90 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] |
251 | Truth has the supreme value, for both gods and men [Plato] |
10073 | There cannot be a set theory which is complete [Smith,P] |
10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
10074 | A 'total function' maps every element to one element in another set [Smith,P] |
10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
10605 | Two functions are the same if they have the same extension [Smith,P] |
10075 | A 'partial function' maps only some elements to another set [Smith,P] |
10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
10613 | No nice theory can define truth for its own language [Smith,P] |
10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
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] |
263 | The only worthwhile life is one devoted to physical and moral perfection [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] |
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] |
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] |
273 | Movement is transmitted through everything, and it must have started with self-generated motion [Plato] |
17366 | Virtually all modern views of speciation rest on relational rather than intrinsic features [Okasha] |
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] |