73 ideas
5486 | Essentialism says metaphysics can't be done by analysing unreliable language [Ellis] |
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
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] |
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] |
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] |
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] |
5468 | Properties are 'dispositional', or 'categorical' (the latter as 'block' or 'intrinsic' structures) [Ellis, by PG] |
5469 | The passive view of nature says categorical properties are basic, but others say dispositions [Ellis] |
5456 | Redness is not a property as it is not mind-independent [Ellis] |
5481 | Properties have powers; they aren't just ways for logicians to classify objects [Ellis] |
5458 | Nearly all fundamental properties of physics are dispositional [Ellis] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
5443 | Kripke and others have made essentialism once again respectable [Ellis] |
5444 | 'Individual essences' fix a particular individual, and 'kind essences' fix the kind it belongs to [Ellis] |
5462 | Essential properties are usually quantitatively determinate [Ellis] |
5448 | 'Real essence' makes it what it is; 'nominal essence' makes us categorise it a certain way [Ellis] |
5477 | One thing can look like something else, without being the something else [Ellis] |
5479 | Scientific essentialists say science should define the limits of the possible [Ellis] |
5483 | Essentialists deny possible worlds, and say possibilities are what is compatible with the actual world [Ellis] |
5447 | Metaphysical necessities are true in virtue of the essences of things [Ellis] |
5476 | Essentialists say natural laws are in a new category: necessary a posteriori [Ellis] |
5478 | Imagination tests what is possible for all we know, not true possibility [Ellis] |
5482 | Possible worlds realism is only needed to give truth conditions for modals and conditionals [Ellis] |
5453 | Essentialists mostly accept the primary/secondary qualities distinction [Ellis] |
5466 | Primary qualities are number, figure, size, texture, motion, configuration, impenetrability and (?) mass [Ellis] |
5485 | Emeralds are naturally green, and only an external force could turn them blue [Ellis] |
5484 | Essentialists don't infer from some to all, but from essences to necessary behaviour [Ellis] |
5457 | Predicates assert properties, values, denials, relations, conventions, existence and fabrications [Ellis, by PG] |
5488 | Regularity theories of causation cannot give an account of human agency [Ellis] |
5489 | Humans have variable dispositions, and also power to change their dispositions [Ellis] |
5490 | Essentialism fits in with Darwinism, but not with extreme politics of left or right [Ellis] |
6004 | The cardinal virtues are theoretical (based on knowledge), and others are 'non-theoretical' [Hecato, by Dorandi] |
5472 | Natural kinds are of objects/substances, or events/processes, or intrinsic natures [Ellis] |
5471 | Essentialism says natural kinds are fundamental to nature, and determine the laws [Ellis] |
5446 | For essentialists two members of a natural kind must be identical [Ellis] |
5480 | The whole of our world is a natural kind, so all worlds like it necessarily have the same laws [Ellis] |
5445 | Essentialists regard inanimate objects as genuine causal agents [Ellis] |
5463 | Essentialists believe causation is necessary, resulting from dispositions and circumstances [Ellis] |
5491 | A general theory of causation is only possible in an area if natural kinds are involved [Ellis] |
5442 | For 'passivists' behaviour is imposed on things from outside [Ellis] |
5473 | The laws of nature imitate the hierarchy of natural kinds [Ellis] |
5474 | Laws of nature tend to describe ideal things, or ideal circumstances [Ellis] |
5475 | We must explain the necessity, idealisation, ontology and structure of natural laws [Ellis] |
5460 | Causal relations cannot be reduced to regularities, as they could occur just once [Ellis] |
5459 | Essentialists say dispositions are basic, rather than supervenient on matter and natural laws [Ellis] |
5461 | The essence of uranium is its atomic number and its electron shell [Ellis] |
5464 | For essentialists, laws of nature are metaphysically necessary, being based on essences of natural kinds [Ellis] |
5487 | Essentialism requires a clear separation of semantics, epistemology and ontology [Ellis] |