47 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] |
| 4444 | One moderate nominalist view says that properties and relations exist, but they are particulars [Armstrong] |
| 4445 | If properties and relations are particulars, there is still the problem of how to classify and group them [Armstrong] |
| 4448 | Should we decide which universals exist a priori (through words), or a posteriori (through science)? [Armstrong] |
| 4446 | It is claimed that some universals are not exemplified by any particular, so must exist separately [Armstrong] |
| 4440 | 'Resemblance Nominalism' finds that in practice the construction of resemblance classes is hard [Armstrong] |
| 4439 | 'Resemblance Nominalism' says properties are resemblances between classes of particulars [Armstrong] |
| 4431 | 'Predicate Nominalism' says that a 'universal' property is just a predicate applied to lots of things [Armstrong] |
| 4433 | Concept and predicate nominalism miss out some predicates, and may be viciously regressive [Armstrong] |
| 4432 | 'Concept Nominalism' says a 'universal' property is just a mental concept applied to lots of things [Armstrong] |
| 4436 | 'Class Nominalism' may explain properties if we stick to 'natural' sets, and ignore random ones [Armstrong] |
| 4434 | 'Class Nominalism' says that properties or kinds are merely membership of a set (e.g. of white things) [Armstrong] |
| 4435 | 'Class Nominalism' cannot explain co-extensive properties, or sets with random members [Armstrong] |
| 4437 | 'Mereological Nominalism' sees whiteness as a huge white object consisting of all the white things [Armstrong] |
| 4438 | 'Mereological Nominalism' may work for whiteness, but it doesn't seem to work for squareness [Armstrong] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 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] |