52 ideas
| 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] |
| 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] |
| 13047 | It is knowing 'why' that gives scientific understanding, not knowing 'that' [Salmon] |
| 13065 | Understanding is an extremely vague concept [Salmon] |
| 1556 | By nature people are close to one another, but culture drives them apart [Hippias] |
| 13054 | Correlations can provide predictions, but only causes can give explanations [Salmon] |
| 13067 | For the instrumentalists there are no scientific explanations [Salmon] |
| 13055 | Good induction needs 'total evidence' - the absence at the time of any undermining evidence [Salmon] |
| 13046 | Scientific explanation is not reducing the unfamiliar to the familiar [Salmon] |
| 13058 | Why-questions can seek evidence as well as explanation [Salmon] |
| 13064 | The three basic conceptions of scientific explanation are modal, epistemic, and ontic [Salmon] |
| 13050 | The 'inferential' conception is that all scientific explanations are arguments [Salmon] |
| 13059 | Ontic explanations can be facts, or reports of facts [Salmon] |
| 13049 | We must distinguish true laws because they (unlike accidental generalizations) explain things [Salmon] |
| 13051 | Deductive-nomological explanations will predict, and their predictions will explain [Salmon] |
| 13053 | A law is not enough for explanation - we need information about what makes a difference [Salmon] |
| 13061 | Flagpoles explain shadows, and not vice versa, because of temporal ordering [Salmon] |
| 13045 | Explanation at the quantum level will probably be by entirely new mechanisms [Salmon] |
| 13062 | Does an item have a function the first time it occurs? [Salmon] |
| 13063 | Explanations reveal the mechanisms which produce the facts [Salmon] |
| 13060 | Can events whose probabilities are low be explained? [Salmon] |
| 13056 | Statistical explanation needs relevance, not high probability [Salmon] |
| 13057 | Think of probabilities in terms of propensities rather than frequencies [Salmon] |