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| 18395 | Sets are mereological sums of the singletons of their members |
| Full Idea: Lewis pointed out that many-membered classes are nothing more than the mereological wholes of the classes formed by taking the singleton of each member. | |||
| From: report of David Lewis (Parts of Classes [1991]) by David M. Armstrong - Truth and Truthmakers 09.4 | |||
| A reaction: You can't combine members to make the class, because the whole and the parts are of different type, but here the parts and whole are both sets, so they combine like waterdrops. |
| 15496 | We can build set theory on singletons: classes are then fusions of subclasses, membership is the singleton |
| Full Idea: The notion of a singleton, or unit set, can serve as the distinctive primitive of set theory. The rest is mereology: a class is the fusion of its singleton subclasses, something is a member of a class iff its singleton is part of that class. | |||
| From: David Lewis (Parts of Classes [1991], Pref) | |||
| A reaction: This is a gloriously bold proposal which I immediately like, because it cuts out the baffling empty set (which many people think 'exists'!), and gets mathematics back to being about the real world of entities (as the Greeks thought). |
| 15500 | Classes divide into subclasses in many ways, but into members in only one way |
| Full Idea: A class divides exhaustively into subclasses in many different ways; whereas a class divides exhaustively into members in only one way. | |||
| From: David Lewis (Parts of Classes [1991], 1.2) |
| 15499 | A subclass of a subclass is itself a subclass; a member of a member is not in general a member |
| Full Idea: Just as a part of a part is itself a part, so a subclass of a subclass is itself a subclass; whereas a member of a member is not in general a member. | |||
| From: David Lewis (Parts of Classes [1991], 1.2) | |||
| A reaction: Lewis is showing the mereological character of sets, but this is a key distinction in basic set theory. When the members of members are themselves members, the set is said to be 'transitive'. |
| 15498 | We can accept the null set, but there is no null class of anything |
| Full Idea: There is no such class 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. | |||
| From: David Lewis (Parts of Classes [1991], 1.2) | |||
| A reaction: The point is that set theory is a formal system which can do what it likes, but classes are classes 'of' things. Everyone assumes that sets are classes, reserving 'proper classes' for the tricky cases up at the far end. |
| 15502 | There are four main reasons for asserting that there is an empty set |
| Full Idea: The null set is a denotation of last resort for class-terms that fail to denote classes, an intersection of x and y where they have no members in common, the class of all self-members, and the real numbers such that x^2+1=0. This is all mere convenience. | |||
| From: David Lewis (Parts of Classes [1991], 1.4) | |||
| A reaction: A helpful catalogue of main motivations for the existence of the null set in set theory. Lewis aims to undermine these reasons, and dispense with the wretched thing. |
| 15503 | We needn't accept this speck of nothingness, this black hole in the fabric of Reality! |
| Full Idea: Must 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 really. | |||
| From: David Lewis (Parts of Classes [1991], 1.4) | |||
| A reaction: We can only dream of reaching the level of confidence that Lewis reached, to make such beautiful fun of a highly counterintuitive idea that is rooted in the modern techniques of philosophy. |
| 15506 | If we don't understand the singleton, then we don't understand classes |
| Full Idea: Our utter ignorance about the nature of the singletons amounts to sheer ignorance about the nature of classes generally. | |||
| From: David Lewis (Parts of Classes [1991], 2.1) |
| 15497 | We can replace the membership relation with the member-singleton relation (plus mereology) |
| Full Idea: Given the theory of part and whole, the member-singleton relation may replace membership generally as the primitive notion of set theory. | |||
| From: David Lewis (Parts of Classes [1991], Pref) | |||
| A reaction: An obvious question is to ask what the member-singleton relation is if it isn't membership. |
| 15511 | If singleton membership is external, why is an object a member of one rather than another? |
| Full Idea: Suppose the relation of member to singleton is external. Why must Possum be a member of one singleton rather than another? Why isn't it contingent which singleton is his? | |||
| From: David Lewis (Parts of Classes [1991], 2.2) | |||
| A reaction: He cites Van Inwagen for raising this question, and answers it in terms of counterparts. So is the relation internal or external? I think of sets as pairs of curly brackets, not existing entities, so the question doesn't bother me. |
| 15513 | Maybe singletons have a structure, of a thing and a lasso? |
| Full Idea: Maybe the singleton of something x is not an atom, but consists of x plus a lasso. That gives a singleton an internal structure. ...But what do we know of the nature of the lasso, or how it fits? We are no better off. | |||
| From: David Lewis (Parts of Classes [1991], 2.5) | |||
| A reaction: [second bit on p.45] |
| 15507 | Set theory has some unofficial axioms, generalisations about how to understand it |
| Full Idea: Set theory has its unofficial axioms, traditional remarks about the nature of classes. They are never argued, but are passed heedlessly from one author to another. One of these says that the classes are nowhere: they are outside space and time. | |||
| From: David Lewis (Parts of Classes [1991], 2.1) | |||
| A reaction: Why don't the people who write formal books on set theory ever say things like this? |
| 10191 | Set theory reduces to a mereological theory with singletons as the only atoms |
| Full Idea: Lewis has shown that set theory may be reduced to a mereological theory in which singletons are the only atoms. | |||
| From: report of David Lewis (Parts of Classes [1991]) by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' p.80 | |||
| A reaction: Presumably the axiom of extensionality, that a set is no more than its members, translates into unrestricted composition, that any parts will make an object. Difficult territory, but I suspect that this is of great importance in metaphysics. |
| 15508 | If singletons are where their members are, then so are all sets |
| Full Idea: If every singleton was where its member was, then, in general, classes would be where there members were. | |||
| From: David Lewis (Parts of Classes [1991], 2.1) | |||
| A reaction: There seems to be a big dislocation of understanding of the nature of sets, between 'pure' set theory, and set theory with ur-elements. I take the pure to be just an 'abstraction' from the more located one. The empty set has a puzzling location. |
| 15523 | Set theory isn't innocent; it generates infinities from a single thing; but mathematics needs it |
| Full Idea: Set theory is not innocent. Its trouble is that when we have one thing, then somehow we have another wholly distinct thing, the singleton. And another, and another....ad infinitum. But that's the price for mathematical power. Pay it. | |||
| From: David Lewis (Parts of Classes [1991], 3.6) |
| 15514 | A huge part of Reality is only accepted as existing if you have accepted set theory |
| Full Idea: The preponderant part of Reality must consist of unfamiliar, unobserved things, whose existence would have gone unsuspected but for our acceptance of set theory. | |||
| From: David Lewis (Parts of Classes [1991], 2.6) | |||
| A reaction: He is referring to the enormous sets at the far end of set theory, of a size that had never been hitherto conceived. Excellent. Daft to believe in something entirely because you have accepted set theory, with no other basis. |
| 15525 | Plural quantification lacks a complete axiom system |
| Full Idea: There is an irremediable lack of a complete axiom system for plural quantification. | |||
| From: David Lewis (Parts of Classes [1991], 4.7) |
| 15518 | I like plural quantification, but am not convinced of its connection with second-order logic |
| Full Idea: I agree fully with Boolos on substantive questions about plural quantification, though I would make less than he does of the connection with second-order logic. | |||
| From: David Lewis (Parts of Classes [1991], 3.2 n2) | |||
| A reaction: Deep matters, but my inclination is to agree with Lewis, as I have never been able to see why talk of plural quantification led straight on to second-order logic. A plural is just some objects, not some higher-order entity. |
| 15524 | Zermelo's model of arithmetic is distinctive because it rests on a primitive of set theory |
| Full Idea: What sets Zermelo's modelling of arithmetic apart from von Neumann's and all the rest is that he identifies the primitive of arithmetic with an appropriately primitive notion of set theory. | |||
| From: David Lewis (Parts of Classes [1991], 4.6) | |||
| A reaction: Zermelo's model is just endlessly nested empty sets, which is a very simple structure. I gather that connoisseurs seem to prefer von Neumann's model (where each number contains its predecessor number). |
| 15517 | Giving up classes means giving up successful mathematics because of dubious philosophy |
| Full Idea: Renouncing classes means rejecting mathematics. That will not do. Mathematics is an established, going concern. Philosophy is as shaky as can be. | |||
| From: David Lewis (Parts of Classes [1991], 2.8) | |||
| A reaction: This culminates in his famous 'Who's going to tell the mathematicians? Not me!'. He has just given four examples of mathematics that seems to entirely depend on classes. This idea sounds like G.E. Moore's common sense against scepticism. |
| 15515 | To be a structuralist, you quantify over relations |
| Full Idea: To be a structuralist, you quantify over relations. | |||
| From: David Lewis (Parts of Classes [1991], 2.6) |
| 15520 | Existence doesn't come in degrees; once asserted, it can't then be qualified |
| Full Idea: Existence cannot be a matter of degree. If you say there is something that exists to a diminished degree, once you've said 'there is' your game is up. | |||
| From: David Lewis (Parts of Classes [1991], 3.5) | |||
| A reaction: You might have thought that this was so obvious as to be not worth saying, but as far as I can see it is a minority view in contemporary philosophy. It was Quine's view, and it is mine. |
| 15504 | Atomless gunk is an individual whose parts all have further proper parts |
| Full Idea: A blob can represent atomless gunk: an individual whose parts all have further proper parts. | |||
| From: David Lewis (Parts of Classes [1991], 1.8) | |||
| A reaction: This is not the same as 'stuff', since gunk is a precise fusion of all those parts, whereas there is no such precision about stuff. Stuff is neutral as to whether it has atoms, or is endlessly divisible. My love of stuff grows. Laycock is a hero. |
| 15501 | We have no idea of a third sort of thing, that isn't an individual, a class, or their mixture |
| Full Idea: As yet we have no idea of any third sort of thing that is neither individual nor class nor mixture of the two. | |||
| From: David Lewis (Parts of Classes [1991], 1.2) | |||
| A reaction: You can see that Lewis was a pupil of Quine. I quote this to show how little impression 'stuff' makes on the modern radar. His defence is that stuff may not be a 'thing', but then he seems to think that 'things' exhaust reality (top p.8 and 9). |
| 15516 | A property is any class of possibilia |
| Full Idea: A property is any class of possibilia. | |||
| From: David Lewis (Parts of Classes [1991], 2.7) |
| 14748 | The many are many and the one is one, so they can't be identical |
| Full Idea: What is true of the many is not exactly what is true of the one. After all they are many while it is one. The number of the many is six, whereas the number of the fusion is one. The singletons of the many are distinct from the singleton of the one. | |||
| From: David Lewis (Parts of Classes [1991], 3.6) | |||
| A reaction: I wouldn't take this objection to be conclusive. 'Some pebbles' seem to be many, but a 'handful of pebbles' seem to be one, where the physical situation might be identical. If they are not identical, then the non-identity is purely conceptual. |
| 6129 | Lewis affirms 'composition as identity' - that an object is no more than its parts |
| Full Idea: Lewis says that the parts of a thing are identical with the whole they compose, calling his view 'composition as identity', which is the claim that a physical object is 'nothing over and above its parts'. | |||
| From: report of David Lewis (Parts of Classes [1991], p.84-7) by Trenton Merricks - Objects and Persons §I.IV | |||
| A reaction: The ontological economy of this view is obviously attractive, but I don't agree with it. You certainly can't say that all identity consists entirely of composition by parts, because the parts need identity to get the view off the ground. |
| 15512 | In mereology no two things consist of the same atoms |
| Full Idea: It is a principle of mereology that no two things consist of exactly the same atoms. | |||
| From: David Lewis (Parts of Classes [1991], 2.3) | |||
| A reaction: The problem with this is screamingly obvious - that the same atoms might differ in structure. Lewis did refer to this problem, but seems to try to wriggle out of it, in Idea 15444. |
| 15519 | Trout-turkeys exist, despite lacking cohesion, natural joints and united causal power |
| Full Idea: A trout-turkey is inhomogeneous, disconnected, not in contrast with its surroundings. It is not cohesive, not causally integrated, not a causal unit in its impact on the rest of the world. It is not carved at the joints. That doesn't affect its existence. | |||
| From: David Lewis (Parts of Classes [1991], 3.5) | |||
| A reaction: A nice pre-emptive strike against all the reasons why anyone might think more is needed for unity than a mereological fusion. |
| 15521 | Given cats, a fusion of cats adds nothing further to reality |
| Full Idea: Given a prior commitment to cats, a commitment to cat-fusions is not a further commitment. The fusion is nothing over and above the cats that compose it. It just is them. They just are it. Together or separately, the cats are the same portion of Reality. | |||
| From: David Lewis (Parts of Classes [1991], 3.6) | |||
| A reaction: The two extremes of ontology are that there are no objects, or that every combination is an object. Until reading this I thought Lewis was in the second camp, but this sounds like object-nihilism, as in Van Inwagen and Merricks. |
| 15522 | The one has different truths from the many; it is one rather than many, one rather than six |
| Full Idea: What's true of the many is not exactly what's true of the one. After all they are many while it is one. The number of the many is six, whereas the number of the fusion is one. | |||
| From: David Lewis (Parts of Classes [1991], 3.6) | |||
| A reaction: Together with Idea 15521, this nicely illustrates the gulf between commitment to ontology and commitment to truths. The truths about a fusion change, while its ontology remains the same. Possibly this is the key to all of metaphysics. |
| 10566 | Lewis prefers giving up singletons to giving up sums |
| Full Idea: In the face of the conflict between mereology and set theory, Lewis has advocated giving up the existence of singletons rather than sums. | |||
| From: report of David Lewis (Parts of Classes [1991]) by Kit Fine - Replies on 'Limits of Abstraction' 1 |
| 14244 | Lewis only uses fusions to create unities, but fusions notoriously flatten our distinctions |
| Full Idea: Lewis employs mereological fusion as his sole method of making one thing out of many, and fusion is notorious for the way it flattens out and thereby obliterates distinctions. | |||
| From: comment on David Lewis (Parts of Classes [1991]) by Oliver,A/Smiley,T - What are Sets and What are they For? 3.1 | |||
| A reaction: I take this to be a key point in the discussion of mereology in ontological contexts. As a defender of intrinsic structural essences, I have no use for mereological fusions, and look for a quite different identity for 'wholes'. |
| 10660 | A commitment to cat-fusions is not a further commitment; it is them and they are it |
| Full Idea: Given a prior commitment to cats, a commitment to cat-fusions is not a further commitment. The fusion is nothing over and above the cats that compose it. It just is them. They just are it. | |||
| From: David Lewis (Parts of Classes [1991], p.81), quoted by Achille Varzi - Mereology 4.3 | |||
| A reaction: I take this to make Lewis a nominalist, saying the same thing that Goodman said about Utah in Idea 10657. Any commitment to cat-fusions being more than the cats, or Utah being more than its counties, strikes me as crazy. |
| 15509 | Some say qualities are parts of things - as repeatable universals, or as particulars |
| Full Idea: Some philosophers propose that things have their qualities by having them as parts, either as repeatable universals (Goodman), or as particulars (Donald Williams). | |||
| From: David Lewis (Parts of Classes [1991], 2.1 n2) | |||
| A reaction: He refers to 'qualities' rather than 'properties', presumably because this view makes them all intrinsic to the object. Is being 'handsome' a part of a person? |