79 ideas
| 17663 | If you know what it is, investigation is pointless. If you don't, investigation is impossible [Armstrong] |
| Full Idea: Paradox of Analysis:if we ask what sort of thing an X is, then either we know what an X is or we do not. If we know then there is no need to ask the question. If we do not know then there is no way to begin the investigation. It's pointless or impossible | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 01.2) | |
| A reaction: [G.E. Moore is the source of this, somewhere] Plato worried that to get to know something you must already know it. Solving this requires the concept of a 'benign' circularity. |
| 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'. |
| 17688 | Negative facts are supervenient on positive facts, suggesting they are positive facts [Armstrong] |
| Full Idea: Negative facts appear to be supervenient upon the positive facts, which suggests that they are nothing more than the positive facts. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 10.3) |
| 17691 | Nothing is genuinely related to itself [Armstrong] |
| Full Idea: I believe that nothing is genuinely related to itself. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 10.7) |
| 17679 | All instances of some property are strictly identical [Armstrong] |
| Full Idea: A property ...is something which is strictly identical, strictly the same, in all its different instances. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.2) | |
| A reaction: Some is gravitation one property, or an infinity of properties, for each of its values? What is the same between objects of different mass. I sort of believe in all the masses, but I'm not sure what 'mass' is. Abstraction, say I. |
| 12677 | Armstrong holds that all basic properties are categorical [Armstrong, by Ellis] |
| Full Idea: I am against Armstrong's strong categoricalism, that is, the thesis that all basic properties are categorical. | |
| From: report of David M. Armstrong (What is a Law of Nature? [1983]) by Brian Ellis - The Metaphysics of Scientific Realism 3 | |
| A reaction: I certainly agree with this, as I cannot see where the power would come from to get the whole thing off the ground. Armstrong depends on universals to necessitate what happens, which I find very peculiar. |
| 17666 | Actualism means that ontology cannot contain what is merely physically possible [Armstrong] |
| Full Idea: Actualism ...debars us from admitting into our ontology the merely possible, not only the merely logically possible, but also the merely physically possible. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 01.3) | |
| A reaction: This is the big metaphysical question for fans (like myself) of 'powers' in nature. Armstrong declares himself an Actualist. I take it as obvious that the actual world contains powers, but how are we to characterise them? |
| 17667 | Dispositions exist, but their truth-makers are actual or categorical properties [Armstrong] |
| Full Idea: It is not denied that statements attributing dispositions and/or powers to objects are often true. But the truth-makers or ontological ground for such statements must always be found in the actual, or categorical, properties of the objects involved. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 01.3) | |
| A reaction: This is the big debate in the topic of powers. I love powers, but you always think there must be 'something' which has the power. Could reality entirely consist of powers? See Fetzer. |
| 17687 | If everything is powers there is a vicious regress, as powers are defined by more powers [Armstrong] |
| Full Idea: I believe reducing all universals to powers is involved in vicious regress. The power is what it is by the sort of actualisations it gives rise to in suitable sorts of circumstances. But they themselves can be nothing but powers... | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 08.3) | |
| A reaction: [compressed wording] I don't see this problem. Anything postulated as fundamental is going to be baffling. Why are categorical properties superior to powers? Postulate basic powers (or basic empowered stuff), then build up. |
| 17678 | Universals are just the repeatable features of a world [Armstrong] |
| Full Idea: Universals can be brought into the spatio-temporal world, becoming simply the repeatable features of that world. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.2) | |
| A reaction: I wish Armstrong wouldn't use the word 'universal', which has so much historical baggage. The world obviously has repeatable features, but does that mean that our ontology must include things called 'features'? Hm. |
| 17669 | Realist regularity theories of laws need universals, to pick out the same phenomena [Armstrong] |
| Full Idea: A Realistic version of a Regularity theory of laws will have to postulate universals. How else will it be possible to say that the different instances of a certain uniformity are all instances of objectively the same phenomenon? | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 02.4) | |
| A reaction: I disagree. We may (or may not) need properties, but they can be have a range. We just need stable language. We use one word 'red', even when the shade of redness varies. Non-realists presumably refer to sense-data. |
| 17677 | Past, present and future must be equally real if universals are instantiated [Armstrong] |
| Full Idea: Past, present and future I take to be all and equally real. A universal need not be instantiated now. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.2) | |
| A reaction: This is the price you must pay for saying that you only believe in universals which are instantiated. |
| 15442 | Universals are abstractions from their particular instances [Armstrong, by Lewis] |
| Full Idea: Armstrong takes universals generally, and structural universals along with the rest, to be abstractions from their particular instances. | |
| From: report of David M. Armstrong (What is a Law of Nature? [1983], p.83-4) by David Lewis - Against Structural Universals 'The pictorial' | |
| A reaction: To me, 'abstracted' implies a process of human psychology, a way of thinking about the instances. I don't see how there can be an 'abstracted' relation which is a part of the external world. That makes his laws of nature human creations. |
| 17686 | Universals are abstractions from states of affairs [Armstrong] |
| Full Idea: Universals are abstractions from states of affairs. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 7) | |
| A reaction: I'm getting confused about Armstrong's commitments. He bases his whole theory on the existence of universals (repeatable features), but now says those are 'abstracted' from something else. Abstracted by us? |
| 17668 | It is likely that particulars can be individuated by unique conjunctions of properties [Armstrong] |
| Full Idea: For each particular it is likely that there exists at least one individuating conjunction of properties, that is, a conjunction of properties such that the particular instantiates this conjunction and nothing else does. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 02.3) | |
| A reaction: Armstrong commits to a famous Leibniz view, but I don't see his grounds for it. There is nothing incoherent about nature churning out perfect replicas of things, such as quarks and electrons. Would we care if two pens were perfectly identical? |
| 17680 | The identity of a thing with itself can be ruled out as a pseudo-property [Armstrong] |
| Full Idea: There is reason to rule out as pseudo-properties such things as the identity of a thing with itself. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.2) | |
| A reaction: Good on you, David. |
| 17693 | The necessary/contingent distinction may need to recognise possibilities as real [Armstrong] |
| Full Idea: It may be that the necessary/contingent distinction is tied to a metaphysics which recognises possibility as a real something wider than actuality. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 11.2) | |
| A reaction: Armstrong responds by trying to give an account of possibility in terms of 'combinations' from actuality. I think powers offer a much better strategy. |
| 17685 | Induction aims at 'all Fs', but abduction aims at hidden or theoretical entities [Armstrong] |
| Full Idea: Many philosophers of science have distinguished between 'simple induction' - the argument from observed Fs to all Fs - and the argument to hidden or theoretical entities (Peirce's 'abduction'). | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.7) | |
| A reaction: 'Abduction' is (roughly) the same is inference to the best explanation, of which I am a great fan. |
| 17683 | Science suggests that the predicate 'grue' is not a genuine single universal [Armstrong] |
| Full Idea: It is plausible to say, on the basis of total science, that 'grue' is a predicate to which no genuine, that is, unitary, universal corresponds. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.7) |
| 17675 | Unlike 'green', the 'grue' predicate involves a time and a change [Armstrong] |
| Full Idea: The predicate 'grue' involves essential reference to a particular time, which 'green' does not. Also on the 'grue' hypothesis a change occurs in emeralds in a way that change does not occur on the 'green' hypothesis. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 04.5) | |
| A reaction: I'm inclined to think that comparing 'grue' with 'green' is a category mistake. 'Grue' is a behaviour. Armstrong says this is no objection, because Goodman's argument is purely formal. |
| 17674 | The raven paradox has three disjuncts, confirmed by confirming any one of them [Armstrong] |
| Full Idea: We could rewrite the generalisation as For all x, ((x is a raven and x is black) v (x is not a raven and x is black) v (x is not a raven and x is not black)). Instances of any one of the three disjuncts will do as confirmation. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 04.3) | |
| A reaction: A nice clarification. |
| 17672 | A good reason for something (the smoke) is not an explanation of it (the fire) [Armstrong] |
| Full Idea: A good reason for P is not necessarily an explanation of P. The presence of smoke is a good reason for thinking that fire is present. But it is not an explanation of the presence of fire. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 04.2) | |
| A reaction: This may be an equivocation on 'the reason for'. Smoke is a reason for thinking there is a fire, but no one would propose it as a reason for the fire. If the reason for the fire was arson, that would seem to explain it as well. |
| 17684 | To explain observations by a regular law is to explain the observations by the observations [Armstrong] |
| Full Idea: Given the Regularity theory, the explanatory element seems to vanish. For to say that all the observed Fs are Gs because all the Fs are Gs involves explaining the observations in terms of themselves. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.7) | |
| A reaction: This point cries out, it is so obvious (once spotted). Tigers are ferocious because all tigers are ferocious (see?). |
| 17676 | Best explanations explain the most by means of the least [Armstrong] |
| Full Idea: The best explanation explains the most by means of the least. Explanation unifies. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 05.4) | |
| A reaction: To get unification, you need to cite the diversity of what is explained, and not the mere quantity. The force of gravity unifies because it applies to such a diversity of things. |
| 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? |
| 17664 | Each subject has an appropriate level of abstraction [Armstrong] |
| Full Idea: To every subject, its appropriate level of abstraction. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 01.2) | |
| A reaction: Mathematics rises through many levels of abstraction. Economics can be very concrete or very abstract. It think it is clearer to talk of being 'general', rather than 'abstract'. |
| 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. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 17692 | We can't deduce the phenomena from the One [Armstrong] |
| Full Idea: No serious and principled deduction of the phenomena from the One has ever been given, or looks likely to be given. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 11) | |
| A reaction: This seems to pick out the best reason why hardly anybody (apart from Jonathan Schaffer) takes the One seriously. |
| 17689 | Absences might be effects, but surely not causes? [Armstrong] |
| Full Idea: Lacks and absences could perhaps by thought of as effects, but we ought to be deeply reluctant to think of them as causes. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 10.4) | |
| A reaction: Odd. So we allow that they exist (as effects), but then deny that they have any causal powers? |
| 17682 | A universe couldn't consist of mere laws [Armstrong] |
| Full Idea: A universe could hardly consist of laws and nothing else. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.4) | |
| A reaction: Hm. Discuss. How does a universe come into existence, if there are no laws to guide its creation? |
| 17662 | Science depends on laws of nature to study unobserved times and spaces [Armstrong] |
| Full Idea: The scientist trying to establish the geography and history of the unobserved portion of the universe must depend upon what he takes to be the laws of the universe. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 01.1) | |
| A reaction: This does seem to be the prime reason why we wish to invoke 'laws', but we could just as well say that we have to rely on induction. Spot patterns, then expect more of the same. Spot necessities? Mathematics is very valuable here, of course. |
| 17690 | Oaken conditional laws, Iron universal laws, and Steel necessary laws [Armstrong, by PG] |
| Full Idea: Three degress of law: 1) 'Oaken laws' where all Fs that aren't Hs are Gs; 2) 'Iron' laws where all Fs are Gs; and 3) 'Steel' laws where all Fs must be Gs. | |
| From: report of David M. Armstrong (What is a Law of Nature? [1983], 10.4) by PG - Db (ideas) | |
| A reaction: [My summary of Armstrong's distinction] One response is to say that all laws are actually Oaken - see Mumfor and Mumford/Lill Anjum. It's all ceteris paribus. |
| 17670 | Newton's First Law refers to bodies not acted upon by a force, but there may be no such body [Armstrong] |
| Full Idea: Newton's First Law of Motion tells us what happens to a body which is not acted upon by a force. Yet it may be that the antecedent of the law is never instantiated. It may be that every body that there is, is acted upon by some force. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 02.7) |
| 8582 | Regularities are lawful if a second-order universal unites two first-order universals [Armstrong, by Lewis] |
| Full Idea: Armstrong's theory holds that what makes certain regularities lawful are second-order states of affairs N(F,G) in which the two ordinary first-order universals F and G are related by a certain dyadic second-order universal N. | |
| From: report of David M. Armstrong (What is a Law of Nature? [1983]) by David Lewis - New work for a theory of universals 'Laws and C' | |
| A reaction: [see Lewis's footnote] I take the view (from Shoemaker and Ellis) that laws of nature are just plain regularities which arise from the hierarchy of natural kinds. We don't need a commitment to 'universals'. |
| 17671 | A naive regularity view says if it never occurs then it is impossible [Armstrong] |
| Full Idea: It is a Humean uniformity that no race of ravens is white-feathered. Hence, if the Naive Regularity analysis of law is correct, it is a law that no race of ravens is white-feathered, that is, such a race is physically impossible. A most unwelcome result. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 02.6) | |
| A reaction: Chapters 2-4 of Armstrong are a storming attack on the regularity view of laws of nature, and this idea is particularly nice. Laws must refer to what could happen, not what happens to happen. |
| 17681 | The laws of nature link properties with properties [Armstrong] |
| Full Idea: There is an utterly natural idea that the laws of nature link properties with properties. | |
| From: David M. Armstrong (What is a Law of Nature? [1983], 06.3) | |
| A reaction: Put it this way: given that properties are expressions of invariant powers, the interaction of two properties will (ceteris paribus) be invariant, and laws are just invariances in natural behaviour. |
| 16246 | Rather than take necessitation between universals as primitive, just make laws primitive [Maudlin on Armstrong] |
| Full Idea: My own view is simple: the laws of nature ought to be accepted as ontologically primitive. …They are preferable in point of familiarity to such necessitation relations between universals. | |
| From: comment on David M. Armstrong (What is a Law of Nature? [1983]) by Tim Maudlin - The Metaphysics within Physics 1.4 | |
| A reaction: I think you make natures of things primitive, and reduce laws to regularities and universals to resemblances. Job done. Natures are even more 'familiar' as primitives than laws are. |
| 9480 | Armstrong has an unclear notion of contingent necessitation, which can't necessitate anything [Bird on Armstrong] |
| Full Idea: The two criticisms levelled against Armstrong are that it is unclear what his relation of contingent necessitation is, and that it is unclear how it is able to necessitate anything. | |
| From: comment on David M. Armstrong (What is a Law of Nature? [1983]) by Alexander Bird - Nature's Metaphysics 3.1.2 | |
| A reaction: I suppose someone has to explore the middle ground between the mere contingencies of Humean regularities and the strong necessities of scientific essentialism. The area doesn't, however, look promising. |
| 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 |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |
| 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'. |