57 ideas
12667 | Metaphysics aims at the simplest explanation, without regard to testability [Ellis] |
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] |
12666 | We can base logic on acceptability, and abandon the Fregean account by truth-preservation [Ellis] |
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] |
12688 | Mathematics is the formal study of the categorical dimensions of things [Ellis] |
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] |
12683 | Objects and substances are a subcategory of the natural kinds of processes [Ellis] |
12670 | A physical event is any change of distribution of energy [Ellis] |
12673 | Physical properties are those relevant to how a physical system might act [Ellis] |
12665 | I support categorical properties, although most people only want causal powers [Ellis] |
12682 | Essentialism needs categorical properties (spatiotemporal and numerical relations) and dispositions [Ellis] |
12684 | Spatial, temporal and numerical relations have causal roles, without being causal [Ellis] |
12672 | Properties and relations are discovered, so they can't be mere sets of individuals [Ellis] |
12676 | Causal powers can't rest on things which lack causal power [Ellis] |
23781 | Categoricals exist to influence powers. Such as structures, orientations and magnitudes [Ellis, by Williams,NE] |
12686 | Causal powers are a proper subset of the dispositional properties [Ellis] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
12685 | Categorical properties depend only on the structures they represent [Ellis] |
12679 | A real essence is a kind's distinctive properties [Ellis] |
12668 | Metaphysical necessity holds between things in the world and things they make true [Ellis] |
12687 | Metaphysical necessities are those depending on the essential nature of things [Ellis] |
7413 | Without confidence in our beliefs, how should we actually live? [Tuck] |
12669 | Science aims to explain things, not just describe them [Ellis] |
12681 | There are natural kinds of processes [Ellis] |
12680 | Natural kind structures go right down to the bottom level [Ellis] |
12675 | Laws of nature are just descriptions of how things are disposed to behave [Ellis] |
12671 | I deny forces as entities that intervene in causation, but are not themselves causal [Ellis] |
12674 | Energy is the key multi-valued property, vital to scientific realism [Ellis] |
12689 | Simultaneity can be temporal equidistance from the Big Bang [Ellis] |
12690 | The present is the collapse of the light wavefront from the Big Bang [Ellis] |