13 ideas
| 15970 | People generalise because it is easier to understand, and that is mistaken for deep philosophy [Feynman] |
| Full Idea: The topic of the laws of nature has a tendency to become too philosophical because it becomes too general, and a person talks in such generalities, that everybody can understand him. It is then considered to be some deep philosophy. | |
| From: Richard P. Feynman (The Character of Physical Law [1965], 1) | |
| A reaction: Feynman was famously anti-philosophical, but this is a good challenge. I like philosophy because I want to know broad general truths about my world, but I may just be gravitating towards what is easier. The challenge is to get true generalities. |
| 10476 | The idea that groups of concepts could be 'implicitly defined' was abandoned [Hodges,W] |
| Full Idea: Late nineteenth century mathematicians said that, although plus, minus and 0 could not be precisely defined, they could be partially 'implicitly defined' as a group. This nonsense was rejected by Frege and others, as expressed in Russell 1903. | |
| From: Wilfrid Hodges (Model Theory [2005], 2) | |
| A reaction: [compressed] This is helpful in understanding what is going on in Frege's 'Grundlagen'. I won't challenge Hodges's claim that such definitions are nonsense, but there is a case for understanding groups of concepts together. |
| 10478 | Since first-order languages are complete, |= and |- have the same meaning [Hodges,W] |
| Full Idea: In first-order languages the completeness theorem tells us that T |= φ holds if and only if there is a proof of φ from T (T |- φ). Since the two symbols express the same relationship, theorist often just use |- (but only for first-order!). | |
| From: Wilfrid Hodges (Model Theory [2005], 3) | |
| A reaction: [actually no spaces in the symbols] If you are going to study this kind of theory of logic, the first thing you need to do is sort out these symbols, which isn't easy! |
| 10477 | |= in model-theory means 'logical consequence' - it holds in all models [Hodges,W] |
| Full Idea: If every structure which is a model of a set of sentences T is also a model of one of its sentences φ, then this is known as the model-theoretic consequence relation, and is written T |= φ. Not to be confused with |= meaning 'satisfies'. | |
| From: Wilfrid Hodges (Model Theory [2005], 3) | |
| A reaction: See also Idea 10474, which gives the other meaning of |=, as 'satisfies'. The symbol is ALSO used in propositional logical, to mean 'tautologically implies'! Sort your act out, logicians. |
| 10474 | |= should be read as 'is a model for' or 'satisfies' [Hodges,W] |
| Full Idea: The symbol in 'I |= S' reads that if the interpretation I (about word meaning) happens to make the sentence S state something true, then I 'is a model for' S, or I 'satisfies' S. | |
| From: Wilfrid Hodges (Model Theory [2005], 1) | |
| A reaction: Unfortunately this is not the only reading of the symbol |= [no space between | and =!], so care and familiarity are needed, but this is how to read it when dealing with models. See also Idea 10477. |
| 10473 | Model theory studies formal or natural language-interpretation using set-theory [Hodges,W] |
| Full Idea: Model theory is the study of the interpretation of any language, formal or natural, by means of set-theoretic structures, with Tarski's truth definition as a paradigm. | |
| From: Wilfrid Hodges (Model Theory [2005], Intro) | |
| A reaction: My attention is caught by the fact that natural languages are included. Might we say that science is model theory for English? That sounds like Quine's persistent message. |
| 10475 | A 'structure' is an interpretation specifying objects and classes of quantification [Hodges,W] |
| Full Idea: A 'structure' in model theory is an interpretation which explains what objects some expressions refer to, and what classes some quantifiers range over. | |
| From: Wilfrid Hodges (Model Theory [2005], 1) | |
| A reaction: He cites as examples 'first-order structures' used in mathematical model theory, and 'Kripke structures' used in model theory for modal logic. A structure is also called a 'universe'. |
| 10481 | Models in model theory are structures, not sets of descriptions [Hodges,W] |
| Full Idea: The models in model-theory are structures, but there is also a common use of 'model' to mean a formal theory which describes and explains a phenomenon, or plans to build it. | |
| From: Wilfrid Hodges (Model Theory [2005], 5) | |
| A reaction: Hodges is not at all clear here, but the idea seems to be that model-theory offers a set of objects and rules, where the common usage offers a set of descriptions. Model-theory needs homomorphisms to connect models to things, |
| 10480 | First-order logic can't discriminate between one infinite cardinal and another [Hodges,W] |
| Full Idea: First-order logic is hopeless for discriminating between one infinite cardinal and another. | |
| From: Wilfrid Hodges (Model Theory [2005], 4) | |
| A reaction: This seems rather significant, since mathematics largely relies on first-order logic for its metatheory. Personally I'm tempted to Ockham's Razor out all these super-infinities, but mathematicians seem to make use of them. |
| 14494 | Epiphenomenalism is like a pointless nobleman, kept for show, but soon to be abolished [Alexander,S] |
| Full Idea: Epiphenomenalism supposes something to exist in nature which has nothing to do, no purpose to serve, a species of noblesse which depends on the work of its inferiors, but is kept for show and might as well, and undoubtedly would in time be abolished. | |
| From: Samuel Alexander (Space, Time and Deity (2 vols) [1927], 2:8), quoted by Jaegwon Kim - Nonreductivist troubles with ment.causation IV | |
| A reaction: Wonderful! Kim quotes this, and labels the implicit slogan (to be real is to have causal powers) 'Alexander's Dictum'. All the examples given of epiphenomena are only causally inert within a defined system, but they act causally outside the system. |
| 9410 | Physical Laws are rhythms and patterns in nature, revealed by analysis [Feynman] |
| Full Idea: There is a rhythm and a pattern between the phenomena of nature which is not apparent to the eye, but only to the eye of analysis; and it is these rhythms and patterns which we call Physical Laws. | |
| From: Richard P. Feynman (The Character of Physical Law [1965], Ch.1) |
| 18530 | Nobody understands quantum mechanics [Feynman] |
| Full Idea: I think I can safely say the nobody understands quantum mechanics. | |
| From: Richard P. Feynman (The Character of Physical Law [1965], 6) | |
| A reaction: It is really important that philosophers grasp this point! |
| 17707 | We should regard space as made up of many tiny pieces [Feynman, by Mares] |
| Full Idea: Feynman claims that we should regard space as made up of many tiny pieces, which have positive length, width and depth. | |
| From: report of Richard P. Feynman (The Character of Physical Law [1965], p.166) by Edwin D. Mares - A Priori 06.7 | |
| A reaction: The idea seems to be these are the minimum bits of space in which something can happen. |