50 ideas
19404 | Necessities rest on contradiction, and contingencies on sufficient reason [Leibniz] |
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
19215 | Arguers often turn the opponent's modus ponens into their own modus tollens [Merricks] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
19205 | 'Snow is white' only contingently expresses the proposition that snow is white [Merricks] |
19209 | Simple Quantified Modal Logc doesn't work, because the Converse Barcan is a theorem [Merricks] |
19208 | The Converse Barcan implies 'everything exists necessarily' is a consequence of 'necessarily, everything exists' [Merricks] |
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] |
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] |
19207 | Sentence logic maps truth values; predicate logic maps objects and sets [Merricks] |
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] |
19214 | In twinning, one person has the same origin as another person [Merricks] |
19217 | I don't accept that if a proposition is directly about an entity, it has a relation to the entity [Merricks] |
19203 | A sentence's truth conditions depend on context [Merricks] |
19200 | Propositions are standardly treated as possible worlds, or as structured [Merricks] |
19206 | 'Cicero is an orator' represents the same situation as 'Tully is an orator', so they are one proposition [Merricks] |
19202 | Propositions are necessary existents which essentially (but inexplicably) represent things [Merricks] |
19204 | True propositions existed prior to their being thought, and might never be thought [Merricks] |
19210 | The standard view of propositions says they never change their truth-value [Merricks] |
19201 | Propositions can be 'about' an entity, but that doesn't make the entity a constituent of it [Merricks] |
19211 | Early Russell says a proposition is identical with its truthmaking state of affairs [Merricks] |
19212 | Unity of the proposition questions: what unites them? can the same constituents make different ones? [Merricks] |
19213 | We want to explain not just what unites the constituents, but what unites them into a proposition [Merricks] |
19403 | Each of the infinite possible worlds has its own laws, and the individuals contain those laws [Leibniz] |