75 ideas
19693 | There is practical wisdom (for action), and theoretical wisdom (for deep understanding) [Aristotle, by Whitcomb] |
15053 | If metaphysics can't be settled, it hardly matters whether it makes sense [Fine,K] |
15054 | 'Quietist' says abandon metaphysics because answers are unattainable (as in Kant's noumenon) [Fine,K] |
1575 | For Aristotle logos is essentially the ability to talk rationally about questions of value [Roochnik on Aristotle] |
1589 | Aristotle is the supreme optimist about the ability of logos to explain nature [Roochnik on Aristotle] |
8200 | Aristotelian definitions aim to give the essential properties of the thing defined [Aristotle, by Quine] |
4385 | Aristotelian definition involves first stating the genus, then the differentia of the thing [Aristotle, by Urmson] |
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] |
13282 | Aristotle relativises the notion of wholeness to different measures [Aristotle, by Koslicki] |
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] |
4730 | For Aristotle, the subject-predicate structure of Greek reflected a substance-accident structure of reality [Aristotle, by O'Grady] |
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] |
15007 | If you make 'grounding' fundamental, you have to mention some non-fundamental notions [Sider on Fine,K] |
15006 | Something is grounded when it holds, and is explained, and necessitated by something else [Fine,K, by Sider] |
15055 | Grounding relations are best expressed as relations between sentences [Fine,K] |
15050 | Reduction might be producing a sentence which gets closer to the logical form [Fine,K] |
15051 | Reduction might be semantic, where a reduced sentence is understood through its reduction [Fine,K] |
15052 | Reduction is modal, if the reductions necessarily entail the truth of the target sentence [Fine,K] |
15056 | The notion of reduction (unlike that of 'ground') implies the unreality of what is reduced [Fine,K] |
15047 | What is real can only be settled in terms of 'ground' [Fine,K] |
15046 | Reality is a primitive metaphysical concept, which cannot be understood in other terms [Fine,K] |
15060 | Why should what is explanatorily basic be therefore more real? [Fine,K] |
15048 | In metaphysics, reality is regarded as either 'factual', or as 'fundamental' [Fine,K] |
13276 | The unmoved mover and the soul show Aristotelian form as the ultimate mereological atom [Aristotle, by Koslicki] |
13277 | The 'form' is the recipe for building wholes of a particular kind [Aristotle, by Koslicki] |
5991 | For Aristotle, knowledge is of causes, and is theoretical, practical or productive [Aristotle, by Code] |
11239 | The notion of a priori truth is absent in Aristotle [Aristotle, by Politis] |
15061 | Although colour depends on us, we can describe the world that way if it picks out fundamentals [Fine,K] |
23312 | Aristotle is a rationalist, but reason is slowly acquired through perception and experience [Aristotle, by Frede,M] |
16111 | Aristotle wants to fit common intuitions, and therefore uses language as a guide [Aristotle, by Gill,ML] |
16971 | Plato says sciences are unified around Forms; Aristotle says they're unified around substance [Aristotle, by Moravcsik] |
11243 | Aristotelian explanations are facts, while modern explanations depend on human conceptions [Aristotle, by Politis] |
3320 | Aristotle's standard analysis of species and genus involves specifying things in terms of something more general [Aristotle, by Benardete,JA] |
15059 | Grounding is an explanation of truth, and needs all the virtues of good explanations [Fine,K] |
12000 | Aristotle regularly says that essential properties explain other significant properties [Aristotle, by Kung] |
15057 | Ultimate explanations are in 'grounds', which account for other truths, which hold in virtue of the grounding [Fine,K] |
23300 | Aristotle and the Stoics denied rationality to animals, while Platonists affirmed it [Aristotle, by Sorabji] |
15058 | A proposition ingredient is 'essential' if changing it would change the truth-value [Fine,K] |
11240 | The notion of analytic truth is absent in Aristotle [Aristotle, by Politis] |
6559 | Aristotle never actually says that man is a rational animal [Aristotle, by Fogelin] |
11150 | It is the mark of an educated mind to be able to entertain an idea without accepting it [Aristotle] |
3037 | Aristotle said the educated were superior to the uneducated as the living are to the dead [Aristotle, by Diog. Laertius] |
8660 | There are potential infinities (never running out), but actual infinity is incoherent [Aristotle, by Friend] |
12058 | Aristotle's matter can become any other kind of matter [Aristotle, by Wiggins] |
22729 | The concepts of gods arose from observing the soul, and the cosmos [Aristotle, by Sext.Empiricus] |