107 ideas
17016 | Philosophy must abstract from the senses [Newton] |
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] |
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] |
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] |
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] |
18079 | Newton developed a kinematic approach to geometry [Newton, by Kitcher] |
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] |
18082 | Quantities and ratios which continually converge will eventually become equal [Newton] |
10618 | All numbers are related to zero by the ancestral of the successor relation [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] |
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] |
458 | Nothing could come out of nothing, and existence could never completely cease [Empedocles] |
5112 | Empedocles says things are at rest, unless love unites them, or hatred splits them [Empedocles, by Aristotle] |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
17011 | I suspect that each particle of bodies has attractive or repelling forces [Newton] |
13209 | There is no coming-to-be of anything, but only mixing and separating [Empedocles, by Aristotle] |
17028 | Particles mutually attract, and cohere at short distances [Newton] |
17014 | The place of a thing is the sum of the places of its parts [Newton] |
457 | Substance is not created or destroyed in mortals, but there is only mixing and exchange [Empedocles] |
462 | One vision is produced by both eyes [Empedocles] |
17546 | If you changed one of Newton's concepts you would destroy his whole system [Heisenberg on Newton] |
17027 | Science deduces propositions from phenomena, and generalises them by induction [Newton] |
17022 | We should admit only enough causes to explain a phenomenon, and no more [Newton] |
17021 | Natural effects of the same kind should be assumed to have the same causes [Newton] |
17026 | From the phenomena, I can't deduce the reason for the properties of gravity [Newton] |
22765 | Wisdom and thought are shared by all things [Empedocles] |
1524 | For Empedocles thinking is almost identical to perception [Empedocles, by Theophrastus] |
552 | Empedocles said good and evil were the basic principles [Empedocles, by Aristotle] |
589 | 'Nature' is just a word invented by people [Empedocles] |
6421 | Newton's four fundamentals are: space, time, matter and force [Newton, by Russell] |
21823 | The principle of 'Friendship' in Empedocles is the One, and is bodiless [Empedocles, by Plotinus] |
2680 | Empedocles said that there are four material elements, and two further creative elements [Empedocles, by Aristotle] |
6002 | Empedocles says bone is water, fire and earth in ratio 2:4:2 [Empedocles, by Inwood] |
13207 | Fire, Water, Air and Earth are elements, being simple as well as homoeomerous [Empedocles, by Aristotle] |
459 | All change is unity through love or division through hate [Empedocles] |
13218 | The elements combine in coming-to-be, but how do the elements themselves come-to-be? [Aristotle on Empedocles] |
13225 | Love and Strife only explain movement if their effects are distinctive [Aristotle on Empedocles] |
460 | If the one Being ever diminishes it would no longer exist, and what could ever increase it? [Empedocles] |
13470 | Mass is central to matter [Newton, by Hart,WD] |
17020 | An attraction of a body is the sum of the forces of their particles [Newton] |
23012 | Newtonian causation is changes of motion resulting from collisions [Newton, by Baron/Miller] |
17008 | You have discovered that elliptical orbits result just from gravitation and planetary movement [Newton, by Leibniz] |
17010 | We have given up substantial forms, and now aim for mathematical laws [Newton] |
17023 | I am not saying gravity is essential to bodies [Newton] |
15866 | Newton reclassified vertical motion as violent, and unconstrained horizontal motion as natural [Newton, by Harré] |
15958 | Inertia rejects the Aristotelian idea of things having natural states, to which they return [Newton, by Alexander,P] |
17017 | 1: Bodies rest, or move in straight lines, unless acted on by forces [Newton] |
17018 | 2: Change of motion is proportional to the force [Newton] |
20968 | Newton's Third Law implies the conservation of momentum [Newton, by Papineau] |
17019 | 3: All actions of bodies have an equal and opposite reaction [Newton] |
17547 | Newton's idea of force acting over a long distance was very strange [Heisenberg on Newton] |
20966 | Newton introduced forces other than by contact [Newton, by Papineau] |
20967 | Newton's laws cover the effects of forces, but not their causes [Newton, by Papineau] |
16708 | Newton's forces were accused of being the scholastics' real qualities [Pasnau on Newton] |
13153 | I am studying the quantities and mathematics of forces, not their species or qualities [Newton] |
12724 | The aim is to discover forces from motions, and use forces to demonstrate other phenomena [Newton] |
13593 | Newton showed that falling to earth and orbiting the sun are essentially the same [Newton, by Ellis] |
20969 | Early Newtonians could not formulate conservation of energy, having no concept of potential energy [Newton, by Papineau] |
17013 | Absolute space is independent, homogeneous and immovable [Newton] |
22915 | Newton needs intervals of time, to define velocity and acceleration [Newton, by Le Poidevin] |
22893 | Newton thought his laws of motion needed absolute time [Newton, by Bardon] |
17012 | Time exists independently, and flows uniformly [Newton] |
14012 | Absolute time, from its own nature, flows equably, without relation to anything external [Newton] |
22954 | Newtonian mechanics does not distinguish negative from positive values of time [Newton, by Coveney/Highfield] |
17015 | If there is no uniform motion, we cannot exactly measure time [Newton] |
5090 | Maybe bodies are designed by accident, and the creatures that don't work are destroyed [Empedocles, by Aristotle] |
466 | God is pure mind permeating the universe [Empedocles] |
461 | God is a pure, solitary, and eternal sphere [Empedocles] |
17025 | If a perfect being does not rule the cosmos, it is not God [Newton] |
1719 | In Empedocles' theory God is ignorant because, unlike humans, he doesn't know one of the elements (strife) [Aristotle on Empedocles] |
17024 | The elegance of the solar system requires a powerful intellect as designer [Newton] |
1522 | It is wretched not to want to think clearly about the gods [Empedocles] |