17 ideas
| 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. |
| 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. |
| 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'? |
| 21500 | We rely on memory for empirical beliefs because they mutually support one another [Lewis,CI] |
| Full Idea: When the whole range of empirical beliefs is taken into account, all of them more or less dependent on memorial knowledge, we find that those which are most credible can be assured by their mutual support, or 'congruence'. | |
| From: C.I. Lewis (An Analysis of Knowledge and Valuation [1946], 334), quoted by Erik J. Olsson - Against Coherence 3.1 | |
| A reaction: Lewis may be over-confident about this, and is duly attacked by Olson, but it seems to me roughly correct. How do you assess whether some unusual element in your memory was a dream or a real experience? |
| 21501 | If we doubt memories we cannot assess our doubt, or what is being doubted [Lewis,CI] |
| Full Idea: To doubt our sense of past experience as founded in actuality, would be to lose any criterion by which either the doubt itself or what is doubted could be corroborated. | |
| From: C.I. Lewis (An Analysis of Knowledge and Valuation [1946], 358), quoted by Erik J. Olsson - Against Coherence 3.3.1 | |
| A reaction: Obviously scepticism about memory can come in degrees, but total rejection of short-term and clear memories looks like a non-starter. What could you put in its place? Hyper-rationalism? Even maths needs memory. |
| 6556 | If anything is to be probable, then something must be certain [Lewis,CI] |
| Full Idea: If anything is to be probable, then something must be certain. | |
| From: C.I. Lewis (An Analysis of Knowledge and Valuation [1946], 186), quoted by Robert Fogelin - Walking the Tightrope of Reason Intro | |
| A reaction: Lewis makes this comment when facing infinite regress problems. It is a very nice slogan for foundationalism, which embodies the slippery slope view. Personally I feel the emotional pull of foundations, but acknowledge the very strong doubts about them. |
| 21498 | Congruents assertions increase the probability of each individual assertion in the set [Lewis,CI] |
| Full Idea: A set of statements, or a set of supposed facts asserted, will be said to be congruent if and only if they are so related that the antecedent probability of any one of them will be increased if the remainder of the set can be assumed as given premises. | |
| From: C.I. Lewis (An Analysis of Knowledge and Valuation [1946], 338), quoted by Erik J. Olsson - Against Coherence 2.2 | |
| A reaction: This thesis is vigorously attacked by Erik Olson, who works through the probability calculations. There seems an obvious problem without that. How else do you assess 'congruence', other than by evidence of mutual strengthening? |
| 5828 | Extension is the class of things, intension is the correct definition of the thing, and intension determines extension [Lewis,CI] |
| Full Idea: "The denotation or extension of a term is the class of all actual or existent things which the term correctly applies to or names; the connotation or intension of a term is delimited by any correct definition of it." ..And intension determines extension. | |
| From: C.I. Lewis (An Analysis of Knowledge and Valuation [1946]), quoted by Stephen P. Schwartz - Intro to Naming,Necessity and Natural Kinds §II | |
| A reaction: The last part is one of the big ideas in philosophy of language, which was rejected by Putnam and co. If you were to reverse the slogan, though, (to extension determines intension) how would you identify the members of the extension? |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| Full Idea: Philosophers are the forefathers of heretics. | |
| From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
| 6610 | I believe because it is absurd [Tertullian] |
| Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
| From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
| A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |