63 ideas
17651 | Without words or other symbols, we have no world [Goodman] |
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
17652 | Truth is irrelevant if no statements are involved [Goodman] |
17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
15510 | Classes are a host of ethereal, platonic, pseudo entities [Goodman] |
17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
17803 | Limitation of size is part of the very conception of a set [Mayberry] |
9920 | Two objects can apparently make up quite distinct arrangements in sets [Goodman, by Burgess/Rosen] |
10657 | The counties of Utah, and the state, and its acres, are in no way different [Goodman] |
17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
12394 | If the result is bad, we change the rule; if we like the rule, we reject the result [Goodman] |
17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
17656 | Being primitive or prior always depends on a constructional system [Goodman] |
17661 | We don't recognise patterns - we invent them [Goodman] |
17659 | Reality is largely a matter of habit [Goodman] |
17657 | We build our world, and ignore anything that won't fit [Goodman] |
17654 | A world can be full of variety or not, depending on how we sort it [Goodman] |
14292 | Dispositions seem more ethereal than behaviour; a non-occult account of them would be nice [Goodman] |
7956 | If all and only red things were round things, we would need to specify the 'respect' of the resemblance [Goodman, by Macdonald,C] |
7957 | Without respects of resemblance, we would collect blue book, blue pen, red pen, red clock together [Goodman, by Macdonald,C] |
7952 | If we apply the same word to different things, it is only because we are willing to do so [Goodman, by Macdonald,C] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
17653 | Things can only be judged the 'same' by citing some respect of sameness [Goodman] |
12191 | Counterfactuals are true if logical or natural laws imply the consequence [Goodman, by McFetridge] |
17660 | Discovery is often just finding a fit, like a jigsaw puzzle [Goodman] |
17658 | Users of digital thermometers recognise no temperatures in the gaps [Goodman] |
17650 | We lack frames of reference to transform physics, biology and psychology into one another [Goodman] |
6027 | From the fact that some men die, we cannot infer that they all do [Philodemus] |
18749 | Goodman argued that the confirmation relation can never be formalised [Goodman, by Horsten/Pettigrew] |
17646 | Goodman showed that every sound inductive argument has an unsound one of the same form [Goodman, by Putnam] |
17655 | Grue and green won't be in the same world, as that would block induction entirely [Goodman] |
20440 | Art is a referential activity, hence indefinable, but it has a set of symptoms [Goodman] |
20439 | Artistic symbols are judged by the fruitfulness of their classifications [Goodman, by Giovannelli] |
8113 | Art is like understanding a natural language, and needs a grasp of a symbol system [Goodman, by Gardner] |
20438 | A performance is only an instance of a work if there is not a single error [Goodman] |
20437 | A copy only becomes an 'instance' of an artwork if there is a system of notation [Goodman] |
22241 | Don't fear god or worry about death; the good is easily got and the terrible easily cured [Philodemus] |
17649 | If the world is one it has many aspects, and if there are many worlds they will collect into one [Goodman] |
4794 | We don't use laws to make predictions, we call things laws if we make predictions with them [Goodman] |