78 ideas
| 5896 | Speak the truth, for this alone deifies man [Pythagoras, by Porphyry] |
| 3051 | Pythagoras discovered the numerical relation of sounds on a string [Pythagoras, by Diog. Laertius] |
| 21360 | Unobservant thinkers tend to dogmatise using insufficient facts [Aristotle] |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| 7485 | For Pythagoreans 'one' is not a number, but the foundation of numbers [Pythagoras, by Watson] |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| 13212 | Infinity is only potential, never actual [Aristotle] |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| 13221 | Existence is either potential or actual [Aristotle] |
| 16100 | True change is in a thing's logos or its matter, not in its qualities [Aristotle] |
| 16101 | A change in qualities is mere alteration, not true change [Aristotle] |
| 12133 | If the substratum persists, it is 'alteration'; if it doesn't, it is 'coming-to-be' or 'passing-away' [Aristotle] |
| 13213 | All comings-to-be are passings-away, and vice versa [Aristotle] |
| 12134 | Matter is the substratum, which supports both coming-to-be and alteration [Aristotle] |
| 16572 | Does the pure 'this' come to be, or the 'this-such', or 'so-great', or 'somewhere'? [Aristotle] |
| 16573 | Philosophers have worried about coming-to-be from nothing pre-existing [Aristotle] |
| 13214 | The substratum changing to a contrary is the material cause of coming-to-be [Aristotle] |
| 13215 | If a perceptible substratum persists, it is 'alteration'; coming-to-be is a complete change [Aristotle] |
| 16717 | Which of the contrary features of a body are basic to it? [Aristotle] |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| 3053 | Pythagoras taught that virtue is harmony, and health, and universal good, and God [Pythagoras, by Diog. Laertius] |
| 5244 | For Pythagoreans, justice is simply treating all people the same [Pythagoras, by Aristotle] |
| 638 | Pythagoreans define timeliness, justice and marriage in terms of numbers [Pythagoras, by Aristotle] |
| 553 | Pythagoreans think mathematical principles are the principles of all of nature [Pythagoras, by Aristotle] |
| 375 | When musical harmony and rhythm were discovered, similar features were seen in bodily movement [Pythagoras, by Plato] |
| 554 | Pythagoreans say things imitate numbers, but Plato says things participate in numbers [Pythagoras, by Aristotle] |
| 644 | For Pythagoreans the entire universe is made of numbers [Pythagoras, by Aristotle] |
| 13216 | Matter is the limit of points and lines, and must always have quality and form [Aristotle] |
| 17994 | The primary matter is the substratum for the contraries like hot and cold [Aristotle] |
| 13224 | There couldn't be just one element, which was both water and air at the same time [Aristotle] |
| 16594 | The Four Elements must change into one another, or else alteration is impossible [Aristotle] |
| 13223 | Fire is hot and dry; Air is hot and moist; Water is cold and moist; Earth is cold and dry [Aristotle] |
| 13220 | Bodies are endlessly divisible [Aristotle] |
| 13210 | Wood is potentially divided through and through, so what is there in the wood besides the division? [Aristotle] |
| 13211 | If a body is endlessly divided, is it reduced to nothing - then reassembled from nothing? [Aristotle] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 13228 | There is no time without movement [Aristotle] |
| 16595 | If each thing can cease to be, why hasn't absolutely everything ceased to be long ago? [Aristotle] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| 13227 | Being is better than not-being [Aristotle] |
| 13226 | An Order controls all things [Aristotle] |
| 7467 | The modern idea of an immortal soul was largely created by Pythagoras [Pythagoras, by Watson] |