16 ideas
| 13451 | The two best understood conceptions of set are the Iterative and the Limitation of Size [Rayo/Uzquiano] |
| Full Idea: The two best understood conceptions of set are the Iterative Conception and the Limitation of Size Conception. | |
| From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2) |
| 17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
| Full Idea: Zermelo's paper sets out to show that the standard set-theoretic axioms (what he calls the 'constitutive axioms', thus the ZF axioms minus the axiom of infinity) have an unending sequence of different models, thus that they are non-categorical. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1209 | |
| A reaction: Hallett says later that Zermelo is working with second-order set theory. The addition of an Axiom of Infinity seems to have aimed at addressing the problem, and the complexities of that were pursued by Gödel. |
| 13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
| Full Idea: Zermelo included Replacement in 1930, after it was noticed that the sequence of power sets was needed, and Replacement gave the ordinal form of the well-ordering theorem, and justification for transfinite recursion. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Penelope Maddy - Believing the Axioms I §1.8 | |
| A reaction: Maddy says that this axiom suits the 'limitation of size' theorists very well, but is not so good for the 'iterative conception'. |
| 13452 | Some set theories give up Separation in exchange for a universal set [Rayo/Uzquiano] |
| Full Idea: There are set theories that countenance exceptions to the Principle of Separation in exchange for a universal set. | |
| From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2) |
| 13449 | We could have unrestricted quantification without having an all-inclusive domain [Rayo/Uzquiano] |
| Full Idea: The possibility of unrestricted quantification does not immediately presuppose the existence of an all-inclusive domain. One could deny an all-inclusive domain but grant that some quantifications are sometimes unrestricted. | |
| From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.1) | |
| A reaction: Thus you can quantify over anything you like, but only from what is available. Eat what you like (in this restaurant). |
| 13450 | Absolute generality is impossible, if there are indefinitely extensible concepts like sets and ordinals [Rayo/Uzquiano] |
| Full Idea: There are doubts about whether absolute generality is possible, if there are certain concepts which are indefinitely extensible, lacking definite extensions, and yielding an ever more inclusive hierarchy. Sets and ordinals are paradigm cases. | |
| From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.1) |
| 13453 | Perhaps second-order quantifications cover concepts of objects, rather than plain objects [Rayo/Uzquiano] |
| Full Idea: If one thought of second-order quantification as quantification over first-level Fregean concepts [note: one under which only objects fall], talk of domains might be regimented as talk of first-level concepts, which are not objects. | |
| From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2) | |
| A reaction: That is (I take it), don't quantify over objects, but quantify over concepts, but only those under which known objects fall. One might thus achieve naďve comprehension without paradoxes. Sound like fun. |
| 17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
| Full Idea: Two opposite tendencies of thought, the idea of creative advance and of collection and completion (underlying the Kantian 'antinomies') find their symbolic representation and their symbolic reconciliation in the transfinite numbers based on well-ordering. | |
| From: Ernst Zermelo (On boundary numbers and domains of sets [1930], §5) | |
| A reaction: [a bit compressed] It is this sort of idea, from one of the greatest set-theorists, that leads philosophers to think that the philosophy of mathematics may offer solutions to metaphysical problems. As an outsider, I am sceptical. |
| 17488 | Empiricist theories are sets of laws, which give explanations and reductions [Glennan] |
| Full Idea: In the empiricist tradition theories were understood to be deductive closures of sets of laws, explanations were understood as arguments from covering laws, and reduction was understood as a deductive relationship between laws of different theories. | |
| From: Stuart Glennan (Mechanisms [2008], 'Intro') | |
| A reaction: A lovely crisp summary of the whole tradition of philosophy of science from Comte through to Hempel. Mechanism and essentialism are the new players in the game. |
| 17493 | Modern mechanism need parts with spatial, temporal and function facts, and diagrams [Glennan] |
| Full Idea: Modern champions of mechanisms say models should identify both the parts and their spatial, temporal and functional organisation, ...and the practical importance of diagrams in addition to or in place of linguistic representations of mechanisms. | |
| From: Stuart Glennan (Mechanisms [2008], 'Discover') | |
| A reaction: Apparently chemists obtain much more refined models by using mathematics than they did by diagrams or 3D models (let alone verbal descriptions). For that reason, I'm thinking that 'model' might be a better term than 'mechanism'. |
| 17489 | Mechanisms are either systems of parts or sequences of activities [Glennan] |
| Full Idea: There are two sorts of mechanisms: systems consist of collections of parts that interact to produce some behaviour, and processes are sequences of activities which produce some outcome. | |
| From: Stuart Glennan (Mechanisms [2008], 'Intro') | |
| A reaction: [compressed] The second one is important because it is more generic, and under that account all kinds the features of the world that need to be explained can be subsumed. E.g. hyperinflation in an economy is a 'mechanism'. |
| 17487 | Mechanistic philosophy of science is an alternative to the empiricist law-based tradition [Glennan] |
| Full Idea: To a significant degree, a mechanistic philosophy of science can be seen as an alternative to an earlier logical empiricist tradition in philosophy of science that gave pride of place to laws of nature. | |
| From: Stuart Glennan (Mechanisms [2008], 'Intro') | |
| A reaction: Lovely! Someone who actually spells out what's going on here. Most philosophers are far too coy about explaining what their real game is. Mechanism is fine in chemistry and biology. How about in 'mathematical' physics, or sociology? |
| 17490 | 17th century mechanists explained everything by the kinetic physical fundamentals [Glennan] |
| Full Idea: 17th century mechanists said that interactions governed by chemical, electrical or gravitational forces would have to be explicable in terms of the operation of some atomistic (or corpuscular) kinetic mechanism. | |
| From: Stuart Glennan (Mechanisms [2008], 'Intro') | |
| A reaction: Glennan says science has rejected this, so modern mechanists do not reduce mechanisms to anything in particular. |
| 17491 | Unlike the lawlike approach, mechanistic explanation can allow for exceptions [Glennan] |
| Full Idea: One of the advantages of the move from nomological to mechanistic modes of explanation is that the latter allows for explanations involving exception-ridden generalizations. | |
| From: Stuart Glennan (Mechanisms [2008], 'regular') | |
| A reaction: The lawlike approach has endless problems with 'ceteris paribus' ('all things being equal') laws, where specifying all the other 'things' seems a bit tricky. |
| 13448 | The domain of an assertion is restricted by context, either semantically or pragmatically [Rayo/Uzquiano] |
| Full Idea: We generally take an assertion's domain of discourse to be implicitly restricted by context. [Note: the standard approach is that this restriction is a semantic phenomenon, but Kent Bach (2000) argues that it is a pragmatic phenomenon] | |
| From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.1) | |
| A reaction: I think Kent Bach is very very right about this. Follow any conversation, and ask what the domain is at any moment. The reference of a word like 'they' can drift across things, with no semantics to guide us, but only clues from context and common sense. |
| 17494 | Since causal events are related by mechanisms, causation can be analysed in that way [Glennan] |
| Full Idea: Causation can be analyzed in terms of mechanisms because (except for fundamental causal interactions) causally related events will be connected by intervening mechanisms. | |
| From: Stuart Glennan (Mechanisms [2008], 'causation') | |
| A reaction: This won't give us the metaphysics of causation (which concerns the fundamentals), but this strikes me as a very coherent and interesting proposal. He mentions electron interaction as non-mechanistic causation. |