17 ideas
| 10807 | Mathematics reduces to set theory, which reduces, with some mereology, to the singleton function [Lewis] |
| Full Idea: It is generally accepted that mathematics reduces to set theory, and I argue that set theory in turn reduces, with some aid of mereology, to the theory of the singleton function. | |
| From: David Lewis (Mathematics is Megethology [1993], p.03) |
| 10809 | We can accept the null set, but not a null class, a class lacking members [Lewis] |
| Full Idea: In my usage of 'class', there is no such things as the null class. I don't mind calling some memberless thing - some individual - the null set. But that doesn't make it a memberless class. Rather, that makes it a 'set' that is not a class. | |
| From: David Lewis (Mathematics is Megethology [1993], p.05) | |
| A reaction: Lewis calls this usage 'idiosyncratic', but it strikes me as excellent. Set theorists can have their vital null class, and sensible people can be left to say, with Lewis, that classes of things must have members. |
| 10811 | The null set plays the role of last resort, for class abstracts and for existence [Lewis] |
| Full Idea: The null set serves two useful purposes. It is a denotation of last resort for class abstracts that denote no nonempty class. And it is an individual of last resort: we can count on its existence, and fearlessly build the hierarchy of sets from it. | |
| From: David Lewis (Mathematics is Megethology [1993], p.09) | |
| A reaction: This passage assuages my major reservation about the existence of the null set, but at the expense of confirming that it must be taken as an entirely fictional entity. |
| 10812 | The null set is not a little speck of sheer nothingness, a black hole in Reality [Lewis] |
| Full Idea: Should we accept the null set as a most extraordinary individual, a little speck of sheer nothingness, a sort of black hole in the fabric of Reality itself? Not that either, I think. | |
| From: David Lewis (Mathematics is Megethology [1993], p.09) | |
| A reaction: Correct! |
| 10813 | What on earth is the relationship between a singleton and an element? [Lewis] |
| Full Idea: A new student of set theory has just one thing, the element, and he has another single thing, the singleton, and not the slightest guidance about what one thing has to do with the other. | |
| From: David Lewis (Mathematics is Megethology [1993], p.12) |
| 10814 | Are all singletons exact intrinsic duplicates? [Lewis] |
| Full Idea: Are all singletons exact intrinsic duplicates? | |
| From: David Lewis (Mathematics is Megethology [1993], p.13) |
| 10806 | Megethology is the result of adding plural quantification to mereology [Lewis] |
| Full Idea: Megethology is the result of adding plural quantification, as advocated by George Boolos, to the language of mereology. | |
| From: David Lewis (Mathematics is Megethology [1993], p.03) |
| 10816 | We can use mereology to simulate quantification over relations [Lewis] |
| Full Idea: We can simulate quantification over relations using megethology. Roughly, a quantifier over relations is a plural quantifier over things that encode ordered pairs by mereological means. | |
| From: David Lewis (Mathematics is Megethology [1993], p.18) | |
| A reaction: [He credits this idea to Burgess and Haven] The point is to avoid second-order logic, which quantifies over relations as ordered n-tuple sets. |
| 10808 | Mathematics is generalisations about singleton functions [Lewis] |
| Full Idea: We can take the theory of singleton functions, and hence set theory, and hence mathematics, to consist of generalisations about all singleton functions. | |
| From: David Lewis (Mathematics is Megethology [1993], p.03) | |
| A reaction: At first glance this sounds like a fancy version of the somewhat discredited Greek idea that mathematics is built on the concept of a 'unit'. |
| 10815 | We don't need 'abstract structures' to have structural truths about successor functions [Lewis] |
| Full Idea: We needn't believe in 'abstract structures' to have general structural truths about all successor functions. | |
| From: David Lewis (Mathematics is Megethology [1993], p.16) |
| 10810 | I say that absolutely any things can have a mereological fusion [Lewis] |
| Full Idea: I accept the principle of Unrestricted Composition: whenever there are some things, no matter how many or how unrelated or how disparate in character they may be, they have a mereological fusion. ...The trout-turkey is part fish and part fowl. | |
| From: David Lewis (Mathematics is Megethology [1993], p.07) | |
| A reaction: This nicely ducks the question of when things form natural wholes and when they don't, but I would have thought that that might be one of the central issues of metaphysicals, so I think I'll give Lewis's principle a miss. |
| 21513 | We can no more expect a precise definition of coherence than we can of the moral ideal [Ewing] |
| Full Idea: I think it is wrong to tie down the advocates of the coherence theory to a precise definition. ...It would be altogether unreasonable to demand that the moral ideal should be exhaustively defined, and the same may be true of the ideal of thought. | |
| From: A.C. Ewing (Idealism: a critical survey [1934], p.231), quoted by Erik J. Olsson - Against Coherence 7.6 | |
| A reaction: I strongly agree. It is not a council of despair. I think the criteria of coherence can be articulated quite well (e.g by Thagard), and the virtues of enquiry can also be quite well specified (e.g. by Zagzebski). Very dissimilar evidence must cohere. |
| 21497 | If undetailed, 'coherence' is just a vague words that covers all possible arguments [Ewing] |
| Full Idea: Without a detailed account, coherence is reduced to the mere muttering of the word 'coherence', which can be interpreted so as to cover all arguments, but only by making its meaning so wide as to rob it of almost all significance. | |
| From: A.C. Ewing (Idealism: a critical survey [1934], p.246), quoted by Erik J. Olsson - Against Coherence 2.2 | |
| A reaction: I'm a fan of coherence, but it is a placeholder, involving no intrinsic or detailed theory. I just think it points to the reality of how we make judgements, especially practical ones. We can categorise the inputs, and explain the required virtues. |
| 20508 | How people vote should be on public record, so they can be held accountable [Mill, by Wolff,J] |
| Full Idea: Mill believed in an open vote. People should be held accountable for how they vote, and therefore it should be a matter of public record. | |
| From: report of John Stuart Mill (Representative Government [1861]) by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 3 'Representative' | |
| A reaction: Nowadays it is a mantra that voting should be secret, because coercion is an obvious problem, but MPs vote publicly, and are held accountable for their voting records. People like the mafia seem to make open public voting impossible. |
| 20507 | Voting is a strict duty, like jury service, and must only be aimed at the public good [Mill] |
| Full Idea: The citizen's vote is not a thing in which he has an option; it has no more to do with his personal wishes than the verdict of a juryman. ...he is bound to give it according to his best and most conscientious opinion of the public good. | |
| From: John Stuart Mill (Representative Government [1861], p.299), quoted by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 3 'Representative' | |
| A reaction: Mill was also concerned that voters might pursue 'class interest' (which they currently do, big time). |
| 20505 | Direct democracy is inexperience judging experience, and ignorance judging knowledge [Mill] |
| Full Idea: At its best [direct democracy] is inexperience sitting in judgement on experience, ignorance on knowledge. | |
| From: John Stuart Mill (Representative Government [1861], p.232), quoted by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 4 'Representative' | |
| A reaction: Recent experiments have suggested that inexperienced people can become very good at making large decisions, if they are allowed to consult experts when they want to. See Van Reybrouck's 'Against Elections'. |
| 20504 | People can only participate in decisions in small communities, so representatives are needed [Mill] |
| Full Idea: Since all cannot, in a community exceeding a single small town, participate personally in any but some very minor portions of the public business, it follows that the ideal type of a perfect government must be representative. | |
| From: John Stuart Mill (Representative Government [1861], p.217-8), quoted by Jonathan Wolff - An Introduction to Political Philosophy (Rev) 4 'Representative' | |
| A reaction: Wolff offers Mill as the principal spokesman for representative democracy. It is not only the difficulty of achieving participation, but also the slowness of decision-making. Modern technology may be changing all of this. |