16 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. |
| 8983 | If 'red' is vague, then membership of the set of red things is vague, so there is no set of red things [Sainsbury] |
| Full Idea: Sets have sharp boundaries, or are sharp objects; an object either definitely belongs to a set, or it does not. But 'red' is vague; there objects which are neither definitely red nor definitely not red. Hence there is no set of red things. | |
| From: Mark Sainsbury (Concepts without Boundaries [1990], §2) | |
| A reaction: Presumably that will entail that there IS a set of things which can be described as 'definitely red'. If we describe something as 'definitely having a hint of red about it', will that put it in a set? In fact will the applicability of 'definitely' do? |
| 8986 | We should abandon classifying by pigeon-holes, and classify around paradigms [Sainsbury] |
| Full Idea: We must reject the classical picture of classification by pigeon-holes, and think in other terms: classifying can be, and often is, clustering round paradigms. | |
| From: Mark Sainsbury (Concepts without Boundaries [1990], §8) | |
| A reaction: His conclusion to a discussion of the problem of vagueness, where it is identified with concepts which have no boundaries. Pigeon-holes are a nice exemplar of the Enlightenment desire to get everything right. I prefer Aristotle's categories, Idea 3311. |
| 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'? |
| 8982 | Vague concepts are concepts without boundaries [Sainsbury] |
| Full Idea: If a word is vague, there are or could be borderline cases, but non-vague expressions can also have borderline cases. The essence of vagueness is to be found in the idea vague concepts are concepts without boundaries. | |
| From: Mark Sainsbury (Concepts without Boundaries [1990], Intro) | |
| A reaction: He goes on to say that vague concepts are not embodied in clear cut sets, which is what gives us our notion of a boundary. So what is vague is 'membership'. You are either a member of a club or not, but when do you join the 'middle-aged'? |
| 8984 | If concepts are vague, people avoid boundaries, can't spot them, and don't want them [Sainsbury] |
| Full Idea: Vague concepts are boundaryless, ...and the manifestations are an unwillingness to draw any such boundaries, the impossibility of identifying such boundaries, and needlessness and even disutility of such boundaries. | |
| From: Mark Sainsbury (Concepts without Boundaries [1990], §5) | |
| A reaction: People have a very fine-tuned notion of whether the sharp boundary of a concept is worth discussing. The interesting exception are legal people, who are often forced to find precision where everyone else hates it. Who deserves to inherit the big house? |
| 8985 | Boundaryless concepts tend to come in pairs, such as child/adult, hot/cold [Sainsbury] |
| Full Idea: Boundaryless concepts tend to come in systems of contraries: opposed pairs like child/adult, hot/cold, weak/strong, true/false, and complex systems of colour terms. ..Only a contrast with 'adult' will show what 'child' excludes. | |
| From: Mark Sainsbury (Concepts without Boundaries [1990], §5) | |
| A reaction: This might be expected. It all comes down to the sorites problem, of when one thing turns into something else. If it won't merge into another category, then presumably the isolated concept stays applicable (until reality terminates it? End of sheep..). |
| 15877 | The aim of science is just to create a comprehensive, elegant language to describe brute facts [Poincaré, by Harré] |
| Full Idea: In Poincaré's view, we try to construct a language within which the brute facts of experience are expressed as comprehensively and as elegantly as possible. The job of science is the forging of a language precisely suited to that purpose. | |
| From: report of Henri Poincaré (The Value of Science [1906], Pt III) by Rom Harré - Laws of Nature 2 | |
| A reaction: I'm often struck by how obscure and difficult our accounts of self-evident facts can be. Chairs are easy, and the metaphysics of chairs is hideous. Why is that? I'm a robust realist, but I like Poincaré's idea. He permits facts. |