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] |
9476 | If dispositions are more fundamental than causes, then they won't conceptually reduce to them [Bird on Lewis] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
3016 | Even the gods cannot strive against necessity [Pittacus, by Diog. Laertius] |
8425 | For true counterfactuals, both antecedent and consequent true is closest to actuality [Lewis] |
8424 | Determinism says there can't be two identical worlds up to a time, with identical laws, which then differ [Lewis] |
8420 | A proposition is a set of possible worlds where it is true [Lewis] |
8405 | A theory of causation should explain why cause precedes effect, not take it for granted [Lewis, by Field,H] |
8427 | I reject making the direction of causation axiomatic, since that takes too much for granted [Lewis] |
10392 | It is just individious discrimination to pick out one cause and label it as 'the' cause [Lewis] |
8419 | The modern regularity view says a cause is a member of a minimal set of sufficient conditions [Lewis] |
8421 | Regularity analyses could make c an effect of e, or an epiphenomenon, or inefficacious, or pre-empted [Lewis] |
17525 | The counterfactual view says causes are necessary (rather than sufficient) for their effects [Lewis, by Bird] |
17524 | Lewis has basic causation, counterfactuals, and a general ancestral (thus handling pre-emption) [Lewis, by Bird] |
8397 | Counterfactual causation implies all laws are causal, which they aren't [Tooley on Lewis] |
8423 | My counterfactual analysis applies to particular cases, not generalisations [Lewis] |
8426 | One event causes another iff there is a causal chain from first to second [Lewis] |
4795 | Lewis's account of counterfactuals is fine if we know what a law of nature is, but it won't explain the latter [Cohen,LJ on Lewis] |