111 ideas
1597 | Thales was the first western thinker to believe the arché was intelligible [Roochnik on Thales] |
1887 | You cannot divide anything into many parts, because after the first division you are no longer dividing the original [Sext.Empiricus] |
22764 | Ordinary speech is not exact about what is true; we say we are digging a well before the well exists [Sext.Empiricus] |
22752 | Reasoning is impossible without a preconception [Sext.Empiricus] |
1885 | Proof moves from agreed premises to a non-evident inference [Sext.Empiricus] |
6021 | It is only when we say a proposition that we speak truly or falsely [Sext.Empiricus] |
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
12196 | A valid hypothetical syllogism is 'that which does not begin with a truth and end with a falsehood' [Sext.Empiricus] |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
6020 | 'Man is a rational mortal animal' is equivalent to 'if something is a man, that thing is a rational mortal animal' [Sext.Empiricus] |
1902 | Since Socrates either died when he was alive (a contradiction) or died when he was dead (meaningless), he didn't die [Sext.Empiricus] |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
22744 | Parts are not parts if their whole is nothing more than the parts [Sext.Empiricus] |
22762 | Some properties are inseparable from a thing, such as the length, breadth and depth of a body [Sext.Empiricus] |
3013 | Nothing is stronger than necessity, which rules everything [Thales, by Diog. Laertius] |
1889 | If an argument has an absurd conclusion, we should not assent to the absurdity, but avoid the absurd argument [Sext.Empiricus] |
1871 | Whether honey is essentially sweet may be doubted, as it is a matter of judgement rather than appearance [Sext.Empiricus] |
1883 | How can the intellect know if sensation is reliable if it doesn't directly see external objects? [Sext.Empiricus] |
22748 | Some say motion is perceived by sense, but others say it is by intellect [Sext.Empiricus] |
1890 | We distinguish ambiguities by seeing what is useful [Sext.Empiricus] |
22759 | Fools, infants and madmen may speak truly, but do not know [Sext.Empiricus] |
20795 | Some things are their own criterion, such as straightness, a set of scales, or light [Sext.Empiricus] |
22760 | Madmen are reliable reporters of what appears to them [Sext.Empiricus] |
1870 | The basis of scepticism is the claim that every proposition has an equal opposing proposition [Sext.Empiricus] |
1882 | The necks of doves appear different in colour depending on the angle of viewing [Sext.Empiricus] |
1881 | The same oar seems bent in water and straight when out of it [Sext.Empiricus] |
1872 | The same tower appears round from a distance, but square close at hand [Sext.Empiricus] |
1873 | If we press the side of an eyeball, objects appear a different shape [Sext.Empiricus] |
20794 | How can sceptics show there is no criterion? Weak without, contradiction with [Sext.Empiricus] |
1874 | How can we judge between our impressions and those of other animals, when we ourselves are involved? [Sext.Empiricus] |
1878 | Water that seems lukewarm can seem very hot on inflamed skin [Sext.Empiricus] |
1880 | Some actions seem shameful when sober but not when drunk [Sext.Empiricus] |
1877 | If we had no hearing or sight, we would assume no sound or sight exists, so there may be unsensed qualities [Sext.Empiricus] |
1879 | Sickness is perfectly natural to the sick, so their natural perceptions should carry some weight [Sext.Empiricus] |
1876 | If we enjoy different things, presumably we receive different impressions [Sext.Empiricus] |
1911 | Even if all known nations agree on a practice, there may be unknown nations which disagree [Sext.Empiricus] |
1910 | With us it is shameful for men to wear earrings, but among Syrians it is considered noble [Sext.Empiricus] |
6026 | How can you investigate without some preconception of your object? [Sext.Empiricus] |
1886 | If you don't view every particular, you may miss the one which disproves your universal induction [Sext.Empiricus] |
22746 | If we try to conceive of a line with no breadth, it ceases to exist, and so has no length [Sext.Empiricus] |
22741 | The incorporeal is not in the nature of body, and so could not emerge from it [Sext.Empiricus] |
1884 | If we utter three steps of a logical argument, they never exist together [Sext.Empiricus] |
22763 | We can only dream of a winged man if we have experienced men and some winged thing [Sext.Empiricus] |
22754 | Saying the good is useful or choiceworth or happiness-creating is not the good, but a feature of it [Sext.Empiricus] |
22755 | Like a warming fire, what is good by nature should be good for everyone [Sext.Empiricus] |
22756 | If a desire is itself desirable, then we shouldn't desire it, as achieving it destroys it [Sext.Empiricus] |
6032 | Right actions, once done, are those with a reasonable justification [Sext.Empiricus] |
1517 | The tektraktys (1+2+3+4=10) is the 'fount of ever-flowing nature' [Sext.Empiricus] |
1494 | Thales said water is the first principle, perhaps from observing that food is moist [Thales, by Aristotle] |
1894 | Some say that causes are physical, some say not [Sext.Empiricus] |
1898 | Cause can't exist before effect, or exist at the same time, so it doesn't exist [Sext.Empiricus] |
1896 | If there were no causes then everything would have been randomly produced by everything [Sext.Empiricus] |
1897 | Knowing an effect results from a cause means knowing that the cause belongs with the effect, which is circular [Sext.Empiricus] |
1895 | Causes are either equal to the effect, or they link equally with other causes, or they contribute slightly [Sext.Empiricus] |
1900 | If time and place are infinitely divided, it becomes impossible for movement ever to begin [Sext.Empiricus] |
1899 | Does the original self-mover push itself from behind, or pull itself from in front? [Sext.Empiricus] |
1901 | If all atoms, times and places are the same, everything should move with equal velocity [Sext.Empiricus] |
1713 | Thales must have thought soul causes movement, since he thought magnets have soul [Thales, by Aristotle] |
22747 | A man walking backwards on a forwards-moving ship is moving in a fixed place [Sext.Empiricus] |
1903 | If motion and rest are abolished, so is time [Sext.Empiricus] |
1904 | Time must be unlimited, but past and present can't be non-existent, and can't be now, so time does not exist [Sext.Empiricus] |
22749 | Time doesn't end with the Universe, because tensed statements about destruction remain true [Sext.Empiricus] |
22750 | Time is divisible, into past, present and future [Sext.Empiricus] |
1905 | How can time be divisible if we can't compare one length of time with another? [Sext.Empiricus] |
22742 | Socrates either dies when he exists (before his death) or when he doesn't (after his death) [Sext.Empiricus] |
22751 | If the present is just the limit of the past or the future, it can't exist because they don't exist [Sext.Empiricus] |
22730 | All men agree that God is blessed, imperishable, happy and good [Sext.Empiricus] |
1891 | How can we agree on the concept of God, unless we agree on his substance or form or place? [Sext.Empiricus] |
22739 | God must suffer to understand suffering [Sext.Empiricus] |
22738 | The Divine must lack the virtues of continence and fortitude, because they are not needed [Sext.Empiricus] |
22734 | God is defended by agreement, order, absurdity of denying God, and refutations [Sext.Empiricus] |
22736 | God's sensations imply change, and hence perishing, which is absurd, so there is no such God [Sext.Empiricus] |
22740 | God without virtue is absurd, but God's virtues will be better than God [Sext.Empiricus] |
1892 | The existence of God can't be self-evident or everyone would have agreed on it, so it needs demonstration [Sext.Empiricus] |
22735 | The original substance lacked motion or shape, and was given these by a cause [Sext.Empiricus] |
22732 | The perfections of God were extrapolations from mankind [Sext.Empiricus] |
22728 | Gods were invented as watchers of people's secret actions [Sext.Empiricus] |
22737 | An incorporeal God could do nothing, and a bodily god would perish, so there is no God [Sext.Empiricus] |
22731 | It is mad to think that what is useful to us, like lakes and rivers, are gods [Sext.Empiricus] |
1742 | Thales said the gods know our wrong thoughts as well as our evil actions [Thales, by Diog. Laertius] |
1893 | If God foresaw evil he would presumably prevent it, and if he only foresees some things, why those things? [Sext.Empiricus] |