56 ideas
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] |
10405 | In the iterative conception of sets, they form a natural hierarchy [Swoyer] |
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] |
10407 | Logical Form explains differing logical behaviour of similar sentences [Swoyer] |
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] |
14592 | Some abstract things have a beginning and end, so may exist in time (though not space) [Swoyer] |
10421 | Supervenience is nowadays seen as between properties, rather than linguistic [Swoyer] |
14594 | Ontologists seek existence and identity conditions, and modal and epistemic status for a thing [Swoyer] |
10410 | Anti-realists can't explain different methods to measure distance [Swoyer] |
10399 | If a property such as self-identity can only be in one thing, it can't be a universal [Swoyer] |
10416 | Can properties have parts? [Swoyer] |
14595 | Can properties exemplify other properties? [Swoyer] |
10417 | There are only first-order properties ('red'), and none of higher-order ('coloured') [Swoyer] |
10413 | The best-known candidate for an identity condition for properties is necessary coextensiveness [Swoyer] |
10402 | Various attempts are made to evade universals being wholly present in different places [Swoyer] |
10400 | Conceptualism says words like 'honesty' refer to concepts, not to properties [Swoyer] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
10403 | If properties are abstract objects, then their being abstract exemplifies being abstract [Swoyer] |
14593 | Quantum field theory suggests that there are, fundamentally, no individual things [Swoyer] |
594 | Speusippus suggested underlying principles for every substance, and ended with a huge list [Speussipus, by Aristotle] |
10406 | One might hope to reduce possible worlds to properties [Swoyer] |
10404 | Extreme empiricists can hardly explain anything [Swoyer] |
10408 | Intensions are functions which map possible worlds to sets of things denoted by an expression [Swoyer] |
10409 | Research suggests that concepts rely on typical examples [Swoyer] |
10401 | The F and G of logic cover a huge range of natural language combinations [Swoyer] |
10420 | Maybe a proposition is just a property with all its places filled [Swoyer] |
10412 | If laws are mere regularities, they give no grounds for future prediction [Swoyer] |
10411 | Two properties can have one power, and one property can have two powers [Swoyer] |
2632 | Speusippus said things were governed by some animal force rather than the gods [Speussipus, by Cicero] |