50 ideas
| 7490 | Because of Darwin, wisdom as a definite attainable state has faded [Watson] |
| 7461 | The three key ideas are the soul, Europe, and the experiment [Watson] |
| 7464 | The big idea: imitation, the soul, experiments, God, heliocentric universe, evolution? [Watson] |
| 7465 | Babylonian thinking used analogy, rather than deduction or induction [Watson] |
| 21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
| 17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
| 17886 | The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner] |
| 10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
| 19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
| 10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
| 17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
| 10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
| 17453 | The meaning of a number isn't just the numerals leading up to it [Heck] |
| 17457 | A basic grasp of cardinal numbers needs an understanding of equinumerosity [Heck] |
| 17448 | In counting, numerals are used, not mentioned (as objects that have to correlated) [Heck] |
| 17455 | Is counting basically mindless, and independent of the cardinality involved? [Heck] |
| 17456 | Counting is the assignment of successively larger cardinal numbers to collections [Heck] |
| 7466 | Mesopotamian numbers applied to specific things, and then became abstract [Watson] |
| 17450 | Understanding 'just as many' needn't involve grasping one-one correspondence [Heck] |
| 17451 | We can know 'just as many' without the concepts of equinumerosity or numbers [Heck] |
| 17459 | Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck] |
| 3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
| 10072 | First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P] |
| 9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman] |
| 11069 | Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna] |
| 10118 | First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman] |
| 10122 | Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman] |
| 10611 | There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P] |
| 10867 | 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg] |
| 17454 | Children can use numbers, without a concept of them as countable objects [Heck] |
| 17458 | Equinumerosity is not the same concept as one-one correspondence [Heck] |
| 17449 | We can understand cardinality without the idea of one-one correspondence [Heck] |
| 8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
| 3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
| 7477 | Modern democracy is actually elective oligarchy [Watson] |
| 7478 | Greek philosophers invented the concept of 'nature' as their special subject [Watson] |
| 7462 | DNA mutation suggests humans and chimpanzees diverged 6.6 million years ago [Watson] |
| 7470 | During the rise of civilizations, the main gods changed from female to male [Watson] |
| 7474 | Hinduism has no founder, or prophet, or creed, or ecclesiastical structure [Watson] |
| 7479 | Modern Judaism became stabilised in 200 CE [Watson] |
| 7481 | The Israelites may have asserted the uniqueness of Yahweh to justify land claims [Watson] |
| 7480 | Monotheism was a uniquely Israelite creation within the Middle East [Watson] |
| 7471 | The Gathas (hymns) of Zoroastrianism date from about 1000 BCE [Watson] |
| 7473 | Zoroaster conceived the afterlife, judgement, heaven and hell, and the devil [Watson] |
| 7484 | Jesus never intended to start a new religion [Watson] |
| 7483 | Paul's early writings mention few striking episodes from Jesus' life [Watson] |
| 7475 | Confucius revered the spiritual world, but not the supernatural, or a personal god, or the afterlife [Watson] |
| 7476 | Taoism aims at freedom from the world, the body, the mind, and nature [Watson] |
| 7463 | The three basic ingredients of religion are: the soul, seers or priests, and ritual [Watson] |
| 7468 | In ancient Athens the souls of the dead are received by the 'upper air' [Watson] |