98 ideas
| 5486 | Essentialism says metaphysics can't be done by analysing unreliable language [Ellis] |
| Full Idea: The new essentialism leads to a turning away from semantic analysis as a fundamental tool for the pursuit of metaphysical aims, ..since there is no reason to think that the language we speak accurately reflects the kind of world we live in. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: The last part of that strikes me as false. We have every reason to think that a lot of our language very accurately reflects reality. It had better, because we have no plan B. We should analyse our best concepts, but not outdated, culture-laden ones. |
| 3969 | There are no ultimate standards of rationality, since we only assess others by our own standard [Davidson] |
| Full Idea: It makes no sense to speak of comparing or agreeing on ultimate standards of rationality, since it is our own standards in each case to which we must turn in interpreting others. This is not a failure of objectivity, but where 'questions come to an end'. | |
| From: Donald Davidson (Davidson on himself [1994], p.232) | |
| A reaction: This seems wrong, given the commitment to truth and charity in interpretation. He could have said the same about perception, but I doubt if he would. |
| 3972 | Truth and objectivity depend on a community of speakers to interpret what they mean [Davidson] |
| Full Idea: The basis on which the concepts of truth and objectivity depend for application is a community of understanding, agreement among speakers on how each is to be understood. | |
| From: Donald Davidson (Davidson on himself [1994], p.233) | |
| A reaction: Obviously all understanding is, in practice, an interpretation by a community, but that isn't what 'truth' means. We mean 'true independently of any community'. |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 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'. |
| 5468 | Properties are 'dispositional', or 'categorical' (the latter as 'block' or 'intrinsic' structures) [Ellis, by PG] |
| Full Idea: 'Dispositional' properties involve behaviour, and 'categorical properties' are structures in two or more dimensions. 'Block' structures (e.g. molecules) depend on other things, and 'instrinsic' structures (e.g. fields) involve no separate parts. | |
| From: report of Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.4) by PG - Db (ideas) | |
| A reaction: This is an essentialist approach to properties, and sounds correct to me. The crucial preliminary step to understanding properties is to eliminate secondary qualities (e.g. colour), which are not properties at all, and cause confusion. |
| 5469 | The passive view of nature says categorical properties are basic, but others say dispositions [Ellis] |
| Full Idea: 'Categorical realism' is the most widely accepted theory of dispositional properties, because passivists can accept it, ..that is, that dispositions supervene on categorical properties; ..the opposite would imply nature is active and reactive. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.4) | |
| A reaction: Essentialists believe 'the opposite' - i.e. that dispositions are fundamental, and hence that the essence of nature is active. See 5468 for explanations of the distinctions. I am with the essentialists on this one. |
| 5456 | Redness is not a property as it is not mind-independent [Ellis] |
| Full Idea: Redness is not a property, because it has no mind-independent existence. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: Well said. Secondary qualities are routinely cited in discussions of properties, and they shouldn't be. Redness causes nothing to happen in the physical world, unless a consciousness experiences it. |
| 5481 | Properties have powers; they aren't just ways for logicians to classify objects [Ellis] |
| Full Idea: One cannot think of a property as just a set of objects in a domain (as Fregean logicians do), as though the property has no powers, but is just a way of classifying objects. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: I agree. It is sometimes suggested that properties are what 'individuate' objects, but how could they do that if they didn't have some power? If properties are known by their causal role, why do they have that causal role? |
| 5458 | Nearly all fundamental properties of physics are dispositional [Ellis] |
| Full Idea: With few, if any, exceptions, the fundamental properties of physical theory are dispositional properties of the things that have them. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: He is denying that they are passive (as Locke saw primary qualities), and says they are actively causal, or else capacities or propensities. Sounds right to me. |
| 5443 | Kripke and others have made essentialism once again respectable [Ellis] |
| Full Idea: The revival of essentialism owes much to the work of Saul Kripke and Hilary Putnam, who made belief in essences once again respectable, with Harré and Madden arguing that there were real causal powers in nature. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Intro) | |
| A reaction: It seems to me important to separate two stages of this: 1) causation results from essences, and 2) essences can never change. The first seems persuasive to me. For the second, see METAPHYSICS/IDENTITY/COUNTERPARTS. |
| 5444 | 'Individual essences' fix a particular individual, and 'kind essences' fix the kind it belongs to [Ellis] |
| Full Idea: The new essentialism retains Aristotelian ideas about essential properties, but it distinguishes more clearly between 'individual essences' and 'kind essences'; the former define a particular individual, the latter what kind it belongs to. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.1) | |
| A reaction: This might actually come into conflict with Aristotle, who seems to think that my personal essence is largely a human nature I share with everyone else. The new distinction is trying to keep the Kantian individual on the stage. |
| 5462 | Essential properties are usually quantitatively determinate [Ellis] |
| Full Idea: Most of the essential properties of things are quantitatively determinate properties. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: This makes the essential nature of the world very much the province of science, which deals in quantities and equations. Essentialists must deal with mental events, as well as basic physics. |
| 5448 | 'Real essence' makes it what it is; 'nominal essence' makes us categorise it a certain way [Ellis] |
| Full Idea: The 'real essence' of a thing is that set of its properties or structures in virtue of which it is a thing of that kind; its 'nominal essence' is the properties or structures in virtue of which it is described as a thing of that kind. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.1) | |
| A reaction: I like this distinction, because it is the kind made by realists like me who are fighting to make philosophers keep their epistemology and their ontology separate. |
| 5477 | One thing can look like something else, without being the something else [Ellis] |
| Full Idea: In considering questions of real possibility, it is important to keep the distinction between what a thing is and what it looks like clearly in mind. There is a possible world containing a horse that could then look like a cow, but it wouldn't BE a horse. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.6) | |
| A reaction: This is an interesting test assertion of the notion that there are essences (although Ellis does not allow that animals actually have essences - how could you, given evolution?). His point is a good one. |
| 5479 | Scientific essentialists say science should define the limits of the possible [Ellis] |
| Full Idea: Scientific essentialists hold that one of the primary aims of science is to define the limits of the possible. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.6) | |
| A reaction: I'm not sure working scientists will go along with that, but I like the claim that philosophy is very much part of the same enterprise as practical science (and NOT subservient to it!). I think of metaphysics as very high level physics. |
| 5483 | Essentialists deny possible worlds, and say possibilities are what is compatible with the actual world [Ellis] |
| Full Idea: Essentialists are modal realists; ..what is really possible, they say, is what is compatible with the natures of things in this world (and this does not commit them to the existence of any world other than the actual world). | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: This introduces something like 'compatibilities' into our ontology. That must rest on some kind of idea of a 'natural contradiction'. We can discuss the possibilities resulting from essences, but what are the possible variations in the essences? |
| 5447 | Metaphysical necessities are true in virtue of the essences of things [Ellis] |
| Full Idea: Metaphysical necessities are propositions that are true in virtue of the essences of things. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.1) | |
| A reaction: I am cautious about this. It sounds like huge Leibnizian metaphysical claims riding in on the back of a rather sensible new view of the laws of science. How can we justify equating natural necessity with metaphysical necessity? |
| 5476 | Essentialists say natural laws are in a new category: necessary a posteriori [Ellis] |
| Full Idea: Essentialists do not accept the standard position, which says necessity is a priori, and contingency is a posteriori. They have a radically new category: the necessary a posteriori. The laws of nature are, for example, both necessary and a posteriori. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.6) | |
| A reaction: Based on Kripke. I'm cautious about this. Presumably God, who would know the essences, could therefore infer the laws a priori. The laws may follow of necessity from the essences, but the essences can't be known a posteriori to be necessary. |
| 5478 | Imagination tests what is possible for all we know, not true possibility [Ellis] |
| Full Idea: The imaginability test of possibility confuses what is really or metaphysically possible with what is only epistemically possible. ..The latter is just what is possible for all we know. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.6) |
| 5482 | Possible worlds realism is only needed to give truth conditions for modals and conditionals [Ellis] |
| Full Idea: The main trouble with possible worlds realism is that the only reason anyone has, or ever could have, to believe in other possible worlds (other than this one) is that they are needed, apparently, to provide truth conditions for modals and conditionals. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: This attacks Lewis. Ellis makes this sound like a trivial technicality, but if our metaphysics is going to make sense it must cover modals and conditionals. What do they actually mean? Lewis has a theory, at least. |
| 5453 | Essentialists mostly accept the primary/secondary qualities distinction [Ellis] |
| Full Idea: Essentialists mostly accept the distinction between primary and secondary qualities, ..where the primary qualities of things are those that are intrinsic to the objects that have them. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: One reason I favour essentialism is because I have always thought that the primary/secondary distinction was a key to understanding the world. 'Primary' gets at the ontology, 'secondary' shows us the epistemology. |
| 5466 | Primary qualities are number, figure, size, texture, motion, configuration, impenetrability and (?) mass [Ellis] |
| Full Idea: For Boyle, Locke and Newton, the qualities inherent in bodies were just the primary qualities, namely number, figure, size, texture, motion and configuration of parts, impenetrability and, perhaps, body (or mass). | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.4) | |
| A reaction: It is nice to have a list. Ellis goes on to say these are too passive, and urges dispositions as primary. Even so, the original seventeenth century insight seems to me a brilliant step forward in our understanding of the world. |
| 5485 | Emeralds are naturally green, and only an external force could turn them blue [Ellis] |
| Full Idea: Emeralds cannot all turn blue in 2050 (as Nelson Goodman envisaged), because to do so they would have to have an extrinsically variable nature. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: I was never very impressed by the 'grue' problem, probably for this reason, but also because Goodman probably thought predicates and properties are the same thing, which they aren't (Idea 5457). |
| 5484 | Essentialists don't infer from some to all, but from essences to necessary behaviour [Ellis] |
| Full Idea: For essentialists the problem of induction reduces to discovering what natural kinds there are, and identifying their essential problems and structures. We then know how they must behave in any world, and there is no inference from some to all. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: The obvious question is how you would determine the essences if you are not allowed to infer 'from some to all'. Personally I don't see induction as a problem, because it is self-evidently rational in a stable world. Hume was right to recommend caution. |
| 3960 | There are no such things as minds, but people have mental properties [Davidson] |
| Full Idea: There are no such things as minds, but people have mental properties. | |
| From: Donald Davidson (Davidson on himself [1994], p.231) | |
| A reaction: I think this is right. It fits with Searle's notion of consciousness as a property, like the liquidity of water. I don't panic if I think "I have no mind, but I have extraordinary properties". |
| 3964 | If the mind is an anomaly, this makes reduction of the mental to the physical impossible [Davidson] |
| Full Idea: If there are no strict psychophysical laws, this rules out reductionism, either by definition of mental predicates in physical terms, or by way of bridging laws. | |
| From: Donald Davidson (Davidson on himself [1994], p.231) | |
| A reaction: But it is by no means clear that there are no psycho-physical laws. How could this be known a priori? |
| 3961 | Obviously all mental events are causally related to physical events [Davidson] |
| Full Idea: All mental events are causally related to physical events. ..This seems obvious. | |
| From: Donald Davidson (Davidson on himself [1994], p.231) | |
| A reaction: All mental events are physically caused. Some bodily physical events result from mental events. Probably all mental events have some effect of other mental events (all of which are in some sense physical). |
| 3963 | There are no strict psychophysical laws connecting mental and physical events [Davidson] |
| Full Idea: There are no strict psychophysical laws (that is, laws connecting mental events under their mental descriptions with physical events under their physical descriptions). | |
| From: Donald Davidson (Davidson on himself [1994], p.231) | |
| A reaction: This is clearly open to question. It may be just that no human mind could ever grasp such laws, given their probable complexity. |
| 3965 | Mental entities do not add to the physical furniture of the world [Davidson] |
| Full Idea: Mental entities do not add to the physical furniture of the world. | |
| From: Donald Davidson (Davidson on himself [1994], p.231) | |
| A reaction: This seems to me clearly true, however we propose to characterise mental events. |
| 3966 | The correct conclusion is ontological monism combined with conceptual dualism [Davidson] |
| Full Idea: My basic premises lead to the conclusion of ontological monism coupled with conceptual dualism (like Spinoza, except that he denied mental causation). | |
| From: Donald Davidson (Davidson on himself [1994], p.231) | |
| A reaction: 'Conceptual dualism' implies no real difference, but 'property dualism' is better, suggesting different properties when viewed from different angles. |
| 3967 | Absence of all rationality would be absence of thought [Davidson] |
| Full Idea: To imagine a totally irrational animal is to imagine an animal without thought. | |
| From: Donald Davidson (Davidson on himself [1994], p.232) | |
| A reaction: This wouldn't be so clear without the theory of evolution, which suggests that only the finders of truth last long enough to breed. |
| 3974 | Our meanings are partly fixed by events of which we may be ignorant [Davidson] |
| Full Idea: What we mean by what we say is partly fixed by events of which we may be ignorant. | |
| From: Donald Davidson (Davidson on himself [1994], p.235) | |
| A reaction: There is 'strict and literal meaning', which is fixed by the words, even if I don't know what I am saying. But 'speaker's meaning' is surely a pure matter of a state of mind? |
| 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. |
| 5457 | Predicates assert properties, values, denials, relations, conventions, existence and fabrications [Ellis, by PG] |
| Full Idea: As well as properties, predicates can assert evaluation, denial, relations, conventions, existence or fabrication. | |
| From: report of Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) by PG - Db (ideas) | |
| A reaction: This seems important, in order to disentangle our ontological commitments from our language, which was a confusion that ran throughout twentieth-century philosophy. A property is a real thing in the world, not a linguistic convention. |
| 3968 | Propositions explain nothing without an explanation of how sentences manage to name them [Davidson] |
| Full Idea: The idea of a proposition is unhelpful, until it is explained how exactly the words in the contained sentence manage to name or describe a proposition (which even Frege failed to achieve). | |
| From: Donald Davidson (Davidson on himself [1994], p.232) | |
| A reaction: It seems obvious to me that there are brain events best labelled as propositions, even if their fit with language is puzzling. |
| 3970 | Thought is only fully developed if we communicate with others [Davidson] |
| Full Idea: We would have no fully-fledge thoughts if we were not in communication with others. | |
| From: Donald Davidson (Davidson on himself [1994], p.233) | |
| A reaction: This seems a plausible empirical observation, though I would doubt any a priori proof of it. If animals could speak, they would become intellectuals? |
| 3971 | There is simply no alternative to the 'principle of charity' in interpreting what others do [Davidson] |
| Full Idea: The 'principle of charity' is a misleading term, since there is no alternative if we want to make sense of the attitudes and actions of the agents around us. | |
| From: Donald Davidson (Davidson on himself [1994], p.233) | |
| A reaction: I suppose so, but only with a background of evolutionary theory. I would necessarily assume charity if a robot spoke to me. |
| 5488 | Regularity theories of causation cannot give an account of human agency [Ellis] |
| Full Idea: A Humean theory of causation (as observed regularities) makes it very difficult for anyone even to suggest a plausible theory of human agency. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: I'm not quite sure what a 'theory' of human agency would look like. Hume himself said we only get to understand our mental powers from repeated experience (Idea 2220). How do we learn about the essence of our own will? |
| 5489 | Humans have variable dispositions, and also power to change their dispositions [Ellis] |
| Full Idea: It seems that human beings not only have variable dispositional properties, as most complex systems have, but also meta-powers: powers to change their own dispositional properties. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: This seems to me a key to how we act, and also to morality. 'What dispositions do you want to have?' is the central question of virtue theory. Humans are essentially multi-level thinkers. Irony is the window into the soul. |
| 5490 | Essentialism fits in with Darwinism, but not with extreme politics of left or right [Ellis] |
| Full Idea: The extremes of left and right in politics have much more reason than Darwinists to be threatened by the 'new essentialism', because it must reinstate the concept of human nature. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: The point being that political extremes go against the grain of our nature. Personally I am favour of essentialism, and human nature. I notice that Steven Pinker is now defending human nature, from a background of linguistics and psychology. |
| 3973 | Without a teacher, the concept of 'getting things right or wrong' is meaningless [Davidson] |
| Full Idea: Without a 'teacher', nothing would give content to the idea that there is a difference between getting things right and getting them wrong. | |
| From: Donald Davidson (Davidson on himself [1994], p.234) | |
| A reaction: Seems right. A group of speculators with no one in the role of 'teacher' would seem to be paralysed with uncertain (except where judgements are very obvious). |
| 5472 | Natural kinds are of objects/substances, or events/processes, or intrinsic natures [Ellis] |
| Full Idea: Natural kinds appear to be of objects or substances, or of events or processes, or of the intrinsic nature of things; hence there should be laws of nature specific to each of these categories. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.5) | |
| A reaction: It is nice to see someone actually discussing what sort of natural kinds there are, instead of getting bogged down in how natural kinds terms get their meaning or reference. Ellis recognises that 'intrinsic nature' needs some discussion. |
| 5471 | Essentialism says natural kinds are fundamental to nature, and determine the laws [Ellis] |
| Full Idea: According to essentialists, the world is wholly structured at the most fundamental level into natural kinds, and the laws of nature are all determined by those kinds. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.5) | |
| A reaction: I am a fan of this view, despite being cautious about claims that natural kinds have necessary identity. Why are the essences active? That is the old Greek puzzle about the origin of movement. And why are natural kinds stable? |
| 5446 | For essentialists two members of a natural kind must be identical [Ellis] |
| Full Idea: Modern essentialists would insist that any two members of the same natural kind must be identical in all essential respects. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.1) | |
| A reaction: For this reason, animals no longer qualify as natural kinds, but electrons, gold atoms, and water molecules do. My sticking point is when anyone asserts that an electron necessarily has (say) its mass. Why no close counterpart of electrons? |
| 5480 | The whole of our world is a natural kind, so all worlds like it necessarily have the same laws [Ellis] |
| Full Idea: It is plausible to suppose that the world is an instance of a natural kind, ..and what is naturally necessary in our world is what must be true in any world of the same natural kind. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.6) | |
| A reaction: This is putting an awful lot of metaphysical weight on the concept of a 'natural kind', so it had better be a secure one. If we accept that natural laws necessarily follow from essences, why shouldn't the whole of our world have an essence, as water does? |
| 3962 | Cause and effect relations between events must follow strict laws [Davidson] |
| Full Idea: If two events are related as cause and effect, there is a strict law under which they may be subsumed. | |
| From: Donald Davidson (Davidson on himself [1994], p.231) | |
| A reaction: Davidson admits that this is open to challenge (though Hume and Kant supported it). It does seem to be central to our understanding of nature. |
| 5445 | Essentialists regard inanimate objects as genuine causal agents [Ellis] |
| Full Idea: Essentialist suppose that the inanimate objects of nature are genuine causal agents: things capable of acting or interacting. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Intro) | |
| A reaction: I have no idea how one might demonstrate such a fact, even though it seems to stare us in the face. This is where science bumps into philosophy. I find myself intuitively taking the essentialist side quite strongly. |
| 5463 | Essentialists believe causation is necessary, resulting from dispositions and circumstances [Ellis] |
| Full Idea: Essentialists believe elementary causal relations involve necessary connections between events, namely between the displays of dispositional properties and the circumstances that give rise to them. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.4) | |
| A reaction: I like essentialism, but I feel a Humean caution about talk of 'natural necessity'. Let's just say that causation seems to be entirely the result of the nature of how things are. How things could be is a large topic for little mites like us. |
| 5491 | A general theory of causation is only possible in an area if natural kinds are involved [Ellis] |
| Full Idea: A general theory of causation in an area is possible only if the kinds of entities under investigation can reasonably be assumed to belong to natural kinds. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: Human beings will be a problem, and also different levels of natural kinds (e.g. a chemical and an organism). 'Natural kind' is a very loose concept. He is referring to scientific, rather than philosophical, theories, I presume. |
| 5442 | For 'passivists' behaviour is imposed on things from outside [Ellis] |
| Full Idea: A 'passivist' believes that the tendencies of things to behave as they do can never be inherent in the things themselves; they must always be imposed on them from the outside. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Intro) | |
| A reaction: This is the medieval view, inherited by Newton and Hume, which makes miracles a possibility, and makes the laws of nature contingent. Essentialism disagree. I think I am with the essentialists. |
| 5473 | The laws of nature imitate the hierarchy of natural kinds [Ellis] |
| Full Idea: If the natural kinds are divided into hierarchical categories, then essentialists would expect the laws of nature also to divide up into these categories, with the same hierarchy. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.5) | |
| A reaction: This seems to me a real step forwards in our understanding of nature, and hence a nice example of the contribution which philosophy can make, instead of just physics. |
| 5474 | Laws of nature tend to describe ideal things, or ideal circumstances [Ellis] |
| Full Idea: Most of the propositions we think of as being (or as expressing) genuine laws of nature seem to describe only the behaviour of ideal kinds of things, or of things in ideal circumstances. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.5) | |
| A reaction: Ellis this suggests that this phenomenon is because science aims at broad understanding instead of strict prediction. Do we simplify because we are a bit dim? Or is it because generalisation wouldn't exist without idealisation and abstraction? |
| 5475 | We must explain the necessity, idealisation, ontology and structure of natural laws [Ellis] |
| Full Idea: There are four major problems about the laws of nature: a necessity problem (must they be true?), an idealisation problem (why is this preferable?), an ontological problem (their grounds), and a structural problem (their relationships). | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.5) | |
| A reaction: One might also ask why the laws (or their underlying essences) are the way they are, and not some other way, though the prospects of answering that don't look good. I don't think we should be satisfied with saying all of these questions are hopeless. |
| 5460 | Causal relations cannot be reduced to regularities, as they could occur just once [Ellis] |
| Full Idea: Causal relations cannot be reduced to mere regularities, as Hume supposed, as they could exist as a singular case, even if it never happened on more than one occasions. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: This seems to be the key reason for modern views moving away from Hume. The suspicion is that regularity is a test for or symptom of causation, but we are deeply committed to the real nature of causation being whatever creates the regularities. |
| 5459 | Essentialists say dispositions are basic, rather than supervenient on matter and natural laws [Ellis] |
| Full Idea: Essentialists say that dispositional properties may be fundamental, whereas for a passivist such qualities are not primary, but supervene on the primary qualities of matter, and on the laws of nature. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: I am strongly in favour of this view of nature. Without essentialism, we have laws of nature arising out of a total void (or God), and arbitrarily imposing themselves on matter. What are the 'primary qualities of matter', if not dispositions? |
| 5461 | The essence of uranium is its atomic number and its electron shell [Ellis] |
| Full Idea: The essential properties of uranium are its atomic number, and the common electron shell structure for all uranium atoms. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.3) | |
| A reaction: For those who deny essences (e.g. Quineans) this is a nice challenge. You might have to add accounts of the essences of the various particles that make up the atoms. There is nothing arbitrary or conventional about what makes something uranium. |
| 5464 | For essentialists, laws of nature are metaphysically necessary, being based on essences of natural kinds [Ellis] |
| Full Idea: Essentialist believe the laws of nature are metaphysically necessary, because anything that belongs to a natural kind is logically required (or is necessarily disposed) to behave as its essential properties dictate. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.4) | |
| A reaction: What a thrillingly large claim. Best approached with caution.. If we say 'essences make laws, and essences are necessary', we might wonder whether a natural kind essence could be SLIGHTLY different (a counterpart) in another world. |
| 5487 | Essentialism requires a clear separation of semantics, epistemology and ontology [Ellis] |
| Full Idea: Scientific essentialism requires that philosophers distinguish clearly between semantic issues, epistemological issues, and ontological issues. | |
| From: Brian Ellis (The Philosophy of Nature: new essentialism [2002], Ch.7) | |
| A reaction: Music to my ears - but then I think everyone should require that of philosophers, because it where they get themselves most confused. The trouble is that ontology is only obtainable epistemologically, and only expressible semantically. |
| 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'. |