29 ideas
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
| A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
| A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics. |
| 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]) |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth. | |
| From: Penelope Maddy (Defending the Axioms [2011], 5.3ii) |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
| Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40) |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.4) | |
| A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics. |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.3) | |
| A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor. |
| 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]) |
| 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]) |
| 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]) |
| 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]) |
| 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]) |
| 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]) |
| 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]) |
| 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]) |
| 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]) |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 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]) |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |
| 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]) |