60 ideas
5988 | Anaximander produced the first philosophy book (and maybe the first book) [Anaximander, by Bodnár] |
1496 | The earth is stationary, because it is in the centre, and has no more reason to move one way than another [Anaximander, by Aristotle] |
17774 | Definitions make our intuitions mathematically useful [Mayberry] |
12585 | Most people can't even define a chair [Peacocke] |
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] |
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] |
17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [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] |
14874 | Anaximander saw the contradiction in the world - that its own qualities destroy it [Anaximander, by Nietzsche] |
17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
12581 | Perceptual concepts causally influence the content of our experiences [Peacocke] |
12579 | Perception has proto-propositions, between immediate experience and concepts [Peacocke] |
12586 | Consciousness of a belief isn't a belief that one has it [Peacocke] |
12608 | Concepts are distinguished by roles in judgement, and are thus tied to rationality [Peacocke] |
18568 | Philosophy should merely give necessary and sufficient conditions for concept possession [Peacocke, by Machery] |
18571 | Peacocke's account of possession of a concept depends on one view of counterfactuals [Peacocke, by Machery] |
18572 | Peacocke's account separates psychology from philosophy, and is very sketchy [Machery on Peacocke] |
17722 | The concept 'red' is tied to what actually individuates red things [Peacocke] |
11127 | If concepts just are mental representations, what of concepts we may never acquire? [Peacocke] |
12577 | Possessing a concept is being able to make judgements which use it [Peacocke] |
12578 | A concept is just what it is to possess that concept [Peacocke] |
12587 | Employing a concept isn't decided by introspection, but by making judgements using it [Peacocke] |
12605 | A sense is individuated by the conditions for reference [Peacocke] |
12607 | Fregean concepts have their essence fixed by reference-conditions [Peacocke] |
12609 | Concepts have distinctive reasons and norms [Peacocke] |
12584 | An analysis of concepts must link them to something unconceptualized [Peacocke] |
12604 | Any explanation of a concept must involve reference and truth [Peacocke] |
9335 | Concepts are constituted by their role in a group of propositions to which we are committed [Peacocke, by Greco] |
9336 | A concept's reference is what makes true the beliefs of its possession conditions [Peacocke, by Horwich] |
12610 | Encountering novel sentences shows conclusively that meaning must be compositional [Peacocke] |
13222 | The Boundless cannot exist on its own, and must have something contrary to it [Aristotle on Anaximander] |
1495 | Anaximander introduced the idea that the first principle and element of things was the Boundless [Anaximander, by Simplicius] |
404 | Things begin and end in the Unlimited, and are balanced over time according to justice [Anaximander] |
405 | The essential nature, whatever it is, of the non-limited is everlasting and ageless [Anaximander] |
1746 | The parts of all things are susceptible to change, but the whole is unchangeable [Anaximander, by Diog. Laertius] |