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] |
| 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] |
| 22427 | To explain object qualities, primary qualities must be more than mere sources of experience [McGinn] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 22413 | Being red simply consists in looking red [McGinn] |
| 22415 | Relativity means differing secondary perceptions are not real disagreements [McGinn] |
| 22416 | Phenomenalism is correct for secondary qualities, so scepticism is there impossible [McGinn] |
| 22422 | Maybe all possible sense experience must involve both secondary and primary qualities [McGinn] |
| 22428 | You understood being red if you know the experience involved; not so with thngs being square [McGinn] |
| 22414 | You don't need to know how a square thing looks or feels to understand squareness [McGinn] |
| 22423 | Touch doesn't provide direct experience of primary qualities, because touch feels temperature [McGinn] |
| 22426 | We can perceive objectively, because primary qualities are not mind-created [McGinn] |
| 22412 | Lockean secondary qualities (unlike primaries) produce particular sensory experiences [McGinn] |
| 22421 | Could there be a mind which lacked secondary quality perception? [McGinn] |
| 22424 | Secondary qualities contain information; their variety would be superfluous otherwise [McGinn] |
| 22425 | The utility theory says secondary qualities give information useful to human beings [McGinn] |
| 7629 | We see objects 'directly' by representing them [McGinn] |
| 22420 | The indexical perspective is subjective, incorrigible and constant [McGinn] |
| 18410 | Indexical thought is in relation to my self-consciousness [McGinn] |
| 22417 | Indexicals do not figure in theories of physics, because they are not explanatory causes [McGinn] |
| 18402 | Indexical concepts are indispensable, as we need them for the power to act [McGinn] |
| 22418 | I can know indexical truths a priori, unlike their non-indexical paraphrases [McGinn] |
| 22489 | 'Good' is an attributive adjective like 'large', not predicative like 'red' [Geach, by Foot] |