35 ideas
| 1642 | We must fight fiercely for knowledge, understanding and intelligence [Plato] |
| 1645 | The desire to split everything into its parts is unpleasant and unphilosophical [Plato] |
| 287 | Good analysis involves dividing things into appropriate forms without confusion [Plato] |
| 1644 | Dialectic should only be taught to those who already philosophise well [Plato] |
| 20478 | In discussion a person's opinions are shown to be in conflict, leading to calm self-criticism [Plato] |
| 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] |
| 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] |
| 8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
| 11278 | What does 'that which is not' refer to? [Plato] |
| 1643 | If statements about non-existence are logically puzzling, so are statements about existence [Plato] |
| 7022 | To be is to have a capacity, to act on other things, or to receive actions [Plato] |
| 1641 | Some alarming thinkers think that only things which you can touch exist [Plato] |
| 10784 | Whenever there's speech it has to be about something [Plato] |
| 16122 | Good thinkers spot forms spread through things, or included within some larger form [Plato] |
| 10422 | The not-beautiful is part of the beautiful, though opposed to it, and is just as real [Plato] |
| 15855 | If we see everything as separate, we can then give no account of it [Plato] |
| 12716 | The concept of forces or powers best reveals the true concept of substance [Leibniz] |
| 1637 | A soul without understanding is ugly [Plato] |
| 3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
| 1636 | Wickedness is an illness of the soul [Plato] |
| 1638 | Didactic education is hard work and achieves little [Plato] |