96 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] |
| 9641 | Definitions should be replaceable by primitives, and should not be creative [Brown,JR] |
| 251 | Truth has the supreme value, for both gods and men [Plato] |
| 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
| 9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
| 10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
| 9613 | Naïve set theory assumed that there is a set for every condition [Brown,JR] |
| 9615 | Nowadays conditions are only defined on existing sets [Brown,JR] |
| 9617 | The 'iterative' view says sets start with the empty set and build up [Brown,JR] |
| 9642 | A flock of birds is not a set, because a set cannot go anywhere [Brown,JR] |
| 9605 | If a proposition is false, then its negation is true [Brown,JR] |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| 9649 | Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR] |
| 9638 | Berry's Paradox finds a contradiction in the naming of huge numbers [Brown,JR] |
| 9604 | Mathematics is the only place where we are sure we are right [Brown,JR] |
| 9622 | 'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR] |
| 10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
| 9648 | π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR] |
| 9621 | Mathematics represents the world through structurally similar models. [Brown,JR] |
| 9646 | There is no limit to how many ways something can be proved in mathematics [Brown,JR] |
| 9647 | Computers played an essential role in proving the four-colour theorem of maps [Brown,JR] |
| 10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| 10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
| 9643 | Set theory may represent all of mathematics, without actually being mathematics [Brown,JR] |
| 9644 | When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR] |
| 10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
| 10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
| 10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
| 10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
| 10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
| 10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
| 10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
| 9625 | To see a structure in something, we must already have the idea of the structure [Brown,JR] |
| 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
| 10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
| 9628 | Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR] |
| 10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
| 9606 | The irrationality of root-2 was achieved by intellect, not experience [Brown,JR] |
| 9612 | There is an infinity of mathematical objects, so they can't be physical [Brown,JR] |
| 9610 | Numbers are not abstracted from particulars, because each number is a particular [Brown,JR] |
| 9620 | Empiricists base numbers on objects, Platonists base them on properties [Brown,JR] |
| 9639 | Does some mathematics depend entirely on notation? [Brown,JR] |
| 9629 | For nomalists there are no numbers, only numerals [Brown,JR] |
| 9630 | The most brilliant formalist was Hilbert [Brown,JR] |
| 9608 | There are no constructions for many highly desirable results in mathematics [Brown,JR] |
| 9645 | Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR] |
| 9619 | David's 'Napoleon' is about something concrete and something abstract [Brown,JR] |
| 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
| 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] |
| 9611 | 'Abstract' nowadays means outside space and time, not concrete, not physical [Brown,JR] |
| 9609 | The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars [Brown,JR] |
| 9640 | A term can have not only a sense and a reference, but also a 'computational role' [Brown,JR] |
| 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] |
| 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] |
| 253 | Every crime is the result of excessive self-love [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] |
| 242 | The best people are produced where there is no excess of wealth or poverty [Plato] |
| 256 | Virtue and great wealth are incompatible [Plato] |
| 245 | Totalitarian states destroy friendships and community spirit [Plato] |
| 239 | Education in virtue produces citizens who are active but obedient [Plato] |
| 1402 | Friendship is impossible between master and slave, even if they are made equal [Plato] |
| 262 | Men and women should qualify equally for honours on merit [Plato] |
| 236 | Sound laws achieve the happiness of those who observe them [Plato] |
| 259 | Justice is granting the equality which unequals deserve [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] |
| 257 | Mathematics has the widest application of any subject on the curriculum [Plato] |
| 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] |
| 9635 | Given atomism at one end, and a finite universe at the other, there are no physical infinities [Brown,JR] |
| 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] |
| 21261 | Self-moving soul has to be the oldest thing there is [Plato] |
| 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] |
| 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] |