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] |
| 17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
| 17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [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] |
| 13258 | The 'aggregative' objections says mereology gets existence and location of objects wrong [Koslicki] |
| 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] |
| 13288 | Consequence is truth-preserving, either despite substitutions, or in all interpretations [Koslicki] |
| 14506 | 'Roses are red; therefore, roses are colored' seems truth-preserving, but not valid in a system [Koslicki] |
| 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] |
| 14505 | Some questions concern mathematical entities, rather than whole structures [Koslicki] |
| 13289 | Structures have positions, constituent types and number, and some invariable parts [Koslicki] |
| 14501 | 'Categorical' properties exist in the actual world, and 'hypothetical' properties in other worlds [Koslicki] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 14495 | I aim to put the notion of structure or form back into the concepts of part, whole and object [Koslicki] |
| 13264 | If a whole is just a structure, a dinner party wouldn't need the guests to turn up [Koslicki] |
| 14497 | The clay is just a part of the statue (its matter); the rest consists of its form or structure [Koslicki] |
| 13280 | Statue and clay differ in modal and temporal properties, and in constitution [Koslicki] |
| 14496 | Structure or form are right at the centre of modern rigorous modes of enquiry [Koslicki] |
| 13279 | There are at least six versions of constitution being identity [Koslicki] |
| 14498 | For three-dimensionalist parthood must be a three-place relation, including times [Koslicki] |
| 13283 | The parts may be the same type as the whole, like a building made of buildings [Koslicki] |
| 13266 | Wholes in modern mereology are intended to replace sets, so they closely resemble them [Koslicki] |
| 14500 | Wholes are entities distinct from their parts, and have different properties [Koslicki] |
| 13281 | Wholes are not just their parts; a whole is an entity distinct from the proper parts [Koslicki] |
| 20239 | Unlike us, the early Greeks thought envy was a good thing, and hope a bad thing [Hesiod, by Nietzsche] |
| 14504 | The Kripke/Putnam approach to natural kind terms seems to give them excessive stability [Koslicki] |
| 13285 | Natural kinds support inductive inferences, from previous samples to the next one [Koslicki] |
| 13287 | Concepts for species are either intrinsic structure, or relations like breeding or ancestry [Koslicki] |
| 13284 | Should vernacular classifications ever be counted as natural kind terms? [Koslicki] |
| 13286 | There are apparently no scientific laws concerning biological species [Koslicki] |