39 ideas
| 4036 | What matters is not how many entities we postulate, but how many kinds of entities [Armstrong, by Mellor/Oliver] |
| Full Idea: Armstrong argues that what matters is not how few entities we postulate (quantitative economy), but how few kinds of entities (qualitative economy). | |
| From: report of David M. Armstrong (Properties [1992]) by DH Mellor / A Oliver - Introduction to 'Properties' §9 | |
| A reaction: Is this what Ockham meant? Armstrong is claiming that the notion of a 'property' is needed to identify kinds. See also Idea 7038. |
| 10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
| Full Idea: For a set to be 'well-ordered' it is required that every subset of the set has a first element. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10857 | Set theory made a closer study of infinity possible [Clegg] |
| Full Idea: Set theory made a closer study of infinity possible. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
| Full Idea: The idea of the 'power set' means that it is always possible to generate a bigger one using only the elements of that set, namely the set of all its subsets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
| Full Idea: Axiom of Extension: Two sets are equal if and only if they have the same elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
| Full Idea: Axiom of Pairing: For any two sets there exists a set to which they both belong. So you can make a set out of two other sets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
| Full Idea: Axiom of Unions: For every collection of sets there exists a set that contains all the elements that belong to at least one of the sets in the collection. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
| Full Idea: Axiom of Infinity: There exists a set containing the empty set and the successor of each of its elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This is rather different from the other axioms because it contains the notion of 'successor', though that can be generated by an ordering procedure. |
| 10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
| Full Idea: Axiom of Powers: For each set there exists a collection of sets that contains amongst its elements all the subsets of the given set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: Obviously this must include the whole of the base set (i.e. not just 'proper' subsets), otherwise the new set would just be a duplicate of the base set. |
| 10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
| Full Idea: Axiom of Choice: For every set we can provide a mechanism for choosing one member of any non-empty subset of the set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This axiom is unusual because it makes the bold claim that such a 'mechanism' can always be found. Cohen showed that this axiom is separate. The tricky bit is choosing from an infinite subset. |
| 10871 | Axiom of Existence: there exists at least one set [Clegg] |
| Full Idea: Axiom of Existence: there exists at least one set. This may be the empty set, but you need to start with something. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
| Full Idea: Axiom of Specification: For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true. So the concept 'number is even' produces a set from the integers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: What if the condition won't apply to the set? 'Number is even' presumably won't produce a set if it is applied to a set of non-numbers. |
| 10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
| Full Idea: Three views of mathematics: 'pure' mathematics, where it doesn't matter if it could ever have any application; 'real' mathematics, where every concept must be physically grounded; and 'applied' mathematics, using the non-real if the results are real. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.17) | |
| A reaction: Very helpful. No one can deny the activities of 'pure' mathematics, but I think it is undeniable that the origins of the subject are 'real' (rather than platonic). We do economics by pretending there are concepts like the 'average family'. |
| 10860 | An ordinal number is defined by the set that comes before it [Clegg] |
| Full Idea: You can think of an ordinal number as being defined by the set that comes before it, so, in the non-negative integers, ordinal 5 is defined as {0, 1, 2, 3, 4}. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
| Full Idea: With ordinary finite numbers ordinals and cardinals are in effect the same, but beyond infinity it is possible for two sets to have the same cardinality but different ordinals. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
| Full Idea: The 'transcendental numbers' are those irrationals that can't be fitted to a suitable finite equation, of which π is far and away the best known. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
| Full Idea: The realisation that brought 'i' into the toolkit of physicists and engineers was that you could extend the 'number line' into a new dimension, with an imaginary number axis at right angles to it. ...We now have a 'number plane'. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.12) |
| 10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
| Full Idea: It is a chicken-and-egg problem, whether the lack of zero forced forced classical mathematicians to rely mostly on a geometric approach to mathematics, or the geometric approach made 0 a meaningless concept, but the two remain strongly tied together. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
| Full Idea: As far as Kronecker was concerned, Cantor had built a whole structure on the irrational numbers, and so that structure had no foundation at all. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
| Full Idea: Paul Cohen showed that the Continuum Hypothesis is independent of the axioms of set theory. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
| Full Idea: The 'continuum hypothesis' says that aleph-one is the cardinality of the rational and irrational numbers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 15754 | Without properties we would be unable to express the laws of nature [Armstrong] |
| Full Idea: The ontological correlates of true law-statements must involve properties. How else can one pick our the uniformities which the law-statements entail? | |
| From: David M. Armstrong (Properties [1992], 1) | |
| A reaction: I'm unconvinced about the 'laws', but I have to admit that it is hard to know how to describe the relevant bits of nature without some family of concepts covered by the word 'property'. I'm in favour of taking some of the family into care, though. |
| 4034 | Whether we apply 'cold' or 'hot' to an object is quite separate from its change of temperature [Armstrong] |
| Full Idea: Evading properties by means of predicates is implausible when things change. If a cold thing becomes hot, first 'cold' applies, and then 'hot', but what have predicates to do with the temperature of an object? | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: A clear illustration of why properties are part of nature, not just part of language. But some applications of predicates are more arbitrary than this (ugly, cool) |
| 8535 | To the claim that every predicate has a property, start by eliminating failure of application of predicate [Armstrong] |
| Full Idea: Upholders of properties have been inclined to postulate a distinct property corresponding to each distinct predicate. We could start by eliminating all those properties where the predicate fails to apply, is not true, of anything. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: This would leave billions of conjunctional, disjunctional and gerrymandered properties where the predicate applies very well. We are all 'on the same planet as New York'. Am I allowed to say that I 'wish' that a was F? He aims for 'sparse' properties. |
| 8537 | Tropes fall into classes, because exact similarity is symmetrical and transitive [Armstrong] |
| Full Idea: Exact similarity is a symmetrical and transitive relation. (Less than exact similarity is not transitive, even for tropes). So the relation of exact similarity is an equivalence relation, partitioning the field of tropes into equivalence classes. | |
| From: David M. Armstrong (Properties [1992], §2) | |
| A reaction: Armstrong goes on the explore the difficulties for trope theory of less than exact similarity, which is a very good line of discussion. Unfortunately it is a huge problem for everyone, apart from the austere nominalist. |
| 8538 | Trope theory needs extra commitments, to symmetry and non-transitivity, unless resemblance is exact [Armstrong] |
| Full Idea: Trope theory needs extra ontological baggage, the Axioms of Resemblance. There is a principle of symmetry, and there is the failure of transitivity - except in the special case of exact resemblance. | |
| From: David M. Armstrong (Properties [1992], §2) | |
| A reaction: [see text for fuller detail] Is it appropriate to describe such axioms as 'ontological' baggage? Interesting, though I suspect that any account of properties and predicates will have a similar baggage of commitments. |
| 8539 | Universals are required to give a satisfactory account of the laws of nature [Armstrong] |
| Full Idea: A reason why I reject trope theory is that universals are required to give a satisfactory account of the laws of nature. | |
| From: David M. Armstrong (Properties [1992], §2) | |
| A reaction: This is the key thought in Armstrong's defence of universals. Issues about universals may well be decided on such large playing fields. I think he is probably wrong, and I will gradually explain why. Watch this space as the story unfolds... |
| 8529 | Deniers of properties and relations rely on either predicates or on classes [Armstrong] |
| Full Idea: The great deniers of properties and relations are of two sorts: those who put their faith in predicates and those who appeal to sets (classes). | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: This ignores the Quine view, which is strictly for ostriches. Put like this, properties and relations seem undeniable. Predicates are too numerous (gerrymandering) or too few (colour shades). Classes can have arbitrary members. |
| 8532 | Resemblances must be in certain 'respects', and they seem awfully like properties [Armstrong] |
| Full Idea: If a resembles b, in general, they resemble in certain respects, and fail to resemble in other respects. But respects are uncomfortably close to properties, which the Resemblance theory proposes to do without. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: This is a good objection. I think it is plausible to build a metaphysics around the idea of respects, and drop properties. Shall we just talk of 'respects' for categorising, and 'powers' for causation and explanation? Respects only exist in comparisons. |
| 8530 | Change of temperature in objects is quite independent of the predicates 'hot' and 'cold' [Armstrong] |
| Full Idea: To appreciate the implausibility of the predicate view, consider where a thing's properties change. 'Hot' becomes applicable when 'cold' ceases to, ..but the change in the object would have occurred if the predicates had never existed. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: They keep involving secondary qualities! Armstrong is taking a strongly realist view (fine by me), but anti-realists can ignore his argument. I take predicate nominalism to be a non-starter. |
| 8536 | We want to know what constituents of objects are grounds for the application of predicates [Armstrong] |
| Full Idea: The properties that are of ontological interest are those constituents of objects, of particulars, which serve as the ground in the objects for the application of predicates. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: Good. This is a reversal of the predicate nominalist approach, and is a much healthier attitude to the relationship between ontology and language. Value judgements will be an interesting case. Does this allow us to invent new predicates? |
| 8531 | In most sets there is no property common to all the members [Armstrong] |
| Full Idea: Most sets are uninteresting because they are utterly heterogeneous, that is, the members have nothing in common. For most sets there is no common property F, such that the set is the set of all the Fs. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: One might link the interesting sets together by resemblance, without invoking the actual existence of an item F which all the members carry (like freemasons' briefcases). Personally I am only really interested in 'natural' sets. |
| 15753 | Essences might support Resemblance Nominalism, but they are too coarse and ill-defined [Armstrong] |
| Full Idea: A sophisticated Resemblance theory can appeal to the natures of the resembling things, from which the resemblances flow. The natures are suitably internal, but are as coarse as the things themselves (and perhaps are the things themselves). | |
| From: David M. Armstrong (Properties [1992], 1) | |
| A reaction: Note that this is essentialism as an underpinning for Resemblance Nominalism. His objection is that he just can't believe in essences, because they are too 'coarse' - which I take to mean that we cannot distinguish the boundaries of an essence. |
| 8533 | Predicates need ontological correlates to ensure that they apply [Armstrong] |
| Full Idea: Must there not be something quite specific about the thing which allows, indeed ensures, that predicates like 'underneath' and 'hot' apply? The predicates require ontological correlates. | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: An interesting proposal, that in addition to making use of predicates, we should 'ensure that they apply'. Sounds verificationist. Obvious problem cases would be speculative, controversial or metaphorical predicates. "He's beneath contempt". |
| 4035 | There must be some explanation of why certain predicates are applicable to certain objects [Armstrong] |
| Full Idea: When we have said that predicates apply to objects, we have surely not said enough. The situation cries out for an explanation. Must there not be something specific about the things which allows, indeed ensures, that these predicates apply? | |
| From: David M. Armstrong (Properties [1992], §1) | |
| A reaction: A nice challenge to any philosopher who places too much emphasis on language. A random and arbitrary (nominalist?) language simply wouldn't work. Nature has joints. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 8541 | Regularities theories are poor on causal connections, counterfactuals and probability [Armstrong] |
| Full Idea: Regularity theories make laws molecular, with no inner causal connections; also, only some cosmic regularities are manifestations of laws; molecular states can't sustain counterfactuals; and probabilistic laws are hard to accommodate. | |
| From: David M. Armstrong (Properties [1992], §2) | |
| A reaction: [very compressed] A helpful catalogue of difficulties. The first difficulty is the biggest one - that regularity theories have nothing to say about why there is a regularity. They offer descriptions instead of explanations. |
| 8540 | The introduction of sparse properties avoids the regularity theory's problem with 'grue' [Armstrong] |
| Full Idea: Regularity theories of laws face the grue problem. That, I think, can only be got over by introducing properties, sparse properties, into one's ontology. | |
| From: David M. Armstrong (Properties [1992], §2) | |
| A reaction: The problem is, roughly, that regularities have to be described in language, which is too arbitrary in character. Armstrong rightly tries to break the rigid link to language. See his Idea 8536, which puts reality before language. |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |