51 ideas
1597 | Thales was the first western thinker to believe the arché was intelligible [Roochnik on Thales] |
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
7791 | The simplest of the logics based on possible worlds is Lewis's S5 [Lewis,CI, by Girle] |
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] |
9358 | There are several logics, none of which will ever derive falsehoods from truth [Lewis,CI] |
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] |
9357 | Excluded middle is just our preference for a simplified dichotomy in experience [Lewis,CI] |
9364 | Names represent a uniformity in experience, or they name nothing [Lewis,CI] |
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] |
11002 | Equating necessity with informal provability is the S4 conception of necessity [Lewis,CI, by Read] |
9362 | Necessary truths are those we will maintain no matter what [Lewis,CI] |
7803 | Modal logic began with translation difficulties for 'If...then' [Lewis,CI, by Girle] |
3013 | Nothing is stronger than necessity, which rules everything [Thales, by Diog. Laertius] |
9365 | We can maintain a priori principles come what may, but we can also change them [Lewis,CI] |
21500 | We rely on memory for empirical beliefs because they mutually support one another [Lewis,CI] |
21501 | If we doubt memories we cannot assess our doubt, or what is being doubted [Lewis,CI] |
6556 | If anything is to be probable, then something must be certain [Lewis,CI] |
21498 | Congruents assertions increase the probability of each individual assertion in the set [Lewis,CI] |
5828 | Extension is the class of things, intension is the correct definition of the thing, and intension determines extension [Lewis,CI] |
9361 | We have to separate the mathematical from physical phenomena by abstraction [Lewis,CI] |
1494 | Thales said water is the first principle, perhaps from observing that food is moist [Thales, by Aristotle] |
9363 | Science seeks classification which will discover laws, essences, and predictions [Lewis,CI] |
1713 | Thales must have thought soul causes movement, since he thought magnets have soul [Thales, by Aristotle] |
1742 | Thales said the gods know our wrong thoughts as well as our evil actions [Thales, by Diog. Laertius] |