13 ideas
| 17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
| Full Idea: Zermelo's paper sets out to show that the standard set-theoretic axioms (what he calls the 'constitutive axioms', thus the ZF axioms minus the axiom of infinity) have an unending sequence of different models, thus that they are non-categorical. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1209 | |
| A reaction: Hallett says later that Zermelo is working with second-order set theory. The addition of an Axiom of Infinity seems to have aimed at addressing the problem, and the complexities of that were pursued by Gödel. |
| 13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
| Full Idea: Zermelo included Replacement in 1930, after it was noticed that the sequence of power sets was needed, and Replacement gave the ordinal form of the well-ordering theorem, and justification for transfinite recursion. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Penelope Maddy - Believing the Axioms I §1.8 | |
| A reaction: Maddy says that this axiom suits the 'limitation of size' theorists very well, but is not so good for the 'iterative conception'. |
| 17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
| Full Idea: Two opposite tendencies of thought, the idea of creative advance and of collection and completion (underlying the Kantian 'antinomies') find their symbolic representation and their symbolic reconciliation in the transfinite numbers based on well-ordering. | |
| From: Ernst Zermelo (On boundary numbers and domains of sets [1930], §5) | |
| A reaction: [a bit compressed] It is this sort of idea, from one of the greatest set-theorists, that leads philosophers to think that the philosophy of mathematics may offer solutions to metaphysical problems. As an outsider, I am sceptical. |
| 8440 | Maybe an event's time of occurrence is essential to it [Bennett] |
| Full Idea: It has been argued that an event's time of occurrence is essential to it. | |
| From: Jonathan Bennett (Event Causation: counterfactual analysis [1987], p.221) | |
| A reaction: [He cites Lawrence Lombard] This sound initially implausible, particularly if a rival event happened, say, .1 of a second later than the actual event. It might depend on one's view about determinism. Interesting. |
| 8439 | Maybe each event has only one possible causal history [Bennett] |
| Full Idea: Perhaps it is impossible that an event should have had a causal history different from the one that it actually had. | |
| From: Jonathan Bennett (Event Causation: counterfactual analysis [1987], p.220) | |
| A reaction: [He cites van Inwagen for this] The idea is analagous to baptismal accounts of reference. Individuate an event by its history. It might depend (as Davidson implies) on how you describe the event. |
| 7346 | Jeremiah implied a link between weakness and goodness, and the evil of the state [Jeremiah, by Johnson,P] |
| Full Idea: Jeremiah was the first to perceive the possibility that powerlessness and goodness were somehow linked; ...he comes close to the notion that the state itself was inherently evil. | |
| From: report of Jeremiah (24: Book of Jeremiah [c.570 BCE]) by Paul Johnson - The History of the Jews Pt II | |
| A reaction: This looks like the first seeds of the anarchist idea. You abandon the state for something 'higher'. 'Perceive' rather begs the question of whether he is right. This is the full 'inversion of values' of Nietzsche. |
| 8441 | Delaying a fire doesn't cause it, but hastening it might [Bennett] |
| Full Idea: Although you cannot cause a fire by delaying something's burning, you can cause a fire by hastening something's burning. | |
| From: Jonathan Bennett (Event Causation: counterfactual analysis [1987], p.223) | |
| A reaction: A very nice observation which brings out all sorts of problems about identifying causes. Bennett is criticising the counterfactual account. It is part of the problem of pre-emption, where causes are queueing up to produce a given effect. |
| 8436 | Either cause and effect are subsumed under a conditional because of properties, or it is counterfactual [Bennett] |
| Full Idea: We must choose between subsumption and counterfactual analyses of causal statements. The former means that cause and effect have some properties that enables them to be subsumed under a conditional. The latter is just 'if no-c then no-e'. | |
| From: Jonathan Bennett (Event Causation: counterfactual analysis [1987], p.217) | |
| A reaction: I have an immediate preference for the former account, which seems to potentially connect it with physics and features of the world which make one thing lead to another. The counterfactual account seems very thin, and is more like mere semantics. |
| 8435 | Causes are between events ('the explosion') or between facts/states of affairs ('a bomb dropped') [Bennett] |
| Full Idea: Theories of causation are split between event and fact/state of affairs theories. The first have the form 'the explosion caused the fire' (perfect nominals) and the second have the form 'the fire started because a bomb dropped' (sentential clauses). | |
| From: Jonathan Bennett (Event Causation: counterfactual analysis [1987]) | |
| A reaction: Surely events must have priority? The form which uses facts is drifting off into explanation, and is much more likely to involve subjective human elements and interpretations. Events are closer to the physics, and the mechanics of what happens. |
| 8437 | The full counterfactual story asserts a series of events, because counterfactuals are not transitive [Bennett] |
| Full Idea: The refinement of a simple counterfactual analysis is to say that cause and effect depend on a series of events. This must be asserted because counterfactual conditionals are well known not to be transitive. | |
| From: Jonathan Bennett (Event Causation: counterfactual analysis [1987]) | |
| A reaction: This fills out the theory, but offers another target for critics. If the glue that binds the series is not in the counterfactuals, is it just in the mind of the speaker? How do you decide what's in the series? Cf. deciding offside in football (soccer!). |
| 8438 | A counterfactual about an event implies something about the event's essence [Bennett] |
| Full Idea: Any counterfactual about a particular event implies or presupposes something about the event's essence. | |
| From: Jonathan Bennett (Event Causation: counterfactual analysis [1987], p.219) | |
| A reaction: This is where the counterfactual theory suddenly becomes more interesting, instead of just being a rather bare account of the logical structure of causation. (Bennett offers some discussion of possible essential implications). |
| 22920 | Do I not fill heaven and earth? saith the Lord [Jeremiah] |
| Full Idea: Can any hide himself in secret places that I shall not see him? saith the Lord. Do I not fill heaven and earth? | |
| From: Jeremiah (24: Book of Jeremiah [c.570 BCE], 23:24), quoted by Robin Le Poidevin - Travels in Four Dimensions 03 'Where' | |
| A reaction: If the Lord is omnipresent, then He must be present in each one of us. But does the Lord interact with each of us? |
| 22089 | Am I a God afar off, and not a God close at hand? [Jeremiah] |
| Full Idea: Am I a God afar off, and not a God close at hand? Do I not fill heaven and earth? | |
| From: Jeremiah (24: Book of Jeremiah [c.570 BCE], 23:23), quoted by Clare Carlisle - Kierkegaard: a guide for the perplexed 3 | |
| A reaction: I assume this was often quoted by eighteenth century divines, against the rise of deism. |