49 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] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 5673 | If we have a pain, we are strongly aware of the bodily self [Cassam] |
| 5670 | Knowledge of thoughts covers both their existence and their contents [Cassam] |
| 5671 | Outer senses are as important as introspection in the acquisition of self-knowledge [Cassam] |
| 5672 | Is there a mode of self-awareness that isn't perception, and could it give self-knowledge? [Cassam] |
| 5675 | Neither self-consciousness nor self-reference require self-knowledge [Cassam] |
| 5674 | We can't introspect ourselves as objects, because that would involve possible error [Cassam] |
| 7352 | Jesus said learning was unnecessary, and only the spirit of the Law was needed [Jesus, by Johnson,P] |
| 6289 | Love your enemies [Jesus] |
| 6292 | Love thy neighbour as thyself [Jesus] |
| 5356 | Treat others as you would have them treat you [Jesus] |
| 6286 | Blessed are the merciful: for they shall obtain mercy [Jesus] |
| 6290 | Except ye become as little children, ye shall not enter heaven [Jesus] |
| 6287 | If you lust after a woman, you have committed adultery [Jesus] |
| 6285 | Blessed are the meek; for they shall inherit the earth [Jesus] |
| 6288 | Don't resist evil, but turn the other cheek [Jesus] |
| 6293 | It is almost impossible for the rich to go to heaven [Jesus] |
| 6291 | No one is good except God [Jesus] |
| 7351 | Jesus turned the ideas of Hillel into a theology reduced to its moral elements [Jesus, by Johnson,P] |