33 ideas
| 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] |
| 10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
| 10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
| 10860 | An ordinal number is defined by the set that comes before it [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] |
| 7508 | Good reductionism connects fields of knowledge, but doesn't replace one with another [Pinker] |
| 7510 | Connectionists say the mind is a general purpose learning device [Pinker] |
| 7513 | Is memory stored in protein sequences, neurons, synapses, or synapse-strengths? [Pinker] |
| 7509 | Roundworms live successfully with 302 neurons, so human freedom comes from our trillions [Pinker] |
| 7511 | Neural networks can generalise their training, e.g. truths about tigers apply mostly to lions [Pinker] |
| 7512 | There are five types of reasoning that seem beyond connectionist systems [Pinker, by PG] |
| 7876 | Even if we identify pain with neural events, we can't explain why those neurons cause that feeling [Levine, by Papineau] |
| 7877 | Only phenomenal states have an explanatory gap; water is fully explained by H2O [Levine, by Papineau] |
| 7878 | Materialism won't explain phenomenal properties, because the latter aren't seen in causal roles [Papineau on Levine] |
| 7505 | Many think that accepting human nature is to accept innumerable evils [Pinker] |
| 7516 | In 1828, the stuff of life was shown to be ordinary chemistry, not a magic gel [Pinker] |
| 7515 | All the evidence says evolution is cruel and wasteful, not intelligent [Pinker] |
| 7514 | Intelligent Design says that every unexplained phenomenon must be design, by default [Pinker] |