92 ideas
| 23183 | Different abilities are needed for living in an incomplete and undogmatic system [Nietzsche] |
| 23188 | Bad writers use shapeless floating splotches of concepts [Nietzsche] |
| 23212 | A text has many interpretations, but no 'correct' one [Nietzsche] |
| 6161 | Structuralism is neo-Kantian idealism, with language playing the role of categories of understanding [Rowlands] |
| 23199 | What is the search for truth if it isn't moral? [Nietzsche] |
| 23202 | Like all philosophers, I love truth [Nietzsche] |
| 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] |
| 23196 | Logic is a fiction, which invents the view that one thought causes another [Nietzsche] |
| 6163 | If bivalence is rejected, then excluded middle must also be rejected [Rowlands] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [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] |
| 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] |
| 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] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [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] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [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] |
| 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] |
| 23186 | Numbers enable us to manage the world - to the limits of counting [Nietzsche] |
| 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] |
| 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] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 23211 | Events are just interpretations of groups of appearances [Nietzsche] |
| 6155 | Supervenience is a one-way relation of dependence or determination between properties [Rowlands] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 6154 | It is argued that wholes possess modal and counterfactual properties that parts lack [Rowlands] |
| 6157 | Tokens are dated, concrete particulars; types are their general properties or kinds [Rowlands] |
| 23201 | The 'I' does not think; it is a construction of thinking, like other useful abstractions [Nietzsche] |
| 23207 | Appearance is the sole reality of things, to which all predicates refer [Nietzsche] |
| 6159 | Strong idealism is the sort of mess produced by a Cartesian separation of mind and world [Rowlands] |
| 23197 | Memory is essential, and is only possible by means of abbreviation signs [Nietzsche] |
| 23206 | Schematic minds think thoughts are truer if they slot into a scheme [Nietzsche] |
| 23209 | Each of our personal drives has its own perspective [Nietzsche] |
| 23184 | The mind is a simplifying apparatus [Nietzsche] |
| 6152 | Minds are rational, conscious, subjective, self-knowing, free, meaningful and self-aware [Rowlands] |
| 6173 | Content externalism implies that we do not have privileged access to our own minds [Rowlands] |
| 6174 | If someone is secretly transported to Twin Earth, others know their thoughts better than they do [Rowlands] |
| 23190 | Consciousness is our awareness of our own mental life [Nietzsche] |
| 23191 | Minds have an excluding drive to scare things off, and a selecting one to filter facts [Nietzsche] |
| 23213 | The greatest drive of life is to discharge strength, rather than preservation [Nietzsche] |
| 23210 | That all events are necessary does not mean they are compelled [Nietzsche] |
| 6158 | Supervenience of mental and physical properties often comes with token-identity of mental and physical particulars [Rowlands] |
| 6168 | The content of a thought is just the meaning of a sentence [Rowlands] |
| 23189 | Concepts are rough groups of simultaneous sensations [Nietzsche] |
| 23192 | Concepts don’t match one thing, but many things a little bit [Nietzsche] |
| 23187 | Whatever their origin, concepts survive by being useful [Nietzsche] |
| 23205 | Thought starts as ambiguity, in need of interpretation and narrowing [Nietzsche] |
| 6167 | Action is bodily movement caused by intentional states [Rowlands] |
| 23198 | Aesthetics can be more basic than morality, in our pleasure in certain patterns of experience [Nietzsche] |
| 6177 | Moral intuition seems unevenly distributed between people [Rowlands] |
| 23208 | Caesar and Napoleon point to the future, when they pursue their task regardless of human sacrifice [Nietzsche] |
| 23193 | Napoleon was very focused, and rightly ignored compassion [Nietzsche] |
| 23214 | For the strongest people, nihilism gives you wings! [Nietzsche] |
| 23203 | The great question is approaching, of how to govern the earth as a whole [Nietzsche] |
| 23200 | The controlling morality of aristocracy is the desire to resemble their ancestors [Nietzsche] |
| 23194 | People feel united as a nation by one language, but then want a common ancestry and history [Nietzsche] |
| 23204 | To be someone you need property, and wanting more is healthy [Nietzsche] |
| 6156 | The 17th century reintroduced atoms as mathematical modes of Euclidean space [Rowlands] |
| 6170 | Natural kinds are defined by their real essence, as in gold having atomic number 79 [Rowlands] |
| 23195 | Laws of nature are actually formulas of power relations [Nietzsche] |
| 23185 | In chemistry every substance pushes, and thus creates new substances [Nietzsche] |
| 6178 | It is common to see the value of nature in one feature, such as life, diversity, or integrity [Rowlands] |