13 ideas
| 6602 | Philosophy is like a statue which is worshipped but never advances [Bacon] |
| Full Idea: Philosophy and the intellectual sciences stand like statues, worshipped and celebrated, but not moved or advanced. | |
| From: Francis Bacon (Preface to Great Instauration (Renewal) [1620], Vol.4.14), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5 | |
| A reaction: Still the view of most scientists, I suspect. Personally I disagree, because I think philosophy has made enormous advances, in accurate analysis of arguments. The trouble is there is so much of it that it is hard to discern, and we don't live long enough. |
| 21959 | Metaphysics is the most general attempt to make sense of things [Moore,AW] |
| Full Idea: Metaphysics is the most general attempt to make sense of things. | |
| From: A.W. Moore (The Evolution of Modern Metaphysics [2012], Intro) | |
| A reaction: This is the first sentence of Moore's book, and a touchstone idea all the way through. It stands up well, because it says enough without committing to too much. I have to agree with it. It implies explanation as the key. I like generality too. |
| 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. |
| 21958 | Appearances are nothing beyond representations, which is transcendental ideality [Moore,AW] |
| Full Idea: Appearances in general are nothing outside our representations, which is just what we mean by transcendental ideality. | |
| From: A.W. Moore (The Evolution of Modern Metaphysics [2012], B535/A507) |
| 16724 | The senses deceive, but also show their own errors [Bacon] |
| Full Idea: It is certain that the senses deceive, but they also testify to their own errors. | |
| From: Francis Bacon (Preface to Great Instauration (Renewal) [1620], p.32), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 22.1 | |
| A reaction: Nice. This is the empiricist view, rather than the rationalist line that reason sorts out the mess created by the senses. Most people know things if you just show them. |
| 6603 | Nature is revealed when we put it under pressure rather than observe it [Bacon] |
| Full Idea: The secrets of nature reveal themselves more readily under the vexations of art than when they go their own way. | |
| From: Francis Bacon (Preface to Great Instauration (Renewal) [1620], Vol.4.95), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5 | |
| A reaction: This is a splendid slogan for the dawn of the age of science, and pinpoints the reason why we have advanced so much further than the Greeks. You can, of course, overdo the 'vexations of art'. It also justifies the critical approach to philosophy. |