77 ideas
| 17311 | Real definitions don't just single out a thing; they must also explain its essence [Koslicki] |
| Full Idea: A statement expressing a real definition must also accomplish more than simply to offer two different ways of singling out the same entity, since the definiens must also be explanatory of the essential nature of the definiendum. | |
| From: Kathrin Koslicki (Varieties of Ontological Dependence [2012], 7.4) | |
| A reaction: This is why Aristotelian definitions are not just short lexicographical definitions, but may be quite length. Effectively, a definition IS an explanation. |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 10373 | Logical form can't dictate metaphysics, as it may propose an undesirable property [Schaffer,J] |
| Full Idea: Logical form should not have the last word in metaphysics, since it might predicate a property that we have theoretical reason to reject. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 1.3.1) | |
| A reaction: These kind of warnings need to be sounded all the time, to prevent logicians and language experts from pitching their tents in the middle of metaphysics. They are welcome guests only, |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 17312 | It is more explanatory if you show how a number is constructed from basic entities and relations [Koslicki] |
| Full Idea: Being the successor of the successor of 0 is more explanatory than being predecessor of 3 of the nature of 2, since it mirrors more closely the method by which 2 is constructed from a basic entity, 0, and a relation (successor) taken as primitive. | |
| From: Kathrin Koslicki (Varieties of Ontological Dependence [2012], 7.4) | |
| A reaction: This assumes numbers are 'constructed', which they are in the axiomatised system of Peano Arithmetic, but presumably the numbers were given in ordinary experience before 'construction' occurred to anyone. Nevertheless, I really like this. |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 17314 | The relata of grounding are propositions or facts, but for dependence it is objects and their features [Koslicki] |
| Full Idea: The relata of the grounding relation are typically taken to be facts or propositions, while the relata of ontological dependence ...are objects and their characteristics, activities, constituents and so on. | |
| From: Kathrin Koslicki (Varieties of Ontological Dependence [2012], 7.5 n25) | |
| A reaction: Interesting. Good riddance to propositions here, but this seems a bit unfair to facts, since I take facts to be in the world. Audi's concept of 'worldly facts' is what we need here. |
| 10367 | There is only one fact - the True [Schaffer,J] |
| Full Idea: It can be argued that if all facts are logically equivalent, then there is only one fact - the True. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 1.1) | |
| A reaction: [he cites Davidson's 'Causal Relations', who cites Frege] This is the sort of bizarre stuff you end up with if you start from formal logic and work out to the world, instead of vice versa. |
| 17313 | Modern views want essences just to individuate things across worlds and times [Koslicki] |
| Full Idea: According to the approach of Plantinga, Forbes and Mackie, the primary job of essences is to individuate the entities whose essences they are across worlds and times at which these entities exist. | |
| From: Kathrin Koslicki (Varieties of Ontological Dependence [2012], 7.4 n13) | |
| A reaction: A helpful simplification of what is going on. I wish those authors would just say this one their first pages. They all get in a right tangle, because individuation is either too easy, or hopeless. 'Tracking' is a good word for this game. |
| 17309 | For Fine, essences are propositions true because of identity, so they are just real definitions [Koslicki] |
| Full Idea: Fine assumes that essences can be identified with collections of propositions that are true in virtue of the identity of a particular object, or objects. ...There is not, on this approach, much of a distinction between essences and real definitions. | |
| From: Kathrin Koslicki (Varieties of Ontological Dependence [2012], 7.4) | |
| A reaction: This won't do, because the essence of a physical object is not a set of propositions, it is some aspects of the object itself, which are described in a definition. Koslicki notes that psuché is an essence, and the soul is hardly a set of propositions! |
| 17315 | We need a less propositional view of essence, and so must distinguish it clearly from real definitions [Koslicki] |
| Full Idea: To make room for a less propositional conception of essence than that assumed by Fine, I urge that we distinguish more firmly between essences and real definitions (which state these essences in the form of propositions). | |
| From: Kathrin Koslicki (Varieties of Ontological Dependence [2012], 7.6) | |
| A reaction: Yes. The idea that essence is just a verbal or conceptual entity would be utterly abhorrent to Aristotle (a hero for Fine), and it is anathema to me too. We intend essences to be in the world (even if we are deceived about that). They explain! |
| 17317 | A good explanation captures the real-world dependence among the phenomena [Koslicki] |
| Full Idea: It is plausible to think that an explanation, when successful, captures or represents (by argument, or a why? question) an underlying real-world relation of dependence which obtains among the phenomena cited. | |
| From: Kathrin Koslicki (Varieties of Ontological Dependence [2012], 7.6) | |
| A reaction: She cites causal dependence as an example. I'm incline to think that 'grounding' is a better word for the target of good explanations than is 'dependence' (which can, surely, be mutual, where ground has the directionality needed for explanation). |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| Full Idea: Cantor (in his exploration of infinities) pushed the bounds of conceivability further than anyone before him. To discover what is conceivable, we have to enquire into the concept. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics 6.5 | |
| A reaction: This remark comes during a discussion of Husserl's phenomenology. Intuitionists challenge Cantor's claim, and restrict what is conceivable to what is provable. Does possibility depend on conceivability? |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| Full Idea: Cantor thought that we abstract a number as something common to all and only those sets any one of which has as many members as any other. ...However one wants to see the logic of the inference. The irony is that set theory lays out this logic. | |
| From: comment on George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: The logic Hart has in mind is the notion of an equivalence relation between sets. This idea sums up the older and more modern concepts of abstraction, the first as psychological, the second as logical (or trying very hard to be!). Cf Idea 9145. |
| 17316 | We can abstract to a dependent entity by blocking out features of its bearer [Koslicki] |
| Full Idea: In 'feature dependence', the ontologically dependent entity may be thought of as the result of a process of abstraction which takes the 'bearer' as its starting point and arrives at the abstracted entity by blocking out all the irrelevant features. | |
| From: Kathrin Koslicki (Varieties of Ontological Dependence [2012], 7.6) | |
| A reaction: She seems unaware that this is traditional abstraction, found in Aristotle, and a commonplace of thought until Frege got his evil hands on abstraction and stole it for other purposes. I'm a fan. |
| 10359 | In causation there are three problems of relata, and three metaphysical problems [Schaffer,J] |
| Full Idea: The questions about causation concern their relata (in space-time, how fine-grained, how many?) and the metaphysics (distinguish causal sequences from others, the direction of causation, selecting causes among pre-conditions?). | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], Intro) | |
| A reaction: A very nice map (which has got me thinking about restructuring this database). I can't think of a better way to do philosophy than this (let's hear it for analysis - but the greatest role models for the approach are Aristotle and Aquinas). |
| 10372 | Causation may not be transitive; the last event may follow from the first, but not be caused by it [Schaffer,J] |
| Full Idea: It is not clear whether causation is transitive. For example, if a boulder roll's towards a hiker's head, causing the hiker to duck, which causes the hiker to survive, it does not seem that the rolling boulder causes the survival of the hiker. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 1.2) | |
| A reaction: Maybe survival is not an event or an effect. How many times have I survived in my life? We could, though, say that the hiker strained a muscle as he or she ducked. But then it is unclear whether the boulder caused the muscle-strain. |
| 10374 | There are at least ten theories about causal connections [Schaffer,J] |
| Full Idea: Theories of causal connection are: nomological subsumption, statistical correlation, counterfactual dependence, agential manipulability, contiguous change, energy flow, physical processes, property transference, primitivism and eliminativism. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 1.3.1) | |
| A reaction: Schaffer reduces these to probability and process. I prefer the latter. The first two are wrong, the third right but superficial, the fourth wrong, the fifth, sixth and seventh on the right lines, the eighth wrong, the ninth tempting, and the last wrong. |
| 10366 | Causation transcends nature, because absences can cause things [Schaffer,J] |
| Full Idea: The main argument for causation being transcendent (rather than being immanent in nature) is that absences can be involved in causal relations. Thus a rock-climber is caused to survive by not falling. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 1.1) | |
| A reaction: I don't like that. The obvious strategy is to redescribe the events. Even being hit with a brick could be described as an 'absence of brick-prevention'. So not being hit by a brick can be described as 'presence of brick prevention'. |
| 10377 | Causation may not be a process, if a crucial part of the process is 'disconnected' [Schaffer,J] |
| Full Idea: One problem case for the process view of causation is 'disconnection'. If a brick breaks a window by being fired from a catapult, a latch is released which was preventing the catapult from firing, so the 'process' is just internal to the catapult. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.1) | |
| A reaction: Schaffer says the normal reply is to deny that the catch-releasing is genuinely causal. I would have thought we should go more fine-grained, and identify linked components of the causal process. |
| 10378 | A causal process needs to be connected to the effect in the right way [Schaffer,J] |
| Full Idea: A problem case for the process view of causation is 'misconnection'. A process may be connected to an effect, without being causal, as when someone watches an act of vandalism in dismay. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.1) | |
| A reaction: This is a better objection to the process view than Idea 10377. If I push a window with increasing force until it breaks, the process is continuous, but it suddenly becomes a cause. |
| 10382 | Causation can't be a process, because a process needs causation as a primitive [Schaffer,J] |
| Full Idea: It might be that if causation is said to be a process, then a process is nothing more than a causal sequence, so that causation is primitive. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2) | |
| A reaction: This again is tempting (as well as the primitivist view of probabilistic causation). If one tries to define a process as mere chronology, then the causal and accidental are indistinguishable. I take the label 'primitive' to be just our failure. |
| 10375 | At least four rivals have challenged the view that causal direction is time direction [Schaffer,J] |
| Full Idea: The traditional view that the direction of causation is the direction of time has been challenged, by the direction of forking, by overdetermination, by independence, and by manipulation, which all seem to be one-directional features. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 1.3.1) | |
| A reaction: Personally I incline to the view that time is prior, and fixes the direction of causation. I'm not sure that 'backward causation' can be stated coherently, even if it is metaphysically or naturally possible. |
| 10389 | Causal order must be temporal, or else causes could be blocked, and time couldn't be explained [Schaffer,J] |
| Full Idea: Reasons for causal order being temporal order are that otherwise the effect might occur but the cause then get prevented, ..and that they must be the same, because the temporal order can only be analysed in terms of the causal order. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.2) | |
| A reaction: If one took both time and causation as primitive, then the second argument would be void. The first argument, though, sounds pretty overwhelming to me. |
| 10390 | Causal order is not temporal, because of time travel, and simultanous, joint or backward causes [Schaffer,J] |
| Full Idea: Reasons for denying that causal order is temporal order are that time travel seems possible, that cause and effect can be simultaneous, because joint effects have temporal order without causal connection, and because backward causation may exist. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.2) | |
| A reaction: The possibility of time travel and backward causation can clearly be doubted, and certainly can't be grounds for one's whole metaphysics. The other two need careful analysis, but I think they can be answered. Causation is temporal. |
| 10380 | Causation is primitive; it is too intractable and central to be reduced; all explanations require it [Schaffer,J] |
| Full Idea: Primitivism arises from our failure to reduce causation, but also from causation being too central to reduce. The probability and process accounts are said to be inevitably circular, as they cannot be understood without reference to causation. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2) | |
| A reaction: This is very tempting. The primitive view, though, must deal with the direction problem, which may suggest that time is even more primitive. Can we have a hierarchy of primitiveness? To be alive is to be causal. |
| 10385 | If causation is just observables, or part of common sense, or vacuous, it can't be primitive [Schaffer,J] |
| Full Idea: The three main objections to causation being primitive are that causation can't be anything more than what we observe, or that such a primitive is too spooky to be acceptable, or that primitivism leads to elimination of causation. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2) | |
| A reaction: [summarised] I don't like the first (Humean) view. I suspect that anything which we finally decide has to be primitive (time, for example) is going to be left looking 'spooky', and I suspect that eliminativism is just Humeanism in disguise. |
| 10387 | The notion of causation allows understanding of science, without appearing in equations [Schaffer,J] |
| Full Idea: The concepts of 'event', 'law', 'cause' and 'explanation' are nomic concepts which serve to allow a systematic understanding of science; they do not themselves appear in the equations. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2) | |
| A reaction: This is a criticism of Russell's attempt to eliminate causation from science. It shows that there has to be something we can call 'metascience', which is the province of philosophers, since scientists don't have much interest in it. |
| 10388 | Causation is utterly essential for numerous philosophical explanations [Schaffer,J] |
| Full Idea: Causation can't be eliminated if it is needed to explain persistence, explanation, disposition, perception, warrant, action, responsibility, mental functional role, conceptual content, and reference. It's elimination would be catastrophic. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2) | |
| A reaction: [compressed list] I think I am going to vote for the view that causation is one of the primitives in the metaphysics of nature, so I have to agree with this. Most of the listed items, though, are controversial, so eliminativists are not defeated. |
| 10384 | If two different causes are possible in one set of circumstances, causation is primitive [Schaffer,J] |
| Full Idea: Causation seems to be primitive if the same laws and patterns of events might embody three different possible causes, as when two magicians cast the same successful spell, each with a 50% chance of success, and who was successful is unclear. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2) | |
| A reaction: I'm cautious when the examples involve magic. It implies that the process that leads to the result will be impossible to observe, but if magic never really happens, then the patterns of events will always be different. |
| 10386 | If causation is primitive, it can be experienced in ourselves, or inferred as best explanation [Schaffer,J] |
| Full Idea: The view that causation is primitive can be defended against Humean critics by saying that causation can be directly observed in the will or our bodies, or that it can be inferred as the best explanation. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2) | |
| A reaction: I like both views, and have just converted myself to the primitivist view of causation! I can't know the essence of a tree, because I am not a tree, but I can know the essence of causation. The Greek fascination with explaining movement is linked. |
| 10361 | Events are fairly course-grained (just saying 'hello'), unlike facts (like saying 'hello' loudly) [Schaffer,J] |
| Full Idea: Events are relatively coarse-grained, unlike facts; so the event of John's saying 'hello' seems to be the same event as John's saying 'hello' loudly, while they seem to be different facts. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 1) | |
| A reaction: The example seems good support for facts, since saying 'hello' loudly could have quite different effects from just saying 'hello'. I also incline temperamentally towards a fine-grained account, because it is more reductivist. |
| 10360 | Causal relata are events - or facts, features, tropes, states, situations or aspects [Schaffer,J] |
| Full Idea: The standard view make causal relata events (Davidson, Kim, Lewis), but there is considerable support for facts (Bennett, Mellor), and occasional support for features (Dretske), tropes (Campbell), states of affairs (Armstrong), and situations and aspects. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 1) | |
| A reaction: An event is presumed to be concrete, while a fact is more abstract (a proposition, perhaps). I'm always drawn to 'processes' (because they are good for discussing the mind), so an event, as a sort of natural process, looks good. |
| 10362 | One may defend three or four causal relata, as in 'c causes e rather than e*' [Schaffer,J] |
| Full Idea: The view that there are two causal relata is widely assumed but seldom defended. But the account based on 'effectual difference' says the form is 'c causes e rather than e*'. One might defend four relata, in 'c rather than c* causes e rather than e*'. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 1) | |
| A reaction: [compressed] This doesn't sound very plausible to me. How do you decide which is e*? If I lob a brick into the crowd, it hits Jim rather than - who? |
| 10368 | If causal relata must be in nature and fine-grained, neither facts nor events will do [Schaffer,J] |
| Full Idea: Theorists who reject both events and facts as causal relata do so because the relata must be immanent in nature, and thus not facts, but also fine-grained and thus not events. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 1.2) | |
|
A reaction:
Kim, however, offers a fine-grained account of events (as |
| 10383 | The relata of causation (such as events) need properties as explanation, which need causation! [Schaffer,J] |
| Full Idea: The primitivist about causation might say that the notion of an event (or other relata) cannot be understood without reference to causation, because properties themselves are individuated by their causal role. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2) | |
| A reaction: Having enthusiastically embraced the causal view of properties (see Shoemaker and Ellis), I suddenly realise that I seem required to embrace primitivism about causation, which I hadn't anticipated! I've no immediate problem with that. |
| 10393 | Our selection of 'the' cause is very predictable, so must have a basis [Schaffer,J] |
| Full Idea: The main argument against saying that there is no basis for selecting the one cause of an event is that our selections are too predictable to be without a basis. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.3) | |
| A reaction: The problem is that we CAN, if we wish, whimsically pick out any pre-condition of an event for discussion (e.g. the railways before WW1). I would say that sensitivity to nature leads us to a moderately correct selection of 'the' cause. |
| 10394 | Selecting 'the' cause must have a basis; there is no causation without such a selection [Schaffer,J] |
| Full Idea: Another argument against the view that there is no basis for selecting 'the' cause is that we have no concept of causation without such a selection. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.3) | |
| A reaction: Good. Otherwise we could only state the conditions preceding an event, and then every event that occurred at any given moment in a region would have the same cause. How can 'the' cause be necessary, and yet capricious? |
| 10376 | The actual cause may make an event less likely than a possible more effective cause [Schaffer,J] |
| Full Idea: If Pam threw the brick that broke the window, then Bob (who refrained) might be a more reliable vandal, so that Pam's throw might have made the shattering less likely, so probability-raising is not necessary for causation. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1) | |
| A reaction: That objection looks pretty conclusive to me. I take the probabilistic view to be a non-starter. |
| 10381 | All four probability versions of causation may need causation to be primitive [Schaffer,J] |
| Full Idea: All four probability versions of causation may need causation to be primitive: nomological - to distinguish laws from generalizations; statistical - to decide background; counterfactual - decide background; agent intervention - to understand intervention. | |
| From: Jonathan Schaffer (The Metaphysics of Causation [2007], 2.1.2) | |
| A reaction: I don't need much convincing that the probabilistic view is wrong. To just accept causation as primitive seems an awful defeat for philosophy. We should be able to characterise it, even if we cannot know its essence. |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |