10 ideas
3340 | Von Neumann defines each number as the set of all smaller numbers [Neumann, by Blackburn] |
Full Idea: Von Neumann defines each number as the set of all smaller numbers. | |
From: report of John von Neumann (works [1935]) by Simon Blackburn - Oxford Dictionary of Philosophy p.280 |
3355 | Von Neumann wanted mathematical functions to replace sets [Neumann, by Benardete,JA] |
Full Idea: Von Neumann suggested that functions be pressed into service to replace sets. | |
From: report of John von Neumann (works [1935]) by José A. Benardete - Metaphysics: the logical approach Ch.23 |
22716 | Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone] |
Full Idea: At age twenty, Von Neumann devised the formal definition of ordinal numbers that is used today: an ordinal number is the set of all smaller ordinal numbers. | |
From: report of John von Neumann (works [1935]) by William Poundstone - Prisoner's Dilemma 02 'Sturm' | |
A reaction: I take this to be an example of an impredicative definition (not predicating something new), because it uses 'ordinal number' in the definition of ordinal number. I'm guessing the null set gets us started. |
12354 | A 'categorial' property is had by virtue of being or having an item from a category [Wedin] |
Full Idea: A 'categorial' property is a property something has by virtue of being or having an item from one of the categories. | |
From: Michael V. Wedin (Aristotle's Theory of Substance [2000], V.5) | |
A reaction: I deny that these are 'properties'. A thing is categorised according to its properties. To denote the category as a further property is the route to madness (well, to a regress). |
12358 | Substance is a principle and a kind of cause [Wedin] |
Full Idea: Substance [ousia] is a principle [arché] and a kind of cause [aitia]. | |
From: Michael V. Wedin (Aristotle's Theory of Substance [2000], 1041a09) | |
A reaction: The fact that substance is a cause is also the reason why substance is the ultimate explanation. It is here that I take the word 'power' to capture best what Aristotle has in mind. |
12346 | Form explains why some matter is of a certain kind, and that is explanatory bedrock [Wedin] |
Full Idea: The form of a thing (of a given kind) explains why certain matter constitutes a thing of that kind, and with this, Aristotle holds, we have reached explanatory bedrock. | |
From: Michael V. Wedin (Aristotle's Theory of Substance [2000], Intro) | |
A reaction: We must explain an individual tiger which is unusually docile. It must have an individual form which makes it a tiger, but also an individual form which makes it docile. |
19553 | Commitment to 'I have a hand' only makes sense in a context where it has been doubted [Hawthorne] |
Full Idea: If I utter 'I know I have a hand' then I can only be reckoned a cooperative conversant by my interlocutors on the assumption that there was a real question as to whether I have a hand. | |
From: John Hawthorne (The Case for Closure [2005], 2) | |
A reaction: This seems to point to the contextualist approach to global scepticism, which concerns whether we are setting the bar high or low for 'knowledge'. |
19551 | How can we know the heavyweight implications of normal knowledge? Must we distort 'knowledge'? [Hawthorne] |
Full Idea: Those who deny skepticism but accept closure will have to explain how we know the various 'heavyweight' skeptical hypotheses to be false. Do we then twist the concept of knowledge to fit the twin desiderata of closue and anti-skepticism? | |
From: John Hawthorne (The Case for Closure [2005], Intro) | |
A reaction: [He is giving Dretske's view; Dretske says we do twist knowledge] Thus if I remember yesterday, that has the heavyweight implication that the past is real. Hawthorne nicely summarises why closure produces a philosophical problem. |
19552 | We wouldn't know the logical implications of our knowledge if small risks added up to big risks [Hawthorne] |
Full Idea: Maybe one cannot know the logical consequences of the proposition that one knows, on account of the fact that small risks add up to big risks. | |
From: John Hawthorne (The Case for Closure [2005], 1) | |
A reaction: The idea of closure is that the new knowledge has the certainty of logic, and each step is accepted. An array of receding propositions can lose reliability, but that shouldn't apply to logic implications. Assuming monotonic logic, of course. |
19554 | Denying closure is denying we know P when we know P and Q, which is absurd in simple cases [Hawthorne] |
Full Idea: How could we know that P and Q but not be in a position to know that P (as deniers of closure must say)? If my glass is full of wine, we know 'g is full of wine, and not full of non-wine'. How can we deny that we know it is not full of non-wine? | |
From: John Hawthorne (The Case for Closure [2005], 2) | |
A reaction: Hawthorne merely raises this doubt. Dretske is concerned with heavyweight implications, but how do you accept lightweight implications like this one, and then suddenly reject them when they become too heavy? [see p.49] |