78 ideas
| 15447 | We shouldn't always follow where the argument leads! [Lewis on Plato] |
| 243 | It is foolish to quarrel with the mind's own reasoning processes [Plato] |
| 241 | We ought to follow where the argument leads us [Plato] |
| 21264 | Mortals are incapable of being fully rational [Plato] |
| 17774 | Definitions make our intuitions mathematically useful [Mayberry] |
| 17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
| 251 | Truth has the supreme value, for both gods and men [Plato] |
| 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] |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| 21259 | To grasp a thing we need its name, its definition, and what it really is [Plato] |
| 21260 | Soul is what is defined by 'self-generating motion' [Plato] |
| 276 | My individuality is my soul, which carries my body around [Plato] |
| 249 | People who value beauty above virtue insult the soul by placing the body above it [Plato] |
| 265 | An action is only just if it is performed by someone with a just character and outlook [Plato] |
| 269 | Attempted murder is like real murder, but we should respect the luck which avoided total ruin [Plato] |
| 240 | It would be strange if the gods rewarded those who experienced the most pleasure in life [Plato] |
| 264 | The conquest of pleasure is the noblest victory of all [Plato] |
| 253 | Every crime is the result of excessive self-love [Plato] |
| 4332 | Virtue is a concord of reason and emotion, with pleasure and pain trained to correct ends [Plato] |
| 248 | A serious desire for moral excellence is very rare indeed [Plato] |
| 263 | The only worthwhile life is one devoted to physical and moral perfection [Plato] |
| 235 | Virtue is the aim of all laws [Plato] |
| 277 | The Guardians must aim to discover the common element in the four cardinal virtues [Plato] |
| 254 | Excessive laughter and tears must be avoided [Plato] |
| 266 | Injustice is the mastery of the soul by bad feelings, even if they do not lead to harm [Plato] |
| 256 | Virtue and great wealth are incompatible [Plato] |
| 242 | The best people are produced where there is no excess of wealth or poverty [Plato] |
| 245 | Totalitarian states destroy friendships and community spirit [Plato] |
| 239 | Education in virtue produces citizens who are active but obedient [Plato] |
| 262 | Men and women should qualify equally for honours on merit [Plato] |
| 1402 | Friendship is impossible between master and slave, even if they are made equal [Plato] |
| 236 | Sound laws achieve the happiness of those who observe them [Plato] |
| 259 | Justice is granting the equality which unequals deserve [Plato] |
| 257 | Mathematics has the widest application of any subject on the curriculum [Plato] |
| 238 | Children's games should channel their pleasures into adult activity [Plato] |
| 260 | Control of education is the key office of state, and should go to the best citizen [Plato] |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| 4331 | Education is channelling a child's feelings into the right course before it understands why [Plato] |
| 250 | The best way to educate the young is not to rebuke them, but to set a good example [Plato] |
| 275 | Creation is not for you; you exist for the sake of creation [Plato] |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| 273 | Movement is transmitted through everything, and it must have started with self-generated motion [Plato] |
| 8004 | In 'The Laws', to obey the law is to be obey god [Plato, by MacIntyre] |
| 21257 | Self-generating motion is clearly superior to all other kinds of motion [Plato] |
| 21258 | The only possible beginning for the endless motions of reality is something self-generated [Plato] |
| 21261 | Self-moving soul has to be the oldest thing there is [Plato] |
| 274 | Soul must be the cause of all the opposites, such as good and evil or beauty and ugliness [Plato] |
| 21263 | If all the motions of nature reflect calculations of reason, then the best kind of soul must direct it [Plato] |
| 278 | If astronomical movements are seen as necessary instead of by divine will, this leads to atheism [Plato] |
| 21265 | The heavens must be full of gods, controlling nature either externally or from within [Plato] |
| 21262 | There must be at least two souls controlling the cosmos, one doing good, the other the opposite [Plato] |