45 ideas
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
12619 | We have no successful definitions, because they all use indefinable words [Fodor] |
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] |
12620 | If 'exist' is ambiguous in 'chairs and numbers exist', that mirrors the difference between chairs and numbers [Fodor] |
12613 | Empiricists use dispositions reductively, as 'possibility of sensation' or 'possibility of experimental result' [Fodor] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
12617 | Associationism can't explain how truth is preserved [Fodor] |
12615 | Mental representations are the old 'Ideas', but without images [Fodor] |
6650 | Fodor is now less keen on the innateness of concepts [Fodor, by Lowe] |
12618 | It is essential to the concept CAT that it be satisfied by cats [Fodor] |
12614 | I prefer psychological atomism - that concepts are independent of epistemic capacities [Fodor] |
12621 | Definable concepts have constituents, which are necessary, individuate them, and demonstrate possession [Fodor] |
12622 | Many concepts lack prototypes, and complex prototypes aren't built from simple ones [Fodor] |
12623 | The theory theory can't actually tell us what concepts are [Fodor] |
12616 | English has no semantic theory, just associations between sentences and thoughts [Fodor] |
1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |