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Monday, July 26, 2010

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Wouldn't it be something along the lines of making one-to-one mappings?

In other words, for case 1 you take every individual you can find who calls himself "Hatfield", and associate him with some individual who calls himself "McCoy". If you run out of McCoys while still having some Hatfields left, then the Hatfields outnumber the McCoys.

For case 2, you just map each Hatfield onto one of the ordinal numbers until you run out of Hatfields.

Deosn't that do the trick? I may be missing something.

Hi Malcolm,

I don't think you appreciate the question I have been discussing with William. What you have given us is a method of determining whether the propositions are true.

But our question cuts deeper and concerns what the propositions are saying quite apart from whehter they are true or false. We are wondering about the correct analysis of such sentences and whether a correct analysis requires us to posit entities such as sets.

I claim that 'The Hatfields' refers to an entity distinct from each of the human beings who bear that name, and that the property of being 100 strong is a property of that entity. William denies this. So I challenge him: provide analyses of (1) and (2) which are adequate but do not involve the postulation of entities other than concrete particulars.

As a New Yorker you must have an opinion about the Ground Zero mosque. Feel free to tell us what it is in the 'mosque' thread.

Sorry, Bill, I hadn't seen the earlier post on this topic. I'll read it before commenting further. I will confess I am guilty of nominalist leanings myself, but I did indeed misunderstand the question.

And as a New Yorker who watched the towers fall on 9/11 -- and spent the whole day wondering if his daughter, who was at Stuyvesant High three blocks away, was dead or alive -- I most certainly do have an opinion about that mosque, and I'll bet you can guess what it is.

I'll go look at that thread. If I have anything productive to add, I will.

Hi Malcolm. Good seeing you again.

Well 2 is easy if I am allowed 'There are 100 Hatfields'.

1 is more difficult, as is well-known. When we replace a number-word or sign with a variable it becomes more difficult for the nominalist, as we have to quantify over numbers. Thus

For some m, n: There are m Hatfields and n McCoys and m > n

Clearly I am not positing a 'collective entity'. But I am quantifying over numbers. Is that an acceptable solution? You are not the first person to have pointed out this difficulty.

Actually there may be a simpler solution. The concept of 'same number' is captured by bijection (as Malcolm correctly notes - Hi Malcolm). A bijective function f is from one set X to another set Y such that for every y in Y, there is exactly one x in X such that f(x) = y and no unmapped element exists in either X or Y. To get 'greater than' or 'less than' we just suppose there are unmapped elements.

This requires sets. But nominalists are allowed pluralities, and can quantify over plural X's and Y's. Thus, for some X's, for some Y's, there is a function from the X's to the Y's such that for each y that is one of the Y's, there is exactly one x among the X's such that etc...

Will that do? I am worrying that perhaps the notion of a function carries some collectional, realist concept. I will have to think about it.

You paraphrase
1. The Hatfields outnumber the McCoys
as
P. For some m, n: There are m Hatfields and n McCoys and m > n

That paraphrase does not serve the nominalist cause. As you point out, there is quantification over numbers. But also, 'Hatfields' and 'McCoys' occur in the analysans. What does 'Hatfields' refer to? It is a plural expression and refers to some kind of collection. If that reference is merely apparent, then you owe me a paraphrase that makes it clear that the reference is merely apparent.

Obviously, I am not going to give you 'There are 100 Hatfields.' You need to paraphrase away the apparent reference to a collection in that sentence.

Would this do?

For each McCoy, there is exactly one matching Hatfield. No Hatfield matches more than one McCoy. But there are some Hatfields who do not match a McCoy.

(My set theory is a bit rusty).

>>But nominalists are allowed pluralities. . .<< But musn't even a plurality be something in addition to its members? Doesn't 'its' suggest that it is? What makes the Hatfields and the McCoys two pluralities rather than one? Why aren't all pluralities one plurality?

Functions should also give a nominalist trouble. A function is a relation, and a relation is a set of ordered n-tuples, where an ordered n-tuple is an ordered set, order being captured by the Wiener-Kuratowski procedure.

No sets, no functions, as mathematicians ordinatrily understand 'function.' So you have the task of defining functions without recourse to sets.

I'd have to think about it, but you may also hit a snag if you try to represent order without set theory.

>>For each McCoy, there is exactly one matching Hatfield. No Hatfield matches more than one McCoy. But there are some Hatfields who do not match a McCoy.<<

That does seem to do the trick, except for one problem. Quantification is relative to a domain of entities. In this example, the domain is persons. But what is a domain? You can't say it is a set. But it must be a collection of some sort, in which case you must admit a collective entity.

>>Obviously, I am not going to give you 'There are 100 Hatfields.' You need to paraphrase away the apparent reference to a collection in that sentence.

I don't see the problem here. The name 'Hatfield' is clearly being used as a common noun, as when we say 'Manchester is no London' or that someone is 'an Achilles'. Do you have the same problem with 'there are a hundred people in jail'?

>>Functions should also give a nominalist trouble.

I've already signalled that. However, you have ignored the point (see McKay's book) that anti-realists can allow plural quantification, i.e. quantification where the relevant instances are not single entities, but numbers of entities. As in 'any people who surround the building will be arrested'.

I will have to think about examples, but it seems plausible we can replace any set-theoretical expression, such is 'x is in Y' with the pluralist equivalent

x is one of the Ys

where 'the Ys' does not refer to a singular entity like a set, containing elements, but the elements themselves.

>>But musn't even a plurality be something in addition to its members?

No, not according to the singularist anti-realist. The pluralist allows plural quantification only. The singularist allows singular quantification, but only where it is grammatical only, as in 'take any dozen ...'. "'the planets', refers plurally to Mercury, Venus, Mars, Earth, Jupiter, Saturn, Uranus, Neptune and Pluto, and the predicate attributes the property of being nine to that plurality. We should not be misled here by the fact that the expression 'that plurality' is grammatically singular in form. This is a mere idiosyncrasy of idiom and does not signify that there is some further thing, 'the plurality of the planets', in addition to the planets themselves. " (Jonathan Lowe).

>>Doesn't 'its' suggest that it is?

No. The singular reference is a grammatical feature only. What is referred to, or rather, what are referred to, is/are plural.

>>Quantification is relative to a domain of entities. In this example, the domain is persons. But what is a domain? You can't say it is a set. But it must be a collection of some sort, in which case you must admit a collective entity.

A domain is a kind of container. Alternatively, we can get rid of domain talk and stipulate that x ranges over all F's, where 'F' is a description. Quantification is relative to a description.

You should take some time to look at McKay's book where he covers many of the examples and objections you mention. If you find something you do object to, raise it here so we can discuss.

I am going to the library today to pick up McKay's book. No time to respond today. Tomorrow perhaps. Very interesting discussion we are having.

HI

I heard that the price went down recently. You can buy it!

Consider:
The Hatfields are surrounding the building.
They are co-operating.

The domain is the Hatfields (and the building). 'The Hatfields' and 'They' refer to them (not to some other thing, a plurality). Standard first-order logic doesn't allow any simple representation that reflects this because standard first-order logic is thoroughly singularist. This is a limit of the logic that can be overcome, though.

The numerical claims are other examples of such non-distributive predication.

Tom McKay

I give citations for the medieval examples (Lambert, Aquinas, Ockham) here

http://ocham.blogspot.com/2010/07/plural-quantification-and-scholasticism.html

Hi Peter and William, and thanks for the friendly greetings.

This is a very interesting discussion, and I should probably just lie low, as I have only modest familiarity with the mathematical and philosophical arcana of set theory.

Before Security escorts me back to the bleachers, though, I must say that my sentiments are solidly with the nominalist camp: I find it hard to see how we can insist that there is really anything else in the world above and beyond those individual Hatfields and McCoys. That we have found it convenient to be able to refer to them, on occasion, in the aggregate does not, it seems to me, pack a whole lot of ontological oomph; looking around, all I see are individual rifle-totin' hillbillies.

I also wonder, as an aside, just how these meta-objects might spring into actual existence if no mind ever defines them -- but this is probably covered in Arcana of Sets 201, which I probably ought to have taken.

So I will enjoy this thread from the sidelines, I think.

Professor McKay,

I have a maxim, "Never buy a book you haven't read." But I got hold of a library copy yesterday and began reading it this morning. I'm into the second chapter and am finding it excellent and extremely interesting. It's definitely worth buying! Congratulations on writing such a fine book.

Perhaps you could refer us to some reviews.

>>Consider:
The Hatfields are surrounding the building.
They are co-operating.<<

This is important linguistic data. Obviously, a (mathematical) set cannot surround a building (assuming that sets are abstract). Sets as abstract are not located in space or spread out through space. But what about mereological sums? I'll have to see what you say about sums.

As for co-operation, no single thing, whether concrete or abstract, co-operates. Suppose Mutt and Jeff co-operate in the mopunting of a tire. One guy lifts the tire, the other guy screws on the lug nuts. Here a relational analysis suggests itself. You of course argue againts this. I'll have to study exactly what you say.

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