32 ideas
| 13011 | New axioms are being sought, to determine the size of the continuum [Maddy] |
| Full Idea: In current set theory, the search is on for new axioms to determine the size of the continuum. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §0) | |
| A reaction: This sounds the wrong way round. Presumably we seek axioms that fix everything else about set theory, and then check to see what continuum results. Otherwise we could just pick our continuum, by picking our axioms. |
| 13013 | The Axiom of Extensionality seems to be analytic [Maddy] |
| Full Idea: Most writers agree that if any sense can be made of the distinction between analytic and synthetic, then the Axiom of Extensionality should be counted as analytic. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.1) | |
| A reaction: [Boolos is the source of the idea] In other words Extensionality is not worth discussing, because it simply tells you what the world 'set' means, and there is no room for discussion about that. The set/class called 'humans' varies in size. |
| 13014 | Extensional sets are clearer, simpler, unique and expressive [Maddy] |
| Full Idea: The extensional view of sets is preferable because it is simpler, clearer, and more convenient, because it individuates uniquely, and because it can simulate intensional notions when the need arises. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.1) | |
| A reaction: [She cites Fraenkel, Bar-Hillet and Levy for this] The difficulty seems to be whether the extensional notion captures our ordinary intuitive notion of what constitutes a group of things, since that needs flexible size and some sort of unity. |
| 13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy] |
| Full Idea: The Axiom of Infinity is a simple statement of Cantor's great breakthrough. His bold hypothesis that a collection of elements that had lurked in the background of mathematics could be infinite launched modern mathematics. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.5) | |
| A reaction: It also embodies one of those many points where mathematics seems to depart from common sense - but then most subjects depart from common sense when they get more sophisticated. Look what happened to art. |
| 13022 | Infinite sets are essential for giving an account of the real numbers [Maddy] |
| Full Idea: If one is interested in analysis then infinite sets are indispensable since even the notion of a real number cannot be developed by means of finite sets alone. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.5) | |
| A reaction: [Maddy is citing Fraenkel, Bar-Hillel and Levy] So Cantor's great breakthrough (Idea 13021) actually follows from the earlier acceptance of the real numbers, so that's where the departure from common sense started. |
| 13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy] |
| Full Idea: The Power Set Axiom is indispensable for a set-theoretic account of the continuum, ...and in so far as those attempts are successful, then the power-set principle gains some confirmatory support. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.6) | |
| A reaction: The continuum is, of course, notoriously problematic. Have we created an extra problem in our attempts at solving the first one? |
| 13024 | Efforts to prove the Axiom of Choice have failed [Maddy] |
| Full Idea: Jordain made consistent and ill-starred efforts to prove the Axiom of Choice. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |
| A reaction: This would appear to be the fate of most axioms. You would presumably have to use a different system from the one you are engaged with to achieve your proof. |
| 13025 | Modern views say the Choice set exists, even if it can't be constructed [Maddy] |
| Full Idea: Resistance to the Axiom of Choice centred on opposition between existence and construction. Modern set theory thrives on a realistic approach which says the choice set exists, regardless of whether it can be defined, constructed, or given by a rule. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |
| A reaction: This seems to be a key case for the ontology that lies at the heart of theory. Choice seems to be an invaluable tool for proofs, so it won't go away, so admit it to the ontology. Hm. So the tools of thought have existence? |
| 13026 | A large array of theorems depend on the Axiom of Choice [Maddy] |
| Full Idea: Many theorems depend on the Axiom of Choice, including that a countable union of sets is countable, and results in analysis, topology, abstract algebra and mathematical logic. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |
| A reaction: The modern attitude seems to be to admit anything if it leads to interesting results. It makes you wonder about the modern approach of using mathematics and logic as the cutting edges of ontological thinking. |
| 13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy] |
| Full Idea: The Iterative Conception (Zermelo 1930) says everything appears at some stage. Given two objects a and b, let A and B be the stages at which they first appear. Suppose B is after A. Then the pair set of a and b appears at the immediate stage after B. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.3) | |
| A reaction: Presumably this all happens in 'logical time' (a nice phrase I have just invented!). I suppose we might say that the existence of the paired set is 'forced' by the preceding sets. No transcendental inferences in this story? |
| 13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy] |
| Full Idea: The 'limitation of size' is a vague intuition, based on the idea that being too large may generate the paradoxes. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.3) | |
| A reaction: This is an intriguing idea to be found right at the centre of what is supposed to be an incredibly rigorous system. |
| 15105 | F(x) walked into a bar. The barman said.. [Sommers,W] |
| Full Idea: F(x) walked into a bar. The barman said, 'Sorry, we don't cater for functions'. | |
| From: Will Sommers (talk [2019]) |
| 12408 | Sartre to Waitress: Coffee with no cream, please... [Sommers,W] |
| Full Idea: Sartre to Waitress: Coffee with no cream, please. Waitress: Sorry, we're out of cream; would no milk do? | |
| From: Will Sommers (talk [2019]) |
| 12397 | Said Plato: 'The things that we feel... [Sommers,W] |
| Full Idea: Said Plato: 'The things that we feel/ Are not ontologically real,/ But just the excrescence/ Of numinous essence/ Our senses can never reveal.' [Basil Ransome-Davis] | |
| From: Will Sommers (talk [2019]) |
| 12399 | There was a young student called Fred... [Sommers,W] |
| Full Idea: There was a young student called Fred,/ Who was questioned on Descartes and said:/ 'It's perfectly clear/ That I'm not really here,/ For I haven't a thought in my head.' [V.R. Ormerod] | |
| From: Will Sommers (talk [2019]) |
| 12407 | Barman to Descartes: Would you like another drink?... [Sommers,W] |
| Full Idea: Barman to Descartes: Would you like another drink? Descartes: I think not (...and promptly vanishes) | |
| From: Will Sommers (talk [2019]) |
| 20963 | A philosopher and his wife are out for a drive... [Sommers,W] |
| Full Idea: A philosopher and his wife are out for a drive in the country. 'Oh look!' she says, 'Those sheep have been shorn.' 'Yes', says the philosopher, 'on this side'. | |
| From: Will Sommers (talk [2019]) |
| 12402 | ..But if he's a student of Berkeley... [Sommers,W] |
| Full Idea: (continued from 12401) ..But if he's a student of Berkeley,/ One thing will emerge, rather starkly,/ That he ought to believe/ What his senses perceive,/ No matter how dimly or darkly. [Leslie Johnson] | |
| From: Will Sommers (talk [2019]) |
| 12409 | The philosopher Berkeley once said.. [Sommers,W] |
| Full Idea: The philosopher Berkeley once said/ In the dark to a maid in his bed:/ 'No perception, my dear,/ Means I'm not really here,/ But only a thought in your head.' [P.W.R. Foot] | |
| From: Will Sommers (talk [2019]) |
| 12403 | There once was a man who said: 'God... [Sommers,W] |
| Full Idea: There once was a man who said: 'God/ Must think it exceedingly odd/ If he finds that this tree/ Continues to be,/ When there's no-one about in the Quad.' [Ronald Knox] (reply in 12404) | |
| From: Will Sommers (talk [2019]) |
| 12404 | Dear Sir, Your astonishment's odd.... [Sommers,W] |
| Full Idea: (reply to 12403) Dear Sir, Your astonishment's odd:/ I am always about in the Quad./ And that's why the tree/ Will continue to be,/ Since observed by Yours faithfully, God.' [anon] | |
| From: Will Sommers (talk [2019]) |
| 14694 | "My dog's got synaesthesia." How does he smell? ..... [Sommers,W] |
| Full Idea: "My dog's got synaesthesia." How does he smell? "Purple." | |
| From: Will Sommers (talk [2019]) |
| 12401 | A toper who spies in the distance... [Sommers,W] |
| Full Idea: A toper who spies in the distance,/ Striped tigers, will get some assistance/ From reading Descartes,/ Who holds that it's part/ Of his duty to doubt their existence. ... [Leslie Johnson] - (continued in 12402) | |
| From: Will Sommers (talk [2019]) |
| 12410 | There once was a man who said 'Damn!... [Sommers,W] |
| Full Idea: There once was a man who said 'Damn!/ It is borne in upon me I am/ An engine that moves/ In predestinate grooves:/ I'm not even a bus, I'm a tram.' [M.E. Hare] | |
| From: Will Sommers (talk [2019]) |
| 9392 | How do behaviourists greet each other? [Sommers,W] |
| Full Idea: How do behaviourists greet each other? Hi - you're fine, how am I? | |
| From: Will Sommers (talk [2019]) |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 12405 | 'If you're aristocratic,' said Nietzsche... [Sommers,W] |
| Full Idea: 'If you're aristocratic,' said Nietzsche,/ 'It's thumbs up, you're OK. Pleased to mietzsche./ If you're working-class bores,/ It's thumbs down and up yours!/ If you don't know your place, then I'll tietzsche.' [Gerry Hamill] | |
| From: Will Sommers (talk [2019]) |
| 9391 | Why do anarchists drink herbal tea? [Sommers,W] |
| Full Idea: Why do anarchists drink herbal tea? Because proper tea is theft. | |
| From: Will Sommers (talk [2019]) |
| 12400 | Cries the maid: 'You must marry me Hume!'... [Sommers,W] |
| Full Idea: Cries the maid: 'You must marry me Hume!'/ A statement that made David fume./ He said: 'In cause and effect,/ There is a defect;/ That it's mine you can only assume.' [P.W.R. Foot] | |
| From: Will Sommers (talk [2019]) |
| 16527 | Causation - we all thought we knew it/ Till Hume came along and saw through it/…. [Sommers,W] |
| Full Idea: Causation - we all thought we knew it / Till Hume came along and saw through it / We notice that A / Follows B every day / And frankly that's all there is to it. | |
| From: Will Sommers (talk [2019]) |
| 17592 | The barman called 'Time!', and Augustine said..... [Sommers,W] |
| Full Idea: The barman called 'Time!'. Augustine: 'I don't know what you mean, though I did before you said that'. | |
| From: Will Sommers (talk [2019]) |
| 15208 | The past, present and future walked into a bar.... [Sommers,W] |
| Full Idea: The past, present and future walked into a bar. It was tense. | |
| From: Will Sommers (talk [2019]) |