14 ideas
14239 | The empty set is usually derived from Separation, but it also seems to need Infinity [Oliver/Smiley] |
Full Idea: The empty set is usually derived via Zermelo's axiom of separation. But the axiom of separation is conditional: it requires the existence of a set in order to generate others as subsets of it. The original set has to come from the axiom of infinity. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
A reaction: They charge that this leads to circularity, as Infinity depends on the empty set. |
14240 | The empty set is something, not nothing! [Oliver/Smiley] |
Full Idea: Some authors need to be told loud and clear: if there is an empty set, it is something, not nothing. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
A reaction: I'm inclined to think of a null set as a pair of brackets, so maybe that puts it into a metalanguage. |
14241 | We don't need the empty set to express non-existence, as there are other ways to do that [Oliver/Smiley] |
Full Idea: The empty set is said to be useful to express non-existence, but saying 'there are no Us', or ¬∃xUx are no less concise, and certainly less roundabout. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) |
14242 | Maybe we can treat the empty set symbol as just meaning an empty term [Oliver/Smiley] |
Full Idea: Suppose we introduce Ω not as a term standing for a supposed empty set, but as a paradigm of an empty term, not standing for anything. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
A reaction: This proposal, which they go on to explore, seems to mean that Ω (i.e. the traditional empty set symbol) is no longer part of set theory but is part of semantics. |
14243 | The unit set may be needed to express intersections that leave a single member [Oliver/Smiley] |
Full Idea: Thomason says with no unit sets we couldn't call {1,2}∩{2,3} a set - but so what? Why shouldn't the intersection be the number 2? However, we then have to distinguish three different cases of intersection (common subset or member, or disjoint). | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 2.2) |
14234 | If you only refer to objects one at a time, you need sets in order to refer to a plurality [Oliver/Smiley] |
Full Idea: A 'singularist', who refers to objects one at a time, must resort to the language of sets in order to replace plural reference to members ('Henry VIII's wives') by singular reference to a set ('the set of Henry VIII's wives'). | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
A reaction: A simple and illuminating point about the motivation for plural reference. Null sets and singletons give me the creeps, so I would personally prefer to avoid set theory when dealing with ontology. |
14237 | We can use plural language to refer to the set theory domain, to avoid calling it a 'set' [Oliver/Smiley] |
Full Idea: Plurals earn their keep in set theory, to answer Skolem's remark that 'in order to treat of 'sets', we must begin with 'domains' that are constituted in a certain way'. We can speak in the plural of 'the objects', not a 'domain' of objects. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
A reaction: [Skolem 1922:291 in van Heijenoort] Zermelo has said that the domain cannot be a set, because every set belongs to it. |
14245 | Logical truths are true no matter what exists - but predicate calculus insists that something exists [Oliver/Smiley] |
Full Idea: Logical truths should be true no matter what exists, so true even if nothing exists. The classical predicate calculus, however, makes it logically true that something exists. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) |
14246 | If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley] |
Full Idea: If mathematics was purely concerned with mathematical objects, there would be no room for applied mathematics. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) | |
A reaction: Love it! Of course, they are using 'objects' in the rather Fregean sense of genuine abstract entities. I don't see why fictionalism shouldn't allow maths to be wholly 'pure', although we have invented fictions which actually have application. |
14247 | Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley] |
Full Idea: Identifying numbers with sets may mean one of three quite different things: 1) the sets represent the numbers, or ii) they are the numbers, or iii) they replace the numbers. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.2) | |
A reaction: Option one sounds the most plausible to me. I will take numbers to be patterns embedded in nature, and sets are one way of presenting them in shorthand form, in order to bring out what is repeated. |
7458 | The reliability of witnesses depends on whether they benefit from their observations [Laplace, by Hacking] |
Full Idea: The credibility of a witness is in part a function of the story being reported. When the story claims to have infinite value, the temptation to lie for personal benefit is asymptotically infinite. | |
From: report of Pierre Simon de Laplace (Philosophical Essay on Probability [1820], Ch.XI) by Ian Hacking - The Emergence of Probability Ch.8 | |
A reaction: Laplace seems to especially have reports of miracles in mind. This observation certainly dashes any dreams one might have of producing a statistical measure of the reliability of testimony. |
3441 | If a supreme intellect knew all atoms and movements, it could know all of the past and the future [Laplace] |
Full Idea: An intelligence knowing at an instant the whole universe could know the movement of the largest bodies and atoms in one formula, provided his intellect were powerful enough to subject all data to analysis. Past and future would be present to his eyes. | |
From: Pierre Simon de Laplace (Philosophical Essay on Probability [1820]), quoted by Mark Thornton - Do we have free will? p.70 |
16713 | Philosophers are the forefathers of heretics [Tertullian] |
Full Idea: Philosophers are the forefathers of heretics. | |
From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
6610 | I believe because it is absurd [Tertullian] |
Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |