11 ideas
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| Full Idea: Cantor's development of set theory began with his discovery of the progression 0, 1, ....∞, ∞+1, ∞+2, ..∞x2, ∞x3, ...∞^2, ..∞^3, ...∞^∞, ...∞^∞^∞..... | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite VIII.2 |
| 10429 | It is best to say that a name designates iff there is something for it to designate [Sainsbury] |
| Full Idea: It is better to say that 'For all x ("Hesperus" stands for x iff x = Hesperus)', than to say '"Hesperus" stands for Hesperus', since then the expression can be a name with no bearer (e.g. "Vulcan"). | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.2) | |
| A reaction: In cases where it is unclear whether the name actually designates something, it seems desirable that the name is at least allowed to function semantically. |
| 10425 | Definite descriptions may not be referring expressions, since they can fail to refer [Sainsbury] |
| Full Idea: Almost everyone agrees that intelligible definite descriptions may lack a referent; this has historically been a reason for not counting them among referring expressions. | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.2) | |
| A reaction: One might compare indexicals such as 'I', which may be incapable of failing to refer when spoken. However 'look at that!' frequently fails to communicate reference. |
| 10438 | Definite descriptions are usually rigid in subject, but not in predicate, position [Sainsbury] |
| Full Idea: Definite descriptions used with referential intentions (usually in subject position) are normally rigid, ..but in predicate position they are normally not rigid, because there is no referential intention. | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.5) | |
| A reaction: 'The man in the blue suit is the President' seems to fit, but 'The President is the head of state' doesn't. Seems roughly right, but language is always too complex for philosophers. |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence). | |
| From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 17945 | Forms are not a theory of universals, but an attempt to explain how predication is possible [Nehamas] |
| Full Idea: The theory of Forms is not a theory of universals but a first attempt to explain how predication, the application of a single term to many objects - now considered one of the most elementary operations of language - is possible. | |
| From: Alexander Nehamas (Introduction to 'Virtues of Authenticity' [1999], p.xxvii) |
| 17946 | Only Tallness really is tall, and other inferior tall things merely participate in the tallness [Nehamas] |
| Full Idea: Only Tallness and nothing else really is tall; everything else merely participates in the Forms and, being excluded from the realm of Being, belongs to the inferior world of Becoming. | |
| From: Alexander Nehamas (Introduction to 'Virtues of Authenticity' [1999], p.xxviii) | |
| A reaction: This is just as weird as the normal view (and puzzle of participation), but at least it makes more sense of 'metachein' (partaking). |
| 17944 | 'Episteme' is better translated as 'understanding' than as 'knowledge' [Nehamas] |
| Full Idea: The Greek 'episteme' is usually translated as 'knowledge' but, I argue, closer to our notion of understanding. | |
| From: Alexander Nehamas (Introduction to 'Virtues of Authenticity' [1999], p.xvi) | |
| A reaction: He agrees with Julia Annas on this. I take it to be crucial. See the first sentence of Aristotle's 'Metaphysics'. It is explanation which leads to understanding. |
| 10432 | A new usage of a name could arise from a mistaken baptism of nothing [Sainsbury] |
| Full Idea: A baptism which, perhaps through some radical mistake, is the baptism of nothing, is as good a propagator of a new use as a baptism of an object. | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.3) | |
| A reaction: An obvious example might be the Loch Ness Monster. There is something intuitively wrong about saying that physical objects are actually part of linguistic meaning or reference. I am not a meaning! |
| 10434 | Even a quantifier like 'someone' can be used referentially [Sainsbury] |
| Full Idea: A large range of expressions can be used with referential intentions, including quantifier phrases (as in 'someone has once again failed to close the door properly'). | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.5) | |
| A reaction: This is the pragmatic aspect of reference, where it can be achieved by all sorts of means. But are quantifiers inherently referential in their semantic function? Some of each, it seems. |
| 10431 | Things are thought to have a function, even when they can't perform them [Sainsbury] |
| Full Idea: On one common use of the notion of a function, something can possess a function which it does not, or even cannot, perform. A malformed heart is to pump blood, even if such a heart cannot in fact pump blood. | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.2) | |
| A reaction: One might say that the heart in a dead body had the function of pumping blood, but does it still have that function? Do I have the function of breaking the world 100 metres record, even though I can't quite manage it? Not that simple. |