47 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] |
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] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
14637 | Only individuals have essences, so numbers (as a higher type based on classes) lack them [McMichael] |
14636 | Essences are the interesting necessary properties resulting from a thing's own peculiar nature [McMichael] |
14640 | Maybe essential properties have to be intrinsic, as well as necessary? [McMichael] |
14638 | Essentialism is false, because it implies the existence of necessary singular propositions [McMichael] |
19527 | We don't acquire evidence and then derive some knowledge, because evidence IS knowledge [Williamson] |
19528 | Knowledge is prior to believing, just as doing is prior to trying to do [Williamson] |
19529 | Belief explains justification, and knowledge explains belief, so knowledge explains justification [Williamson] |
19530 | A neutral state of experience, between error and knowledge, is not basic; the successful state is basic [Williamson] |
19531 | Internalism about mind is an obsolete view, and knowledge-first epistemology develops externalism [Williamson] |
19536 | Knowledge-first says your total evidence IS your knowledge [Williamson] |
19526 | Surely I am acquainted with physical objects, not with appearances? [Williamson] |
19534 | How does inferentialism distinguish the patterns of inference that are essential to meaning? [Williamson] |
19535 | Internalist inferentialism has trouble explaining how meaning and reference relate [Williamson] |
19533 | Inferentialist semantics relies on internal inference relations, not on external references [Williamson] |
19532 | Truth-conditional referential semantics is externalist, referring to worldly items [Williamson] |
14639 | Individuals enter into laws only through their general qualities and relations [McMichael] |