100 ideas
| 7001 | If you begin philosophy with language, you find yourself trapped in it [Heil] |
| Full Idea: If you start with language and try to work your way outwards, you will never get outside language. | |
| From: John Heil (From an Ontological Point of View [2003], Pref) | |
| A reaction: This voices my pessimism about the linguistic approach to philosophy (and I don't just mean analysis of ordinary language), though I wonder if the career of (say) John Searle is a counterexample. |
| 7037 | Parsimony does not imply the world is simple, but that our theories should try to be [Heil] |
| Full Idea: A commitment to parsimony is not a commitment to a conception of the world as simple. The idea, rather, is that we should not complicate our theories about the world unnecessarily. | |
| From: John Heil (From an Ontological Point of View [2003], 13.6) | |
| A reaction: In other words, Ockham's Razor is about us, not about the world. It would be absurd to make the a priori assumption that the world has to be simple. Are we, though, creating bad theories by insisting that they should be simple? |
| 7038 | A theory with few fundamental principles might still posit a lot of entities [Heil] |
| Full Idea: It could well turn out that a simpler theory - a theory with fewer fundamental principles - posits more entities than a more complex competitor. | |
| From: John Heil (From an Ontological Point of View [2003], 13.6) | |
| A reaction: See also Idea 4036. The point here is that you can't simply translate Ockham as 'keep it simple', as there are different types of simplicity. The best theory will negotiate a balance between entities and principles. |
| 10528 | Definitions concern how we should speak, not how things are [Fine,K] |
| Full Idea: Our concern in giving a definition is not to say how things are by to say how we wish to speak | |
| From: Kit Fine (Precis of 'Limits of Abstraction' [2005], p.310) | |
| A reaction: This sounds like an acceptable piece of wisdom which arises out of analytical and linguistic philosophy. It puts a damper on the Socratic dream of using definition of reveal the nature of reality. |
| 7004 | The view that truth making is entailment is misguided and misleading [Heil] |
| Full Idea: I argue that the widely held view that truth making is to be understood as entailment is misguided in principle and potentially misleading. | |
| From: John Heil (From an Ontological Point of View [2003], Intro) | |
| A reaction: If reality was just one particle, what would entail the truths about it? Suppose something appears to be self-evident true about reality, but no one can think of any entailments to derive it? Do we assume a priori that they are possible? |
| 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 |
| 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). |
| 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 |
| 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. |
| 7035 | God does not create the world, and then add the classes [Heil] |
| Full Idea: It is hard to see classes as an 'addition of being'; God does not create the world, and then add the classes. | |
| From: John Heil (From an Ontological Point of View [2003], 13.4 n6) | |
| A reaction: This seems right. We may be tempted into believing in the reality of classes when considering maths, but it seems utterly implausible when considering trees or cows. |
| 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. |
| 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 |
| 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 |
| 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 |
| 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. |
| 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. |
| 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. |
| 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 |
| 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? |
| 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? |
| 10529 | If Hume's Principle can define numbers, we needn't worry about its truth [Fine,K] |
| Full Idea: Neo-Fregeans have thought that Hume's Principle, and the like, might be definitive of number and therefore not subject to the usual epistemological worries over its truth. | |
| From: Kit Fine (Precis of 'Limits of Abstraction' [2005], p.310) | |
| A reaction: This seems to be the underlying dream of logicism - that arithmetic is actually brought into existence by definitions, rather than by truths derived from elsewhere. But we must be able to count physical objects, as well as just counting numbers. |
| 10530 | Hume's Principle is either adequate for number but fails to define properly, or vice versa [Fine,K] |
| Full Idea: The fundamental difficulty facing the neo-Fregean is to either adopt the predicative reading of Hume's Principle, defining numbers, but inadequate, or the impredicative reading, which is adequate, but not really a definition. | |
| From: Kit Fine (Precis of 'Limits of Abstraction' [2005], p.312) | |
| A reaction: I'm not sure I understand this, but the general drift is the difficulty of building a system which has been brought into existence just by definition. |
| 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'. |
| 7017 | The reductionist programme dispenses with levels of reality [Heil] |
| Full Idea: The reductionist programme dispenses with levels of reality. | |
| From: John Heil (From an Ontological Point of View [2003], 04.3) | |
| A reaction: Fodor, for example, claims that certain causal laws only operate at high levels of reality. I agree with Heil's idea - the notion that there are different realities around here that don't connect properly to one another is philosopher's madness. |
| 7003 | There are levels of organisation, complexity, description and explanation, but not of reality [Heil] |
| Full Idea: We should accept levels of organisation, levels of complexity, levels of description, and levels of explanation, but not the levels of reality favoured by many anti-reductionists. The world is then ontologically, but not analytically, reductive. | |
| From: John Heil (From an Ontological Point of View [2003], Intro) | |
| A reaction: This sounds right to me. The crunch questions seem to be whether the boundaries at higher levels of organisation exist lower down, and whether the causal laws of the higher levels can be translated without remainder into lower level laws. |
| 7045 | Realism says some of our concepts 'cut nature at the joints' [Heil] |
| Full Idea: Realism is sometimes said to involve a commitment to the idea that certain of our concepts, those with respect to which we are realists, 'carve reality at the joints'. | |
| From: John Heil (From an Ontological Point of View [2003], 14.11) | |
| A reaction: Clearly not all concepts cut nature at the joints (e.g. we have concepts of things we know to be imaginary). Personally I am committed to this view of realism. I try very hard to use concepts that cut accurately; why shouldn't I sometimes succeed? |
| 7065 | Anti-realists who reduce reality to language must explain the existence of language [Heil] |
| Full Idea: Anti-realist philosophers, and those who hope to reduce metaphysics to (or replace it with) the philosophy of language, owe the rest of us an account of the ontology of language. | |
| From: John Heil (From an Ontological Point of View [2003], 20.6) | |
| A reaction: A nice turning-the-tables question. In all accounts of relativism, x is usually said to be relative to y. You haven't got proper relativism if you haven't relativised both x and y. But relativised them to what? Nietzsche's 'perspectivism' (Idea 4420)? |
| 7020 | Concepts don't carve up the world, which has endless overlooked or ignored divisions [Heil] |
| Full Idea: Concepts do not 'carve up' the world; the world already contains endless divisions, most of which we remain oblivious to or ignore. | |
| From: John Heil (From an Ontological Point of View [2003], 05.3) | |
| A reaction: Concepts could still carve up the world, without ever aspiring to do a complete job. We carve up the aspects that interest us, but the majority of the carving is in response to natural divisions, not whimsical conventions. |
| 7007 | I think of properties as simultaneously dispositional and qualitative [Heil] |
| Full Idea: Some philosophers who accept that properties are intrinsic features of objects regard them as pure powers, pure dispositionalities; I prefer to think of properties as simultaneously dispositional and qualitative. | |
| From: John Heil (From an Ontological Point of View [2003], Intro) | |
| A reaction: I am uneasy about 'qualitative' as a category, and am inclined to reduce it to being a dispositional power to cause primary and secondary qualities in observers. Roughness is only a power, not a quality, if there are no observers. |
| 7015 | A predicate applies truly if it picks out a real property of objects [Heil] |
| Full Idea: When a predicate applies truly to an object, it does so in virtue of designating a property possessed by that object and by every object to which the predicate truly applies (or would apply). | |
| From: John Heil (From an Ontological Point of View [2003], 03.3) | |
| A reaction: I am sympathetic to Heil's aim of shifting our attention from arbitrary predicates to natural properties, but it won't avoid Fodor's problem (Idea 7014) that all kinds of whimsical predicates will apply 'truly', but fail to pick out anything significant. |
| 7042 | A theory of universals says similarity is identity of parts; for modes, similarity is primitive [Heil] |
| Full Idea: The friend of universals has an account of similarity relations as relations of identity and partial identity; the friend of modes must regard similarity relations as primitive and irreducible. | |
| From: John Heil (From an Ontological Point of View [2003], 14.5) | |
| A reaction: We always seem to be able to ask 'in what respect' a similarity occurs. If similarity is 'primitive and irreducible', we should not be able to analyse and explain a similarity, yet we seem able to. I conclude that Heil is wrong. |
| 7023 | Powers or dispositions are usually seen as caused by lower-level qualities [Heil] |
| Full Idea: The modern default position on dispositionality is that powers or dispositions are higher-level properties objects possess by virtue of those objects' possession of lower-level qualitative (categorical) properties. | |
| From: John Heil (From an Ontological Point of View [2003], 09.2) | |
| A reaction: The new idea which is being floated by Heil, and which I prefer, is that dispositions or powers are basic. A 'quality' is a much more dubious entity than a power. |
| 7025 | Are a property's dispositions built in, or contingently added? [Heil] |
| Full Idea: There is a dispute over whether a property's dispositionality is built into the property or whether it is a contingent add-on. | |
| From: John Heil (From an Ontological Point of View [2003], 09.4) | |
| A reaction: Put that way, the idea that it is built in seems much more plausible. If it is an add-on, an explanation of why that disposition is added to that particular property seems required. If it is built in, it seems legitimate to accept it as a brute fact. |
| 7034 | Universals explain one-over-many relations, and similar qualities, and similar behaviour [Heil] |
| Full Idea: Universals can explain the one-over-many problem, and easily explain similarity relations between objects, and explain the similar behaviour of similar objects. | |
| From: John Heil (From an Ontological Point of View [2003], 13.1) | |
| A reaction: A useful summary. If you accept it, you seem to be faced with a choice between Plato (who has universals existing independently of particulars) and Armstrong (who makes them real, but existing only in particulars). |
| 7039 | How could you tell if the universals were missing from a world of instances? [Heil] |
| Full Idea: Imagine a pair of worlds, one in which there are the universals and their instances and one in which there are just the instances (a world of modes). How would the absence of universals make itself felt? | |
| From: John Heil (From an Ontological Point of View [2003], 13.7) | |
| A reaction: A nice question for Plato, very much in the spirit of Aristotle's string of questions. Compare 'suppose the physics remained, but someone removed the laws'. Either chaos ensues, or you realise they were redundant. Same with Forms. |
| 7009 | Similarity among modes will explain everthing universals were for [Heil] |
| Full Idea: My contention is that similarity among modes can do the job universals are conventionally postulated to do. | |
| From: John Heil (From an Ontological Point of View [2003], Intro) | |
| A reaction: See Idea 4441 for Russell's nice objection to this view. The very process by which we observes similarities (as assess their degrees) needs to be explained by any adequate theory of properties or universals. |
| 7041 | Similar objects have similar properties; properties are directly similar [Heil] |
| Full Idea: Objects are similar by virtue of possessing similar properties; properties, in contrast, are not similar in virtue of anything. | |
| From: John Heil (From an Ontological Point of View [2003], 14.2) | |
| A reaction: I am not sure if I can understand the concept of similarity if there is no answer to the question 'In what respect?' I suppose David Hume is happy to take resemblance as given and basic, but it could be defined as 'sharing identical properties'. |
| 7032 | Objects join sets because of properties; the property is not bestowed by set membership [Heil] |
| Full Idea: The set of red objects is the set of objects possessing a property: being red. Objects are members of the set in virtue of possessing this property; they do not possess the property in virtue of belonging to the set. | |
| From: John Heil (From an Ontological Point of View [2003], 12.2) | |
| A reaction: This seems to be a very effective denial of the claim that universals are sets. However, if 'being a Londoner' counts as a property, you can only have it by joining the London set. Being tall is more fundamental than being a Londoner. |
| 7008 | Trope theorists usually see objects as 'bundles' of tropes [Heil] |
| Full Idea: Philosophers identifying themselves as trope theorists have, by and large, accepted some form of the 'bundle theory' of objects: an object is a bundle of compresent tropes. | |
| From: John Heil (From an Ontological Point of View [2003], Intro) | |
| A reaction: This view eliminates anything called 'matter' or 'substance' or a 'bare particular'. I think I agree with Heil that this doesn't give a coherent picture, as properties seem to be 'of' something, and bundles always raise the question of what unites them. |
| 7018 | Objects are substances, which are objects considered as the bearer of properties [Heil] |
| Full Idea: I think of objects as substances, and a substance is an object considered as a bearer of properties. | |
| From: John Heil (From an Ontological Point of View [2003], 04.2) | |
| A reaction: This is an area of philosophy I always find disconcerting, where an account of how we should see objects seems to have no connection at all to what physicists report about objects. 'Considered as' seems to make substances entirely conventional. |
| 7019 | Maybe there is only one substance, space-time or a quantum field [Heil] |
| Full Idea: It would seem distinctly possible that there is but a single substance: space-time or some all-encompassing quantum field. | |
| From: John Heil (From an Ontological Point of View [2003], 05.2) | |
| A reaction: This would at least meet my concern that philosophers' 'substances' don't seem to connect to what physicists talk about. I wonder if anyone knows what a 'quantum field' is? The clash between relativity and quantum theory is being alluded to. |
| 7046 | Rather than 'substance' I use 'objects', which have properties [Heil] |
| Full Idea: I prefer the more colloquial 'object' to the traditional term 'substance'. An object can be regarded as a possessor of properties: as something that is red, spherical and pungent, for instance. | |
| From: John Heil (From an Ontological Point of View [2003], 15.3) | |
| A reaction: A nice move, but it seems to beg the question of 'what is it that has the properties?' Objects and substances do two different jobs in our ontology. Heil is just refusing to discuss what it is that has properties. |
| 7047 | Statues and bronze lumps have discernible differences, so can't be identical [Heil] |
| Full Idea: Applications of the principle of the indiscernibility of identicals apparently obliges us to distinguish the statue and the lump of bronze making it up. | |
| From: John Heil (From an Ontological Point of View [2003], 16.3) | |
| A reaction: In other words, statues and lumps of bronze have different properties. It is a moot point, though, whether there are any discernible differences between that statue at time t and its constituting lump of bronze at time t. |
| 7048 | Do we reduce statues to bronze, or eliminate statues, or allow statues and bronze? [Heil] |
| Full Idea: Must we choose between reductionism (the statue is the lump of bronze), eliminativism (there are no statues, only statue-shaped lumps of bronze), and a commitment to coincident objects? | |
| From: John Heil (From an Ontological Point of View [2003], 16.5) | |
| A reaction: (Heil goes on to offer his own view). Coincident objects sounds the least plausible view. Modern statues are only statues if we see them that way, but a tree is definitely a tree. Trenton Merricks is good on eliminativism. |
| 7028 | If properties were qualities without dispositions, they would be undetectable [Heil] |
| Full Idea: A pure quality, a property altogether lacking in dispositionality, would be undetectable and would, in one obvious sense, make no difference to its possessor. | |
| From: John Heil (From an Ontological Point of View [2003], 11.4) | |
| A reaction: This seems to be a very forceful and simple reason why we cannot view properties simply as qualities of things. Heil wants properties to be dispositions and qualities; personally I would vote for them just being dispositions or powers. |
| 7029 | Can we distinguish the way a property is from the property? [Heil] |
| Full Idea: It is not clear to me that we easily distinguish ways a property is from the property itself. | |
| From: John Heil (From an Ontological Point of View [2003], 11.6) | |
| A reaction: To defend properties as qualities, he is confusing ontology and epistemology. Presumably he means by 'ways a property is' what I would prefer to call 'ways a property seems to be'. I don't believe a smell is simply what it seems to be. |
| 7030 | Properties don't possess ways they are, because that just is the property [Heil] |
| Full Idea: Objects possess properties, but I am sceptical of the idea that properties possess properties; just as a property is a way some object is, a property of a property would be a way a property is, but that is just the property itself. | |
| From: John Heil (From an Ontological Point of View [2003], 12.1) | |
| A reaction: This is quite a good defence of the idea that properties are qualities as well as dispositions. However, if we make the qualities of properties into secondary qualities, and the dispositions into primary qualities, the absurdity melts away. |
| 7051 | Objects only have secondary qualities because they have primary qualities [Heil] |
| Full Idea: Secondary qualities are not distinct from primary qualities: an object's possession of a given secondary quality is a matter of its possession of certain complex primary qualities. | |
| From: John Heil (From an Ontological Point of View [2003], 17.3) | |
| A reaction: The bottom line here is that, if essentialism is right, colours are not properties at all (see Idea 5456). Heil wants to subsume secondary properties within primary properties. I think we should sharply distinguish them. |
| 7044 | Secondary qualities are just primary qualities considered in the light of their effect on us [Heil] |
| Full Idea: Secondary qualities are just ordinary properties - roughly, Locke's primary qualities - considered in the light of their effects on us. | |
| From: John Heil (From an Ontological Point of View [2003], 14.10) | |
| A reaction: Unconvincing. If they only acquire their ontological status as primary qualities if they have to be considered in relation to something (us), then that is not a primary quality. |
| 7052 | Colours aren't surface properties, because of radiant sources and the colour of the sky [Heil] |
| Full Idea: Theories that take colours to be properties of the surfaces of objects have difficulty accounting for a host of phenomena including coloured light emitted by radiant sources and so-called film colours (the colour of the sky, for instance). | |
| From: John Heil (From an Ontological Point of View [2003], 17.4) | |
| A reaction: Personally I never thought that colours might be actual properties of surfaces, but it is nice to have spelled out a couple of instances that make it very implausible. Neon and sodium lights I take to be examples of the first case. |
| 7053 | Treating colour as light radiation has the implausible result that tomatoes are not red [Heil] |
| Full Idea: Theories that tie colours to features of light radiation deal with radiant and diffused colours, but yield implausible results for objects; tomatoes are not red, on such a view, but merely reflect red light. | |
| From: John Heil (From an Ontological Point of View [2003], 17.4) | |
| A reaction: I see absolutely no problem with the philosophical denial that tomatoes are actually red, while continuing to use 'red' of tomatoes in the normal way. When we analyse our processes of knowledge acquisition, we must give up 'common sense'. |
| 7066 | If the world is just texts or social constructs, what are texts and social constructs? [Heil] |
| Full Idea: For those who regard the world as text or a social construct, are texts and social constructs real entities? If they are, what are they? | |
| From: John Heil (From an Ontological Point of View [2003], 20.6) | |
| A reaction: A nice turn-the-tables question. The oldest attacks of all on scepticism and relativism consist of showing that the positions themselves rest on knowledge or truth. Nietzsche may be the best model for relativists. E.g. Idea 4420. |
| 7021 | If the world is theory-dependent, the theories themselves can't be theory-dependent [Heil] |
| Full Idea: If the world is somehow theory-dependent, this implies, on pain of a regress, that theories are not theory-dependent. | |
| From: John Heil (From an Ontological Point of View [2003], 06.4) | |
| A reaction: I am not sure where this puts the ontology of theories, but this is a nice question, of a type which never seems to occur to your more simple-minded relativist. |
| 7026 | Science is sometimes said to classify powers, neglecting qualities [Heil] |
| Full Idea: The sciences are sometimes said to be in the business of identifying and classifying powers; the mass of an electron, its spin and charge, could be regarded as powers possessed by the electron; science is silent on an electron's qualities. | |
| From: John Heil (From an Ontological Point of View [2003], 11.2) | |
| A reaction: Heil raises the possibility that qualities are real, despite the silence of science; he wants colour to be a real quality. I like the simpler version of science. Qualities are the mental effects of powers; there exist substances, powers and effects. |
| 7060 | One form of explanation is by decomposition [Heil] |
| Full Idea: One form of explanation is by decomposition. | |
| From: John Heil (From an Ontological Point of View [2003], 19.8) | |
| A reaction: This is a fancy word for taking it apart, presumably to see how it works, which implies a functional explanation, rather than to see what it is made of, which seeks an ontological explanation. Simply 'decomposing' something wouldn't in itself explain. |
| 7010 | Dispositionality provides the grounding for intentionality [Heil] |
| Full Idea: Dispositionality provides the grounding for intentionality. | |
| From: John Heil (From an Ontological Point of View [2003], Intro) | |
| A reaction: This is a view with which I am sympathetic, though I am not sure if it explains anything. It would be necessary to identify a disposition of basic matter that could be built up into the disposition of a brain to think about things. |
| 7054 | Intentionality now has internalist (intrinsic to thinkers) and externalist (environment or community) views [Heil] |
| Full Idea: Nowadays philosophers concerned with intentionality divide into two camps. Internalists epitomise a traditional approach to thought, as intrinsic features of thinkers; externalists say it depends on contextual factors (environment or community). | |
| From: John Heil (From an Ontological Point of View [2003], 18.2) | |
| A reaction: This is basic to understanding modern debates (those that grow out of Putnam's Twin Earth). Externalism is fashionable, but I am reluctant to shake off my quaint internalism. Start by separating strict and literal meaning from speaker's meaning. |
| 7011 | Qualia are not extra appendages, but intrinsic ingredients of material states and processes [Heil] |
| Full Idea: Properties of conscious experience, the so-called qualia, are not dangling appendages to material states and processes but intrinsic ingredients of those states and processes. | |
| From: John Heil (From an Ontological Point of View [2003], Intro) | |
| A reaction: Personally I am inclined to the view that qualia are intrinsic to the processes and NOT to the 'states'. Heil must be right, though. I am sure qualia are not just epiphenomena - they are too useful. |
| 7061 | Philosophers' zombies aim to show consciousness is over and above the physical world [Heil] |
| Full Idea: Philosophers' zombies (invented by Robert Kirk) differ from the zombies of folklore; they are intended to make clear the idea that consciousness is an addition of being, something 'over and above' the physical world. | |
| From: John Heil (From an Ontological Point of View [2003], 20.1 n1) | |
| A reaction: The famous defender of zombies is David Chalmers. You can't believe in zombies if you believe (as I do) that 'the physical entails the mental'. Could there be redness without something that is red? If consciousness is extra, what is conscious? |
| 7063 | Zombies are based on the idea that consciousness relates contingently to the physical [Heil] |
| Full Idea: The possibility of zombies is founded on the idea that consciousness is related contingently to physical states and processes. | |
| From: John Heil (From an Ontological Point of View [2003], 20.3) | |
| A reaction: The question is, how do you decide whether the relationship is contingent or necessary? Hence the interest in whether conceivability entails possibility. Kripke attacks the idea of contingent identity, pointing towards necessity, and away from zombies. |
| 7064 | Functionalists deny zombies, since identity of functional state means identity of mental state [Heil] |
| Full Idea: Functionalists deny that zombies are possible since states of mind (including conscious states) are purely functional states. If two agents are in the same functional state, regardless of qualitative difference, they are in the same mental state. | |
| From: John Heil (From an Ontological Point of View [2003], 20.5) | |
| A reaction: In its 'brief' form this idea begins to smell of tautology. Only the right sort of functional state would entail a mental state, and how else can that functional state be defined, apart from its leading to a mental state? |
| 7027 | Functionalists say objects can be the same in disposition but differ in quality [Heil] |
| Full Idea: A central tenet of functionalism is that objects can be dispositionally indiscernible but differ qualitatively as much as you please. | |
| From: John Heil (From an Ontological Point of View [2003], 11.3) | |
| A reaction: This refers to the multiple realisability of functions. Presumably we reconcile essentialism with the functionalist view by saying that dispositions result from combinations of qualities. A unique combination of qualities will necessitate a disposition. |
| 7062 | Functionalism cannot explain consciousness just by functional organisation [Heil] |
| Full Idea: Functionalism has been widely criticized on the grounds that it is implausible to think that functional organization alone could suffice for conscious experience. | |
| From: John Heil (From an Ontological Point of View [2003], 20.2) | |
| A reaction: He cites Block's 'Chinese Mind' as an example. The obvious reply is that you can't explain consciousness with a lump of meat, or with behaviour, or with an anomalous property, or even with a non-physical substance. |
| 7059 | The 'explanatory gap' is used to say consciousness is inexplicable, at least with current concepts [Heil] |
| Full Idea: The expression 'explanatory gap' was coined by Joseph Levine in 1983. McGinn and Chalmers have invoked it in defence of the view that consciousness is physically inexplicable, and Nagel that it is inexplicable given existing conceptual resources. | |
| From: John Heil (From an Ontological Point of View [2003], 19.8 n14) | |
| A reaction: Coining a few concepts isn't going to help, but discovering more about the brain might. With computer simulations we will 'see' more of the physical end of thought. Psychologists may break thought down into physically more manageable components. |
| 7012 | If a car is a higher-level entity, distinct from its parts, how could it ever do anything? [Heil] |
| Full Idea: If we regard a Volvo car as a higher-level entity with its own independent reality, something distinct from its constituents (arranged in particular ways and variously connected to other things), we render mysterious how Volvos could do anything at all. | |
| From: John Heil (From an Ontological Point of View [2003], 02.3) | |
| A reaction: This seems to me perhaps the key reason why we have to be reductionists. The so-called 'bridge laws' from mind to brain are not just needed to explain the mind, they are also essential to show how a mind would cause behaviour. |
| 7043 | Multiple realisability is actually one predicate applying to a diverse range of properties [Heil] |
| Full Idea: Cases of multiple realisability are typically cases in which some predicate ('is red', 'is in pain') applies to an object in virtue of that object's possession of any of a diverse range of properties. | |
| From: John Heil (From an Ontological Point of View [2003], 14.8) | |
| A reaction: If the properties are diverse, why does one predicate apply to them? I take it that in the case of the pain, the predicate is ambiguous in applying to the behaviour or the phenomenal property. Same behaviour is possible with many qualia. |
| 7058 | Externalism is causal-historical, or social, or biological [Heil] |
| Full Idea: Some externalists focus on causal-historical connections, others emphasise social matters (especially thinkers' linguistic communities), still others focus on biological function. | |
| From: John Heil (From an Ontological Point of View [2003], 18.5 n6) | |
| A reaction: Helpful. The social view strikes me as the one to take most seriously (allowing for contextual views of justification, and for the social role of experts). The problem is to combine the social view with realism and a robust view of truth. |
| 7057 | Intentionality is based in dispositions, which are intrinsic to agents, suggesting internalism [Heil] |
| Full Idea: I suggest that intentionality is grounded in the dispositionalities of agents. Dispositions are intrinsic to agents, so this places me on the side of the internalists and against the externalists. | |
| From: John Heil (From an Ontological Point of View [2003], 18.4) | |
| A reaction: I think this is a key idea, and the right view. The key question is whether we see intentionality as active or passive. The externalist view seems to see the brain as a passive organ which the world manipulates. If the brain is active, what is it doing? |
| 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. |
| 10527 | An abstraction principle should not 'inflate', producing more abstractions than objects [Fine,K] |
| Full Idea: If an abstraction principle is going to be acceptable, then it should not 'inflate', i.e. it should not result in there being more abstracts than there are objects. By this mark Hume's Principle will be acceptable, but Frege's Law V will not. | |
| From: Kit Fine (Precis of 'Limits of Abstraction' [2005], p.307) | |
| A reaction: I take this to be motivated by my own intuition that abstract concepts had better be rooted in the world, or they are not worth the paper they are written on. The underlying idea this sort of abstraction is that it is 'shared' between objects. |
| 7013 | The Picture Theory claims we can read reality from our ways of speaking about it [Heil] |
| Full Idea: The theory of language which I designate the 'Picture Theory' says that language pictures reality in roughly the sense that we can 'read off' features of reality from our ways of speaking about it. | |
| From: John Heil (From an Ontological Point of View [2003], 03.2) | |
| A reaction: Heil, quite rightly, attacks this view very strongly. I think of it as the great twentieth century philosophical heresy, that leads to shocking views like relativism and anti-realism. |
| 7002 | If propositions are states of affairs or sets of possible worlds, these lack truth values [Heil] |
| Full Idea: When pressed, philosophers will describe propositions as states of affairs or sets of possible worlds. But wait! Neither sets of possible worlds nor states of affairs - electrons being negatively charged, for instance - have truth values. | |
| From: John Heil (From an Ontological Point of View [2003], Intro) | |
| A reaction: I'm not sure that I see a problem. A pure proposition, expressed as, say "there is a giraffe on the roof" only acquires a truth value at the point where you assert it or believe it. There IS a possible world where there is a giraffe on the roof. |
| 7016 | The standard view is that causal sequences are backed by laws, and between particular events [Heil] |
| Full Idea: The notion that every causal sequence if backed by a law, like the idea that causation is a relation among particular events, forms a part of philosophy's Humean heritage. | |
| From: John Heil (From an Ontological Point of View [2003], 04.3) | |
| A reaction: This nicely pinpoints a view that needs to come under attack. I take the view that there are no 'laws' - other than the regularities in behaviour that result from the interaction of essential dispositional properties. Essences don't need laws. |
| 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 |
| 7036 | The real natural properties are sparse, but there are many complex properties [Heil] |
| Full Idea: I am sympathetic to the idea that the real properties are 'sparse'; ...but if, in counting kinds of property, we include complex properties as well as simple properties, the image of sparseness evaporates. | |
| From: John Heil (From an Ontological Point of View [2003], 13.4) | |
| A reaction: This seems right to me, and invites the obvious question of which are the sparse real properties. Presumably we let the physicists tell us that, though Heil wants to include qualities like phenomenal colour, which physicists ignore. |
| 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'. |