41 ideas
| 21463 | Hamann, Herder and Jacobi were key opponents of the Enlightenment [Gardner] |
| 21459 | Kant halted rationalism, and forced empiricists to worry about foundations [Gardner] |
| 21460 | Only Kant and Hegel have united nature, morals, politics, aesthetics and religion [Gardner] |
| 21443 | Transcendental proofs derive necessities from possibilities (e.g. possibility of experiencing objects) [Gardner] |
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
| 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
| 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
| 13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
| 13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
| 13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
| 13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
| 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
| 13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| 13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
| 13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
| 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
| 21444 | Modern geoemtry is either 'pure' (and formal), or 'applied' (and a posteriori) [Gardner] |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 21453 | Leibnizian monads qualify as Kantian noumena [Gardner] |
| 7352 | Jesus said learning was unnecessary, and only the spirit of the Law was needed [Jesus, by Johnson,P] |
| 8108 | Aesthetics presupposes a distinctive sort of experience, and a unified essence for art [Gardner] |
| 8112 | Art works originate in the artist's mind, and appreciation is re-creating this mental object [Gardner] |
| 8111 | Aesthetic objectivists must explain pleasure being essential, but not in the object [Gardner] |
| 8109 | Aesthetic judgements necessarily require first-hand experience, unlike moral judgements [Gardner] |
| 6289 | Love your enemies [Jesus] |
| 6292 | Love thy neighbour as thyself [Jesus] |
| 5356 | Treat others as you would have them treat you [Jesus] |
| 6286 | Blessed are the merciful: for they shall obtain mercy [Jesus] |
| 6290 | Except ye become as little children, ye shall not enter heaven [Jesus] |
| 6287 | If you lust after a woman, you have committed adultery [Jesus] |
| 6285 | Blessed are the meek; for they shall inherit the earth [Jesus] |
| 6288 | Don't resist evil, but turn the other cheek [Jesus] |
| 6293 | It is almost impossible for the rich to go to heaven [Jesus] |
| 6291 | No one is good except God [Jesus] |
| 7351 | Jesus turned the ideas of Hillel into a theology reduced to its moral elements [Jesus, by Johnson,P] |