88 ideas
16604 | Philosophy consists of choosing between Plato, Aristotle and Democritus [Pasnau] |
16606 | Original philosophers invariably seek inspiration from past thinkers [Pasnau] |
16586 | The commentaries of Averroes were the leading guide to Aristotle [Pasnau] |
16568 | Modernity begins in the late 12th century, with Averroes's commentaries on Aristotle [Pasnau] |
16653 | Once accidents were seen as real, 'Categories' became the major text for ontology [Pasnau] |
16704 | In 1347, the Church effectively stopped philosophy for the next 300 years [Pasnau] |
16605 | After c.1450 all of Plato was available. Before that, only the first half of 'Timaeus' was known [Pasnau] |
16607 | Renaissance Platonism is peripheral [Pasnau] |
16715 | Plato only made an impact locally in 15th century Italy [Pasnau] |
16610 | Philosophy could easily have died in 17th century, if it weren't for Descartes [Pasnau] |
16781 | The 17th century is a metaphysical train wreck [Pasnau] |
16677 | Anti-Razor: if you can't account for a truth, keep positing things until you can [Pasnau] |
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] |
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] |
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] |
16598 | Priority was a major topic of dispute for scholastics [Pasnau] |
16727 | In mixtures, the four elements ceased to exist, replaced by a mixed body with a form [Pasnau] |
16732 | 17th C qualities are either microphysical, or phenomenal, or powers [Pasnau] |
16733 | 17th century authors only recognised categorical properties, never dispositions [Pasnau] |
16662 | The biggest question for scholastics is whether properties are real, or modes of substances [Pasnau] |
16767 | There is no centralised power, but we still need essence for a metaphysical understanding [Pasnau] |
16788 | Instead of adding Aristotelian forms to physical stuff, one could add dispositions [Pasnau] |
16738 | Scholastics reject dispositions, because they are not actual, as forms require [Pasnau] |
16649 | Scholastics say there is a genuine thing if it is 'separable' [Pasnau] |
16785 | If you reject essences, questions of individuation become extremely difficult [Pasnau] |
16680 | Scholastics thought Quantity could be the principle of individuation [Pasnau] |
16628 | Corpuscularianism promised a decent account of substance [Pasnau] |
16617 | Corpuscularian critics of scholasticism say only substances exist [Pasnau] |
16741 | Scholastics wanted to treat Aristotelianism as physics, rather than as metaphysics [Pasnau] |
16777 | If crowds are things at all, they seem to be Substances, since they bear properties [Pasnau] |
16615 | Scholastics use 'substantia' for thick concrete entities, and for thin metaphysical ones [Pasnau] |
16775 | For corpuscularians, a substance is just its integral parts [Pasnau] |
16769 | If clay survives destruction of the statue, the statue wasn't a substance, but a mere accident [Pasnau] |
16602 | Corpuscularianism rejected not only form, but also the dependence of matter on form [Pasnau] |
16612 | Hylomorphism may not be a rival to science, but an abstract account of unity and endurance [Pasnau] |
16613 | Hylomorphism declined because scholastics made it into a testable physical theory [Pasnau] |
16747 | Scholastics made forms substantial, in a way unintended by Aristotle [Pasnau] |
16759 | Scholastics began to see substantial form more as Aristotle's 'efficient' cause [Pasnau] |
16748 | Aquinas says a substance has one form; Scotists say it has many forms [Pasnau] |
16671 | Scholastic Quantity either gives a body parts, or spreads them out in a unified way [Pasnau] |
16579 | There may be different types of substrate, or temporary substrates [Pasnau] |
16596 | A substratum can't be 'bare', because it has a job to do [Pasnau] |
16584 | If a substrate gives causal support for change, quite a lot of the ingredients must endure [Pasnau] |
16580 | A substrate may be 'prime matter', which endures through every change [Pasnau] |
16749 | Aristotelians deny that all necessary properties are essential [Pasnau] |
16694 | Typical successive things are time and motion [Pasnau] |
16583 | Weak ex nihilo says it all comes from something; strong version says the old must partly endure [Pasnau] |
16736 | Explanation is generally to deduce it from something better known, which comes in degrees [Boyle] |
16783 | Essences must explain, so we can infer them causally from the accidents [Pasnau] |
16737 | The best explanations get down to primary basics, but others go less deep [Boyle] |
16609 | Atomists say causation is mechanical collisions, and all true qualities are microscopic [Pasnau] |
16603 | In the 17th C matter became body, and was then studied by science [Pasnau] |
16592 | Atomism is the commonest version of corpuscularianism, but isn't required by it [Pasnau] |
16750 | If there are just arrangements of corpuscles, where are the boundaries between substances? [Pasnau] |
16722 | Scholastic causation is by changes in the primary qualities of hot, cold, wet, dry [Pasnau] |
16760 | Substantial forms were a step towards scientific essentialism [Pasnau] |
16581 | Scholastic authors agree that matter was created by God, out of nothing [Pasnau] |
16642 | Transubstantion says accidents of bread and wine don't inhere in the substance [Pasnau] |