53 ideas
| 15716 | If axioms and their implications have no contradictions, they pass my criterion of truth and existence [Hilbert] |
| 7791 | The simplest of the logics based on possible worlds is Lewis's S5 [Lewis,CI, by Girle] |
| 9358 | There are several logics, none of which will ever derive falsehoods from truth [Lewis,CI] |
| 18739 | Three stages of philosophical logic: syntactic (1905-55), possible worlds (1963-85), widening (1990-) [Horsten/Pettigrew] |
| 9357 | Excluded middle is just our preference for a simplified dichotomy in experience [Lewis,CI] |
| 18844 | You would cripple mathematics if you denied Excluded Middle [Hilbert] |
| 18741 | Logical formalization makes concepts precise, and also shows their interrelation [Horsten/Pettigrew] |
| 9364 | Names represent a uniformity in experience, or they name nothing [Lewis,CI] |
| 18744 | Models are sets with functions and relations, and truth built up from the components [Horsten/Pettigrew] |
| 17963 | The facts of geometry, arithmetic or statics order themselves into theories [Hilbert] |
| 17966 | Axioms must reveal their dependence (or not), and must be consistent [Hilbert] |
| 8717 | Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Hilbert, by Friend] |
| 12456 | I aim to establish certainty for mathematical methods [Hilbert] |
| 12461 | We believe all mathematical problems are solvable [Hilbert] |
| 13472 | Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD] |
| 9633 | No one shall drive us out of the paradise the Cantor has created for us [Hilbert] |
| 12460 | We extend finite statements with ideal ones, in order to preserve our logic [Hilbert] |
| 12462 | Only the finite can bring certainty to the infinite [Hilbert] |
| 12455 | The idea of an infinite totality is an illusion [Hilbert] |
| 12457 | There is no continuum in reality to realise the infinitely small [Hilbert] |
| 17967 | To decide some questions, we must study the essence of mathematical proof itself [Hilbert] |
| 9546 | Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara] |
| 18742 | Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew] |
| 18217 | Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H] |
| 17965 | The whole of Euclidean geometry derives from a basic equation and transformations [Hilbert] |
| 17964 | Number theory just needs calculation laws and rules for integers [Hilbert] |
| 17697 | The existence of an arbitrarily large number refutes the idea that numbers come from experience [Hilbert] |
| 17698 | Logic already contains some arithmetic, so the two must be developed together [Hilbert] |
| 10113 | The grounding of mathematics is 'in the beginning was the sign' [Hilbert] |
| 10115 | Hilbert substituted a syntactic for a semantic account of consistency [Hilbert, by George/Velleman] |
| 22293 | Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency [Hilbert, by Potter] |
| 12459 | The subject matter of mathematics is immediate and clear concrete symbols [Hilbert] |
| 10116 | Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [Hilbert, by George/Velleman] |
| 18112 | Mathematics divides in two: meaningful finitary statements, and empty idealised statements [Hilbert] |
| 18740 | If 'exist' doesn't express a property, we can hardly ask for its essence [Horsten/Pettigrew] |
| 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] |
| 18745 | A Tarskian model can be seen as a possible state of affairs [Horsten/Pettigrew] |
| 18747 | The 'spheres model' was added to possible worlds, to cope with counterfactuals [Horsten/Pettigrew] |
| 18748 | Epistemic logic introduced impossible worlds [Horsten/Pettigrew] |
| 18746 | Possible worlds models contain sets of possible worlds; this is a large metaphysical commitment [Horsten/Pettigrew] |
| 18750 | Using possible worlds for knowledge and morality may be a step too far [Horsten/Pettigrew] |
| 9636 | My theory aims at the certitude of mathematical methods [Hilbert] |
| 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] |
| 9363 | Science seeks classification which will discover laws, essences, and predictions [Lewis,CI] |
| 17968 | By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge [Hilbert] |