29 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. |
| 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) |
| 16235 | Persistence conditions cannot contradict, so there must be a 'dominant sortal' [Burke,M, by Hawley] |
| Full Idea: Burke says a single object cannot have incompatible persistence conditions, for this would entail that there are events in which the object would both survive and perish. He says one sortal 'dominates' the other (sweater dominates thread). | |
| From: report of Michael Burke (Dion and Theon: an essentialist solution [1994]) by Katherine Hawley - How Things Persist 5.3 | |
| A reaction: This I take to be the most extreme version of sortal essentialism, and strikes me as incredibly gerrymandered and unacceptable. It is just too anthropocentric to count as genuine metaphysics. I may care more about the thread. |
| 14753 | The 'dominant' of two coinciding sortals is the one that entails the widest range of properties [Burke,M, by Sider] |
| Full Idea: Burke claims that the 'dominant' sortal is the one whose satisfaction entails possession of the widest range of properties. For example, the statue (unlike the lump of clay) also possesses aesthetic properties, and hence is dominant. | |
| From: report of Michael Burke (Dion and Theon: an essentialist solution [1994]) by Theodore Sider - Four Dimensionalism 5.4 | |
| A reaction: [there are three papers by Burke on this; see all the quotations from Burke] Presumably one sortal could entail a single very important property, and the other sortal entail a huge range of trivial properties. What does being a 'thing' entail? |
| 16072 | 'The rock' either refers to an object, or to a collection of parts, or to some stuff [Burke,M, by Wasserman] |
| Full Idea: Burke distinguishes three different readings of 'the rock'. It can be a singular description denoting an object, or a plural description denoting all the little pieces of rock, or a mass description the relevant rocky stuff. | |
| From: report of Michael Burke (Dion and Theon: an essentialist solution [1994]) by Ryan Wasserman - Material Constitution 5 | |
| A reaction: Idea 16068 is an objection to the second reading. Only the first reading seems plausible, so we must just get over all the difficulties philosophers have unearthed about knowing exactly what an 'object' is. I offer you essentialism. Rocks have unity. |
| 14751 | Tib goes out of existence when the tail is lost, because Tib was never the 'cat' [Burke,M, by Sider] |
| Full Idea: Burke argues that Tib (the whole cat apart from its tail) goes out of existence when the tail is lost. His essentialist principle is that if something is ever of a particular sort (such as 'cat') then it is always of that sort. Tib is not initially a cat. | |
| From: report of Michael Burke (Dion and Theon: an essentialist solution [1994]) by Theodore Sider - Four Dimensionalism 5.4 | |
| A reaction: This I take to be a souped up version of Wiggins, and I just don't buy that identity conditions are decided by sortals, when it seems obvious that sortals are parasitic on identities. |
| 16071 | Sculpting a lump of clay destroys one object, and replaces it with another one [Burke,M, by Wasserman] |
| Full Idea: On Burke's view, the process of sculpting a lump of clay into a statue destroys one object (a mere lump of clay) and replaces it with another (a statue). | |
| From: report of Michael Burke (Dion and Theon: an essentialist solution [1994]) by Ryan Wasserman - Material Constitution 5 | |
| A reaction: There is something right about this, but how many intermediate objects are created during the transition. It seems to make the notion of an object very conventional. |
| 16234 | Burke says when two object coincide, one of them is destroyed in the process [Burke,M, by Hawley] |
| Full Idea: Michael Burke argues that a sweater is identical with the thread that consitutes it, that both were created at the moment when they began to coincide, and that the original thread was destroyed in the process. | |
| From: report of Michael Burke (Dion and Theon: an essentialist solution [1994]) by Katherine Hawley - How Things Persist 5.3 | |
| A reaction: [Burke's ideas are spread over three articles] It is the thread which is destroyed, because the sweater is the 'dominant sortal' (which strikes me as a particularlyd desperate concept). |
| 13278 | Maybe the clay becomes a different lump when it becomes a statue [Burke,M, by Koslicki] |
| Full Idea: Burke has argued in a series of papers that the lump of clay which constitutes the statue is numerically distinct from the lump of clay which exists before or after the statue exists. The first is a statue, while the second is merely a lump of clay. | |
| From: report of Michael Burke (Dion and Theon: an essentialist solution [1994]) by Kathrin Koslicki - The Structure of Objects | |
| A reaction: Koslicki objects that this introduces radically different persistence conditions from normal. It would mean that a pile of sugar was a different pile of sugar every time a grain moved (even slightly). You couldn't step into the same sugar twice. |
| 14750 | Two entities can coincide as one, but only one of them (the dominant sortal) fixes persistence conditions [Burke,M, by Sider] |
| Full Idea: Michael Burke has given an account that avoids distinguishing coinciding entities. ...The statue/lump satisfies both 'lump' and 'statue', but only the latter determines that object's persistence conditions, and so is that object's 'dominant sortal'. | |
| From: report of Michael Burke (Dion and Theon: an essentialist solution [1994]) by Theodore Sider - Four Dimensionalism 5.4 | |
| A reaction: Presumably a lump on its own can have its own persistance conditions (as a 'lump'), but those would presumably be lost if you shaped it into a statue. Burke concedes that. Can of worms. Using a book as a doorstop... |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
| Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
| From: Herodotus (The Histories [c.435 BCE], 2.123.2) |