hume's empirical attitude to values, is obviously of major importance. For we Aristotelians values are a self-evident aspect of nature. Boulter says philosophy has added to the disenchantment. I agree. |
| 5999 | The good is what is perfect by nature [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined the good as what is perfect by nature. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: This might come close to G.E. Moore's Ideal Utilitarianism, but its dependence on the rather uneasy of concept of 'perfection' makes it questionable. Personally I find it appealing. I wish we had Diogenes' explanation. |
| 6001 | Justice is a disposition to distribute according to desert [Diogenes of Babylon, by Blank] |
| Full Idea: Diogenes of Babylon defined justice as the disposition which distributes to everyone what he deserves. | |
| From: report of Diogenes (Bab) (fragments/reports [c.180 BCE]) by D.L. Blank - Diogenes of Babylon | |
| A reaction: The questions that arise would be 'what does a new-born baby deserve?', and 'what do animals deserve?', and 'does the lowest and worst of criminals deserve anything at all?' |