63 ideas
13786 | Wisdom is called 'beautiful', because it performs fine works [Plato] |
13780 | Good people are no different from wise ones [Plato] |
17651 | Without words or other symbols, we have no world [Goodman] |
13778 | A dialectician is someone who knows how to ask and to answer questions [Plato] |
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
17652 | Truth is irrelevant if no statements are involved [Goodman] |
13776 | Truths say of what is that it is, falsehoods say of what is that it is not [Plato] |
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] |
13777 | A name is a sort of tool [Plato] |
13790 | A name-giver might misname something, then force other names to conform to it [Plato] |
13791 | Things must be known before they are named, so it can't be the names that give us knowledge [Plato] |
13789 | Anyone who knows a thing's name also knows the thing [Plato] |
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] |
2063 | How can beauty have identity if it changes? [Plato] |
17656 | Being primitive or prior always depends on a constructional system [Goodman] |
17661 | We don't recognise patterns - we invent them [Goodman] |
17659 | Reality is largely a matter of habit [Goodman] |
17657 | We build our world, and ignore anything that won't fit [Goodman] |
13775 | We only succeed in cutting if we use appropriate tools, not if we approach it randomly [Plato] |
17654 | A world can be full of variety or not, depending on how we sort it [Goodman] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
13787 | Doesn't each thing have an essence, just as it has other qualities? [Plato] |
13774 | Things don't have every attribute, and essence isn't private, so each thing has an essence [Plato] |
13772 | Is the being or essence of each thing private to each person? [Plato] |
17653 | Things can only be judged the 'same' by citing some respect of sameness [Goodman] |
13788 | If we made a perfect duplicate of Cratylus, there would be two Cratyluses [Plato] |
17660 | Discovery is often just finding a fit, like a jigsaw puzzle [Goodman] |
13792 | There can't be any knowledge if things are constantly changing [Plato] |
17658 | Users of digital thermometers recognise no temperatures in the gaps [Goodman] |
17650 | We lack frames of reference to transform physics, biology and psychology into one another [Goodman] |
17655 | Grue and green won't be in the same world, as that would block induction entirely [Goodman] |
13781 | Soul causes the body to live, and gives it power to breathe and to be revitalized [Plato] |
13785 | 'Arete' signifies lack of complexity and a free-flowing soul [Plato] |
17649 | If the world is one it has many aspects, and if there are many worlds they will collect into one [Goodman] |
13779 | The natural offspring of a lion is called a 'lion' (but what about the offspring of a king?) [Plato] |
13783 | Even the gods love play [Plato] |