18 ideas
| 7823 | Lucretius was rediscovered in 1417 [Grayling] |
| Full Idea: Lucretius's 'De Rerum Natura' was rediscovered in 1417, after languishing forgotten for six centuries. | |
| From: A.C. Grayling (What is Good? [2003], Ch.5) | |
| A reaction: A wonder. Is it the greatest book of the ancient world - because it partially preserves the lost philosophy of great Democritus? |
| 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. |
| 7809 | In an honour code shame is the supreme punishment, and revenge is a duty [Grayling] |
| Full Idea: An honour code is one in which the greatest punishment is shame, and in which revenge is a duty. | |
| From: A.C. Grayling (What is Good? [2003], Ch.2) | |
| A reaction: Is this really what Nietzsche wanted to revive? Shame isn't a private matter - it needs solidarity of values in the community, and contempt for dishonour, so that it becomes everyone's worst fear. |
| 7824 | If suicide is lawful, but assisting suicide is unlawful, powerless people are denied their rights [Grayling] |
| Full Idea: An anomaly created by England's 1961 Suicide Act is that it is lawful to take one's own life, but unlawful to help anyone else to do it. This means anyone unable to commit suicide without help is denied one of their fundamental rights. | |
| From: A.C. Grayling (What is Good? [2003], Ch.8) | |
| A reaction: There is a difference, not really captured either by law or by reason, between tolerating an activity, and encouraging and helping it. I think the test question is "this activity is legal, but would you want your child to do it?" |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |
| 7819 | Religion gives answers, comforts, creates social order, and panders to superstition [Grayling] |
| Full Idea: The four standard explanations given for religion are that it provides answer, that it gives comfort, that it makes for social order, and that it rests on mere superstition. | |
| From: A.C. Grayling (What is Good? [2003], Ch.4) | |
| A reaction: All four of these could be correct, though the first and fourth would be incompatible if religion gives correct answers. Why religion begins might be not the same as the reason why it continues. |
| 7817 | To make an afterlife appealing, this life has to be denigrated [Grayling] |
| Full Idea: It is remarkable how much the life of this world has to be denigrated to make the promise of happiness after death appealing. | |
| From: A.C. Grayling (What is Good? [2003], Ch.4) | |
| A reaction: This seems to be true of most religions, but it could be otherwise. Surely you want such a wonderful life to continue after death? But then you would not be obliged to do anything difficult to achieve immortality. Power comes into it... |
| 7818 | In Greek mythology only heroes can go to heaven [Grayling] |
| Full Idea: In Greek mythology only a hero like Hercules could hope to go to heaven (by becoming a god himself). | |
| From: A.C. Grayling (What is Good? [2003], Ch.4) | |
| A reaction: This illustrates Nietsche's 'inversion of morality' most clearly, because Christianity says that the person most likely to go to heaven is the humblest person. |