32 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) |
| 16771 | A composite is a true unity if all of its parts fall under one essence [Scheibler] |
| Full Idea: A composite entity is a unum per se if the partial entities that are in it are contained under one common essence. …In water, all those parts are contained under one essence of water. | |
| From: Chistoph Scheibler (Metaphysics [1650], I.4.1 n9), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 | |
| A reaction: Water mixed with wine is said to be an 'ens per accidens'. This is an unusual but possible view, that all the water there is is a single thing, united by its compositional essence. When we talk about 'water', we include possible water, and past water. |
| 12790 | Generalisations must be invariant to explain anything [Leuridan] |
| Full Idea: A generalisation is explanatory if and only if it is invariant. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §4) | |
| A reaction: [He cites Jim Woodward 2003] I dislike the idea that generalisations and regularities explain anything at all, but this rule sounds like a bare minimum for being taken seriously in the space of explanations. |
| 12789 | Biological functions are explained by disposition, or by causal role [Leuridan] |
| Full Idea: The main alternative to the dispositional theory of biological functions (which confer a survival-enhancing propensity) is the etiological theory (effects are functions if they play a role in the causal history of that very component). | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §3) | |
| A reaction: [Bigelow/Pargetter 1987 for the first, Mitchell 2003 for the second] The second one sounds a bit circular, but on the whole a I prefer causal explanations to dispositional explanations. |
| 14388 | Mechanisms must produce macro-level regularities, but that needs micro-level regularities [Leuridan] |
| Full Idea: Nothing can count as a mechanism unless it produces some macro-level regular behaviour. To produce macro-level regular behaviour, it has to rely on micro-level regularities. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §5) | |
| A reaction: This is the core of Leuridan's argument that regularities are more basic than mechanisms. It doesn't follow, though, that the more basic a thing is the more explanatory work it can do. I say mechanisms explain more than low-level regularities do. |
| 14386 | Mechanisms are ontologically dependent on regularities [Leuridan] |
| Full Idea: Mechanisms are ontologically dependent on the existence of regularities. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §3) | |
| A reaction: This seems to be the Humean rearguard action in favour of the regularity account of laws. Wrong, but a nice paper. This point shows why only powers (despite their vagueness!) are the only candidate for the bottom level of explanation. |
| 12787 | Mechanisms can't explain on their own, as their models rest on pragmatic regularities [Leuridan] |
| Full Idea: To model a mechanism one must incorporate pragmatic laws. ...As valuable as the concept of mechanism and mechanistic explanation are, they cannot replace regularities nor undermine their relevance for scientific explanation. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1) | |
| A reaction: [See Idea 12786 for 'pragmatic laws'] I just don't see how the observation of a regularity is any sort of explanation. I just take a regularity to be something interesting which needs to be explained. |
| 14384 | We can show that regularities and pragmatic laws are more basic than mechanisms [Leuridan] |
| Full Idea: Summary: mechanisms depend on regularities, there may be regularities without mechanisms, models of mechanisms must incorporate pragmatic laws, and pragmatic laws do not depend epistemologically on mechanistic models. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1) | |
| A reaction: See Idea 14382 for 'pragmatic' laws. I'm quite keen on mechanisms, so this is an arrow close to the heart, but at this point I say that my ultimate allegiance is to powers, not to mechanisms. |
| 14389 | There is nothing wrong with an infinite regress of mechanisms and regularities [Leuridan] |
| Full Idea: I see nothing metaphysically wrong in an infinite ontological regress of mechanisms and regularities. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §5) | |
| A reaction: This is a pretty unusual view, and I can't accept it. My revulsion at this regress is precisely the reason why I believe in powers, as the bottom level of explanation. |
| 14387 | Rather than dispositions, functions may be the element that brought a thing into existence [Leuridan] |
| Full Idea: The dispositional theory of biological functions is not unquestioned. The main alternative is the etiological theory: a component's effect is a function of that component if it has played an essential role in the causal history of its existence. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §3) | |
| A reaction: [He cites S.D. Mitchell 2003] Presumably this account is meant to fit into a theory of evolution in biology. The obvious problem is where something comes into existence for one reason, and then acquires a new function (such as piano-playing). |
| 14382 | Pragmatic laws allow prediction and explanation, to the extent that reality is stable [Leuridan] |
| Full Idea: A generalization is a 'pragmatic law' if it allows of prediction, explanation and manipulation, even if it fails to satisfy the traditional criteria. To this end, it should describe a stable regularity, but not necessarily a universal and necessary one. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1) | |
| A reaction: I am tempted to say of this that all laws are pragmatic, given that it is rather hard to know whether reality is stable. The universal laws consist of saying that IF reality stays stable in certain ways, certain outcomes will ensue necessarily. |
| 14385 | Strict regularities are rarely discovered in life sciences [Leuridan] |
| Full Idea: Strict regularities are rarely if ever discovered in the life sciences. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §2) | |
| A reaction: This is elementary once it is pointed out, but too much philosophy have science has aimed at the model provided by the equations of fundamental physics. Science is a broad church, to employ an entertaining metaphor. |
| 14383 | A 'law of nature' is just a regularity, not some entity that causes the regularity [Leuridan] |
| Full Idea: By 'law of nature' or 'natural law' I mean a generalization describing a regularity, not some metaphysical entity that produces or is responsible for that regularity. | |
| From: Bert Leuridan (Can Mechanisms Replace Laws of Nature? [2010], §1 n1) | |
| A reaction: I take the second version to be a relic of a religious world view, and having no place in a naturalistic metaphysic. The regularity view is then the only player in the field, and the question is, can we do more? Can't we explain regularities? |