35 ideas
| 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. |
| 8525 | Relations need terms, so they must be second-order entities based on first-order tropes [Campbell,K] |
| Full Idea: Because there cannot be relations without terms, in a meta-physic that makes first-order tropes the terms of all relations, relational tropes must belong to a second, derivative order. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §8) | |
| A reaction: The admission that there could be a 'derivative order' may lead to trouble for trope theory. Ostrich Nominalists could say that properties themselves are derivative second-order abstractions from indivisible particulars. Russell makes them first-order. |
| 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'. |
| 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) |
| 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) |
| 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) |
| 8518 | Events are trope-sequences, in which tropes replace one another [Campbell,K] |
| Full Idea: Events are widely acknowledged to be particulars, but they are plainly not ordinary concrete particulars. They are best viewed as trope-sequences, in which one condition gives way to another. They are changes in which tropes replace one another. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §3) | |
| A reaction: If nothing exists except bundles of tropes, it is worth asking WHY one trope would replace another. Some tropes are active (i.e. they are best described as 'powers'). |
| 8513 | Two red cloths are separate instances of redness, because you can dye one of them blue [Campbell,K] |
| Full Idea: If we have two cloths of the very same shade of redness, we can show there are two cloths by burning one and leaving the other unaffected; we show there are two cases of redness in the same way: dye one blue, leaving the other unaffected. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §1) | |
| A reaction: This has to be one of the basic facts of the problem accepted by everyone. If you dye half of one of the pieces, was the original red therefore one instance or two? Has it become two? How many red tropes are there in a red cloth? |
| 8514 | Red could only recur in a variety of objects if it was many, which makes them particulars [Campbell,K] |
| Full Idea: If there are a varied group of red objects, the only element that recurs is the colour. But it must be the colour as a particular (a 'trope') that is involved in the recurrence, for only particulars can be many in the way required for recurrence. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §1) | |
| A reaction: This claim seems to depend on the presupposition that rednesses are countable things, but it is tricky trying to count the number of blue tropes in the sky. |
| 8522 | Tropes solve the Companionship Difficulty, since the resemblance is only between abstract particulars [Campbell,K] |
| Full Idea: The 'companionship difficulty' cannot arise if the members of the resemblance class are tropes rather than whole concrete particulars. The instances of having a heart, as abstract particulars, are quite different from instances of having a kidney. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §6) | |
| A reaction: The companionship difficulty seems worst if you base your account of properties just on being members of a class. Any talk of resemblance eventually has to talk about 'respects' of resemblance. Is a trope a respect? Is a mode an object? |
| 8523 | Tropes solve the Imperfect Community problem, as they can only resemble in one respect [Campbell,K] |
| Full Idea: The 'problem of imperfect community' cannot arise where our resemblance sets are sets of tropes. Tropes, by their very nature and mode of differentiation can only resemble in one respect. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §6) | |
| A reaction: You arrive at very different accounts of what resemblance means according to how you express the problem verbally. We can only find a solution through thinking which transcends language. Heresy! |
| 8524 | Trope theory makes space central to reality, as tropes must have a shape and size [Campbell,K] |
| Full Idea: The metaphysics of abstract particulars gives a central place to space, or space-time, as the frame of the world. ...Tropes are, of their essence, regional, which carries with it the essential presence of shape and size in any trope occurrence. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §7) | |
| A reaction: Trope theory has a problem with Aristotle's example (Idea 557) of what happens when white is mixed with white. Do two tropes become one trope if you paint on a second coat of white? How can particulars merge? How can abstractions merge? |
| 8521 | Nominalism has the problem that without humans nothing would resemble anything else [Campbell,K] |
| Full Idea: The objection to nominalism is its consequence that if there were no human race (or other living things), nothing would be like anything else. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §6) | |
| A reaction: Anti-realists will be unflustered by this difficulty. Personally it strikes me as obvious that some aspects of resemblance are part of reality which we did not contribute. This I take to be a contingent fact, founded on the existence of natural kinds. |
| 8515 | Tropes are basic particulars, so concrete particulars are collections of co-located tropes [Campbell,K] |
| Full Idea: If tropes are basic particulars, then concrete particulars count as dependent realities. They are collections of co-located tropes, depending on these tropes as a fleet does upon its component ships. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §2) | |
| A reaction: If I sail my yacht through a fleet, do I become part of it? Presumably trope theory could avoid a bundle view of objects. A bare substratum could be a magnet which attracts tropes. |
| 8519 | Bundles must be unique, so the Identity of Indiscernibles is a necessity - which it isn't! [Campbell,K] |
| Full Idea: Each individual is distinct from each other individual, so the bundle account of objects requires each bundle to be different from every other bundle. So the Identity of Indiscernibles must be a necessary truth, which, unfortunately, it is not. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §5) | |
| A reaction: Clearly the Identity of Indiscernibles is not a necessary truth (consider just two identical spheres). Location and time must enter into it. Could we not add a further individuation requirement to the necessary existence of a bundle? (Quinton) |
| 4033 | Two pure spheres in non-absolute space are identical but indiscernible [Campbell,K] |
| Full Idea: The Identity of Indiscernibles is not a necessary truth. It fails in possible worlds where there are two identical spheres in a non-absolute space, or worlds without beginning or end where events are exactly cyclically repeated. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §5) | |
| A reaction: The principle was always very suspect, and these seem nice counterexamples. As so often, epistemology and ontology had become muddled. |
| 8130 | Qualities of experience are just representational aspects of experience ('Representationalism') [Harman, by Burge] |
| Full Idea: Harman defended what came to be known as 'representationalism' - the view that qualitative aspects of experience are nothing other than representational aspects. | |
| From: report of Gilbert Harman (The Intrinsic Quality of Experience [1990]) by Tyler Burge - Philosophy of Mind: 1950-2000 p.459 | |
| A reaction: Functionalists like Harman have a fairly intractable problem with the qualities of experience, and this may be clutching at straws. What does 'represent' mean? How is the representation achieved? Why that particular quale? |
| 8512 | Abstractions come before the mind by concentrating on a part of what is presented [Campbell,K] |
| Full Idea: An item is abstract if it is got before the mind by an act of abstraction, that is, by concentrating attention on some, but not all, of what is presented. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §1) | |
| A reaction: I think this point is incredibly important. Pure Fregean semantics tries to leave out the psychological component, and yet all the problems in semantics concern various sorts of abstraction. Imagination is the focus of the whole operation. |
| 8517 | Causal conditions are particular abstract instances of properties, which makes them tropes [Campbell,K] |
| Full Idea: The conditions in causal statements are usually particular cases of properties. A collapse results from the weakness of this cable (not any other). This is specific to a time and place; it is an abstract particular. It is, in short, a trope. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §3) | |
| A reaction: The fan of universals could counter this by saying that the collapse results from this unique combination of universals. Resemblance nominalist can equally build an account on the coincidence of certain types of concrete particulars. |
| 8516 | Davidson can't explain causation entirely by events, because conditions are also involved [Campbell,K] |
| Full Idea: Not all singular causal statements are of Davidson's event-event type. Many involve conditions, so there are condition-event (weakness/collapse), event-condition (explosion/movement), and condition-condition (hot/warming) causal connections. | |
| From: Keith Campbell (The Metaphysic of Abstract Particulars [1981], §3) | |
| A reaction: Fans of Davidson need to reduce conditions to events. The problem of individuation keeps raising its head. Davidson makes it depend on description. Kim looks good, because events, and presumably conditions, reduce to something small and precise. |