78 ideas
| 1695 | Without extensive examination firm statements are hard, but studying the difficulties is profitable [Aristotle] |
| 1697 | The contrary of good is bad, but the contrary of bad is either good or another evil [Aristotle] |
| 1698 | Both sides of contraries need not exist (as health without sickness, white without black) [Aristotle] |
| 8964 | Entities can be multiplied either by excessive categories, or excessive entities within a category [Hoffman/Rosenkrantz] |
| 11034 | The differentiae of genera which are different are themselves different in kind [Aristotle] |
| 18367 | A true existence statement has its truth caused by the existence of the thing [Aristotle] |
| 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] |
| 11033 | Predications of predicates are predications of their subjects [Aristotle] |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| 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] |
| 11044 | One is prior to two, because its existence is implied by two [Aristotle] |
| 11042 | Parts of a line join at a point, so it is continuous [Aristotle] |
| 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] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [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] |
| 11041 | Some quantities are discrete, like number, and others continuous, like lines, time and space [Aristotle] |
| 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] |
| 11286 | Primary being must be more than mere indeterminate ultimate subject of predication [Politis on Aristotle] |
| 1700 | There are six kinds of change: generation, destruction, increase, diminution, alteration, change of place [Aristotle] |
| 1699 | A thing is prior to another if it implies its existence [Aristotle] |
| 18366 | Of interdependent things, the prior one causes the other's existence [Aristotle] |
| 11035 | There are ten basic categories for thinking about things [Aristotle] |
| 3311 | The categories (substance, quality, quantity, relation, action, passion, place, time) peter out inconsequentially [Benardete,JA on Aristotle] |
| 13121 | Substance,Quantity,Quality,Relation,Place,Time,Being-in-a-position,Having,Doing,Being affected [Aristotle, by Westerhoff] |
| 16116 | Aristotle derived categories as answers to basic questions about nature, size, quality, location etc. [Aristotle, by Gill,ML] |
| 21345 | Aristotle said relations are not substances, so (if they exist) they must be accidents [Aristotle, by Heil] |
| 16155 | Aristotle promoted the importance of properties and objects (rather than general and particular) [Aristotle, by Frede,M] |
| 11032 | Some things said 'of' a subject are not 'in' the subject [Aristotle] |
| 11038 | We call them secondary 'substances' because they reveal the primary substances [Aristotle] |
| 16739 | Four species of quality: states, capacities, affects, and forms [Aristotle, by Pasnau] |
| 11037 | Colour must be in an individual body, or it is not embodied [Aristotle] |
| 8962 | 'There are shapes which are never exemplified' is the toughest example for nominalists [Hoffman/Rosenkrantz] |
| 8961 | Nominalists are motivated by Ockham's Razor and a distrust of unobservables [Hoffman/Rosenkrantz] |
| 16154 | Aristotle gave up his earlier notion of individuals, because it relied on universals [Aristotle, by Frede,M] |
| 12351 | Genus and species are substances, because only they reveal the primary substance [Aristotle, by Wedin] |
| 11040 | A single substance can receive contrary properties [Aristotle] |
| 1694 | Substances have no opposites, and don't come in degrees (including if the substance is a man) [Aristotle] |
| 16091 | Is primary substance just an ultimate subject, or some aspect of a complex body? [Aristotle, by Gill,ML] |
| 11280 | Primary being is 'that which lies under', or 'particular substance' [Aristotle, by Politis] |
| 11036 | A 'primary' substance is in each subject, with species or genera as 'secondary' substances [Aristotle] |
| 16140 | Secondary substances do have subjects, so they are not ultimate in the ontology [Aristotle, by Frede,M] |
| 10965 | In earlier Aristotle the substances were particulars, not kinds [Aristotle, by Lawson-Tancred] |
| 8287 | Earlier Aristotle had objects as primary substances, but later he switched to substantial form [Aristotle, by Lowe] |
| 12350 | Things are called 'substances' because they are subjects for everything else [Aristotle] |
| 11039 | A primary substance reveals a 'this', which is an individual unit [Aristotle] |
| 12361 | Primary substances are ontological in 'Categories', and explanatory in 'Metaphysics' [Aristotle, by Wedin] |
| 3315 | Aristotle denigrates the category of relation, but for modern absolutists self-relation is basic [Benardete,JA on Aristotle] |
| 8963 | Four theories of possible worlds: conceptualist, combinatorial, abstract, or concrete [Hoffman/Rosenkrantz] |
| 12349 | Only what can be said of many things is a predicable [Aristotle, by Wedin] |
| 11837 | Some predicates signify qualification of a substance, others the substance itself [Aristotle] |
| 11043 | It is not possible for fire to be cold or snow black [Aristotle] |
| 1696 | Change goes from possession to loss (as in baldness), but not the other way round [Aristotle] |