74 ideas
| 13395 | If an analysis shows the features of a concept, it doesn't seem to 'reduce' the concept [Jubien] |
| Full Idea: An analysis of a concept tells us what the concept is by telling us what its constituents are and how they are combined. ..The features of the concept are present in the analysis, making it surprising the 'reductive' analyses are sought. | |
| From: Michael Jubien (Possibility [2009], 4.5) | |
| A reaction: He says that there are nevertheless reductive analyses, such as David Lewis's analysis of modality. We must disentangle conceptual analysis from causal analysis (e.g. in his example of the physicalist reduction of mind). |
| 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 |
| 9616 | A set is a collection into a whole of distinct objects of our intuition or thought [Cantor] |
| Full Idea: A set is any collection into a whole M of definite, distinct objects m ... of our intuition or thought. | |
| From: George Cantor (The Theory of Transfinite Numbers [1897], p.85), quoted by James Robert Brown - Philosophy of Mathematics Ch.2 | |
| A reaction: This is the original conception of a set, which hit trouble with Russell's Paradox. Cantor's original definition immediately invites thoughts about the status of vague objects. |
| 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. |
| 13378 | It is a mistake to think that the logic developed for mathematics can clarify language and philosophy [Jubien] |
| Full Idea: It has often been uncritically assumed that logic that was initially a tool for clarifying mathematics could be seamlessly and uniformly applied in the effort to clarify ordinary language and philosophy, but this has been a real mistake. | |
| From: Michael Jubien (Possibility [2009], Intro) | |
| A reaction: I'm not saying he's right (since you need stupendous expertise to make that call) but my intuitions are that he has a good point, and he is at least addressing a crucial question which most analytical philosophers avert their eyes from. |
| 13402 | We only grasp a name if we know whether to apply it when the bearer changes [Jubien] |
| Full Idea: We cannot be said to have a full grasp of a name unless we have a definite disposition to apply it or to withhold it under whatever conceivable changes the bearer of the name might come to undergo. | |
| From: Michael Jubien (Possibility [2009], 5.3) | |
| A reaction: This is right, and an excellent counterproposal to the logicians' notion that names have to rigidly designate. As a bare minimum, you are not supposed to deny the identity of your parents because they have grown a bit older, or a damaged painting. |
| 13405 | The baptiser picks the bearer of a name, but social use decides the category [Jubien] |
| Full Idea: The person who introduces a proper name gets to pick its bearer, but its category - and consequently the meaning of the name - is determined by social use. | |
| From: Michael Jubien (Possibility [2009], 7) | |
| A reaction: New 'division of labour'. The idea that a name has some sort of meaning seems right and important. If babies were switched after baptism, social use might fix the name to the new baby. The namer could stipulate the category at the baptism. Too neat. |
| 13399 | Examples show that ordinary proper names are not rigid designators [Jubien] |
| Full Idea: There are plenty of examples to show that ordinary proper names simply are not rigid designators. | |
| From: Michael Jubien (Possibility [2009], 5.1) | |
| A reaction: His examples are the planet Venus and the dust of which it is formed, and a statue made of clay. In other words, for some objects, perhaps under certain descriptions (e.g. functional ones), the baptised matter can change. Rigidity is an extra topping. |
| 13398 | We could make a contingent description into a rigid and necessary one by adding 'actual' to it [Jubien] |
| Full Idea: 'The winner of the Derby' satisfies some horse, but only accidentally. But we could 'rigidify' the description by inserting 'actual' into it, giving 'the actual winner of the Derby'. Winning is a contingent property, but actually winning is necessary. | |
| From: Michael Jubien (Possibility [2009], 5.1) | |
| A reaction: I like this unusual proposal because instead of switching into formal logic in order to capture the ideas we are after, he is drawing on the resources of ordinary language, offering philosophers a way of speaking plain English more precisely. |
| 13392 | Philosophers reduce complex English kind-quantifiers to the simplistic first-order quantifier [Jubien] |
| Full Idea: There is a readiness of philosophers to 'translate' English, with its seeming multitude of kind-driven quantifiers, into first-order logic, with its single wide-open quantifier. | |
| From: Michael Jubien (Possibility [2009], 4.1) | |
| A reaction: As in example he says that reference to a statue involves a 'statue-quantifier'. Thus we say things about the statue that we would not say about the clay, which would involve a 'clay-quantifier'. |
| 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 |
| 15896 | Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine] |
| Full Idea: Cantor grafted the Power Set axiom onto his theory when he needed it to incorporate the real numbers, ...but his theory was supposed to be theory of collections that can be counted, but he didn't know how to count the new collections. | |
| From: report of George Cantor (The Theory of Transfinite Numbers [1897]) by Shaughan Lavine - Understanding the Infinite I | |
| A reaction: I take this to refer to the countability of the sets, rather than the members of the sets. Lavine notes that counting was Cantor's key principle, but he now had to abandon it. Zermelo came to the rescue. |
| 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 |
| 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. |
| 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? |
| 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'. |
| 13404 | To exist necessarily is to have an essence whose own essence must be instantiated [Jubien] |
| Full Idea: For a thing to exist necessarily is for it to have an entity-essence whose own entity-essence entails being instantiated. | |
| From: Michael Jubien (Possibility [2009], 6.4) | |
| A reaction: This is the culmination of a lengthy discussion, and is not immediately persuasive. For Jubien the analysis rests on a platonist view of properties, which doesn't help. |
| 13386 | If objects are just conventional, there is no ontological distinction between stuff and things [Jubien] |
| Full Idea: Under the Quinean (conventional) view of objects, there is no ontological distinction between stuff and things. | |
| From: Michael Jubien (Possibility [2009], 1.5) | |
| A reaction: This is the bold nihilistic account of physical objects, which seems to push all of our ontology into language (English?). We could devise divisions into things that were just crazy, and likely to lead to the rapid extinction of creatures who did it. |
| 13403 | The category of Venus is not 'object', or even 'planet', but a particular class of good-sized object [Jubien] |
| Full Idea: The category of Venus is not 'physical object' or 'mereological sum', but narrower. Surprisingly, it is not 'planet', since it might cease to be a planet and still merit the name 'Venus'. It is something like 'well-integrated, good-sized physical object'. | |
| From: Michael Jubien (Possibility [2009], 5.3) | |
| A reaction: Jubien is illustrating Idea 13402. This is a nice demonstration of how one might go about the task of constructing categories - by showing the modal profiles of things to which names have been assigned. Categories are file names. |
| 13375 | The idea that every entity must have identity conditions is an unfortunate misunderstanding [Jubien] |
| Full Idea: The pervasiveness, throughout philosophy, of the assumption that entities of various kinds need identity conditions is one unfortunate aspect of Quine's important philosophical legacy. | |
| From: Michael Jubien (Possibility [2009], Intro) | |
| A reaction: Lowe seems to be an example of a philosopher who habitually demands individuation conditions for everything that is referred to. Presumably the alternative is to take lots of things as primitive, but this seems to be second best. |
| 13393 | Any entity has the unique property of being that specific entity [Jubien] |
| Full Idea: For any entity of any sort, abstract or concrete, I assume there is a property of being that specific entity. For want of a better term, I will call such properties entity-essences. They are 'singulary' - not instantiable by more than one thing at a time. | |
| From: Michael Jubien (Possibility [2009], 4.2) | |
| A reaction: Baffling. Why would someone who has mocked all sorts of bogus philosophical claims based on logic then go on to assert the existence of such weird things as these? I can't make sense of this property being added to a thing's other properties. |
| 13388 | It is incoherent to think that a given entity depends on its kind for its existence [Jubien] |
| Full Idea: It is simply far-fetched - even incoherent - to think that, given an entity, of whatever kind, its being a single entity somehow consists in its satisfying some condition involving the kind to which it belongs (or concepts related to that kind). | |
| From: Michael Jubien (Possibility [2009], 2.3) | |
| A reaction: Well said. I can't see how philosophers have allowed themselves to drift into such a daft view. Kinds blatantly depend on the individuals that constitute them, so how could the identity of the individuals depend on their kind? |
| 13384 | Objects need conventions for their matter, their temporal possibility, and their spatial possibility [Jubien] |
| Full Idea: We need a first convention to determine what matter constitutes objects, then a second to determine whether there are different temporal possibilities for a given object, then a third for different spatial possibilities. | |
| From: Michael Jubien (Possibility [2009], 1.5) | |
| A reaction: This is building up a Quinean account of objects, as mere matter in regions of spacetime, which are then precisely determined by a set of social conventions. |
| 13385 | Basically, the world doesn't have ready-made 'objects'; we carve objects any way we like [Jubien] |
| Full Idea: There is a certain - very mild - sense in which I don't think the physical world comes with ready-made objects. I think instead that we (conventionally) carve it up into objects, and this can be done any way we like. | |
| From: Michael Jubien (Possibility [2009], 1.5) | |
| A reaction: I have no idea how one could begin to refute such a view. Obviously there are divisions (even if only of physical density) in the world, but nothing obliges us to make divisions at those points. We happily accept objects with gaps in them. |
| 13383 | If the statue is loved and the clay hated, that is about the object first qua statue, then qua clay [Jubien] |
| Full Idea: If a sculptor says 'I love the statue but I really hate that piece of clay - it is way too hard to work with' ...the statement is partly is partly about that object qua statue and partly about that object qua piece of clay. | |
| From: Michael Jubien (Possibility [2009], 1.4) | |
| A reaction: His point is that identity is partly determined by the concept or category under which the thing falls. Plausible. Lots of identity muddles seem to come from our conceptual scheme not being quite up to the job when things change. |
| 13400 | If one entity is an object, a statue, and some clay, these come apart in at least three ways [Jubien] |
| Full Idea: A single entity is a physical object, a piece of clay and a statue. We seem to have that the object could be scattered, but not the other two; the object and the clay could be spherical, but not the statue; and only the object could have different matter. | |
| From: Michael Jubien (Possibility [2009], 5.2) | |
| A reaction: His proposal, roughly, is to reduce object-talk to property-talk, and then see the three views of this object as referring to different sets of properties, rather than to a single thing. Promising, except that he goes platonist about properties. |
| 13401 | The idea of coincident objects is a last resort, as it is opposed to commonsense naturalism [Jubien] |
| Full Idea: I find it surprising that some philosophers accept 'coincident objects'. This notion clearly offends against commonsense 'naturalism' about the world, so it should be viewed as a last resort. | |
| From: Michael Jubien (Possibility [2009], 5.2 n9) | |
| A reaction: I'm not quite clear why he invokes 'naturalism', but I pass on his intuition because it seems right to me. |
| 13380 | Parts seem to matter when it is just an object, but not matter when it is a kind of object [Jubien] |
| Full Idea: When thought of just as an object, the parts of a thing seem definitive and their arrangement seems inconsequential. But when thought of as an object of a familiar kind it is reversed: the arrangement is important and the parts are inessential. | |
| From: Michael Jubien (Possibility [2009], 1.4) | |
| A reaction: This is analogous to the Ship of Theseus, where we say that the tour operator and the museum keeper give different accounts of whether it is the same ship. The 'kind' Jubien refers to is most likely to be a functional kind. |
| 13376 | We should not regard essentialism as just nontrivial de re necessity [Jubien] |
| Full Idea: I argue against the widely accepted characterization of the doctrine of 'essentialism' as the acceptance of nontrivial de re necessity | |
| From: Michael Jubien (Possibility [2009], Intro) | |
| A reaction: I agree entirely. The notion of an essence is powerful if clearly distinguished. The test is: can everything being said about essences be just as easily said by referring to necessities? If so, you are talking about the wrong thing. |
| 13381 | Thinking of them as 'ships' the repaired ship is the original, but as 'objects' the reassembly is the original [Jubien] |
| Full Idea: Thinking about the original ship as a ship, we think we continue to have the 'same ship' as each part is replaced; ...but when we think of them as physical objects, we think the original ship and the outcome of the reassembly are one and the same. | |
| From: Michael Jubien (Possibility [2009], 1.4) | |
| A reaction: It seems to me that you cannot eliminate how we are thinking of the ship as influencing how we should read it. My suggestion is to think of Theseus himself valuing either the repaired or the reassembled version. That's bad for Jubien's account. |
| 13382 | Rearranging the planks as a ship is confusing; we'd say it was the same 'object' with a different arrangement [Jubien] |
| Full Idea: That the planks are rearranged as a ship elevates the sense of mystery, because arrangements matter for ships, but if they had been arranged differently we would have the same intuition - that it still counts as the same object. | |
| From: Michael Jubien (Possibility [2009], 1.4) | |
| A reaction: Implausible. Classic case: can I have my pen back? - smashes it to pieces and hands it over with 'there you are' - that's not my pen! - Jubien says it's the same object! - it isn't my pen, and it isn't the same object either! Where is Shelley's skylark? |
| 13379 | If two objects are indiscernible across spacetime, how could we decide whether or not they are the same? [Jubien] |
| Full Idea: If a bit of matter has a qualitatively indistinguishable object located at a later time, with a path of spacetime connecting them, how could we determine they are identical? Neither identity nor diversity follows from qualitative indiscernibility. | |
| From: Michael Jubien (Possibility [2009], 1.3) | |
| A reaction: All these principles expounded by Leibniz were assumed to be timeless, but for identity over time the whole notion of things retaining identity despite changing has to be rethought. Essentialism to the rescue. |
| 13394 | Entailment does not result from mutual necessity; mutual necessity ensures entailment [Jubien] |
| Full Idea: Typically philosophers say that for P to entail Q is for the proposition that all P's are Q's to be necessary. I think this analysis is backwards, and that necessity rests on entailment, not vice versa. | |
| From: Michael Jubien (Possibility [2009], 4.4) | |
| A reaction: His example is that being a horse and being an animal are such that one entails the other. In other words, necessities arise out of property relations (which for Jubien are necessary because the properties are platonically timeless). Wrong. |
| 13391 | Modality concerns relations among platonic properties [Jubien] |
| Full Idea: I think modality has to do with relations involving the abstract part of the world, specifically with relations among (Platonic) properties. | |
| From: Michael Jubien (Possibility [2009], 3.2) | |
| A reaction: [Sider calls Jubien's the 'governance' view, since abstract relations govern the concrete] I take Jubien here (having done a beautiful demolition job on the possible worlds account of modality) to go spectacularly wrong. Modality starts in the concrete. |
| 13374 | To analyse modality, we must give accounts of objects, properties and relations [Jubien] |
| Full Idea: The ultimate analysis of possibility and necessity depends on two important ontological decisions: the choice of an analysis of the intuitive concept of a physical object, and the other is the positing of properties and relations. | |
| From: Michael Jubien (Possibility [2009], Intro) | |
| A reaction: In the same passage he adopts Quine's view of objects, leading to mereological essentialism, and a Platonic view of properties, based on Lewis's argument for taking some things at face value. One might start with processes and events instead. |
| 13389 | The love of possible worlds is part of the dream that technical logic solves philosophical problems [Jubien] |
| Full Idea: I believe the contemporary infatuation with possible worlds in philosophy stems in part from a tendency to think that technical logic offers silver-bullet solutions to philosophical problems. | |
| From: Michael Jubien (Possibility [2009], 3.2) | |
| A reaction: I would say that the main reason for the infatuation is just novelty. As a technical device it was only invented in the 1960s, so we are in a honeymoon period, as we would be with any new gadget. I can't imagine possible worlds figuring much in 100 years. |
| 13390 | Possible worlds don't explain necessity, because they are a bunch of parallel contingencies [Jubien] |
| Full Idea: The fundamental problem is that in world theory, what passes for necessity is in effect just a bunch of parallel 'contingencies'. | |
| From: Michael Jubien (Possibility [2009], 3.2) | |
| A reaction: Jubien's general complaint is that there is no connection between the possible worlds and the actual world, so they are irrelevant, but this is a nicely different point - that lots of contingent worlds can't add up to necessity. Nice. |
| 13396 | Analysing mental concepts points to 'inclusionism' - that mental phenomena are part of the physical [Jubien] |
| Full Idea: We have (physicalist) 'inclusionism' when the mental is included in the physical, and mental phenomena are to be found among physical phenomena. Only inclusionism is compatible with a genuine physicalist analysis of mental concepts. | |
| From: Michael Jubien (Possibility [2009], 4.5) | |
| A reaction: This isn't the thesis of conceptual dualism (which I like), but an interesting accompaniment for it. Jubien is offering this as an alternative to 'reductive' analysis, translating all the mental concepts into physical language. He extends 'physical'. |
| 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. |
| 13377 | First-order logic tilts in favour of the direct reference theory, in its use of constants for objects [Jubien] |
| Full Idea: First-order logic tilts in favor of the direct reference account of proper names by using individual constants to play the intuitive role of names, and by 'interpreting' the constants simply as the individuals that are assigned to them for truth-values. | |
| From: Michael Jubien (Possibility [2009], Intro) | |
| A reaction: This is the kind of challenge to orthodoxy that is much needed at the moment. We have an orthodoxy which is almost a new 'scholasticism', that logic will clarify our metaphysics. Trying to enhance the logic for the job may be a dead end. |
| 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'. |