109 ideas
9199 | Wisdom for one instant is as good as wisdom for eternity [Chrysippus] |
20853 | Wise men should try to participate in politics, since they are a good influence [Chrysippus, by Diog. Laertius] |
20772 | Three branches of philosophy: first logic, second ethics, third physics (which ends with theology) [Chrysippus] |
5969 | Chrysippus said the uncaused is non-existent [Chrysippus, by Plutarch] |
18369 | There are at least fourteen candidates for truth-bearers [Kirkham] |
21388 | The causes of future true events must exist now, so they will happen because of destiny [Chrysippus, by Cicero] |
20780 | Graspable presentations are criteria of facts, and are molded according to their objects [Chrysippus, by Diog. Laertius] |
20793 | How could you ever know that the presentation is similar to the object? [Sext.Empiricus on Chrysippus] |
19318 | A 'sequence' of objects is an order set of them [Kirkham] |
19319 | If one sequence satisfies a sentence, they all do [Kirkham] |
19320 | If we define truth by listing the satisfactions, the supply of predicates must be finite [Kirkham] |
8077 | Stoic propositional logic is like chemistry - how atoms make molecules, not the innards of atoms [Chrysippus, by Devlin] |
20791 | Chrysippus has five obvious 'indemonstrables' of reasoning [Chrysippus, by Diog. Laertius] |
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] |
19315 | In quantified language the components of complex sentences may not be sentences [Kirkham] |
8078 | Modus ponens is one of five inference rules identified by the Stoics [Chrysippus, by Devlin] |
6023 | Every proposition is either true or false [Chrysippus, by Cicero] |
19317 | An open sentence is satisfied if the object possess that property [Kirkham] |
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] |
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] |
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] |
5992 | Chrysippus says action is the criterion for existence, which must be physical [Chrysippus, by Tieleman] |
21673 | There are simple and complex facts; the latter depend on further facts [Chrysippus, by Cicero] |
19322 | Why can there not be disjunctive, conditional and negative facts? [Kirkham] |
16652 | Stoics categories are Substrate, Quality, Disposition, and Relation [Chrysippus, by Pasnau] |
16058 | Dion and Theon coexist, but Theon lacks a foot. If Dion loses a foot, he ousts Theon? [Chrysippus, by Philo of Alexandria] |
16059 | Change of matter doesn't destroy identity - in Dion and Theon change is a condition of identity [Chrysippus, by Long/Sedley] |
1875 | Dogs show reason in decisions made by elimination [Chrysippus, by Sext.Empiricus] |
20834 | Chrysippus allows evil to say it is fated, or even that it is rational and natural [Plutarch on Chrysippus] |
20833 | A swerve in the atoms would be unnatural, like scales settling differently for no reason [Chrysippus, by Plutarch] |
20835 | Chrysippus is wrong to believe in non-occurring future possibilities if he is a fatalist [Plutarch on Chrysippus] |
20808 | Everything is fated, either by continuous causes or by a supreme rational principle [Chrysippus, by Diog. Laertius] |
20837 | Fate is an eternal and fixed chain of causal events [Chrysippus] |
20836 | The Lazy Argument responds to fate with 'why bother?', but the bothering is also fated [Chrysippus, by Cicero] |
21679 | When we say events are fated by antecedent causes, do we mean principal or auxiliary causes? [Chrysippus] |
5971 | Destiny is only a predisposing cause, not a sufficient cause [Chrysippus, by Plutarch] |
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] |
20787 | A proposition is what can be asserted or denied on its own [Chrysippus] |
20850 | Passions are judgements; greed thinks money is honorable, and likewise drinking and lust [Chrysippus, by Diog. Laertius] |
20869 | The highest degree of morality performs all that is appropriate, omitting nothing [Chrysippus] |
3044 | Stoics say that beauty and goodness are equivalent and linked [Chrysippus, by Diog. Laertius] |
20838 | Fate initiates general causes, but individual wills and characters dictate what we do [Chrysippus] |
20813 | Human purpose is to contemplate and imitate the cosmos [Chrysippus] |
3045 | Stoics say justice is a part of nature, not just an invented principle [Chrysippus, by Diog. Laertius] |
20774 | Only nature is available to guide action and virtue [Chrysippus] |
20864 | Live in agreement, according to experience of natural events [Chrysippus] |
5972 | Living happily is nothing but living virtuously [Chrysippus, by Plutarch] |
1777 | Pleasure is not the good, because there are disgraceful pleasures [Chrysippus, by Diog. Laertius] |
5973 | Justice can be preserved if pleasure is a good, but not if it is the goal [Chrysippus, by Plutarch] |
20845 | There are shameful pleasures, and nothing shameful is good, so pleasure is not a good [Chrysippus, by Diog. Laertius] |
5967 | People need nothing except corn and water [Chrysippus, by Plutarch] |
5966 | All virtue is good, but not always praised (as in not lusting after someone ugly) [Chrysippus] |
20855 | Chrysippus says virtue can be lost (though Cleanthes says it is too secure for that) [Chrysippus, by Diog. Laertius] |
5970 | Chrysippus says nothing is blameworthy, as everything conforms with the best nature [Chrysippus, by Plutarch] |
20842 | Rational animals begin uncorrupted, but externals and companions are bad influences [Chrysippus, by Diog. Laertius] |
20856 | Justice, the law, and right reason are natural and not conventional [Chrysippus, by Diog. Laertius] |
1779 | We don't have obligations to animals as they aren't like us [Chrysippus, by Diog. Laertius] |
20857 | Justice is irrelevant to animals, because they are too unlike us [Chrysippus, by Diog. Laertius] |
20812 | Covers are for shields, and sheaths for swords; likewise, all in the cosmos is for some other thing [Chrysippus] |
21403 | The later Stoics identified the logos with an air-fire compound, called 'pneuma' [Chrysippus, by Long] |
20828 | Fire is a separate element, not formed with others (as was previously believed) [Chrysippus, by Stobaeus] |
5975 | Stoics say earth, air, fire and water are the primary elements [Chrysippus, by Plutarch] |
10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
20819 | The past and the future subsist, but only the present exists [Chrysippus, by Plutarch] |
20818 | The present does not exist, so our immediate experience is actually part past and part future [Chrysippus, by Plutarch] |
20821 | Time is continous and infinitely divisible, so there cannot be a wholly present time [Chrysippus, by Stobaeus] |
13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
3048 | Stoics say that God the creator is the perfection of all animals [Chrysippus, by Diog. Laertius] |
20773 | The origin of justice can only be in Zeus, and in nature [Chrysippus] |
5965 | The source of all justice is Zeus and the universal nature [Chrysippus] |
3042 | Stoics teach that law is identical with right reason, which is the will of Zeus [Chrysippus, by Diog. Laertius] |
1782 | Stoics teach that God is a unity, variously known as Mind, or Fate, or Jupiter [Chrysippus, by Diog. Laertius] |
20830 | Death can't separate soul from body, because incorporeal soul can't unite with body [Chrysippus] |
21404 | There is a rationale in terrible disasters; they are useful to the whole, and make good possible [Chrysippus] |