14 ideas
8000 | He who is ignorant of the history of philosophy is doomed to repeat it [Santayana, by MacIntyre] |
Full Idea: Santayana remarked that he who is ignorant of the history of philosophy is doomed to repeat it. | |
From: report of George Santayana (The Life of Reason [1906]) by Alasdair MacIntyre - A Short History of Ethics Ch.1 | |
A reaction: Santayana's remark seems to have been about history in general, so this is a Macintyre thought. It obviously has a lot of truth, and most great philosophers seem hugely knowledgeable. However, ignorance brings a kind of freedom. |
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) |
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. |
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. |
18521 | The criterion of existence is the possibility of action [Santayana] |
Full Idea: The possibility of action ...is the criterion of existence, and the test of substantiality. | |
From: George Santayana (The Realm of Matter [1930], p.107), quoted by John Heil - The Universe as We Find It | |
A reaction: I rather like this. I think I would say the power is the criterion of existence. |
3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
Full Idea: Anaxarchus said that he was not even sure that he knew nothing. | |
From: report of Anaxarchus (fragments/reports [c.340 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.10.1 |
23060 | The good is not relative, but is rooted in facts about human needs [Santayana] |
Full Idea: The good is by no means relative to opinion, but is rooted in the unconscious and fatal nature of living beings, a nature which predetermines for them the difference between foods and poisons, happiness and misery. | |
From: George Santayana (Platonism and the Spiritual Life [1927], p.3), quoted by John Gray - Seven Types of Atheism 6 | |
A reaction: That is, he concedes that the good is relative to human beings, but that the relevant facts about human beings are not relative. I think he has the correct picture. The key point is that the good is 'rooted' in something, and doesn't just float free. |