9 ideas
| 8758 | We could talk of open sentences, instead of sets [Chihara, by Shapiro] |
| Full Idea: Chihara's programme is to replace talk of sets with talk of open sentences. Instead of speaking of the set of all cats, we talk about the open sentence 'x is a cat'. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2 | |
| A reaction: As Shapiro points out, this is following up Russell's view that sets should be replaced with talk of properties. Chihara is expressing it more linguistically. I'm in favour of any attempt to get rid of sets. |
| 10265 | Chihara's system is a variant of type theory, from which he can translate sentences [Chihara, by Shapiro] |
| Full Idea: Chihara's system is a version of type theory. Translate thus: replace variables of sets of type n with level n variables over open sentences, replace membership/predication with satisfaction, and high quantifiers with constructability quantifiers. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4 |
| 8759 | We can replace type theory with open sentences and a constructibility quantifier [Chihara, by Shapiro] |
| Full Idea: Chihara's system is similar to simple type theory; he replaces each type with variables over open sentences, replaces membership (or predication) with satisfaction, and replaces quantifiers over level 1+ variables with constructability quantifiers. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2 | |
| A reaction: This is interesting for showing that type theory may not be dead. The revival of supposedly dead theories is the bread-and-butter of modern philosophy. |
| 10264 | Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ' [Chihara, by Shapiro] |
| Full Idea: Chihara has proposal a modal primitive, a 'constructability quantifier'. Syntactically it behaves like an ordinary quantifier: Φ is a formula, and x a variable. Then (Cx)Φ is a formula, read as 'it is possible to construct an x such that Φ'. | |
| From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Philosophy of Mathematics 7.4 | |
| A reaction: We only think natural numbers are infinite because we see no barrier to continuing to count, i.e. to construct new numbers. We accept reals when we know how to construct them. Etc. Sounds promising to me (though not to Shapiro). |
| 21386 | We should accept as explanations all the plausible ways in which something could come about [Epicurus] |
| Full Idea: The phases of the Moon could happen in all the ways [at least four] which the phenomena in our experience suggest for the explanation of this kind of thing - as long as one is not so enamoured of unique explanations as to groundlessly reject the others. | |
| From: Epicurus (Letter to Pythocles [c.292 BCE], 94) | |
| A reaction: Very interesting, for IBE. While you want to embrace the 'best', it is irrational to reject all of the other candidates, simply because you want a single explanation, if there are no good grounds for the rejection. |
| 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. |
| 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. |
| 14051 | A cosmos is a collection of stars and an earth, with some sort of boundary, movement and shape [Epicurus] |
| Full Idea: A cosmos is a circumscribed portion of the heavens containing stars and an earth; it is separated from the unlimited, with a boundary which is rare or dense; it is revolving or stationary; it is round or triangular, or some shape. All these are possible. | |
| From: Epicurus (Letter to Pythocles [c.292 BCE], 88) | |
| A reaction: Notice that there seem to exist the 'heavens' which extend beyond the cosmos. See Idea 14036, saying that there are many other cosmoi in the heavens. |
| 1828 | God does not intervene in heavenly movements, but is beyond all action and perfectly happy [Epicurus] |
| Full Idea: Let us beware of making the Deity interpose in heavenly movements, for that being we ought to suppose exempt from all occupation and perfectly happy. | |
| From: Epicurus (Letter to Pythocles [c.292 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 10.25 |