150 ideas
| 12997 | Analysis is the art of finding the middle term [Leibniz] |
| 13009 | A reason is a known truth which leads to assent to some further truth [Leibniz] |
| 12963 | Opposing reason is opposing truth, since reason is a chain of truths [Leibniz] |
| 19360 | General principles, even if unconscious, are indispensable for thinking [Leibniz] |
| 12983 | A nominal definition is of the qualities, but the real definition is of the essential inner structure [Leibniz] |
| 12982 | One essence can be expressed by several definitions [Leibniz] |
| 12976 | If our ideas of a thing are imperfect, the thing can have several unconnected definitions [Leibniz] |
| 12984 | Real definitions, unlike nominal definitions, display possibilities [Leibniz] |
| 12980 | Genus and differentia might be swapped, and 'rational animal' become 'animable rational' [Leibniz] |
| 22094 | Subjective truth can only be sustained by repetition [Kierkegaard, by Carlisle] |
| 13000 | Truth is correspondence between mental propositions and what they are about [Leibniz] |
| 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] |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| 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] |
| 12992 | Logic teaches us how to order and connect our thoughts [Leibniz] |
| 10056 | At bottom eternal truths are all conditional [Leibniz] |
| 12974 | People who can't apply names usually don't understand the thing to which it applies [Leibniz] |
| 13002 | It is always good to reduce the number of axioms [Leibniz] |
| 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] |
| 13008 | Geometry, unlike sensation, lets us glimpse eternal truths and their necessity [Leibniz] |
| 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] |
| 12956 | Only whole numbers are multitudes of units [Leibniz] |
| 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] |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 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] |
| 12937 | We shouldn't just accept Euclid's axioms, but try to demonstrate them [Leibniz] |
| 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] |
| 12932 | The idea of being must come from our own existence [Leibniz] |
| 12966 | Objects of ideas can be divided into abstract and concrete, and then further subdivided [Leibniz] |
| 12993 | Have five categories - substance, quantity, quality, action/passion, relation - and their combinations [Leibniz] |
| 12989 | Our true divisions of nature match reality, but are probably incomplete [Leibniz] |
| 12959 | We discern active power from our minds, so mind must be involved in all active powers [Leibniz] |
| 12967 | I use the word 'entelechy' for a power, to include endeavour, as well as mere aptitude [Leibniz] |
| 12965 | All occurrence in the depth of a substance is spontaneous 'action' [Leibniz] |
| 12999 | Substances are primary powers; their ways of being are the derivative powers [Leibniz] |
| 5056 | Material or immaterial substances cannot be conceived without their essential activity [Leibniz] |
| 12969 | The active powers which are not essential to the substance are the 'real qualities' [Leibniz] |
| 12941 | There cannot be power without action; the power is a disposition to act [Leibniz] |
| 12990 | Real (non-logical) abstract terms are either essences or accidents [Leibniz] |
| 12939 | Wholly uniform things like space and numbers are mere abstractions [Leibniz] |
| 12979 | The only way we can determine individuals is by keeping hold of them [Leibniz] |
| 12971 | If two individuals could be indistinguishable, there could be no principle of individuation [Leibniz] |
| 13098 | We use things to distinguish places and times, not vice versa [Leibniz] |
| 13075 | No two things are quite the same, so there must be an internal principle of distinction [Leibniz] |
| 12953 | Fluidity is basic, and we divide into bodies according to our needs [Leibniz] |
| 12943 | Individuality is in the bond substance gives between past and future [Leibniz] |
| 11855 | Substances cannot be bare, but have activity as their essence [Leibniz] |
| 12970 | We can imagine two bodies interpenetrating, as two rays of light seem to [Leibniz] |
| 12986 | The essence of baldness is vague and imperfect [Leibniz] |
| 12968 | A 'substratum' is just a metaphor for whatever supports several predicates [Leibniz] |
| 12931 | Particular truths are just instances of general truths [Leibniz] |
| 12811 | We can't know individuals, or determine their exact individuality [Leibniz] |
| 12981 | Essence is just the possibility of a thing [Leibniz] |
| 5057 | If you fully understand a subject and its qualities, you see how the second derive from the first [Leibniz] |
| 12987 | For some sorts, a member of it is necessarily a member [Leibniz] |
| 12884 | The same whole ceases to exist if a part is lost [Leibniz] |
| 12975 | We have a distinct idea of gold, to define it, but not a perfect idea, to understand it [Leibniz] |
| 12805 | If two people apply a single term to different resemblances, they refer to two different things [Leibniz] |
| 12806 | Locke needs many instances to show a natural kind, but why not a single instance? [Leibniz, by Jolley] |
| 12972 | Bodies, like Theseus's ship, are only the same in appearance, and never strictly the same [Leibniz] |
| 5055 | No two things are totally identical [Leibniz] |
| 12978 | A perfect idea of an object shows that the object is possible [Leibniz] |
| 17079 | Proofs of necessity come from the understanding, where they have their source [Leibniz] |
| 12998 | Understanding grasps the agreements and disagreements of ideas [Leibniz] |
| 12960 | We understand things when they are distinct, and we can derive necessities from them [Leibniz] |
| 13006 | Certainty is where practical doubt is insane, or at least blameworthy [Leibniz] |
| 12996 | I know more than I think, since I know I think A then B then C [Leibniz] |
| 13003 | The Cogito doesn't prove existence, because 'I am thinking' already includes 'I am' [Leibniz] |
| 21253 | Descartes needs to demonstrate how other people can attain his clear and distinct conceptions [Leibniz] |
| 12933 | Arithmetic and geometry are implicitly innate, awaiting revelation [Leibniz] |
| 12991 | Children learn language fast, with little instruction and few definitions [Leibniz] |
| 12929 | All of our thoughts come from within the soul, and not from the senses [Leibniz] |
| 12940 | What is left of the 'blank page' if you remove the ideas? [Leibniz] |
| 19358 | Colour and pain must express the nature of their stimuli, without exact resemblance [Leibniz] |
| 12948 | A pain doesn't resemble the movement of a pin, but it resembles the bodily movement pins cause [Leibniz] |
| 13005 | Truth arises among sensations from grounding reasons and from regularities [Leibniz] |
| 12947 | We only believe in sensible things when reason helps the senses [Leibniz] |
| 4302 | You may experience a universal truth, but only reason can tell you that it is always true [Leibniz] |
| 12930 | The senses are confused, and necessities come from distinct intellectual ideas [Leibniz] |
| 13001 | Our sensation of green is a confused idea, like objects blurred by movement [Leibniz] |
| 12949 | Light takes time to reach us, so objects we see may now not exist [Leibniz] |
| 5053 | The instances confirming a general truth are never enough to establish its necessity [Leibniz] |
| 12977 | We will only connect our various definitions of gold when we understand it more deeply [Leibniz] |
| 5054 | Animal thought is a shadow of reasoning, connecting sequences of images by imagination [Leibniz] |
| 12944 | It is a serious mistake to think that we are aware of all of our perceptions [Leibniz] |
| 12951 | Abstraction attends to the general, not the particular, and involves universal truths [Leibniz] |
| 19364 | Volition automatically endeavours to move towards what it sees as good (and away from bad) [Leibniz] |
| 12942 | Memory doesn't make identity; a man who relearned everything would still be the same man [Leibniz] |
| 12973 | We know our own identity by psychological continuity, even if there are some gaps [Leibniz] |
| 19368 | The will determines action, by what is seen as good, but it does not necessitate it [Leibniz] |
| 12935 | Every feeling is the perception of a truth [Leibniz] |
| 12938 | An idea is an independent inner object, which expresses the qualities of things [Leibniz] |
| 12945 | Thoughts correspond to sensations, but ideas are independent of thoughts [Leibniz] |
| 12950 | We must distinguish images from exact defined ideas [Leibniz] |
| 19357 | The idea of green seems simple, but it must be compounded of the ideas of blue and yellow [Leibniz] |
| 12995 | The name 'gold' means what we know of gold, and also further facts about it which only others know [Leibniz] |
| 12807 | The word 'gold' means a hidden constitution known to experts, and not just its appearances [Leibniz] |
| 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] |
| 12946 | The idea of the will includes the understanding [Leibniz] |
| 12964 | If would be absurd not to disagree with someone's taste if it was a taste for poisons [Leibniz] |
| 12958 | Love is pleasure in the perfection, well-being or happiness of its object [Leibniz] |
| 12957 | The good is the virtuous, the pleasing, or the useful [Leibniz] |
| 12962 | Pleasure is a sense of perfection [Leibniz] |
| 12934 | We can't want everyone to have more than their share, so a further standard is needed [Leibniz] |
| 22093 | Life is a repetition when what has been now becomes [Kierkegaard] |
| 12936 | There are natural rewards and punishments, like illness after over-indulgence [Leibniz] |
| 11856 | Qualities should be predictable from the nature of the subject [Leibniz] |
| 12994 | Gold has a real essence, unknown to us, which produces its properties [Leibniz] |
| 12808 | Part of our idea of gold is its real essence, which is not known to us in detail [Leibniz] |
| 12985 | Maybe motion is definable as 'change of place' [Leibniz] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 12952 | Space is an order among actual and possible things [Leibniz] |
| 12955 | If there were duration without change, we could never establish its length [Leibniz] |
| 12954 | God's essence is the source of possibilities, and his will the source of existents [Leibniz] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| 1414 | A perfection is a simple quality, which is positive and absolute, and has no limit [Leibniz] |
| 12988 | The universe contains everything possible for its perfect harmony [Leibniz] |
| 21252 | Perfections must have overlapping parts if their incompatibility is to be proved [Leibniz] |
| 19328 | Without the principle of sufficient reason, God's existence could not be demonstrated [Leibniz] |
| 5058 | Animals have thought and sensation, and indestructible immaterial souls [Leibniz] |