29 ideas
| 8368 | A correct definition is what can be substituted without loss of meaning [Ducasse] |
| Full Idea: A definition of a word is correct if the definition can be substituted for the word being defined in an assertion without in the least changing the meaning which the assertion is felt to have. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §1) | |
| A reaction: This sounds good, but a very bland and uninformative rephrasing would fit this account, without offering anything very helpful. The word 'this' could be substituted for a lot of object words. A 'blade' is 'a thing always attached to a knife handle'. |
| 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) |
| 16766 | One thing needs a single thing to unite it; if there were two forms, something must unite them [Aquinas] |
| Full Idea: One thing simpliciter is produced out of many actually existing things only if there is something uniting and tying them to each other. If Socrates were animal and rational by different forms, then to be united they would need something to make them one. | |
| From: Thomas Aquinas (Quaestiones de anima [1269], 11c), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 25.2 | |
| A reaction: This is the reply to the idea that a single thing is just an interesting of many sortal essences. It presumes, of course, that a thing like a horse has something called 'unity'. |
| 8367 | Causation is defined in terms of a single sequence, and constant conjunction is no part of it [Ducasse] |
| Full Idea: The correct definition of the causal relation is to be framed in terms of one single case of sequence, and constancy of conjunction is therefore no part of it. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], Intro) | |
| A reaction: This is the thesis of Ducasse's paper. I immediately warm to it. I take constant conjunction to be a consequence and symptom of causation, not its nature. There is a classic ontology/epistemology confusion to be avoided here. |
| 8372 | We see what is in common between causes to assign names to them, not to perceive them [Ducasse] |
| Full Idea: The part of a generalization concerning what is common to one individual concrete event and the causes of certain other events of the same kind is involved in the mere assigning of a name to the cause and its effect, but not in the perceiving them. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §5) | |
| A reaction: A nice point, that we should keep distinct the recognition of a cause, and the assigning of a general name to it. Ducasse is claiming that we can directly perceive singular causation. |
| 8369 | Causes are either sufficient, or necessary, or necessitated, or contingent upon [Ducasse] |
| Full Idea: There are four causal connections: an event is sufficient for another if it is its cause; an event is necessary for another if it is a condition for it; it is necessitated by another if it is an effect; it is contingent upon another if it is a resultant. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §2) | |
| A reaction: An event could be a condition for another without being necessary. He seems to have missed the indispensable aspect of a necessary condition. |
| 8373 | When a brick and a canary-song hit a window, we ignore the canary if we are interested in the breakage [Ducasse] |
| Full Idea: If a brick and the song of a canary strike a window, which breaks....we can truly say that the song of the canary had nothing to do with it, that is, in so far as what occurred is viewed merely as a case of breakage of window. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §5) | |
| A reaction: This is the germ of Davidson's view, that causation is entirely dependent on the mode of description, rather than being an actual feature of reality. If one was interested in the sound of the breakage, the canary would become relevant. |
| 8370 | A cause is a change which occurs close to the effect and just before it [Ducasse] |
| Full Idea: The cause of the particular change K was such particular change C as alone occurred in the immediate environment of K immediately before. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §3) | |
| A reaction: The obvious immediately difficulty would be overdetermination, as when it rains while I am watering my garden. The other problem would coincidence, as when I clap my hands just before a bomb goes off. |
| 8371 | Recurrence is only relevant to the meaning of law, not to the meaning of cause [Ducasse] |
| Full Idea: The supposition of recurrence is wholly irrelevant to the meaning of cause: that supposition is relevant only to the meaning of law. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §4) | |
| A reaction: This sounds plausible, especially if our notion of laws of nature is built up from a series of caused events. But we could just have an ontology of 'similar events', out of which we build laws, and 'causation' could drop out (á la Russell). |
| 8374 | We are interested in generalising about causes and effects purely for practical purposes [Ducasse] |
| Full Idea: We are interested in causes and effects primarily for practical purposes, which needs generalizations; so the interest of concrete individual facts of causation is chiefly an indirect one, as raw material for generalizations. | |
| From: Curt Ducasse (Nature and Observability of Causal Relations [1926], §6) | |
| A reaction: A nice explanation of why, if causation is fundamentally about single instances, people seem so interested in generalisations and laws. We want to predict, and we want to explain, and we want to intervene. |