89 ideas
| 13734 | Modern Quinean metaphysics is about what exists, but Aristotelian metaphysics asks about grounding [Schaffer,J] |
| 13751 | If you tore the metaphysics out of philosophy, the whole enterprise would collapse [Schaffer,J] |
| 14600 | Analysis aims at secure necessary and sufficient conditions [Schaffer,J] |
| 13743 | We should not multiply basic entities, but we can have as many derivative entities as we like [Schaffer,J] |
| 14603 | 'Reification' occurs if we mistake a concept for a thing [Schaffer,J] |
| 14607 | T adds □p→p for reflexivity, and is ideal for modeling lawhood [Schaffer,J] |
| 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] |
| 10373 | Logical form can't dictate metaphysics, as it may propose an undesirable property [Schaffer,J] |
| 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] |
| 13741 | If 'there are red roses' implies 'there are roses', then 'there are prime numbers' implies 'there are numbers' [Schaffer,J] |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| 14604 | If a notion is ontologically basic, it should be needed in our best attempt at science [Schaffer,J] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 13748 | Grounding is unanalysable and primitive, and is the basic structuring concept in metaphysics [Schaffer,J] |
| 17304 | As causation links across time, grounding links the world across levels [Schaffer,J] |
| 17306 | If ground is transitive and irreflexive, it has a strict partial ordering, giving structure [Schaffer,J] |
| 14599 | Three types of reduction: Theoretical (of terms), Definitional (of concepts), Ontological (of reality) [Schaffer,J] |
| 13747 | Supervenience is just modal correlation [Schaffer,J] |
| 13744 | The cosmos is the only fundamental entity, from which all else exists by abstraction [Schaffer,J] |
| 10367 | There is only one fact - the True [Schaffer,J] |
| 13739 | Maybe categories are just the different ways that things depend on basic substances [Schaffer,J] |
| 14605 | Tropes are the same as events [Schaffer,J] |
| 14601 | Individuation aims to count entities, by saying when there is one [Schaffer,J] |
| 14082 | No sortal could ever exactly pin down which set of particles count as this 'cup' [Schaffer,J] |
| 13742 | There exist heaps with no integral unity, so we should accept arbitrary composites in the same way [Schaffer,J] |
| 13752 | The notion of 'grounding' can explain integrated wholes in a way that mere aggregates can't [Schaffer,J] |
| 14081 | Identities can be true despite indeterminate reference, if true under all interpretations [Schaffer,J] |
| 14606 | Only ideal conceivability could indicate what is possible [Schaffer,J] |
| 13749 | Belief in impossible worlds may require dialetheism [Schaffer,J] |
| 13740 | 'Moorean certainties' are more credible than any sceptical argument [Schaffer,J] |
| 22363 | You have only begun to do real science when you can express it in numbers [Kelvin] |
| 17308 | Explaining 'Adam ate the apple' depends on emphasis, and thus implies a contrast [Schaffer,J] |
| 17305 | I take what is fundamental to be the whole spatiotemporal manifold and its fields [Schaffer,J] |
| 10359 | In causation there are three problems of relata, and three metaphysical problems [Schaffer,J] |
| 10372 | Causation may not be transitive; the last event may follow from the first, but not be caused by it [Schaffer,J] |
| 10374 | There are at least ten theories about causal connections [Schaffer,J] |
| 17307 | Nowadays causation is usually understood in terms of equations and variable ranges [Schaffer,J] |
| 10366 | Causation transcends nature, because absences can cause things [Schaffer,J] |
| 10377 | Causation may not be a process, if a crucial part of the process is 'disconnected' [Schaffer,J] |
| 10378 | A causal process needs to be connected to the effect in the right way [Schaffer,J] |
| 10382 | Causation can't be a process, because a process needs causation as a primitive [Schaffer,J] |
| 10375 | At least four rivals have challenged the view that causal direction is time direction [Schaffer,J] |
| 10389 | Causal order must be temporal, or else causes could be blocked, and time couldn't be explained [Schaffer,J] |
| 10390 | Causal order is not temporal, because of time travel, and simultanous, joint or backward causes [Schaffer,J] |
| 10380 | Causation is primitive; it is too intractable and central to be reduced; all explanations require it [Schaffer,J] |
| 10385 | If causation is just observables, or part of common sense, or vacuous, it can't be primitive [Schaffer,J] |
| 10387 | The notion of causation allows understanding of science, without appearing in equations [Schaffer,J] |
| 10388 | Causation is utterly essential for numerous philosophical explanations [Schaffer,J] |
| 10384 | If two different causes are possible in one set of circumstances, causation is primitive [Schaffer,J] |
| 10386 | If causation is primitive, it can be experienced in ourselves, or inferred as best explanation [Schaffer,J] |
| 10361 | Events are fairly course-grained (just saying 'hello'), unlike facts (like saying 'hello' loudly) [Schaffer,J] |
| 10360 | Causal relata are events - or facts, features, tropes, states, situations or aspects [Schaffer,J] |
| 10362 | One may defend three or four causal relata, as in 'c causes e rather than e*' [Schaffer,J] |
| 10368 | If causal relata must be in nature and fine-grained, neither facts nor events will do [Schaffer,J] |
| 10383 | The relata of causation (such as events) need properties as explanation, which need causation! [Schaffer,J] |
| 10393 | Our selection of 'the' cause is very predictable, so must have a basis [Schaffer,J] |
| 10394 | Selecting 'the' cause must have a basis; there is no causation without such a selection [Schaffer,J] |
| 10376 | The actual cause may make an event less likely than a possible more effective cause [Schaffer,J] |
| 10381 | All four probability versions of causation may need causation to be primitive [Schaffer,J] |
| 20644 | Energy has progressed from a mere formula, to a principle pervading all nature [Kelvin] |