64 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] |
| 458 | Nothing could come out of nothing, and existence could never completely cease [Empedocles] |
| 5112 | Empedocles says things are at rest, unless love unites them, or hatred splits them [Empedocles, by Aristotle] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 16435 | Plantinga proposes necessary existent essences as surrogates for the nonexistent things [Plantinga, by Stalnaker] |
| 14655 | The 'identity criteria' of a name are a group of essential and established facts [Plantinga] |
| 14658 | 'Being Socrates' and 'being identical with Socrates' characterise Socrates, so they are among his properties [Plantinga] |
| 13209 | There is no coming-to-be of anything, but only mixing and separating [Empedocles, by Aristotle] |
| 14656 | Does Socrates have essential properties, plus a unique essence (or 'haecceity') which entails them? [Plantinga] |
| 14654 | Properties are 'trivially essential' if they are instantiated by every object in every possible world [Plantinga] |
| 14653 | X is essentially P if it is P in every world, or in every X-world, or in the actual world (and not ¬P elsewhere) [Plantinga] |
| 14660 | If a property is ever essential, can it only ever be an essential property? [Plantinga] |
| 14661 | Essences are instantiated, and are what entails a thing's properties and lack of properties [Plantinga] |
| 457 | Substance is not created or destroyed in mortals, but there is only mixing and exchange [Empedocles] |
| 14657 | Does 'being identical with Socrates' name a property? I can think of no objections to it [Plantinga] |
| 14652 | 'De re' modality is as clear as 'de dicto' modality, because they are logically equivalent [Plantinga] |
| 14659 | We can imagine being beetles or alligators, so it is possible we might have such bodies [Plantinga] |
| 462 | One vision is produced by both eyes [Empedocles] |
| 22765 | Wisdom and thought are shared by all things [Empedocles] |
| 1524 | For Empedocles thinking is almost identical to perception [Empedocles, by Theophrastus] |
| 552 | Empedocles said good and evil were the basic principles [Empedocles, by Aristotle] |
| 589 | 'Nature' is just a word invented by people [Empedocles] |
| 21823 | The principle of 'Friendship' in Empedocles is the One, and is bodiless [Empedocles, by Plotinus] |
| 2680 | Empedocles said that there are four material elements, and two further creative elements [Empedocles, by Aristotle] |
| 6002 | Empedocles says bone is water, fire and earth in ratio 2:4:2 [Empedocles, by Inwood] |
| 13207 | Fire, Water, Air and Earth are elements, being simple as well as homoeomerous [Empedocles, by Aristotle] |
| 459 | All change is unity through love or division through hate [Empedocles] |
| 13218 | The elements combine in coming-to-be, but how do the elements themselves come-to-be? [Aristotle on Empedocles] |
| 13225 | Love and Strife only explain movement if their effects are distinctive [Aristotle on Empedocles] |
| 460 | If the one Being ever diminishes it would no longer exist, and what could ever increase it? [Empedocles] |
| 5090 | Maybe bodies are designed by accident, and the creatures that don't work are destroyed [Empedocles, by Aristotle] |
| 461 | God is a pure, solitary, and eternal sphere [Empedocles] |
| 466 | God is pure mind permeating the universe [Empedocles] |
| 1719 | In Empedocles' theory God is ignorant because, unlike humans, he doesn't know one of the elements (strife) [Aristotle on Empedocles] |
| 1522 | It is wretched not to want to think clearly about the gods [Empedocles] |