14 ideas
| 23367 | Even pointing a finger should only be done for a reason [Epictetus] |
| Full Idea: Philosophy says it is not right even to stretch out a finger without some reason. | |
| From: Epictetus (fragments/reports [c.57], 15) | |
| A reaction: The key point here is that philosophy concerns action, an idea on which Epictetus is very keen. He rather despise theory. This idea perfectly sums up the concept of the wholly rational life (which no rational person would actually want to live!). |
| 9944 | We understand some statements about all sets [Putnam] |
| Full Idea: We seem to understand some statements about all sets (e.g. 'for every set x and every set y, there is a set z which is the union of x and y'). | |
| From: Hilary Putnam (Mathematics without Foundations [1967], p.308) | |
| A reaction: His example is the Axiom of Choice. Presumably this is why the collection of all sets must be referred to as a 'class', since we can talk about it, but cannot define it. |
| 9937 | I do not believe mathematics either has or needs 'foundations' [Putnam] |
| Full Idea: I do not believe mathematics either has or needs 'foundations'. | |
| From: Hilary Putnam (Mathematics without Foundations [1967]) | |
| A reaction: Agreed that mathematics can function well without foundations (given that the enterprise got started with no thought for such things), the ontology of the subject still strikes me as a major question, though maybe not for mathematicians. |
| 9939 | It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam] |
| Full Idea: I believe that under certain circumstances revisions in the axioms of arithmetic, or even of the propositional calculus (e.g. the adoption of a modular logic as a way out of the difficulties in quantum mechanics), is fully conceivable. | |
| From: Hilary Putnam (Mathematics without Foundations [1967], p.303) | |
| A reaction: One can change the axioms of a system without necessarily changing the system (by swapping an axiom and a theorem). Especially if platonism is true, since the eternal objects reside calmly above our attempts to axiomatise them! |
| 9940 | Maybe mathematics is empirical in that we could try to change it [Putnam] |
| Full Idea: Mathematics might be 'empirical' in the sense that one is allowed to try to put alternatives into the field. | |
| From: Hilary Putnam (Mathematics without Foundations [1967], p.303) | |
| A reaction: He admits that change is highly unlikely. It take hardcore Millian arithmetic to be only changeable if pebbles start behaving very differently with regard to their quantities, which appears to be almost inconceivable. |
| 9941 | Science requires more than consistency of mathematics [Putnam] |
| Full Idea: Science demands much more of a mathematical theory than that it should merely be consistent, as the example of the various alternative systems of geometry dramatizes. | |
| From: Hilary Putnam (Mathematics without Foundations [1967]) | |
| A reaction: Well said. I don't agree with Putnam's Indispensability claims, but if an apparent system of numbers or lines has no application to the world then I don't consider it to be mathematics. It is a new game, like chess. |
| 9943 | You can't deny a hypothesis a truth-value simply because we may never know it! [Putnam] |
| Full Idea: Surely the mere fact that we may never know whether the continuum hypothesis is true or false is by itself just no reason to think that it doesn't have a truth value! | |
| From: Hilary Putnam (Mathematics without Foundations [1967]) | |
| A reaction: This is Putnam in 1967. Things changed later. Personally I am with the younger man all they way, but I reserve the right to totally change my mind. |
| 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. |
| 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. |
| 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. |