51 ideas
| 17774 | Definitions make our intuitions mathematically useful [Mayberry] |
| 17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
| 15063 | Some sentences depend for their truth on worldly circumstances, and others do not [Fine,K] |
| 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] |
| 15078 | There are levels of existence, as well as reality; objects exist at the lowest level in which they can function [Fine,K] |
| 15072 | Bottom level facts are subject to time and world, middle to world but not time, and top to neither [Fine,K] |
| 15071 | Tensed and tenseless sentences state two sorts of fact, which belong to two different 'realms' of reality [Fine,K] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 15075 | Modal features are not part of entities, because they are accounted for by the entity [Fine,K] |
| 15065 | What it is is fixed prior to existence or the object's worldly features [Fine,K] |
| 15076 | Essential features of an object have no relation to how things actually are [Fine,K] |
| 15073 | Self-identity should have two components, its existence, and its neutral identity with itself [Fine,K] |
| 15074 | We would understand identity between objects, even if their existence was impossible [Fine,K] |
| 15064 | Proper necessary truths hold whatever the circumstances; transcendent truths regardless of circumstances [Fine,K] |
| 15070 | It is the nature of Socrates to be a man, so necessarily he is a man [Fine,K] |
| 15069 | Possible worlds may be more limited, to how things might actually turn out [Fine,K] |
| 15068 | The actual world is a totality of facts, so we also think of possible worlds as totalities [Fine,K] |
| 23903 | When we admire a work, we see ourselves as its creator [Weil] |
| 23901 | Relationships depend on equality, so unequal treatment kills them [Weil] |
| 15077 | It is said that in the A-theory, all existents and objects must be tensed, as well as the sentences [Fine,K] |
| 15067 | A-theorists tend to reject the tensed/tenseless distinction [Fine,K] |
| 15066 | B-theorists say tensed sentences have an unfilled argument-place for a time [Fine,K] |
| 23904 | The cruelty of the Old Testament put me off Christianity [Weil] |
| 23902 | I attach little importance to immortality, which is an undecidable fact, and irrelevant to us [Weil] |