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Saturday, July 24, 2010

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You should definitely read Mckay's book. Note that McKay is a 'pluralist': he doesn't like the use of grammatically singular terms to signify pluralities. He doesn't like the expression 'a plurality', for example, because the singular 'a' or 'one' suggests one thing, in his mind. We had a long correspondence about this when he was writing the book, and he called my position 'singularist anti-realism' or something like that. Meaning that I am anti-realist about sets just as he is, but I don't have the problem of the 'one' in 'one dozen' or 'one hundred' - I don't think that 'dozen' or 'hundred' is a noun like 'man' or 'shoe', and so don't believe there is a singular entity or thing corresponding to it.

I don't believe, for example, that in the expression 'three million, two thousand', there are three singular entities (not even abstract entities) corresponding to the 'three' of 'three million', or two entities corresponding to the 'two thousand'. That would be just as absurd as supposing there are entities corresponding to the digits '3' and '2' in the number '3,002,000'. Nor is there a singular entity corresponding to 'a number of things' or 'a set of things'. The only difference between that and 'a dozen' or 'a hundred' is that there is, in effect, a variable, as if I had written

m,00n,000

to represent a variable number of millions and thousands.

McKay prefers examples like 'Any people who surround the building', where the quantification is irreducibly plural. This is because '--- are surrounding the building' is an irreducibly plural predicate. The scholastic logicians also spotted this, giving examples like 'the Apostles are twelve' or 'the sailors are pulling the boat'. Unlike I am just comfortable with 'any group of people who surround the building' because I don't believe that the grammatically singular term 'a group' necessarily refers to a single thing. For grammatical and logical reasons, not for metaphysical reasons, as you originally seemed to think.

Thank you for drawing our attention to that nicely concise statement of Michael Potter, as quoted in:

A set is a collection, and a collection is not the mere manifold of its members: it is "a further entity over and above them" as Michael Potter puts it in Set Theory and its Philosophy (Oxford 2004, p. 22).

I confess that I too, like the previous commenter, am an anti-realist in the theory of sets, though perhaps for different reasons. I am simply unable to suspend my disbelief or to see, beyond the need that set theory has of postulating such sets, a compelling reason for believing that:

There is a set, distinct from and in addition to Quine, of which Quine is the only member;

There is a set, distinct from and in addition to Goodman and Quine, of which Goodman and Quine are the only members; and, above all,

There is a set “with no members, the celebrated null set.”

By way of comparison: if a ball comes crashing through my window, I and many others will, in accordance with some common-sense principle of causality, infer that somewhere outside the window, there is to be found the cause of the ball having so gained entry (or, at least, was to be found one, before he hoofed it). Can someone point me to an analogous principle, one which would lead me, knowing that Quine existed, to infer the existence of the set, distinct from and in addition to Quine, of which Quine is the only member? (Actually, recalling a criticism that some have made of Aquinas’s arguments for the existence of God, there should strictly be two inferences, one that there is at least one such set and the other that there is at most one such set.)

Confession thus made, I’ll close by also thanking you for also drawing our attention to Plural Predication.

Prof. Hennessey,

Your challenge to motivate the existence of sets over and above the existence of their members by means of a principle analogous to the "common sense principle of causality" involved in your ball example may require some clarification. What would be the analogy in such a case?

The ball example clearly belongs to the physical realm and the use of the causality principle in this case is based upon the idea that the entry of the ball is clearly an effect of an antecedent cause. But the relationship between a set and its members is clearly not causal. So any principle that would justify positing the existence of sets over and above their members cannot itself be a principle that in this sense is analogous to the causality principle.

So I wonder whether you could say a bit more about the nature of the analogy to the causal principle that you have in mind from a putative principle that would justify the existence of sets over and above their members?

Suppose we compare

1. The sailors are pulling the boat
with
2. The sailors are surrounding the boat.

(1) obviously does not mean that each sailor is pulling the boat, but it could be interpreted to mean that there is one single thing that is pulling the boat, namely, the sailors working in concert as a team. So I suggest that (1) is not clear evidence of the need for irreducibly plural predication.

(2) is much better evidence, indeed strong evidence. Obviously, (2) does not mean that each sailor surrounds the boat. But neither does it mean that there is some single thing, whether concrete or abstract, that surrounds the boat.

But couldn't we give an analysis of (2) so as to avoid irreducibly plural predication? It would be messy and involve relations. Something like: People surround an object O just in case each person stands in a certain spatial relation to every other person who stands in certain spatial relations to O.

I don't have the patience to work this out, but perhaps you catch my drift.

A further thought about (2) above. Obviously no set surrounds the boat. No mereological sum either. But Quine somewhere speaks of "scattered objects." Could we say that the sailors make up a scattered object that surrounds the boat? Just a thought.

That there is irreducibly plural predication strikes me as very strange.

If no individual sailor surrounds the boat, and no higer order object (set, sum, scattered object) surrounds the boat, then what the hell are we talking about? What is the logical subject of 'surrounds the boat'? What is doing the surrounding?

You're welcome, Richard.

You in effect ask for a reason to posit {Quine} in addition to Quine. Well, if you grant me the sets {Quine, Goodman} and {Quine, Putnam}, then I can argue as follows. For any two sets, there is a set which is its intersection. Now the intersection of the two two-membered sets just listed is {Quine}. Therefore {Quine} exists.

As for the null set, it can be proven as follows. Take any two disjoint sets, e.g. {1, 3, 5} and {2, 4, 6}. If for any two sets there is a set which is its intersection, then the null set exists.

To prove that the null set is unique, we proceed by reductio ad absurdum. Suppose there are two null sets N1 and N2. By the Ax. of Extensionality, two sets are the same iff the have the same members and different iff they difer in a member. N1 and N2 differ in no member, so they are the same. Ergo, the null set is unique.

As for why one must accept any math. sets at all, there is the argument I gave in "Sets, Pluralities,and the Axiom of Pair." http://maverickphilosopher.typepad.com/maverick_philosopher/2010/07/sets-pluralities-and-the-axiom-of-pair.html

I recommend McKay's book. I found an early version he sent me in 2004 where he describes my position as follows.

"There are some cases where we might prefer to take an antirealist position about alleged composite objects. For example, if I buy two dozen onion bagels, I might want to say that I have not bought just two of anything. Even though we say 'a dozen' (or 'a thousand') and 'two dozen' (or 'two thousand'), we are not counting one or two of anything. There is no single thing that is a dozen bagels, there are twelve things that are a dozen bagels. Such an anti-realist position is especially attractive when the phrase ('a dozen' or 'two dozen') is used in counting or measuring.
"We might wonder how far to extend such anti-realism. Perhaps someone would like to avoid ontological excess by taking an antirealist position about decks of cards and suits of cards. Others might extend this to flocks of birds and other groups of things (though this seems to have the counter-intuitive result that a flock cannot survive a small change in membership). Still others (see van Inwagen 1990, for example) might extend this antirealism to composites like tables and chairs (among other things).

"I will not try to say exactly where the line should be drawn (though I will endorse the idea that a dozen bagels is not any single thing). The key thing to see is that this differs radically from the CI (Composition as Identity) view. The antirealist about decks of cards says that no single thing is a deck of cards --- there are only the cards. (This leaves it to the antirealist to give an account of what I am doing when I buy three decks of cards, but perhaps that is not an insurmountable task.) It would seem that the antirealist can only be a strong ally against singularism in the semantics of plurals."

My position now (thanks to Bill and Peter Lupu for clarifying my previously woolly thoughts here) is that the 'three' in 'three packs of cards' is grammatically similar to the '3' in '3,000,000' - which of course we pronounce 'three million'. Three million bagels is not three single things any more than 1 dozen bagels is one single thing.

However this

>> Others might extend this to flocks of birds and other groups of things (though this seems to have the counter-intuitive result that a flock cannot survive a small change in membership).

is an obvious counter-example to the arguments I have given here. 1 dozen clearly cannot survive a small change in membership (taking away 1 leaves 11). A flock of birds certainly can.

William,

Are you saying that 'three decks of cards' does not mean that there are three things; namely, decks of cards? But what else could that mean? Suppose you ask for three decks of cards and you are handed two decks instead. Aren't you going to complain that one deck of cards is missing? What could such a complaint mean? That 54 cards (or are there 52 cards in a deck; I forgot?) are missing. But that won't quite do, since three decks of cards could be missing 54 cards in many different ways. There is a difference between missing one whole deck of cards out of three decks versus missing several cards from each deck.

Peter,

There are 52 cards in a deck. William is saying that the deck is just the 52 cards. Similarly, he is saying that a dozen eggs is just 12 eggs.

But note the difference. 52 playing cards may or may not make up a deck. 52 aces is obviously not a deck of cards. But 12 eggs always makes a dozen.

Does this show that the deck is an entity in addition to the 52 cards? I think it does.

Is this your point?

Bill,

Yes. I gave a similar example regarding a missing deck which is not equivalent to missing 52 cards from three decks.

(Thanks for the info about the number of cards in a deck).

I am not sure about the deck (English 'pack') of cards example for the same reason that a flock of birds cannot be identical with the birds. (If one bird leaves, or another bird joins, I think we would say they are still one and the same flock).

But I take it we are agreed that 'three dozen' does not count three single things any more than 'three million' does. Are we?

The following thoughts occurred to me this morning while walking my canine friend Spooky.

Bill,

"Does this show that the deck is an entity in addition to the 52 cards? I think it does."

To this question I previously said "yes". But now here is a problem. Is the said entity that is over and above the 52 cards a set? Well, construing it as such will create some problems. Sets are not physical, even if their members are. So what do we do with the following sentence: "I hold in my hand a deck of cards", assuming that holding something is some sort of a physical relationship? Perhaps, for things such as decks of cards we need a category of things that features some properties of sets and some properties of something else.

William,

"But I take it we are agreed that 'three dozen' does not count three single things any more than 'three million' does."

Well, suppose that eggs come (as they often do) in cartons of a dozen each. Now consider a supermarket worker who counts how many dozen eggs were delivered this morning. He would say: one dozen, two dozen, three dozen and so on. What is he doing? Doesn't he count one thing, two things, three things, etc.,?

Peter,

You are right: if sets are abstract objects, then one cannot hold a set of 52 cards in one's hand. Therefore, if the deck is distinct from the 52 cards, then it is not a set. Perhaps it is a mereological sum. A sum of physical objects can be said to be located where they are located.

One gross of bagels = 12 dozen bagels = 144 bagels.

If a dozen bagels is a single thing distinct from 12 bagels, then a gross of bagels is a single thing distinct from 12 dozen bagels. Something tells me that will lead to trouble.

Bill-

I actually like 1 above. Examples would include relativism and determinism in ethics. Certain errors come up over and over in conversation, life and philosophy; they are simply garbed in different language. A philosopher's work is never done.

By the way, love your blog.

John,

Thanks for the kind words. Here is (1):

1. Each stupid idea is immortal and is invented by each new generation anew.

I know why you "like" it, but surely it is not true. For example, the idea that 9/11 was an inside job.

>>Well, suppose that eggs come (as they often do) in cartons of a dozen each. Now consider a supermarket worker who counts how many dozen eggs were delivered this morning. He would say: one dozen, two dozen, three dozen and so on. What is he doing? Doesn't he count one thing, two things, three things, etc.,?

I've already discussed this. What if the supermarket worker is counting as follows

100 (utters 'one hundred')
200 ('two hundred')
300 ('three hundred')

If the digit '3' in the written number '300' does not signify a thing (I presume you agree it doesn't), why should the spoken 'three' in the spoken 'three hundred' signify a thing? And if counting in dozens, I can represent the same thing in base 12:

10 ('one dozen')
20 ('two dozen')
30 ('three dozen')

>>For example, the idea that 9/11 was an inside job.

Which is surely a reinvention of an age-old form of conspiracy theory.

Dr. Vallicella writes:

"If a dozen bagels is a single thing distinct from 12 bagels, then a gross of bagels is a single thing distinct from 12 dozen bagels. Something tells me that will lead to trouble."

The nominalist often objects to the reality of sets with the 'number of things in the box' objection. If there are twelve things in the box - he says - then are there really thirteen things there? As we can see, the situation is actually worse. 12 bagels give rise to 2^12 new objects, the members of the power set of the set of 12 bagels.

However, if we claim that sets are on an ontological par with concrete objects in the sense that sets of sets are real just as sets of concrete objects are, the situation becomes still worse. A finite number of concrete objects gives rise to an infinite number of objects. To see it, note that iterating the operation of taking the power set we get a sequence of sets such that their cardinalities are finite and strictly rising.

This does not seem to me a 'nice' consequence, though I can't come up with a good argument against it. The best I can is:

1) It is logically possible that there exist only a finite number of objects.
2) If sets exist (in the unrestricted sense of sets being able to form sets), then a merely finite number of objects cannot exist.
Therefore
3) Sets do not exist in the unrestricted sense.

Premise 1) is quite implausible. In particular, it will be rejected by Platonists with respect to numbers (such as myself). There is another reason why Platonists about numbers will insist on reality of sets. If I remember correctly, Russell defined numbers as equivalence classes of sets w.r.t. the relation of bijectivity. Indeed, the existence of numbers seem to 'stem from' the existence of sets. If twelve is not a property of the set of the apostles, nor of the mereological sum of apostles, then what is it the property of? That is the challenge that has been stated elsewhere.

PS. Dr. Vallicella, thank you for your encouragement to take part in the discussion under a different post. I comment far less than I would like because it takes me very long to write anything worth reading.

Jan,

That is an outstanding comment! It is 'exemplary': an example of what a good comment looks like. Let me add a bit of commentary on your comment.

You are talking about sets as they figure in standard set theory which, I take it, is ZFC. That is the standard from which various 'deviations' are possible. You also, I think, would agree that sets are abstract objects, where 'abstract' is defined as 'causally inert.' You also assume, quite plausibly, that if there are sets, they are just as real as concreta. I.e., they enjoy the same mode and degree of being (existence) as concreta. Jan and {Jan} are equally real -- assuming that there is {Jan}. You are also right that if our twelve bagels form a set, then there is the power set of that set. The cardinality of the power set = 2^12.

So we get an "ontological population explosion." This is not appealing, but it is difficult to come up with a good argument against it. And note that it does not offend against Occam's Razor because the Razor enjoins us to not multiply KINDS of entities. Once the kind 'set' is allowed in, the population explosion of individual sets is no objection. Of course, the nominalist will use the Razor to prevent the admission of the first set, the set of the 12 bagels.

You give an interesting -- I am tempted to say 'brilliant' -- argument:

1) It is logically possible that there exist only a finite number of objects.
2) If sets exist (in the unrestricted sense of sets being able to form sets), then a merely finite number of objects cannot exist.
Therefore
3) Sets do not exist in the unrestricted sense.

Your syllogism could be expressed as an antilogism, where an antilogism is an aporetic triad:

1. Possibly, the number of objects is finite.
2. Necessarily, if sets exist, then the number of objects is not finite.
~3. Sets exist.

The triad is logically inconsistent: the limbs are not all true. To solve the problem we must reject one of the limbs. (2) is rock-solid and non-negotiable. So we must either reject (1) or (~3).

You reject (1) while a nominalist will reject (~3). Trouble is, both propositions are plausible, and if we are not dogmatists, we should admit that we have no really compelling reason to favor one rejection over the other.

This leads me to conjecture that here we have a genuine aporia, a problem that is genuine, but absolutely insoluble.

>>PS. Dr. Vallicella, thank you for your encouragement to take part in the discussion under a different post. I comment far less than I would like because it takes me very long to write anything worth reading.<<

Actually, your comments are as good as any I get. So I should be thanking you. You ought to comment more. We live in a strange world. The people whose contributions are the most thoughtful and worthwhile are often those who hesitate to speak up, while those who contributions are thoughtless and worthless are unaware of the fact and rarely restrain themselves.

Gómez Dávila sometimes wrote more than one version of an aphorism. Here's another version of the aphorism you quoted above: "Among ideas only the stupid ones are immortal" (source).

William maintains that there are no compelling reasons to posit entities other than concrete particulars. For instance, he rejects sets because they are abstract, but he also rejects what Bill calls “collectives” because collectives, if they exist at all, they exist over and above their individual members. It is useful to note in passing that collectives are not strictly speaking sets because they may feature temporal as well as spatial properties.

One example of a collective is sport teams (e.g., the NY Mets). How do we know that? Well, suppose that we follow William’s lead and think of a sports team such as the NY Mets as committing us only to the players that make up the team at a given time. There is no entity, such as a collective, over and above the individual players. Then we face the following difficulty. Imagine that the NY Mets dissolve as a team and all the players are dispersed among the rest of the baseball teams in the National League. Ex hypothesis, all the individual players that played for the Mets prior to the dissolution of the team still exist. Does the team itself still exist? It appears that William is committed to the view that since all the individual players that made up the team prior to the dissolution still exist, the team must exist as well.

But this is just not so. The NY Mets no longer exist as a team, although each individual player does. The team is not listed as a team in the National League; there are no games in which the Mets compete; one cannot be a Mets fan knowing that the Mets as a team dissolved, etc. Hence, there has to be some entity, over and above the players themselves, which can be said to cease to exist, even when all the former players still exist. What is this entity, according to William, that existed up to time t and ceased to exist subsequent to t?

Consider a deck of cards. There are 52 cards in a deck. William insists that when we have a deck of cards we have just 52 things, not 53 things. Suppose I hold a deck of cards in my hand. William asks: Do you hold 52 things in your hand or 53 things? His intuition is that I should answer: “I hold only 52 things in my hand.” This is what Jan calls the nominalist’s “too many things in the box” objection.

But the “too many things in the box” objection is playing fast and loose with the way we count *things*. Clearly, if the manufacturer of cards puts 52 cards in a deck, then there are only 52 things-of-the-sort *cards* in the deck. But this is not to say that there are only 52 things in the world. While there may be only 52 cards in the deck, there can be 53 things in the world: namely, 52 things-of-the-sort *cards* and one thing-of-the-sort *deck*. I can quite legitimately answer William’s question in the previous paragraph as follows: I hold 52 things-of-the-sort *cards* in my hand and one thing-of-the-sort *deck* in my hand.

William thinks that when I say I have 3 dozen eggs in the fridge, then the spoken ‘3’ should be viewed on a par with a similar occurrence of this numeral when it occurs in ‘300’. And since in the later case we do not wish to maintain that ‘3’ signifies three things, why should we think that the ‘3’ in ‘3 dozen eggs’ signifies three things. He says about the deck example: “My position now (thanks to Bill and Peter Lupu for clarifying my previously woolly thoughts here) is that the 'three' in 'three packs of cards' is grammatically similar to the '3' in '3,000,000' - which of course we pronounce 'three million'. Three million bagels is not three single things any more than 1 dozen bagels is one single thing.”

William’s proposal is not intended, and cannot be viewed, as a general proposal for all uses of numerals in an apparent counting position. Suppose a speaker says: “There are three people standing in line.” In cases such as these we are entitled to consider the occurrence of ‘3’ in a counting role because we already accept the existence of people as concrete individuals. The proposal then is intended to demonstrate that when we are disinclined to accept the existence of certain entities; such as deck of cards, teams, etc., then we are not compelled to do so merely because of the use of numerals in apparently counting positions. Thus, the mere use of numerals in apparently counting positions does not force us to ontological commitments to entities which otherwise we are unwilling to countenance because we can construe such occurrences as simply syncategorematic.

But why does William reject the existence of collectives such as decks of cards over and above the member cards, but accepts the existence of people? After all, people can just as reasonably be construed as a bunch of molecules? And why stop there? After all molecules can just as well be construed as a bunch of atoms and atoms as a bunch of more elementary particles, and so on? What are the compelling reasons to draw the ontological line with people, trees, rocks, chairs, houses, etc., rather than with decks, teams, and so on?


Stephen,

Thanks for making me aware of your site. I gave you a plug at the top of the page.

William,

The idea that 9/11 was an inside job is both stupid and new: before that date that particular attack could not have occured. So 'Stupid ideas are immortal' cannot be parsed as 'Every stupid idea is immortal.'

Peter,

Very good. I was thinking along similar lines the other day. The Beatles can't be identical to John, Paul, George, and Ringo because there was a period of time in the '70s after they disbanded when all four men existed but the Beatles did not exist. Ergo, 'The Beatles,' when it referred to something referred to something in addition to John, Paul, Goerge, and Ringo.

William will accept this, but presumably say that a band is a higher-order concrete particular, not an abstract object, and so no threat to nominalism. The Beatles are not just the four guys, but the four guys playing together; but the the four guys playing together is not an abstract object.

Peter makes an important point: a deck of cards is not just 52 cards but 52 cards meeting further requirements---exactly one Ace of Spades, exactly one Queen of Hearts, all cards the same size suitable for holding in the hand, etc. A band is some guys 'banding'. 'Deck' and 'band' imply some structure over and above a mere plurality of individuals. What is exercising us here is the question whether 'set' implies enough extra structure to qualify as a distinct kind of entity. We could ask, What does 'set of Xs' tell us in addition to 'some Xs'? My answer would be Nothing.

Hi David. How things?

Bill,

I am somewhat confused. I no longer understand what constitutes a kosher-nominalist. Can a nominalist accept *any* entities over and above a plurality of concrete particulars? For instance, can a kosher-nominalist accept finite sets? Can a kosher-nominalist accept *higher-order* concrete particulars?

A finite set is a set nonetheless and, hence, and abstract object. And what is meant by a 'higher-order' in the context of the domain of concrete particulars. And what exactly is the difference between higher-order concrete particulars and collectives?

I also asked William previously how can he sanction model theory, since set theory is indispensable for model theory. The issue emerged in various forms in the recent Hatfields-McKoys exchange.

Regarding the (1)-(~3) triad of Bill-Jan. Bill says: "You [Jan] reject (1) while a nominalist will reject (~3). Trouble is, both propositions are plausible, and if we are not dogmatists, we should admit that we have no really compelling reason to favor one rejection over the other."

I do not think that (1) and (~3) are equally plausible. Perhaps, there are no knock-down arguments in favor of rejecting (1) or rejecting (~3), but I certainly think that there are weighty considerations in favor of rejecting the former and accepting the later.

(a) If there are infinitely many numbers, then (1) is false. Are there infinitely many numbers? Very few would deny this. How could they, for then they would have to reject most of mathematics. Incidentally, ZF+~Inf does not deny that there are infinitely many *numbers*: rather it denies the existence of an *infinite set*: that is quite different. So in ZF+~Inf there are in fact infinitely many finite sets except that the universe of such sets does not itself form an infinite set (induction is captured by means of a hereditary procedure of constructing sets, each of which is finite). And I should mention that without a detailed examination of the technical apparatus in the argument that PA is interpretable within ZF+~Inf we cannot really form a rational opinion about the philosophical significance of the proof (In this connection, see the literature on Skolem's Paradox and particularly Bays article in the SEP).

(b) If propositions exist, then there are infinitely many propositions. Are there propositions? Kosher-nominalists obviously will have to deny that propositions exist. Sentences do not express propositions. But, then, what do they express?

(C) Are there sentence types? A nominalist will have to deny the existence of sentence types. But, then, it is difficult to see how any linguistic analysis can be done.


Here is a (perhaps not a complete) list of things the nominalist rejects: a nominalist rejects sets; he rejects collectives (e.g., a deck of cards); he rejects propositions; he rejects universals and properties; he must reject time as a continuum of infinitely many point in time; he rejects sentence-types.

I wonder whether a kosher-nominalist can be forced to reject all of these (and whatever else they reject) all at once. I suspect that too frequently they rely on one or several of these nominalistically unaccepted entities in order to marshal arguments against the existence of the others. What I want to see is a list of nominalistically unacceptable entities and arguments that show that they can account for required phenomena without appeal to any one of these entities on the list.


What are the considerations in favor of rejecting (~3)? Well, sets are abstract so they have no spatio-temporal properties. So what? Does the whole physical universe have spatio-temporal properties? Not according to modern physics.

I think that the consequences of rejecting (1) are grave and require, as usual, a lot of IOUs (too many IOUs, in my opinion). On the other hand, what are the adverse consequences of rejecting (~3)?

(...and if I may be allowed to continue my rant a bit longer...)

Nominalists rely upon the principle "Do not multiply entities beyond necessity." But what kind of constraint is this principle? For instance, what sort of necessity is assumed here? The following reply is often made: do not multiply entities beyond what is required for explanation. Sure! But, what sort of explanations?

Many explanations are within theories. Such explanations already presuppose the existence of various sorts of entities that are countenanced by the theories themselves. Similarly, the kind of phenomena to be explained are also described within the theories themselves. Black holes are not the sort of things we observe on a daily basis. They are not basic data accessible outside of modern physics. The same holds for electrons, quarks, genes, evolution, and so on. Within such explanatory theories, the principle "do not multiply entities beyond necessity" makes sense, but only as a limited guiding methodological principle: i.e., a theory posits certain entities. Explanations within a theory should make use only of such entities as often and as far as possible. However, there is always an escape clause from the stricture: if something resists explanation within the resources of the theory, do not hesitate to introduce new entities as long as they have explanatory value. And this is just the surface of what goes on.


How does the nominalist know antecedently what sort of entities future theories will require? For example, suppose some physical theory requires fields and fields are by their very nature not concrete particulars. What are we to do now? Are we to say that such theories are somehow defective because they postulate entities unacceptable to nominalists?

Does the stricture "do not multiply entities beyond necessity" is to be antecedently interpreted in accordance with nominalist metaphysics and then applied to inquiry or is it simply a methodological advise that allows inquiry itself to interpret it as it proceeds? If it is the later, then there is no a-priori reason to believe that inquiry will conform to a nominalist metaphysics. While nominalists may hope that it does, they have no antecedent reason to think that it will.

On the other hand, what justification is there to first construe the principle in a accordance with nominalist strictures and then view the result so construed as a final and ultimate constraint on all future inquiry? That would be analogous to the principle that all inquiry is fine as long as it preserves the egocentric view of the universe.

I think the nominalist owes us a thorough account of the principle in question.

David,

"We could ask, What does 'set of Xs' tell us in addition to 'some Xs'? My answer would be Nothing."

What does 'some x's' tell us beyond what 'set of x's' tell us? Nothing! Why? Because sets are required in order to explain what 'some x's' means.

Hello again Peter,

How are 'things'? is indeed the question, but very well thank you, and you too I trust.

>> Because sets are required in order to explain what 'some x's' means.

Possibly, but then we would know what 'set' and 'member of' meant and Axiomatic Set Theory would be all about explaining the non-logical terms 'for all' and 'there exists'.

David,

I would not say I am well, but I am surviving.

Well, of course matters are more complicated. Set theory may be useful to explain a certain range of uses of 'some' and exists'. Bill, however, convinced me some time ago that there are compelling reasons to suspect that it can exhaust all that they mean. But explaining some uses already gives set theory and model theory, and hence sets and models, some footing in reality.

You're welcome, Mr. Vallicella. And thank you for the "plug."

In a post dated Tuesday, July 27, 2010 at 11:24 AM, Bill offers the following inconsistent triad of propositions, which he calls ‘aporia’ (this triad is inspired by Jan’s post dated Tuesday, July 27, 2010, at 06:06 AM):

1. Possibly, the number of objects is finite.
2. Necessarily, if sets exist, then the number of objects is not finite.
~3. Sets exist.

Since Bill maintains that (2) is beyond reproach, the only two alternatives left are to reject (1) or reject (~3). Regarding these two alternatives, Bill is willing to accept that “we have no really compelling reason to favor one rejection over the other.”

In response to Bill’s concession, I said the following:

“I do not think that (1) and (~3) are equally plausible. Perhaps, there are no knock-down arguments in favor of rejecting (1) or rejecting (~3), but I certainly think that there are weighty considerations in favor of rejecting the former and accepting the later.” (Wednesday, July 28, 2010 at 06:04 AM)

I have given in that post some of the considerations that should convince us to reject (1). It is worth emphasizing that what I said was that there are “weighty considerations” in favor of rejecting (1) and accepting (~3), not that there are conclusive arguments on behalf of this position. To further this stand, I now wish to examine a bit more closely the content of (1).

(a) Proposition (1) is a modalized proposition; it includes the word ‘possible’. How should we interpret this word? One proposal is to think of (1) as stating that it is *logically possible* that the number of objects is finite (this formulation is stated explicitly by Jan’s original post). So here what we mean is that it is compatible with the laws of logic that the number of objects is finite. But this interpretation raises the following issue: With which laws of logic is the proposition ‘the number of objects is finite’ compatible? If the underlying logic is classical (includes the excluded middle, for instance), then there is no reason to reject classical mathematics as the background system and within such a background the proposition ‘the number of objects is finite’ is simply false. On the other hand, if the underlying logic is intuitionistic (we reject the excluded middle, for instance), then the background mathematics is intuitionistic as well, and the proposition ‘the number of object is finite’ may well be true.

So the first problem I have with (1) is that it is unclear which underlying logic is presupposed. The choice of the underlying logic makes a difference to whether or not one would accept (1) or reject it. The acceptance or rejection of (1) leads fairly quickly to the debate between intuitionistic vs. classical logic (and, hence, about the excluded middle principle). So I wish to emphasize at least that accepting (1) means that you accept some form of non-classical logic.

(b) The following intuition is associated with mathematical truths, where ‘P’ is any mathematical proposition, statement, etc., we like:

(*) If P is true at all, then it is necessarily true.

Clearly, if (1) is true and we recast it in some version of standard possible world semantics, then there is a possible world w in which the number of objects is finite. Given a few additional assumptions (e.g., that numbers are objects), it follows that the number of mathematical objects is finite in w. Let this proposition be P. Since ‘P’ is a mathematical proposition, by (*) it follows that in w (**) is true:

(1*) Necessarily, the number of mathematical objects is finite.

Since this conclusion is fairly general (i.e., it does not depend on the choice of w), (1*) says in effect that in every possible world the number of mathematical objects is finite. And the conclusion that the number of objects in every possible world must be finite is a fairly powerful conclusion. It is certainly stronger than what we thought (1) invites us to accept. So if one accepts the intuition expressed by (*), then whatever compelling reasons Bill might think there are for accepting (1) may no longer be as compelling for accepting (1*). Yet, anyone who relies on those “compelling reason” in order to accept (1) will have to rely on the very same reasons to accept (1*), provided they also accept (*). The only way I can see to block this slippery slope from accepting (1) to accepting (1*) is to reject (*). But rejecting (*) constitutes one more significant concession to the proponents of (1) and a major weakening of our mathematical intuitions; a concession I for one is unwilling to make.

(c) In conversation (and perhaps even in one of his posts) Bill suggested to think of (1) as stating that the sequence of natural numbers, for instance, forms merely a *potentially infinity* rather than it forming an *actually infinite* sequence. Let me mention here a few problems with this view.

(i) There are those who object to the notion that an actual infinite collection of natural numbers exists on the grounds that the very concept of an ‘infinite collection’ is somehow incoherent. But, then, they face the burden of explaining how adding ‘potential’ to an already incoherent concept yields a coherent concept.

(ii) Clearly those who would accept (1) and intend to interpret it along the lines of ‘potential infinity’ do so because they think that it is conceptually incoherent, logically impossible, or metaphysically impossible that an actual infinite collection of objects should exist. So they deal with various examples such as the sequence of natural numbers by introducing the notion of a ‘potential infinite’ sequence. But consider the following example. Everyone agrees that nothing can be simultaneously both a square and a circle. But, now, suppose someone responds to this as follows: “I certainly agree that it is not possible for anything to actually be both a square and a circle simultaneously; but this fact does not rule out the possibility that something can be *potentially* both a square and a circle simultaneously.” Clearly, such a claim is incoherent. And the reason it makes no sense is because it is ruled out by the following principle:

(A-P) If it is not possible (logically, conceptually, metaphysically) for such-and-such to be actually the case, then such-and-such cannot be potentially the case either.

I think that (A-P) is an uncontestable principle. And so those who think that there cannot be an actual infinite series; they must also accept that there cannot be a potentially infinite series either.

(iii) Bill undoubtedly will respond as follows:

“Your argument in (ii) above misinterprets the intended meaning of “potential infinity”. The notion of ‘potentiality’ in the present context is not the same as when we say that an acorn has the potential to become an oak tree. Since there are actual oak trees, an acorn has the potential to become one. But since there are no actual infinite sequences, no finite sequence can be said to be a potentially infinite sequence.”

Bill in one of his posts offered the following alternative construal of the notion of ‘potential infinity’: to say that the natural numbers form a potential infinity means that for every natural number produced, no matter how large, we can always produce another natural number by adding 1. This, Bill maintains, is compatible with denying that there is an actual infinite sequence of natural numbers (to be distinguished from a stronger claim that such a sequence forms an infinite set).

The proposed construal raises several questions. First, produced by whom: me; the whole human race; an ideal mind; God’s mind? Second, what is meant by ‘can’? Third, does the term ‘always’ involves quantifying over a temporal sequence? If so, what is the domain of the bound variable: the infinite sequence of temporal points? Surely, this is incompatible with someone who denies actual infinity. And, finally, what is the domain of the variables involved in the quantified phrase ‘every natural number’? (Note: this point is due to Eric Updike in a conversation with Bill and myself). If the domain is specified in advance to be an infinite set, then the very statement that illuminates what is meant by the statement that there is a potential infinity presupposes an infinite domain. On the other hand, what could it mean to deny that such a domain pre-exists?

Returning to the first two questions, Bill suggested that we should think about the relevant notion of potential infinity in terms of the existence of an ideal mind such that for every natural number, this mind can always produce another natural number by simply adding one. But, what does positing an “ideal” mind purchase and how positing the existence of such a mind is supposed to provide a satisfactory answer to the first two questions posed above? I suspect the idea is this. Clearly neither an actual person nor the whole human race can satisfy the condition that for every natural number, me or some suitable sequence of human beings produces another natural number. The problem is obvious and it has to do with the limitation of natural existence. Regardless of how long I or the whole human race lives, this lifespan is finite and therefore we will eventually reach a number such that it is not the case that I or the human race can produce a number larger by one. But perhaps an “ideal” mind is not hindered by this shortcoming. So the idea of an “ideal” mind is basically to remove the limitation that at some point actual minds cease to exist. But are we now positing a mind which never ceases to exist? And what is the domain of the ‘never’ here? Is this another temporal quantifier that ranges over infinitely many time points? Are we to replace positing an infinite series by positing an “ideal mind” the existence of which might require an infinite temporal series? I do not see the advantages of such a proposal. So the argument on behalf of preserving (1) and rejecting (~3) based on “potential infinity” seems to me fairly weak.

(d) I shall make a final point on behalf of rejecting (1). As I have argued above, accepting (1) presupposes accepting some non-classical logic as a background (e.g., an intuitionistic logic). Such logic rejects the principle of excluded middle thereby rendering truth subservient to a notion such as provability or constructability. But if the suitable logic for mathematics is non-classical and the principle of excluded middle is rejected, then such logic becomes our standard logic in all other areas as well. This seems to be a return to some version of the long discredited verificationist principle. So I suppose one should be aware of some of the presuppositions implicit in the suggestion that rejecting (1) is equally reasonable to rejecting (~3).


Note: This post is inspired by an extensive discussion of these questions by Eric Updike (a colleague), Bill, and I.

Addendum,

As it turns out, in (c)iii above I did not represent Bill’s suggestion correctly. Bill actually said the following:

“So someone who denies that there are infinite sets can say something like this: for any n that you have counted up to, you can alway add 1, and the result will be a nat'l number. We will all agree that the natural numbers are closed under addition. But as far as I can see, it does not straightaway follow that there is a math. set of natural numbers.” (Saturday, July 24, 2010 at 06:21 PM)

So what we actually have is the following. The domain of the first quantifier (i.e., ‘for any n that is counted’) is restricted to range only over those numbers that (some)one actually counted. Presumably the notion of ‘counting’ involved here is either the typical act (covert or overt) of going through the numbers such as saying ‘one’, ‘two’, etc, up to some specific numeral ‘n’ or it is an admissible procedure of generating numbers that is acceptable to those who deny the existence of an infinite collection of natural numbers.

Despite this correction, some of the problems raised in the original post remain. For instance, a normal human being (you!) has a limited lifespan. Therefore, you, I, or even the whole human race will be able to follow Bill’s procedure only up to a given number m. Regardless of how large m happens to be, there are more numbers (infinitely many?) larger than it. So clearly Bill must make use of the “ideal mind” postulate. And now we are back to the same problems I have raised regarding this postulate in my original post.

Addendum,

As it turns out I did not represent Bill’s suggestion correctly in the original post. Bill actually said the following:

“So someone who denies that there are infinite sets can say something like this: for any n that you have counted up to, you can alway add 1, and the result will be a nat'l number. We will all agree that the natural numbers are closed under addition. But as far as I can see, it does not straightaway follow that there is a math. set of natural numbers.” (Saturday, July 24, 2010 at 06:21 PM)

So what we actually have is the following. The domain of the first quantifier (i.e., ‘for any n that is counted’) is restricted to range over only those numbers that one actually *counted*. Presumably the notion of ‘counting’ involved here is either the typical act (covert or overt) of going through the numbers such as saying ‘one’, ‘two’, etc, up to some specific numeral ‘n’ or it is an admissible procedure of generating numbers that is acceptable to those who deny the existence of an infinite collection of natural numbers.

Despite this correction, some of the problems raised in the original post remain. For instance, a normal human being (you!) has a limited lifespan. Therefore, you, I, or even the whole human race will only be able to follow Bill’s procedure only up to a given number m. Regardless of how large m happens to be, there are more numbers larger than it. So clearly Bill must make use of the “ideal mind” postulate. And now we are back to the same problems I have raised regarding this postulate in my original post.

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