14 ideas
| 13489 | Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD] |
| Full Idea: Von Neumann's decision was to start with the ordinals and to treat cardinals as a special sort of ordinal. | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: [see Hart 73-74 for an explication of this] |
| 12336 | A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou] |
| Full Idea: In Von Neumann's definition an ordinal is a transitive set in which all of the elements are transitive. | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Alain Badiou - Briefings on Existence 11 |
| 18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy] |
| Full Idea: For Von Neumann the successor of n is n U {n} (rather than Zermelo's successor, which is {n}). | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 |
| 18180 | Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann] |
| Full Idea: Von Neumann's version of the natural numbers is in fact preferred because it carries over directly to the transfinite ordinals. | |
| From: comment on John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9 |
| 15925 | Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine] |
| Full Idea: Each Von Neumann ordinal number is the set of its predecessors. ...He had shown how to introduce ordinal numbers as sets, making it possible to use them without leaving the domain of sets. | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Shaughan Lavine - Understanding the Infinite V.3 |
| 19544 | Closure says if you know P, and also know P implies Q, then you must know Q [Dretske] |
| Full Idea: Closure is the epistemological principle that if S knows that P is true and knows that P implies Q, then, evidentially speaking, this is enough for S to know that Q is true. Nothing more is needed. | |
| From: Fred Dretske (The Case against Closure (and reply) [2005], p.25) | |
| A reaction: [Dretske was the first to raise this issue] It is 'closure' because it applies to every case of Q, which is every implication of P that is known. The issue is whether we really do know all such Qs. Dretske doubts it. See his zebra case. |
| 19545 | We needn't regret the implications of our regrets; regretting drinking too much implies the past is real [Dretske] |
| Full Idea: One doesn't have to regret everything one knows to be implied by what one regrets. Tom regrets drinking three martinis, but doesn't regret what he knows to be implied by this - that he drank 'something', or that the past is real. | |
| From: Fred Dretske (The Case against Closure (and reply) [2005], p.28) | |
| A reaction: A nice case of analogy! He's right about regret. Perceptual and inferential knowledge have different grounds. To deny inferential knowledge seems to be a denial that modus ponens can be a justification. But MP gives truth, not knowledge. |
| 19547 | Reasons for believing P may not transmit to its implication, Q [Dretske] |
| Full Idea: Some reasons for believing P do not transmit to things, Q, known to be implied by P. | |
| From: Fred Dretske (The Case against Closure (and reply) [2005], p.29) | |
| A reaction: That seems true enough. I see someone limping, but infer that their leg is damaged. The only question is whether I should accept the inference. How can I accept that inference, but then back out of that knowledge? |
| 19546 | Knowing by visual perception is not the same as knowing by implication [Dretske] |
| Full Idea: A way of knowing there are cookies in the jar - visual perception - is not a way of knowing what one knows to be implied by this - that visual appearances are not misleading. | |
| From: Fred Dretske (The Case against Closure (and reply) [2005], p.29) | |
| A reaction: Why is the 'way of knowing' relevant? Isn't the only question that of whether implication of a truth is in infallible route to a truth (modus ponens)? If you know THAT it is true, then you must believe it, and implication is top quality justification. No? |
| 19548 | The only way to preserve our homely truths is to abandon closure [Dretske] |
| Full Idea: The only way to preserve knowledge of homely truths, the truths everyone takes themselves to know, is to abandon closure. | |
| From: Fred Dretske (The Case against Closure (and reply) [2005], p.32) | |
| A reaction: His point is that knowledge of homely truths seems to imply knowledge of the background facts needed to support them, which he takes to be an unreasonable requirement. I recommend pursuing contextualism, rather than abandoning closure. |
| 19549 | P may imply Q, but evidence for P doesn't imply evidence for Q, so closure fails [Dretske] |
| Full Idea: The evidence that gives me knowledge of P (there are cookies in the jar) can exist without evidence for knowing Q (they are not fake), despite my knowing that P implies Q. So closure fails. | |
| From: Fred Dretske (The Case against Closure (and reply) [2005], p.33) | |
| A reaction: His more famous example is the zebra. How can P imply Q if there is no evidence for Q? Maybe 'there are cookies in the jar' does not entail they are not fake, once you disambiguate what is being said? |
| 19550 | We know past events by memory, but we don't know the past is real (an implication) by memory [Dretske] |
| Full Idea: The reality of the past (a 'heavyweight implication') ...is something we know to be implied by things we remember, but it is not itself something we remember. | |
| From: Fred Dretske (The Case against Closure (and reply) [2005], p.35) | |
| A reaction: If I begin to doubt that the past is real, then I must necessarily begin to doubt my ordinary memories. This seems to be the modus tollens of knowledge closure. Doesn't that imply that the modus ponens was valid, and closure is correct? |
| 7861 | Libet says the processes initiated in the cortex can still be consciously changed [Libet, by Papineau] |
| Full Idea: Libet himself points out that the conscious decisions still have the power to 'endorse' or 'cancel', so to speak, the processes initiated by the earlier cortical activity: no action will result if the action's execution is consciously countermanded. | |
| From: report of Benjamin Libet (Unconscious Cerebral Initiative [1985]) by David Papineau - Thinking about Consciousness 1.4 | |
| A reaction: This is why Libet's findings do not imply 'epiphenomenalism'. It seems that part of a decisive action is non-conscious, undermining the all-or-nothing view of consciousness. Searle tries to smuggle in free will at this point (Idea 3817). |
| 6660 | Libet found conscious choice 0.2 secs before movement, well after unconscious 'readiness potential' [Libet, by Lowe] |
| Full Idea: Libet found that a subject's conscious choice to move was about a fifth of a second before movement, and thus later than the onset of the brain's so-called 'readiness potential', which seems to imply that unconscious processes initiates action. | |
| From: report of Benjamin Libet (Unconscious Cerebral Initiative [1985]) by E.J. Lowe - Introduction to the Philosophy of Mind Ch.9 | |
| A reaction: Of great interest to philosophers! It seems to make conscious choices epiphenomenal. The key move, I think, is to give up the idea of consciousness as being all-or-nothing. My actions are still initiated by 'me', but 'me' shades off into unconsciousness. |