29 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. |
| 11115 | 'All horses' either picks out the horses, or the things which are horses [Jubien] |
| Full Idea: Two ways to see 'all horses are animals' are as picking out all the horses (so that it is a 'horse-quantifier'), ..or as ranging over lots of things in addition to horses, with 'horses' then restricting the things to those that satisfy 'is a horse'. | |
| From: Michael Jubien (Analyzing Modality [2007], 2) | |
| A reaction: Jubien says this gives you two different metaphysical views, of a world of horses etc., or a world of things which 'are horses'. I vote for the first one, as the second seems to invoke an implausible categorical property ('being a horse'). Cf Idea 11116. |
| 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) |
| 11116 | Being a physical object is our most fundamental category [Jubien] |
| Full Idea: Being a physical object (as opposed to being a horse or a statue) really is our most fundamental category for dealing with the external world. | |
| From: Michael Jubien (Analyzing Modality [2007], 2) | |
| A reaction: This raises the interesting question of why any categories should be considered to be more 'fundamental' than others. I can only think that we perceive something to be an object fractionally before we (usually) manage to identify it. |
| 11117 | Haecceities implausibly have no qualities [Jubien] |
| Full Idea: Properties of 'being such and such specific entity' are often called 'haecceities', but this term carries the connotation of non-qualitativeness which I don't favour. | |
| From: Michael Jubien (Analyzing Modality [2007], 2) | |
| A reaction: The way he defines it makes it sound as if it was a category, but I take it to be more like a bare individual essence. If it has not qualities then it has no causal powers, so there could be no evidence for its existence. |
| 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. |
| 14633 | How do we tell a table's being contingently plastic from its being essentially plastic? [Jackson] |
| Full Idea: On a friendly reading of Quine, there is nothing to make the difference between a table's being contingently plastic and its being essentially plastic. | |
| From: Frank Jackson (Possible Worlds and Necessary A Posteriori [2010], 5) | |
| A reaction: This is, of course, the dreaded modern usage of 'essential' to just mean 'necessary' and nothing more. In my view, there may be a big problem with knowing whether a problem is necessary, but knowing whether it is essential is much easier. |
| 14635 | An x is essentially F if it is F in every possible world in which it appears [Jackson] |
| Full Idea: On the possible world's account, x's being essentially F is nothing more nor less than x's being F in every world in which it appears. | |
| From: Frank Jackson (Possible Worlds and Necessary A Posteriori [2010], 6) | |
| A reaction: There you go - 'true in every possible world' is the definition of metaphysical necessity, not the definition of essence. Either get back to Aristotle, or stop (forever!) talking about 'essence'! |
| 14632 | Quine may have conflated de re and de dicto essentialism, but there is a real epistemological problem [Jackson] |
| Full Idea: The unfriendly response to Quine's objection to essentialism is that it conflates the de re and the de dicto. The friendly response is that behind that conflation is a real epistemological problem for essentialism. | |
| From: Frank Jackson (Possible Worlds and Necessary A Posteriori [2010], 1) | |
| A reaction: He cites Richard Cartwright 1968 for the friendly response. The epistemological question is how we can know the essentialness of an essence. |
| 11119 | De re necessity is just de dicto necessity about object-essences [Jubien] |
| Full Idea: I suggest that the de re is to be analyzed in terms of the de dicto. ...We have a case of modality de re when (and only when) the appropriate property in the de dicto formulation is an object-essence. | |
| From: Michael Jubien (Analyzing Modality [2007], 5) |
| 11118 | Modal propositions transcend the concrete, but not the actual [Jubien] |
| Full Idea: Where modal propositions may once have seemed to transcend the actual, they now seem only to transcend the concrete. | |
| From: Michael Jubien (Analyzing Modality [2007], 4) | |
| A reaction: This is because Jubien has defended a form of platonism. Personally I take modal propositions to be perceptible in the concrete world, by recognising the processes involved, not the mere static stuff. |
| 11108 | Your properties, not some other world, decide your possibilities [Jubien] |
| Full Idea: The possibility of your having been a playwright has nothing to do with how people are on other planets, whether in our own or in some other realm. It is only to do with you and the relevant property. | |
| From: Michael Jubien (Analyzing Modality [2007], 1) | |
| A reaction: I'm inclined to think that this simple point is conclusive disproof of possible worlds as an explanation of modality (apart from Jubien's other nice points). What we need to understand are modal properties, not other worlds. |
| 11111 | Modal truths are facts about parts of this world, not about remote maximal entities [Jubien] |
| Full Idea: Typical modal truths are just facts about our world, and generally facts about very small parts of it, not facts about some infinitude of complex, maximal entities. | |
| From: Michael Jubien (Analyzing Modality [2007], 1) | |
| A reaction: I think we should embrace this simple fact immediately, and drop all this nonsense about possible worlds, even if they are useful for the semantics of modal logic. |
| 14631 | How can you show the necessity of an a posteriori necessity, if it might turn out to be false? [Jackson] |
| Full Idea: If something is offered as a candidate necessary a posteriori truth, how could we show that it is necessary, in the face of the fact that it takes investigation to show that it is true, and so, in some sense, it might have turned out to be false? | |
| From: Frank Jackson (Possible Worlds and Necessary A Posteriori [2010], 1) | |
| A reaction: This is the topic of his paper, which he compares with how we can know that essences are essential. |
| 11105 | We have no idea how many 'possible worlds' there might be [Jubien] |
| Full Idea: As soon as we start talking about 'possible world', we beg the question of their relevance to our prior notion of possibility. For all we know, there are just two such realms, or twenty-seven, or uncountably many, or even set-many. | |
| From: Michael Jubien (Analyzing Modality [2007], 1) |
| 11107 | If there are no other possible worlds, do we then exist necessarily? [Jubien] |
| Full Idea: Suppose there happen to be no other concrete realms. Would we happily accept the consequence that we exist necessarily? | |
| From: Michael Jubien (Analyzing Modality [2007], 1) |
| 11106 | If all possible worlds just happened to include stars, their existence would be necessary [Jubien] |
| Full Idea: If all of the possible worlds happened to include stars, how plausible is it to think that if this is how things really are, then we've just been wrong to regard the existence of stars as contingent? | |
| From: Michael Jubien (Analyzing Modality [2007], 1) |
| 11112 | Possible worlds just give parallel contingencies, with no explanation at all of necessity [Jubien] |
| Full Idea: In the world theory, what passes for 'necessity' is just a bunch of parallel 'contingencies'. The theory provides no basis for understanding why these contingencies repeat unremittingly across the board (while others do not). | |
| From: Michael Jubien (Analyzing Modality [2007], 1) |
| 11109 | If other worlds exist, then they are scattered parts of the actual world [Jubien] |
| Full Idea: Any other realms that happened to exist would just be scattered parts of the actual world, not entire worlds at all. It would just happen that physical reality was fragmented in this remarkable but modally inconsequential way. | |
| From: Michael Jubien (Analyzing Modality [2007], 1) | |
| A reaction: This is aimed explicitly at Lewis's modal realism, and strikes me as correct. Jubien's key point here is that they are irrelevant to modality, just as foreign countries are irrelevant to the modality of this one. |
| 11113 | Worlds don't explain necessity; we use necessity to decide on possible worlds [Jubien] |
| Full Idea: The suspicion is that the necessity doesn't arise from how worlds are, but rather that the worlds are taken to be as they are in order to capture the intuitive necessity. | |
| From: Michael Jubien (Analyzing Modality [2007], 1) | |
| A reaction: It has always seemed to me rather glaring that you need a prior notion of 'possible' before you can start to talk about 'possible worlds', but I have always been too timid to disagree with the combination of Saul Kripke and David Lewis. Thank you, Jubien! |
| 11110 | We mustn't confuse a similar person with the same person [Jubien] |
| Full Idea: If someone similar to Humphrey won the election, that nicely establishes the possibility of someone's winning who is similar to Humphrey. But we mustn't confuse this possibility with the intuitively different possibility of Humphrey himself winning. | |
| From: Michael Jubien (Analyzing Modality [2007], 1) |