53 ideas
| 17774 | Definitions make our intuitions mathematically useful [Mayberry] |
| 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] |
| 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] |
| 4938 | Prior to language, concepts are universals created by self-mapping of brain activity [Edelman/Tononi] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 4934 | Cultures have a common core of colour naming, based on three axes of colour pairs [Edelman/Tononi] |
| 4924 | A conscious human being rapidly reunifies its mind after any damage to the brain [Edelman/Tononi] |
| 4932 | A conscious state endures for about 100 milliseconds, known as the 'specious present' [Edelman/Tononi] |
| 4931 | Consciousness is a process (of neural interactions), not a location, thing, property, connectivity, or activity [Edelman/Tononi] |
| 4923 | The three essentials of conscious experience are privateness, unity and informativeness [Edelman/Tononi] |
| 4941 | Consciousness can create new axioms, but computers can't do that [Edelman/Tononi] |
| 4930 | Consciousness arises from high speed interactions between clusters of neurons [Edelman/Tononi] |
| 4929 | Dreams and imagery show the brain can generate awareness and meaning without input [Edelman/Tononi] |
| 4940 | Physicists see information as a measure of order, but for biologists it is symbolic exchange between animals [Edelman/Tononi] |
| 4935 | The sensation of red is a point in neural space created by dimensions of neuronal activity [Edelman/Tononi] |
| 4936 | The self is founded on bodily awareness centred in the brain stem [Edelman/Tononi] |
| 4939 | A sense of self begins either internally, or externally through language and society [Edelman/Tononi] |
| 4925 | Brains can initiate free actions before the person is aware of their own decision [Edelman/Tononi] |
| 4933 | Consciousness is a process, not a thing, as it maintains unity as its composition changes [Edelman/Tononi] |
| 4928 | Brain complexity balances segregation and integration, like a good team of specialists [Edelman/Tononi] |
| 4927 | Information-processing views of the brain assume the existence of 'information', and dubious brain codes [Edelman/Tononi] |
| 4922 | Consciousness involves interaction with persons and the world, as well as brain functions [Edelman/Tononi] |
| 5793 | Concepts and generalisations result from brain 'global mapping' by 'reentry' [Edelman/Tononi, by Searle] |
| 4926 | Concepts arise when the brain maps its own activities [Edelman/Tononi] |
| 4937 | Systems that generate a sense of value are basic to the primitive brain [Edelman/Tononi] |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |