52 ideas
19579 | The history of philosophy is just experiments in how to do philosophy [Novalis] |
19583 | Philosophy only begins when it studies itself [Novalis] |
22026 | Philosophy is homesickness - the urge to be at home everywhere [Novalis] |
19588 | The highest aim of philosophy is to combine all philosophies into a unity [Novalis] |
19598 | Philosophy relies on our whole system of learning, and can thus never be complete [Novalis] |
19586 | Philosophers feed on problems, hoping they are digestible, and spiced with paradox [Novalis] |
19587 | Philosophy aims to produce a priori an absolute and artistic world system [Novalis] |
19574 | If man sacrifices truth he sacrifices himself, by acting against his own convictions [Novalis] |
19571 | Delusion and truth differ in their life functions [Novalis] |
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] |
19597 | Logic (the theory of relations) should be applied to mathematics [Novalis] |
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] |
19581 | A problem is a solid mass, which the mind must break up [Novalis] |
13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
19584 | Whoever first counted to two must have seen the possibility of infinite counting [Novalis] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
22025 | Novalis thought self-consciousness cannot disclose 'being', because we are temporal creatures [Novalis, by Pinkard] |
19575 | Refinement of senses increasingly distinguishes individuals [Novalis] |
22067 | Poetry is true idealism, and the self-consciousness of the universe [Novalis] |
19572 | Experiences tests reason, and reason tests experience [Novalis] |
19590 | Empiricists are passive thinkers, given their philosophy by the external world and fate [Novalis] |
19594 | General statements about nature are not valid [Novalis] |
19591 | Desire for perfection is an illness, if it turns against what is imperfect [Novalis] |
19596 | The whole body is involved in the formation of thoughts [Novalis] |
19573 | The seat of the soul is where our inner and outer worlds interpenetrate [Novalis] |
19577 | Everything is a chaotic unity, then we abstract, then we reunify the world into a free alliance [Novalis] |
19585 | Every person has his own language [Novalis] |
19578 | Only self-illuminated perfect individuals are beautiful [Novalis] |
19582 | Morality and philosophy are mutually dependent [Novalis] |
22027 | Life isn't given to us like a novel - we write the novel [Novalis] |
19589 | The whole point of a monarch is that we accept them as a higher-born, ideal person [Novalis] |
21131 | Democracy is competition for support of the people, guided by self-interest on all sides [Posner] |
19580 | If the pupil really yearns for the truth, they only need a hint [Novalis] |
19593 | Persons are shaped by a life history; splendid persons are shaped by world history [Novalis] |
19595 | Nature is a whole, and its individual parts cannot be wholly understood [Novalis] |
19592 | The basic relations of nature are musical [Novalis] |
19576 | Religion needs an intermediary, because none of us can connect directly to a godhead [Novalis] |