85 ideas
| 9108 | From an impossibility anything follows [William of Ockham] |
| 16676 | Why use more things when fewer will do? [William of Ockham] |
| 6806 | Do not multiply entities beyond necessity [William of Ockham] |
| 11223 | Definitions usually have a term, a 'definiendum' containing the term, and a defining 'definiens' [Gupta] |
| 11215 | Notable definitions have been of piety (Plato), God (Anselm), number (Frege), and truth (Tarski) [Gupta] |
| 11225 | A definition needs to apply to the same object across possible worlds [Gupta] |
| 11227 | The 'revision theory' says that definitions are rules for improving output [Gupta] |
| 11221 | A definition can be 'extensionally', 'intensionally' or 'sense' adequate [Gupta] |
| 11224 | Traditional definitions are general identities, which are sentential and reductive [Gupta] |
| 11226 | Traditional definitions need: same category, mention of the term, and conservativeness and eliminability [Gupta] |
| 11217 | Chemists aim at real definition of things; lexicographers aim at nominal definition of usage [Gupta] |
| 11216 | If definitions aim at different ideals, then defining essence is not a unitary activity [Gupta] |
| 11218 | Stipulative definition assigns meaning to a term, ignoring prior meanings [Gupta] |
| 11220 | Ostensive definitions look simple, but are complex and barely explicable [Gupta] |
| 9107 | A proposition is true if its subject and predicate stand for the same thing [William of Ockham] |
| 14965 | Truth rests on Elimination ('A' is true → A) and Introduction (A → 'A' is true) [Gupta] |
| 14968 | A weakened classical language can contain its own truth predicate [Gupta] |
| 16300 | Ockham had an early axiomatic account of truth [William of Ockham, by Halbach] |
| 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] |
| 11222 | The ordered pair <x,y> is defined as the set {{x},{x,y}}, capturing function, not meaning [Gupta] |
| 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] |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| 9106 | The word 'every' only signifies when added to a term such as 'man', referring to all men [William of Ockham] |
| 14964 | The Liar reappears, even if one insists on propositions instead of sentences [Gupta] |
| 14969 | Strengthened Liar: either this sentence is neither-true-nor-false, or it is not true [Gupta] |
| 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] |
| 9113 | Just as unity is not a property of a single thing, so numbers are not properties of many things [William of Ockham] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 9110 | The words 'thing' and 'to be' assert the same idea, as a noun and as a verb [William of Ockham] |
| 16608 | Ockham was an anti-realist about the categories [William of Ockham, by Pasnau] |
| 16654 | Our words and concepts don't always correspond to what is out there [William of Ockham] |
| 18529 | Relations are expressed either as absolute facts, or by a relational concept [William of Ockham] |
| 22132 | Species and genera are individual concepts which naturally signify many individuals [William of Ockham] |
| 9103 | A universal is not a real feature of objects, but only a thought-object in the mind [William of Ockham] |
| 15388 | Universals are single things, and only universal in what they signify [William of Ockham] |
| 16779 | Cut wood doesn't make a new substance, but seems to make separate subjects [William of Ockham] |
| 16757 | Hot water naturally cools down, which is due to the substantial form of the water [William of Ockham] |
| 16599 | Ockham says matter must be extended, so we don't need Quantity [William of Ockham, by Pasnau] |
| 16681 | Matter gets its quantity from condensation and rarefaction, which is just local motion [William of Ockham] |
| 9109 | If essence and existence were two things, one could exist without the other, which is impossible [William of Ockham] |
| 16792 | If parts change, the whole changes [William of Ockham] |
| 9089 | Knowledge is a quality existing subjectively in the soul [William of Ockham] |
| 9091 | Sometimes 'knowledge' just concerns the conclusion, sometimes the whole demonstration [William of Ockham] |
| 9100 | Our intellect only assents to what we believe to be true [William of Ockham] |
| 9090 | Knowledge is certain cognition of something that is true [William of Ockham] |
| 9101 | Abstractive cognition knows universals abstracted from many singulars [William of Ockham] |
| 9102 | If an animal approached from a distance, we might abstract 'animal' from one instance [William of Ockham] |
| 9114 | There are no secure foundations to prove the separate existence of mind, in reason or experience [William of Ockham] |
| 9104 | A universal is the result of abstraction, which is only a kind of mental picturing [William of Ockham] |
| 9105 | Some concepts for propositions exist only in the mind, and in no language [William of Ockham] |
| 16675 | Every extended material substance is composed of parts distant from one another [William of Ockham] |
| 19381 | The past has ceased to exist, and the future does not yet exist, so time does not exist [William of Ockham] |
| 9111 | God is not wise, but more-than-wise; God is not good, but more-than-good [William of Ockham] |
| 8010 | William of Ockham is the main spokesman for God's commands being the source of morality [William of Ockham] |
| 9112 | We could never form a concept of God's wisdom if we couldn't abstract it from creatures [William of Ockham] |
| 9115 | To love God means to love whatever God wills to be loved [William of Ockham] |
| 16679 | Even an angel must have some location [William of Ockham, by Pasnau] |