16 ideas
| 8679 | We perceive the objects of set theory, just as we perceive with our senses [Gödel] |
| Full Idea: We have something like perception of the objects of set theory, shown by the axioms forcing themselves on us as being true. I don't see why we should have less confidence in this kind of perception (i.e. mathematical intuition) than in sense perception. | |
| From: Kurt Gödel (What is Cantor's Continuum Problem? [1964], p.483), quoted by Michčle Friend - Introducing the Philosophy of Mathematics 2.4 | |
| A reaction: A famous strong expression of realism about the existence of sets. It is remarkable how the ingredients of mathematics spread themselves before the mind like a landscape, inviting journeys - but I think that just shows how minds cope with abstractions. |
| 9942 | Gödel proved the classical relative consistency of the axiom V = L [Gödel, by Putnam] |
| Full Idea: Gödel proved the classical relative consistency of the axiom V = L (which implies the axiom of choice and the generalized continuum hypothesis). This established the full independence of the continuum hypothesis from the other axioms. | |
| From: report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by Hilary Putnam - Mathematics without Foundations | |
| A reaction: Gödel initially wanted to make V = L an axiom, but the changed his mind. Maddy has lots to say on the subject. |
| 9038 | We must distinguish what the speaker denotes by a name, from what the name denotes [Evans] |
| Full Idea: There are two related but distinguishable questions concerning proper names: what the speaker denotes (upon an occasion), and what the name denotes. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §I) | |
| A reaction: I don't think any account of language makes sense without this sort of distinction, as in my favourite example: the password is 'swordfish'. So how does language gets its own meanings, independent of what speakers intend? |
| 5824 | How can an expression be a name, if names can change their denotation? [Evans] |
| Full Idea: We need an account of what makes an expression into a name for something that will allow names to change their denotations. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §II) | |
| A reaction: Presumably an example would be 'The Prime Minister is in the building'. Evans proposes to discuss communication, rather than strict meanings and descriptions. |
| 9042 | A private intention won't give a name a denotation; the practice needs it to be made public [Evans] |
| Full Idea: Intentions alone don't bring it about that a name gets a denotation; without the intention being manifest there cannot be the common knowledge required for the practice. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §II) | |
| A reaction: Well, I might have a private name for some hated colleague which I mutter to myself whenever I see her. The way names, and language generally, becomes ossified is by joining the great impersonal sea of the language. ..waves of bones, |
| 9041 | The Causal Theory of Names is wrong, since the name 'Madagascar' actually changed denotation [Evans] |
| Full Idea: Change of denotation is decisive against the Causal Theory of Names. Changes of denotation actually occur: a hearsay report misunderstood by Marco Polo transferred the name 'Madagascar' from a portion of the mainland to the African island. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §I) | |
| A reaction: This doesn't sound decisive, as you could give an intermediate causal account of Marco Polo's mistake. I might take the famous name Winston, and baptise my son with it. And I might have done it because I thought Winston was a German dictator. |
| 18062 | Set-theory paradoxes are no worse than sense deception in physics [Gödel] |
| Full Idea: The set-theoretical paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics. | |
| From: Kurt Gödel (What is Cantor's Continuum Problem? [1964], p.271), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 03.4 |
| 10868 | The Continuum Hypothesis is not inconsistent with the axioms of set theory [Gödel, by Clegg] |
| Full Idea: Gödel proved that the Continuum Hypothesis was not inconsistent with the axioms of set theory. | |
| From: report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15 |
| 13517 | If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Gödel, by Hart,WD] |
| Full Idea: Gödel proved that (if set theory is consistent) we cannot refute the continuum hypothesis, and Cohen proved that (if set theory is consistent) we cannot prove it either. | |
| From: report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by William D. Hart - The Evolution of Logic 10 |
| 10271 | Basic mathematics is related to abstract elements of our empirical ideas [Gödel] |
| Full Idea: Evidently the 'given' underlying mathematics is closely related to the abstract elements contained in our empirical ideas. | |
| From: Kurt Gödel (What is Cantor's Continuum Problem? [1964], Suppl) | |
| A reaction: Yes! The great modern mathematical platonist says something with which I can agree. He goes on to hint at a platonic view of the structure of the empirical world, but we'll let that pass. |
| 5825 | Speakers intend to refer to items that are the source of their information [Evans] |
| Full Idea: In general, a speaker intends to refer to the item that is the dominant source of his associated body of information. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §II) | |
| A reaction: This sounds like a theory of reference which fully preserves the spirit of traditional empiricism. Speakers refer to ideas which connect to the source of their underlying impressions. |
| 5823 | The intended referent of a name needs to be the cause of the speaker's information about it [Evans] |
| Full Idea: A necessary (but not sufficient) condition for x's being the intended referent of S's use of a name is that x should be the source of the causal origin of the body of information that S has associated with the name. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §I) | |
| A reaction: This is Evans's adaptation of Kripke's causal theory of names. This cries out for a counterexample. I say something about General Montgomery, having just listened to 'Monty's Double' give a talk, believing it was Montgomery? |
| 9039 | If descriptions are sufficient for reference, then I must accept a false reference if the descriptions fit [Evans] |
| Full Idea: The strong thesis (that descriptions are sufficient for reference) is outrageous. It would mean that if Mr X is wrongly introduced to me as Mr Y, then I truly say 'this is Mr Y' if X overwhelmingly satisfies descriptions of Y. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §I) | |
| A reaction: [I omit some qualifying phrases] Evans says that probably no one ever held this view. It seems right. In the case of an electron it would seem that all the descriptions could be the same, except space-time location. Same electron as yesterday? |
| 9043 | We use expressions 'deferentially', to conform to the use of other people [Evans] |
| Full Idea: Sometimes we use expressions with the overriding intention to conform to the use made of them by some other person or persons. I shall say we use the expression 'deferentially'; examples might be 'viol' or 'minuet'. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §II) | |
| A reaction: I presume Evans wasn't very musical. This label sounds useful, if you wish to connect Grice's account of meaning with Putnam's externalist account of concepts, where deference to experts is crucial. Is all linguistic usage deferential? |
| 9040 | Charity should minimize inexplicable error, rather than maximising true beliefs [Evans] |
| Full Idea: I think the Principle of Charity (maximise true beliefs) is unacceptable. The acceptable principle enjoins minimizing the attribution of inexplicable error and cannot be operated without a theory of the causation of belief for the creatures investigated. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §I) | |
| A reaction: The normal principle of charity certainly seems on shaky ground if you think you have encountered a fairly normal tribe, when they in fact are in possession of the weirdest belief system on the entire planet. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |