54 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] |
| 1350 | Continuity is needed for existence, otherwise we would say a thing existed after it ceased to exist [Reid] |
| 21322 | We treat slowly changing things as identical for the sake of economy in language [Reid] |
| 21320 | Identity is familiar to common sense, but very hard to define [Reid] |
| 1367 | Identity can only be affirmed of things which have a continued existence [Reid] |
| 23644 | Without memory we could have no concept of duration [Reid] |
| 23643 | We all trust our distinct memories (but not our distinct imaginings) [Reid] |
| 17488 | Empiricist theories are sets of laws, which give explanations and reductions [Glennan] |
| 17493 | Modern mechanism need parts with spatial, temporal and function facts, and diagrams [Glennan] |
| 17487 | Mechanistic philosophy of science is an alternative to the empiricist law-based tradition [Glennan] |
| 17489 | Mechanisms are either systems of parts or sequences of activities [Glennan] |
| 17490 | 17th century mechanists explained everything by the kinetic physical fundamentals [Glennan] |
| 17491 | Unlike the lawlike approach, mechanistic explanation can allow for exceptions [Glennan] |
| 1356 | A person is a unity, and doesn't come in degrees [Reid] |
| 1359 | Personal identity is the basis of all rights, obligations and responsibility [Reid] |
| 21319 | I can hardly care about rational consequence if it wasn't me conceiving the antecedent [Reid] |
| 21323 | The identity of a thief is only known by similarity, but memory gives certainty in our own case [Reid] |
| 21321 | Memory reveals my past identity - but so does testimony of other witnesses [Reid] |
| 21324 | If consciousness is transferable 20 persons can be 1; forgetting implies 1 can be 20 [Reid] |
| 21325 | Boy same as young man, young man same as old man, old man not boy, if forgotten! [Reid] |
| 21327 | If a stolen horse is identified by similitude, its identity is not therefore merely similitude [Reid] |
| 1366 | If consciousness is personal identity, it is continually changing [Reid] |
| 1352 | Thoughts change continually, but the self doesn't [Reid] |
| 17494 | Since causal events are related by mechanisms, causation can be analysed in that way [Glennan] |