17 ideas
| 6859 | Analytic philosophy has much higher standards of thinking than continental philosophy [Williamson] |
| Full Idea: Certain advances in philosophical standards have been made within analytic philosophy, and there would be a serious loss of integrity involved in abandoning them in the way required to participate in current continental philosophy. | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001], p.151) | |
| A reaction: The reply might be to concede the point, but say that the precision and rigour achieved are precisely what debar analytical philosophy from thinking about the really interesting problems. One might as well switch to maths and have done with it. |
| 6862 | Fuzzy logic uses a continuum of truth, but it implies contradictions [Williamson] |
| Full Idea: Fuzzy logic is based on a continuum of degrees of truth, but it is committed to the idea that it is half-true that one identical twin is tall and the other twin is not, even though they are the same height. | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001], p.154) | |
| A reaction: Maybe to be shocked by a contradiction is missing the point of fuzzy logic? Half full is the same as half empty. The logic does not say the twins are different, because it is half-true that they are both tall, and half-true that they both aren't. |
| 10482 | The logic of ZF is classical first-order predicate logic with identity [Boolos] |
| Full Idea: The logic of ZF Set Theory is classical first-order predicate logic with identity. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.121) | |
| A reaction: This logic seems to be unable to deal with very large cardinals, precisely those that are implied by set theory, so there is some sort of major problem hovering here. Boolos is fairly neutral. |
| 10492 | A few axioms of set theory 'force themselves on us', but most of them don't [Boolos] |
| Full Idea: Maybe the axioms of extensionality and the pair set axiom 'force themselves on us' (Gödel's phrase), but I am not convinced about the axioms of infinity, union, power or replacement. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.130) | |
| A reaction: Boolos is perfectly happy with basic set theory, but rather dubious when very large cardinals come into the picture. |
| 10485 | Naïve sets are inconsistent: there is no set for things that do not belong to themselves [Boolos] |
| Full Idea: The naïve view of set theory (that any zero or more things form a set) is natural, but inconsistent: the things that do not belong to themselves are some things that do not form a set. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.127) | |
| A reaction: As clear a summary of Russell's Paradox as you could ever hope for. |
| 10484 | The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first [Boolos] |
| Full Idea: According to the iterative conception, every set is formed at some stage. There is a relation among stages, 'earlier than', which is transitive. A set is formed at a stage if and only if its members are all formed before that stage. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.126) | |
| A reaction: He gives examples of the early stages, and says the conception is supposed to 'justify' Zermelo set theory. It is also supposed to make the axioms 'natural', rather than just being selected for convenience. And it is consistent. |
| 6858 | Formal logic struck me as exactly the language I wanted to think in [Williamson] |
| Full Idea: As soon as I started learning formal logic, that struck me as exactly the language that I wanted to think in. | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001]) | |
| A reaction: It takes all sorts… It is interesting that formal logic might be seen as having the capacity to live up to such an aspiration. I don't think the dream of an ideal formal language is dead, though it will never encompass all of reality. Poetic truth. |
| 10491 | Infinite natural numbers is as obvious as infinite sentences in English [Boolos] |
| Full Idea: The existence of infinitely many natural numbers seems to me no more troubling than that of infinitely many computer programs or sentences of English. There is, for example, no longest sentence, since any number of 'very's can be inserted. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: If you really resisted an infinity of natural numbers, presumably you would also resist an actual infinity of 'very's. The fact that it is unclear what could ever stop a process doesn't guarantee that the process is actually endless. |
| 10483 | Mathematics and science do not require very high orders of infinity [Boolos] |
| Full Idea: To the best of my knowledge nothing in mathematics or science requires the existence of very high orders of infinity. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.122) | |
| A reaction: He is referring to particular high orders of infinity implied by set theory. Personally I want to wield Ockham's Razor. Is being implied by set theory a sufficient reason to accept such outrageous entities into our ontology? |
| 10490 | Mathematics isn't surprising, given that we experience many objects as abstract [Boolos] |
| Full Idea: It is no surprise that we should be able to reason mathematically about many of the things we experience, for they are already 'abstract'. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: He has just given a list of exemplary abstract objects (Idea 10489), but I think there is a more interesting idea here - that our experience of actual physical objects is to some extent abstract, as soon as it is conceptualised. |
| 6863 | Close to conceptual boundaries judgement is too unreliable to give knowledge [Williamson] |
| Full Idea: If one is very close to a conceptual boundary, then one's judgement will be too unreliable to constitute knowledge, and therefore one will be ignorant. | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001], p.156) | |
| A reaction: This is the epistemological rather than ontological interpretation of vagueness. It sounds very persuasive, but I am reluctant to accept that reality is full of very precise boundaries which we cannot quite discriminate. |
| 10488 | It is lunacy to think we only see ink-marks, and not word-types [Boolos] |
| Full Idea: It's a kind of lunacy to think that sound scientific philosophy demands that we think that we see ink-tracks but not words, i.e. word-types. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.128) | |
| A reaction: This seems to link him with Armstrong's mockery of 'ostrich nominalism'. There seems to be some ambiguity with the word 'see' in this disagreement. When we look at very ancient scratches on stones, why don't we always 'see' if it is words? |
| 10487 | I am a fan of abstract objects, and confident of their existence [Boolos] |
| Full Idea: I am rather a fan of abstract objects, and confident of their existence. Smaller numbers, sets and functions don't offend my sense of reality. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.128) | |
| A reaction: The great Boolos is rather hard to disagree with, but I disagree. Logicians love abstract objects, indeed they would almost be out of a job without them. It seems to me they smuggle them into our ontology by redefining either 'object' or 'exists'. |
| 10489 | We deal with abstract objects all the time: software, poems, mistakes, triangles.. [Boolos] |
| Full Idea: We twentieth century city dwellers deal with abstract objects all the time, such as bank balances, radio programs, software, newspaper articles, poems, mistakes, triangles. | |
| From: George Boolos (Must We Believe in Set Theory? [1997], p.129) | |
| A reaction: I find this claim to be totally question-begging, and typical of a logician. The word 'object' gets horribly stretched in these discussions. We can create concepts which have all the logical properties of objects. Maybe they just 'subsist'? |
| 6861 | What sort of logic is needed for vague concepts, and what sort of concept of truth? [Williamson] |
| Full Idea: The problem of vagueness is the problem of what logic is correct for vague concepts, and correspondingly what notions of truth and falsity are applicable to vague statements (does one need a continuum of degrees of truth, for example?). | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001], p.153) | |
| A reaction: This certainly makes vagueness sound like one of the most interesting problems in all of philosophy, though also one of the most difficult. Williamson's solution is that we may be vague, but the world isn't. |
| 6860 | How can one discriminate yellow from red, but not the colours in between? [Williamson] |
| Full Idea: If one takes a spectrum of colours from yellow to red, it might be that given a series of colour samples along that spectrum, each sample is indiscriminable by the naked eye from the next one, though samples at either end are blatantly different. | |
| From: Timothy Williamson (Interview with Baggini and Stangroom [2001], p.151) | |
| A reaction: This seems like a nice variant of the Sorites paradox (Idea 6008). One could demonstrate it with just three samples, where A and C seemed different from each other, but other comparisons didn't. |
| 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? |