16 ideas
17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
A reaction: Each expansion brings a limitation, but then you can expand again. |
17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
14361 | Lewis says indicative conditionals are truth-functional [Lewis, by Jackson] |
Full Idea: Unlike Stalnaker, Lewis holds that indicative conditionals have the truth conditions of material conditionals. | |
From: report of David Lewis (Counterfactuals [1973]) by Frank Jackson - Conditionals 'Further' | |
A reaction: Thus Lewis only uses the possible worlds account for subjunctive conditionals, where Stalnaker uses it for both. Lewis is defending the truth-functional account for the indicative conditionals. |
8434 | In good counterfactuals the consequent holds in world like ours except that the antecedent is true [Lewis, by Horwich] |
Full Idea: According to Lewis, a counterfactual holds when the consequent is true in possible worlds very like our own except for the fact that the antecedent is true. | |
From: report of David Lewis (Counterfactuals [1973]) by Paul Horwich - Lewis's Programme p.213 | |
A reaction: Presumably the world being very like our own would make it unlikely that there would be anything else to cause the consequent, apart from the counterfactual antecedent. |
17498 | In the 'received view' models are formal; the 'semantic view' emphasises representation [Portides, by PG] |
Full Idea: The 'received view' of models is that they are Tarskian formal axiomatic calculi interpreted by meta-mathematical models. The 'semantic' view of models gives equal importance to their representational capacity. | |
From: report of Demetris Portides (Models [2008], 'background') by PG - Db (ideas) | |
A reaction: The Tarskian view is the one covered in my section on Model Theory. Portides favours the semantic account, and I am with him all the way. Should models primarily integrate with formal systems, or with the world? Your choice... |
17501 | Representational success in models depends on success of their explanations [Portides] |
Full Idea: Models are representational, independently of the strength of their relation to theory, depending on how well they achieve the purpose of providing explanations for what occurs in physical systems. | |
From: Demetris Portides (Models [2008], 'Current') | |
A reaction: This doesn't sound quite right. It seems possible to have a perfect representation of a system which remains quite baffling (because too complex, or with obscure ingredients). Does the stylised London tube map explain well but represent badly? |
17502 | The best model of the atomic nucleus is the one which explains the most results [Portides] |
Full Idea: The unified model can be considered a better representation of the atomic nucleus in comparison to the liquid-drop and shell models, because it explains most of the known results about the nucleus. | |
From: Demetris Portides (Models [2008], 'Current') | |
A reaction: The point here is that models are evaluated not just by their accuracy, but by their explanatory power. Presumably a great model is satisfying and illuminating. Do the best models capture the essence of a thing? |
17496 | 'Model' belongs in a family of concepts, with representation, idealisation and abstraction [Portides] |
Full Idea: A better understanding of 'model', as used in science, could be achieved if we examine it as a member of the triad of concepts of representation, idealisation and abstraction. | |
From: Demetris Portides (Models [2008], 'Intro') | |
A reaction: Abstraction seems to have a bad name in philosophy, and yet when you come to discuss things like models, you can't express it any other way. |
17497 | Models are theory-driven, or phenomenological (more empirical and specific) [Portides] |
Full Idea: 'Theory-driven' models are constructed in a systematic theory-regulated way by supplementing the theoretical calculus with locally operative hypotheses. 'Phenomenological' models deploy semi-empirical results, with ad hoc hypotheses, and extra concepts. | |
From: Demetris Portides (Models [2008], 'Intro') | |
A reaction: [compressed] I am not at all clear about this distinction, even after reading his whole article. The first type of model seems more general, while the second seems tuned to particular circumstances. He claims the second type is more explanatory. |
17499 | Theoretical models can represent, by mapping onto the data-models [Portides] |
Full Idea: The semantic approach contends that theoretical models ...are candidates for representing physical systems by virtue of the fact that they stand in mapping relations to corresponding data-models. | |
From: Demetris Portides (Models [2008], 'Current') | |
A reaction: Sounds like a neat and satisfying picture. |
17500 | General theories may be too abstract to actually explain the mechanisms [Portides] |
Full Idea: If theoretical models are highly abstract and idealised descriptions of phenomena, they may only represent general features, and fail to explain the specific mechanisms at work in physical systems. | |
From: Demetris Portides (Models [2008], 'Current') | |
A reaction: [compressed] While there may be an ideal theory that explains everything, it sounds right capturing the actual mechanism (such as the stirrup bone in the ear) is not at all theoretical. |
9419 | A law of nature is a general axiom of the deductive system that is best for simplicity and strength [Lewis] |
Full Idea: A contingent generalization is a law of nature if and only if it appears as a theorem (or axiom) in each of the true deductive systems that achieves a best combination of simplicity and strength. | |
From: David Lewis (Counterfactuals [1973], 3.3) |