48 ideas
| 12596 | Reasoning aims at increasing explanatory coherence [Harman] |
| 12599 | Reason conservatively: stick to your beliefs, and prefer reasoning that preserves most of them [Harman] |
| 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] |
| 12595 | We have a theory of logic (implication and inconsistency), but not of inference or reasoning [Harman] |
| 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] |
| 12597 | I might accept P and Q as likely, but reject P-and-Q as unlikely [Harman] |
| 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] |
| 12598 | Reality is the overlap of true complete theories [Harman] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 12602 | There is no natural border between inner and outer [Harman] |
| 12603 | We can only describe mental attitudes in relation to the external world [Harman] |
| 12601 | The way things look is a relational matter, not an intrinsic matter [Harman] |
| 12592 | Concepts in thought have content, but not meaning, which requires communication [Harman] |
| 12590 | Take meaning to be use in calculation with concepts, rather than in communication [Harman] |
| 12593 | The use theory attaches meanings to words, not to sentences [Harman] |
| 12588 | Meaning from use of thoughts, constructed from concepts, which have a role relating to reality [Harman] |
| 12589 | Some regard conceptual role semantics as an entirely internal matter [Harman] |
| 12600 | The content of thought is relations, between mental states, things in the world, and contexts [Harman] |
| 12594 | If one proposition negates the other, which is the negative one? [Harman] |
| 12591 | Mastery of a language requires thinking, and not just communication [Harman] |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |