39 ideas
14052 | Begin philosophy when you are young, and keep going when you are old [Epicurus] |
18261 | A simplification which is complete constitutes a definition [Kant] |
10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
10857 | Set theory made a closer study of infinity possible [Clegg] |
10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
10871 | Axiom of Existence: there exists at least one set [Clegg] |
10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
22275 | Logic gives us the necessary rules which show us how we ought to think [Kant] |
10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
10860 | An ordinal number is defined by the set that comes before it [Clegg] |
10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
18260 | If we knew what we know, we would be astonished [Kant] |
14062 | Sooner follow mythology, than accept the 'fate' of natural philosophers [Epicurus] |
1837 | We should not refer things to irresponsible necessity, but either to fortune or to our own will [Epicurus] |
1836 | Prudence is more valuable than philosophy, because it avoids confusions of the soul [Epicurus] |
14061 | Our own choices are autonomous, and the basis for praise and blame [Epicurus] |
14053 | It is absurd to fear the pain of death when you are not even facing it [Epicurus] |
14055 | The wisdom that produces a good life also produces a good death [Epicurus] |
14054 | Fearing death is absurd, because we are not present when it occurs [Epicurus] |
1833 | Pleasure is the first good in life [Epicurus] |
14058 | Pleasure is the goal, but as lack of pain and calm mind, not as depraved or greedy pleasure [Epicurus] |
14057 | All pleasures are good, but it is not always right to choose them [Epicurus] |
14063 | Sooner a good decision going wrong, than a bad one turning out for the good [Epicurus] |
14059 | The best life is not sensuality, but rational choice and healthy opinion [Epicurus] |
1835 | True pleasure is not debauchery, but freedom from physical and mental pain [Epicurus] |
14056 | We only need pleasure when we have the pain of desire [Epicurus] |
14060 | Prudence is the greatest good, and more valuable than philosophy, because it produces virtue [Epicurus] |