77 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] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 19597 | Logic (the theory of relations) should be applied to mathematics [Novalis] |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| 10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
| 10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
| 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| 19581 | A problem is a solid mass, which the mind must break up [Novalis] |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| 19584 | Whoever first counted to two must have seen the possibility of infinite counting [Novalis] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
| 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
| 10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 22025 | Novalis thought self-consciousness cannot disclose 'being', because we are temporal creatures [Novalis, by Pinkard] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 12887 | A whole must have one characteristic, an internal relation, and a structure [Rescher/Oppenheim] |
| 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] |
| 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] |