57 ideas
| 4456 | Epistemological Ockham's Razor demands good reasons, but the ontological version says reality is simple [Moreland] |
| 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] |
| 4474 | Existence theories must match experience, possibility, logic and knowledge, and not be self-defeating [Moreland] |
| 4461 | Tropes are like Hume's 'impressions', conceived as real rather than as ideal [Moreland] |
| 4462 | A colour-trope cannot be simple (as required), because it is spread in space, and so it is complex [Moreland] |
| 4463 | In 'four colours were used in the decoration', colours appear to be universals, not tropes [Moreland] |
| 4451 | If properties are universals, what distinguishes two things which have identical properties? [Moreland] |
| 4453 | One realism is one-over-many, which may be the model/copy view, which has the Third Man problem [Moreland] |
| 4464 | Realists see properties as universals, which are single abstract entities which are multiply exemplifiable [Moreland] |
| 4449 | Evidence for universals can be found in language, communication, natural laws, classification and ideals [Moreland] |
| 4450 | The traditional problem of universals centres on the "One over Many", which is the unity of natural classes [Moreland] |
| 4454 | The One-In-Many view says universals have abstract existence, but exist in particulars [Moreland] |
| 4468 | How could 'being even', or 'being a father', or a musical interval, exist naturally in space? [Moreland] |
| 4452 | Maybe universals are real, if properties themselves have properties, and relate to other properties [Moreland] |
| 4467 | A naturalist and realist about universals is forced to say redness can be both moving and stationary [Moreland] |
| 4469 | There are spatial facts about red particulars, but not about redness itself [Moreland] |
| 4472 | Redness is independent of red things, can do without them, has its own properties, and has identity [Moreland] |
| 4459 | Moderate nominalism attempts to embrace the existence of properties while avoiding universals [Moreland] |
| 4458 | Unlike Class Nominalism, Resemblance Nominalism can distinguish natural from unnatural classes [Moreland] |
| 4457 | There can be predicates with no property, and there are properties with no predicate [Moreland] |
| 4471 | We should abandon the concept of a property since (unlike sets) their identity conditions are unclear [Moreland] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 4476 | Most philosophers think that the identity of indiscernibles is false [Moreland] |
| 8840 | There are five possible responses to the problem of infinite regress in justification [Cleve] |
| 8841 | Modern foundationalists say basic beliefs are fallible, and coherence is relevant [Cleve] |
| 4460 | Abstractions are formed by the mind when it concentrates on some, but not all, the features of a thing [Moreland] |
| 4455 | It is always open to a philosopher to claim that some entity or other is unanalysable [Moreland] |
| 4473 | 'Presentism' is the view that only the present moment exists [Moreland] |