49 ideas
| 17774 | Definitions make our intuitions mathematically useful [Mayberry] |
| 17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
| 17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
| 17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [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] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 1212 | Replacing timbers on Theseus' ship was the classic illustration of the problem of growth and change [Plutarch] |
| 5958 | The sun is always bright; it doesn't become bright when it emerges [Plutarch] |
| 8406 | Not all explanations are causal, but if a thing can be explained at all, it can be explained causally [Sanford] |
| 5960 | When the soul is intelligent and harmonious, it is part of god and derives from god [Plutarch] |
| 5959 | Some philosophers say the soul is light [Plutarch] |
| 5952 | Rather than being the whole soul, maybe I am its chief part? [Plutarch] |
| 5951 | If atoms have no qualities, they cannot possibly produce a mind [Plutarch] |
| 5963 | Some say emotion is a sort of reason, and others say virtue concerns emotion [Plutarch] |
| 20796 | Action needs an affinity for a presentation, and an impulse toward the affinity [Plutarch] |
| 1477 | Being manly and brave is the result of convention, not of human nature [Plutarch] |
| 1478 | Animals don't value pleasure, as they cease sexual intercourse after impregnation [Plutarch] |
| 5948 | The good life involves social participation, loyalty, temperance and honesty [Plutarch] |
| 1479 | Animals have not been led into homosexuality, because they value pleasure very little [Plutarch] |
| 5950 | If only atoms exist, how do qualities arise when the atoms come together? [Plutarch] |
| 8407 | A totality of conditions necessary for an occurrence is usually held to be jointly sufficient for it [Sanford] |
| 5974 | People report seeing through rocks, or over the horizon, or impossibly small works [Plutarch] |
| 5957 | Absurd superstitions make people atheist, not disharmony in nature [Plutarch] |
| 5955 | No one will ever find a city that lacks religious practices [Plutarch] |