An Exchange on the Reality of the Past

I have been reviewing pieces from Bill's Time and Change category.  One post from 2020 was in reply to a comment of mine on an earlier piece.

Morning Bill,

I sometimes think our disagreement is about how certain terms are to be used. Compare these claims,

1. The past is in some sense 'there.' Deny that and you are saying that the past is nothing, in which case historians have no object of study.

2. The past was in some sense 'there.' Deny that and you are saying that the past was nothing, in which case historians have no object of study.

I think of 'the past' in this context as collectively referring to multiple things and events. 'The wholly past' denotes a subset of the past. There is also in these sentences an implicit universal quantification over these sets. So I test the truth of these assertions by choosing randomly an element, usually Julius Caesar, making a textual substitution, and asking how comfortable I am with the modified assertions.

1'. JC is in some sense 'there.' Deny that and you are saying that JC is nothing, in which case historians (of JC) have no object of study.

2'. JC was in some sense 'there.' Deny that and you are saying that JC was nothing, in which case historians (of JC) have no object of study.

I find I'm perfectly happy with (2'). Not so with (1'). 'JC' names a thing in time so tenseless 'is' is inapplicable. I have only present tensed 'is'. My presentism says that JC in no sense is 'there'. It follows that JC is nothing, but in my view it doesn't follow from that that historians of JC have no object of study. The problem I have is that I find (2) so innocuously the right thing to say that I'm constantly surprised that you always opt for formulation (1). Is that Problemverlust?

Bill replied:

David B,

Good comment; clear and clarifying. When I speak of the wholly (purely, merely) past, I mean past temporal items that do not overlap the present. A storm that is only half over is then not wholly past, although phases of it are wholly past. A storm is a process. I don't consider myself a process, although this is a debatable point. But I clearly have a past. But I am not wholly past, leastways, not yet. I think we agree on this use of terms.

The difference between (1) and (2) goes to the heart of the matter. You plump for (2) and (2'). JC WAS in some sense 'there.' That suffices for him to be an object of historical study. You can even drop the 'some sense' business and flatly state that JC did exist and that this suffices, etc.

My objection/question is that if JC DID exist, but is now nothing, how can HE now be an object of study? When we investigate JC, the objects of investigation are not causal traces in the present. These traces are pointers to the object of study. We can learn about a burglar from his footprints, but the latter are not what interest us except as a means to identifying the burglar.

>> 'JC' names a thing in time so tenseless 'is' is inapplicable. <<

This is a telescoped or enthymematic argument. I accept the premise, but I think the conclusion is false. Why can't something in time tenselessly exist? 'Tenselessly exist' does not mean 'timelessly exist.' If there are items outside of time, then they tenselessly exist. But it is arguable that an item can be temporal and yet exist tenselessly.

This is one of the crucial issues that needs to be taken up in later posts.

There the exchange ended, attention no doubt moving on to one of those later posts.  But I'd like to answer Bill's question:  if JC DID exist, but is now nothing, how can HE now be an object of study?  My answer is to say that we can study something without getting acquainted with it.  The historian  will look at documents, artifacts, etc, and try to construct an account that is consistent with these vestiges of the past and also with the accounts of JC's times constructed by other historians looking at other sources.  This is not so different from studying an object of the present with which we are not acquainted.  We acquaint ourselves instead with writings, photos, etc, and thus build an understanding of the thing, independent of its continuing to exist.  Perhaps we are back with the recurring problem of reference to the non-existent.

Reference to What is Not

Bill is discussing reference to what is not with Lukas Novak.  In a comment he says,

It now strikes me that what you and I mean by 'nonexistent object' is different. What I mean is an INDIVIDUAL that actually has properties, but does not exist or have any mode of being. The example I have been employing is THE golden mountain. It ACTUALLY, not merely possibly, has exactly two properties, being a mountain and being made of gold. Just those two properties and no properties entailed by them. There can only be one such item; hence THE GM.

But why must there be just one such item?  Are we not in the same domain as Max Black's iron spheres of which there are two?  If Bill insists on just one such individual that has exactly the two properties then I think he must be talking about the description 'is a mountain and is golden'.  The order of the properties is not significant.  But of course, a description has properties in a way different from the way a concrete object has its properties.  We are in the realm of Zaltaian abstract objects.

Propositions solve puzzles

Bill has lately been writing about propositions.  Correspondent Jacques writes that he finds the literature on propositions confusing and frustrating.  Propositions, he feels, have no explanatory value.  But proposition talk seems to make sense to Bill.  At one point he raises Frege's puzzle:

Tom believes that Cicero is a Roman; Cicero is Tully. But Tom does not believe that Tully is a Roman. Is there not a genuine puzzle here the solution to which will involve a theory of propositions?

My thought is that there is a problem here only for certain kinds of theory of the proposition.  Tom's situation is sufficiently common that we ordinarily do not find it puzzling at all.  The explanation is simple:  Tom has heard of a man called 'Cicero' and knows that he is a Roman.  Tom hasn't heard of a man called 'Tully', or if he has then he's heard nothing to suggest that Tully is a Roman.  Tom doesn't know that 'Tully' is another name for Cicero.  Why is it puzzling that he lacks a belief in this Tully's Roman-ness?  Bill has things back to front:  Propositions don't solve this puzzle---they are the source of the puzzle. 

Arguments for anti-presentism?

Bill has been discussing David Benatar's The Human Predicament. Benatar introduces what he calls the Existence Requirement,

(ER) In order for something to be bad for somebody, that person must exist at the time it is bad for him.

Many people find this self-evident. Bill argues here that the reason for this is a tacit commitment to presentism. Bill explains presentism via a somewhat murky notion of 'existence simpliciter', but I think the gist is captured by the following,

Consider Tom Petty who died recently. On mortalism, he no longer exists. On presentism, what no longer exists (i.e., what existed but does not now exist) does not exist at all. So on presentism, Petty does not exist at all. If so, dead Petty cannot be subject to harms or deprivations.

So if we have grounds for rejecting presentism, we can also reject Benatar's Existence Requirement.  But why would we want to say that ceasing to exist does not imply ceasing to exist at all?  What residuum of existence remains after something has ceased to exist?  Bill says,

We don't want to say that a dead man becomes nothing after death since he remains a particular, completely determinate, dead man distinct from others. If the dead become nothing after death then all the dead would be the same. If your dead father and your dead mother are both nothing, then there is nothing to distinguish them.

It's difficult to know what to make of this.  My guess is that Bill is conflating a thing with the idea of a thing.  First, 'particular' and 'completely determinate' do not denote properties of  concrete objects like men.  One can contrast 'I have in mind a particular man' with 'I have in mind a man' but 'particular' here qualifies not 'man' but rather the way of having in mind.  'Completely determinate' functions in a similar way.  What would 'partially determinate man' denote?  A partially determinate idea of a man makes sense, however;  we know some of his properties but not others.  Second, 'dead' is an alienating adjective.  If a man is a living thing and 'dead' means non-living then a 'dead man' is a somewhat contradictory conception.  Better to think of 'is dead' as 'has died'.  A dead man is one who has passed through that final event that all living things inevitably come to, and has ceased to be.  Third, to speak of 'becoming nothing' on death is misleading.  Death is the end of all becoming.  One has finally begone, as it were. It's not that the dead lack something to distinguish them. Rather, they are not there to be distinguished one from another.  But this is not to say that my parents were indistinguishable as objects.  Nor is it to say that my thoughts about my parents are now indistinguishable.  Surely I can say, My mother was short and my father was tall. 

Nor do we want to say that a person who dies goes from being actual to being merely possible. There is clearly a distinction between an actual past individual and a merely possible past individual.  Schopenhauer is an actual past individual; his only son Willy is a merely possible past individual

Once again I'm afraid I can't regard 'being actual' and 'being merely possible' as denoting properties of individuals. How these predications are to be understood is not an easy question.  Suffice it to say that there is clearly a problem with  'Schopenhauer's only son Willy' when the philosopher's only child was a daughter.

On the 'growing block' theory, dead Petty exists. (This is obviously not a present-tensed use of 'exists.') He does not exist at present, but he exists in the sense that he belongs to the actual world.  Once actual, always actual. Is this wholly clear? No, but it is tolerably clear and plausible. After all, we are making singular reference to Petty, a concrete actual individual, as we speak, and this is a good reason to hold that he exists, not at present of course, but simpliciter.

The 'growing block' theory sounds like a kind of four-dimensionalism deriving from the physicist's notion of spacetime as a four-dimensional manifold.  We trace the world-lines of the particles that were ever part of Petty and we find that they form a densely packed blob within a certain spacetime region.  We are tempted to identify the contents of this region with Petty himself.  If we think of the ensemble of worldlines of all material particles as the actual world itself, then yes, the Petty blob seems indeed to belong to the actual world.  But this is a mistake.  The worldline of a particle represents not so much the particle itself but rather its history.  Likewise the blob we take to be Petty represents his biography, in mind-numbing detail.  We are confusing a thing with the life it lived.  Of course Petty belonged to the world---I don't see quite what 'actual' adds here---it's just that he does not belong to it any more.  Perhaps Bill is emphasising that Petty was a real man, not, say, a character in a fiction like Spinal Tap.  There is more than a hint here that Bill is appealing to a theory of direct reference.  Petty has to exist in order that we may refer to him.

In the article, Fiction and Alienans Adjectives, I discuss this and related matters with Bill.

Deducing John McCain?

In the course of Bill's recent discussion on the Univocity of 'Exist(s)', I recalled a much earlier posting where a lack of what we might call 'logical hygiene' leads to much confusion.  This is especially evident in the comments.  Bill offers the following conundrum:

1. (x)(x = x) (Principle of Identity)
Therefore
2. John McCain = John McCain (From 1 by Universal Instantiation)
Therefore
3. (Ex)(x = John McCain) (From 2 by Existential Generalization)
Therefore
4. John McCain exists. (From 3 by translation into ordinary idiom)

Correctly, in my view, Bill identifies the trouble in the move from (1) to (2), though he tables a couple of other panaceas that he says are 'essentially equivalent'.  He says that we need a supplementary premise, namely,

1.5 'John McCain' refers to something that exists.

I would not put it quite that way myself.  The notion of 'reference' is a big topic that we have discussed endlessly at Maverick Philosopher, and the 'something that exists' seems to leave open the possibility that there might be something that doesn't exist, and we will be on the road to Meinongia.  So for simplicity I prefer to replace Bill's (1.5) with my own variant,

DB. There is something named 'John McCain'.

I call this the mention before use rule for proper names. This is just as effective as Bill's proposal for removing the rabbit out of a hat result that the existence of the contingent John McCain 'is validly deducible from an a priori knowable necessary truth of logic', as Bill puts it.  But in this response to my suggestion, Bill seems to reject the idea, saying that I 'seem to be denying the very possibility of coherent fictional discourse' and that he doesn't see the need for my brand of logical hygiene.  We await with interest.

Reference: do we agree?

Bill offers us some distinctions for our consideration.

A. Referring terms versus non-referring terms. 'London' is presumably a referring term as is 'Scollay Square.' 'And,' 'or' and other logical words are presumably not referring terms. Surely not every bit of language plays a referential role. Some terms are syncategorematical or synsemantic.

OK.

B. Purported reference versus successful reference. Asserting 'Scollay Square is in Boston,' I purport to refer to Scollay Square. But I fail: the square no longer exists. But if I say 'Trafalgar Square is in London,' I succeed in referring to Trafalgar Square.

I think there is room for some notion of unsuccessful reference but it isn't what Bill is getting at here.  If I say to you out of the blue, 'I am having dinner with Jane tonight', and you have no idea who Jane is because I haven't introduced the name 'Jane' into the conversation, then I think I have made an unsuccessful reference (or maybe no reference at all).  But Scollay Square was a well-known square that used to be in Boston, so the name 'Scollay Square' hardly fails to refer in this sense. 

C. Guaranteed successful reference versus contingent successful reference. It is not obvious that the first-personal singular pronoun is a referring term. Elizabeth Anscombe, following Wittgenstein, denies that it is. But I say that 'I' is a referring term. Suppose it is. Then it is guaranteed against reference failure: a correct use of 'I' cannot fail to have a referent and it cannot fail to have the right referent. The cooperation of the world is not needed for success in this instance. If I try to make a reference using 'I' I will succeed every time. But the cooperation of the world is needed for successful reference via proper names such as 'Scollay Square.' If I try to make a reference using a proper name I can fail if the name has no (existing) bearer, or if I get hold of the wrong (existing) bearer.

Consider: 'I am Ozymandias, King of Kings.  Look upon my works ye mighty, and despair.'  Is this a case of guaranteed successful reference?

D. Reference versus referents. The referent of a term is not to be confused with its reference. 'Scollay Square' is a referring term and it has a reference, but it has no referent. Not all reference is successful reference.

I don't get the distinction Bill is making here.

E. Mental reference versus linguistic reference. Necessarily, to think is to think of or about. Thinking is object-directed. Thinking is essentially and intrinsically referential. It can occur wordlessly. It is arguably at the basis of all reference, including reference via language. Just as guns don't kill people, but people kill people using guns; words don't refer to things, but people refer to things using words.

Agree with the last clause.  Disagree with the rest, which I think is back to front.

F. Extralinguistic versus purely intralinguistic reference. Consider the following sentence from a piece of pure fiction: 'Tom's wife left him.' The antecedent of the pronoun 'him' is Tom.' This back reference is purely intralinguistic. It is plausible to maintain that the only reference exhibited by 'him' is back reference, and that 'him' does not pick up the extralinguistic reference of 'Tom,' there being no such reference to pick up. Then we would have case of purely intralinguistic reference.

Yes, but 'Tom' could be an intra-linguistic back reference to the indefinite 'a chap' in 'I met a chap called Tom in the pub last night.  Tom's wife has left him'.  And this could all be made up.  Then what?  We need to clarify what extra-linguistic reference is.

G. Extralinguistic per se reference versus extralinguistic per alium reference. 'Max' names (is the name of) one of my two black cats; 'Manny' names (is the name of) the other. These are cases of extralinguistic per se reference. But 'he' in the following sentence, while it refers extralinguistically, refers per alium, 'through another':

Max is sick because he ate too much.

The extralinguistic reference of the pronoun piggy-backs on the the extralinguistic reference of its antecedent. The pronoun has no extralinguistic referential contribution of its own to make.

Leave out 'extralinguistic' and it still makes sense.  'Max' links back to the indefinite 'one of my cats' in the second sentence where the name is introduced.

H. Grammatical pronouns can function pronominally, indexically, and quantificationally. Consider first a sentence featuring a pronoun that has an antecedent:

Peter always calls before he visits.

In this sentence, 'Peter' is the antecedent of the third-person singular pronoun 'he.' It is worth noting that an antecedent needn't come before the term for which it is the antecedent:

After he got home, Peter poured himself a drink.

In this sentence 'Peter' is the antecedent of 'he' despite occurring after 'he' in the order of reading. The antecedency is referential rather than temporal. In both of these cases, the reference of 'he' is supplied by the antecedent. The burden of reference is borne by the antecedent. So there is a clear sense in which the reference of 'he' in both cases is not direct, but mediated by the antecedent. The antecedent is referentially prior to to the pronoun for which it is the antecedent. But suppose I point to Peter and say

He smokes cigarettes.

This is an indexical use of 'he.' Part of what makes it an indexical use is that its reference depends on the context of utterance: I utter a token of 'he' while pointing at Peter, or nodding in his direction. Another part of what makes it an indexical is that it refers directly, not just in the sense that the reference is not routed through a description or sense associated with the use of the pronoun, but also in that there is no need for an antecedent to secure the reference. Now suppose I say

I smoke cigars.

This use of 'I' is clearly indexical, although it is a purely indexical (D. Kaplan) inasmuch as there is no need for a demonstration: I don't need to point to myself when I say 'I smoke cigars.' And like the immediately preceding example, there is no need for an antecedent to nail down the reference of 'I.' Not every pronoun needs an antecedent to do a referential job.

In fact, it seems that no indexical expression, used indexically, has or could have an antecedent. Hector-Neri Castaneda puts it like this:

Whether in oratio recta or in oratio obliqua, (genuine) indicators have no antecedents. ("Indicators and Quasi-Indicators" reprinted in The Phenomeno-Logic of the I, p. 67)

For a quantificational use of a grammatical pronoun, consider

He who hesitates is lost.

(One can imagine Yogi Berra asking, 'You mean Peter?') Clearly, 'he' does not function here pronominally -- there is no antecedent -- not does it function indexically. It functions like the bound variable in

For any person x, if x hesitates, then x is lost.

I am tempted to say that indexical uses have implicit antecedents.  If Bill points to Peter in my presence and says, 'This is Peter. He smokes', then I can perform for myself the introduction, 'There is a man called Peter'. More needs to be said on this.

I. Reference via names, via definite descriptions, via indefinite descriptions. 'Socrates,' 'the wisest Greek philosopher,' 'a famous Greek philosopher of antiquity.' Do they all refer? Or only the first two? Or only the first?

As words none of them refers in Bill's 'extralinguistic' sense.  But I think Bill's distinction between words and uses of words may be useful.  For we could say that Bill refers to Socrates by using these words in certain contexts. I'd like to see this idea developed more.  The key point is that BIll's 'extra-linguistic reference' is something that we can achieve by using words in the real world.  It's not something inherent in the words themselves.

J. Successful reference to the nonexistent versus failed reference to the existent. Does 'Pegasus' fail to refer to something that exists or succeed in referring to what does not exist? Meinongian semantics cannot be dismissed out of hand!

I reject Bill's notion of 'successful/failed reference' and hence the question too.

K. Speaker's reference versus semantic reference. 'The man in the corner drinking champagne is the new dean.' Suppose there is exactly one man in the corner drinking water out of a champagne glass. Has the speaker of the mentioned sentence succeeded in referring to the man in the corner? Presumably yes despite the man's not satisfying the definite description in subject position. Here speaker's reference and semantic reference come apart. This connects up with distinction (E) above. 

I think it connects up with (I).

Purely Fictional Objects

It looks as if Bill's musings on 'same God' have led him back to What Problem Does Literary Fiction Pose?,
a reworking of an earlier post that we have already discussed.  He offers the following aporia:

1. Purely fictional objects do not exist.
2. There are true  sentences about purely fictional objects, e.g., 'Sherlock Holmes is a detective' and 'Sherlock Holmes is purely fictional.'
3. If a sentence of the form Fa is true, then there exists an x such that 'a' refers to x.

For me, alarm bells were ringing before I reached (3). If we conjoin (1) and (2) we get

There are no PFOs yet there are true sentences about them,

and we should ask what the them refers to. Compare with

There are no leprechauns but there are true sentences about them.

So we can reject (1) and (2) without worrying too deeply about what it means to be a PFO.  I have claimed earlier that 'purely fictional', like 'mythological', 'past', 'intentional', and others, are not adjectives in the ordinary sense.

Stromboli exists!

I was reminded recently of Bill's The Stromboli Puzzle Revisited from February this year.  Here is how it begins.

Here is a little puzzle I call the Stromboli Puzzle. An earlier post on this topic was defective. So I return to the topic. The puzzle brings out some of the issues surrounding existence. Consider the following argument.

Stromboli exists.
Stromboli is an island volcano.
Ergo
An island volcano exists.

This is a sound argument: the premises are true and the reasoning is correct. It looks to be an instance of Existential Generalization. How can it fail to be valid? But how can it be valid given the equivocation on 'exists'? 'Exists' in the conclusion is a second-level predicate while 'exists' in the initial premise is a first-level predicate. Although Equivocation is standardly classified as an informal fallacy, it induces a formal fallacy. An equivocation on a term in a syllogism induces the dreaded quaternio terminorum, which is a formal fallacy. Thus the above argument appears invalid because it falls afoul of the Four Term Fallacy.

Objection 1. "The argument is valid without the first premise, and as you yourself have pointed out, a valid argument cannot be made invalid by adding a premise. So the argument is valid. What's your problem?"

Reply 1. The argument without the first premise is not valid. For if the singular term in the argument has no existing referent, then the argument is a non sequitur. If 'Stromboli' has no referent at all, or has only a nonexisting Meinongian referent, then Existential Generalization could not be performed, given, as Quine says, that "Existence is what existential quantification expresses."

Bill tables a second objection, which he also rebuts, and then offers his own account of why the argument is valid, which hinges on a distinction between 'first-level' and 'second-level' uses of exists. I won't comment on these because I think Bill's Objection (1) gets things right.  I reject his Reply (1) as follows.  If 'Stromboli' has no existing referent, has no referent at all, or has only a nonexisting Meinongian referent, then there is no such thing as Stromboli and therefore 'Stromboli is an island volcano' is not true.  It may be false or it may be meaningless, but it certainly can't be true.  In which case the argument remains truth-preserving:  It has not been shown to take us from truth to falsehood.  Bill has not demonstrated that it is invalid.

But this talk of non-existing referents is already verging on the absurd and it would be better to reject Bill's Reply as nonsensical.  After all, an invalid argument is unpersuasive, and yet this argument, even without the first premise, is utterly persuasive.  Consider the following dialogues:

A: Stromboli exists.
B: This tells me nothing more than that there is something called 'Stromboli'.
A: Well, Stromboli is an island volcano.
B: OK.  So there is an island volcano?
A: Yes, an island volcano exists.
B: Good. Glad we understand each other.

A: Stromboli is an island volcano.
B: So an island volcano exists?
A: No.
B: What? You just told me Stromboli was one!
A: Yes, but Stromboli doesn't exist.
B: I don't follow you.

A: Stromboli is an island volcano.
B: OK.
A: And Stromboli exists!
B: Yes, I know, you just told me.  It's an island volcano.

Do these not have the ring of truth?  I think they do.  And this point of view is beautifully simple.  Compare it, for example, with Bill's 'two-level' account.  But I clearly need to say more.  Bill, and others, are happy to talk about 'empty' names, names with no referent, names with no existing referent, names with Meinongian referents, and so on.  So-called 'free' logics (there are at least three kinds) have been invented to accommodate the notion.  I need to say something more about what is going on here.

Let's distinguish the meaning of a name from its referent, if any.  The meaning of a name is what has to be understood about it in order for sentences involving the name to make sense.  This is quite independent of any external referent the name may have, as is shown by the fact that fictional sentences are perfectly understandable despite the proper names within them usually lacking real-world referents.  Now, in properly formulated discourse a name has to be introduced before it can be used.  This is how the name gets its meaning.  If, out of the blue, you say to me 'Stromboli exists!' and I have never been introduced to the name 'Stromboli', then what you say has no meaning for me.  I have no information to attach to the name 'Stromboli' and your statement doesn't change my epistemic state.  Asked to say something about Stromboli, I can tell you nothing.  If, on the other hand, you say 'There is an island volcano called Stromboli',  the name 'Stromboli' acquires a meaning.  If asked about it I can reply 'It's an island volcano, isn't it?'  For me, the name now means this island volcano and you can go on to tell me more about it by using the name.  What you say may be complete fiction---there may be no island volcanoes at all---but it will be meaningful, nevertheless. Notice that the name is introduced on the back of (literally behind) an existential claim:  'There is an island volcano---and it is called Stromboli', and it is open for this claim to be false.  If, out of the blue, you tell me 'Stromboli is an island volcano', and I have never heard of Stromboli, I have to interpret your sentence as a somewhat compressed existential assertion and name introduction.  This is the style adopted by dictionaries.  A subsequent use of the name, such as 'Stromboli is extinct', which has similar form,  does not carry an implicit existential claim, else we would have two entities in play, an island volcano and an extinct thing, rather than one.  Instead, the name means the same thing as was introduced earlier under the same name. 

All this, of course, is Ed Ockham's theory of  relativity of reference  which I think is spot on.  Where Ed and I differ is that Ed, I think, holds that negative free logic is needed to make inferences from sentences using names understood in this way.  I see this as an unnecessary and counter-intuitive complication to a simple and elegant theory.  Logical inference operates on the meanings of sentences not on the real-world objects and properties they might be about.  We can make inferences within fictional discourse just as easily as we do in discourse about the real world, using classical logic.  In Ed's theory all properly introduced names have meanings  and there are no names that lack meaning.  Hence we have no need to resort to free logics.  Ed may say that in classical logic we have no way of expressing negative singular existentials such as 'Vulcan does not exist'.  I answer that this amounts to a rejection of the explicit or implicit existential claim on the back of which the name 'Vulcan' was introduced. See also this post.

Addendum on sets as referencing devices

'6' is the name of 6 and '6' refers to 6.  But we don't tend to think of '6' as an object to be quantified over.  On the other hand, '{6}' is the name of {6}. '{6}' refers to {6}, and {6} refers to 6.  Yet we do tend to think of sets like {6} as objects---primarily because they are the targets of reference: {{6}, 7} refers to {6}---and we do, when doing mathematics, quantify over them.  So sets seem to occupy a kind of no man's land---they both refer and are referred to.  This further confuses their 'ontological status'.  The situation is perhaps analogous to Frege's concept Horse which is primarily a concept, not an object, but when talked about as opposed to with, has to be seen as an object.

Mathematical sets as linguistic reference devices

Bill Vallicella has posted sporadically on the philosophy of set theory.  What exactly are mathematical sets?  Do they exist at all? I had commented occasionally without having a properly developed position of my own.  By the time of Sets, Pluralities, and the Axiom of Pair, to which I attached a brief comment, I'd reached some firmer conclusions. Bill's more recent Does Potential Infinity Presuppose Actual Infinity? revisiting the question of the existence of infinite sets has prompted me to say a little more.  I hope it's interesting and relevant.

I start with the idea that all reference can be pictured as pointing.  A set is therefore a bunch of arrows, each arrow pointing to an object.   Like proper names, being elements of language they are not in the world in the same way as ordinary objects.  I have tried to indicate this in the accompanying diagram.  I appreciate that philosophers are dubious of the place of drawings in philosophy but since I think the essence of reference is pointing, it seems appropriate in this case.  

The world of ordinary objects, 1, 2, 3, etc, is represented by a horizontal plane.  The sets are the bunches of arrows pointing up into the world from outside.  To emphasize the unity of each bunch I place each bunch in a 'quiver'.  This helps in understanding the empty set if the idea of an empty bunch of arrows is resisted.  The empty set perhaps can be seen as a plural analogue of the word 'nothing'.  A singleton set is a single arrow.  There is now no temptation to identify the singleton set {6} with its sole member, 6.

Relations such as membership and inclusion can be understood in terms of arrows:  x is a member of S iff S has an arrow pointing to x (ie, S refers to x).  S is included in T iff every thing pointed to by some arrow in S is pointed to by some arrow in T.  Two sets are equal iff their arrows point to exactly the same things.  Unlike the view of sets as collections there is no sense here that an element is a mereological part of a set.  The extension of a set is clearly distinct from the set itself.  On the view of sets as collections, equal sets are identical  and this gives rise to sense/reference confusions.  Here I can label a set with an intension and happily discover that it has the same extension as a set defined from and labeled with a different intension.  This more accurately reflects how sets are used in proofs and constructions.

The set operations of union, intersection, and difference have simple definitions in terms of arrows.

A set of sets is a bunch of arrows with each arrow pointing to another bunch of arrows. The power set construction follows naturally.  Sets of sets of sets to any required level can be constructed and have a precise meaning.  This is the principal way in which sets extend the plural referencing features of natural languages. The latter are not adequate to the task of tracking the pluralities of objects that can arise in mathematical proof.  They are limited in the complexity they can express and they suffer from ambiguities as evidenced by the discussion in the first of the above posts.  The 'iterative hierarchy' of sets arises in a natural way as tiers of quivers bearing upward pointing arrows.  In the axiomatic presentation of set theory the iterative hierarchy looks a little like a device deliberately introduced to avoid the antinomies.  Here it develops as the need arises in a proof or construction  to refer back to pluralities of objects previously defined, or to construct new pluralities from old.  Enderton, Elements of Set Theory,  says that Cantor was led to set theory through his study of Fourier series:

Cantor's work on Fourier series led him to consider more and more general sets of real numbers.  In 1871 he realised that a certain operation on sets of real numbers (the operation of forming the set of limit points) could be iterated more than a finite number of times; starting with a set P₀ one could form P₀, P₁, P₂,...,P_w, P_w+1,...,P_w+w...

This back referencing means that all arrows point upwards.  That there can be no 'set of all sets' has an obvious geometric interpretation: any putative set of all sets would require an arrow pointing to itself which would thus fail to point upward.

This view of sets changes the complexion of the question of the existence of sets.  If they are artificial linguistic elements akin to proper names then we need not expect to find them in the world itself nor in some Platonic heaven.  Furthermore, merely referring to a plurality of objects is tantamount to their forming a set.  I agree with Bill's remark that 'that from the fact that there ARE many Fs it does not straightaway follow that there IS a single thing comprising these many Fs'.  But on the reference conception of set there is only a weak sense in which its members comprise a set.  Rather it would be a performative inconsistency to refer to the Fs yet to deny the set of the Fs.  Their cardinality is of no consequence.

Ockham sets and Mathematical sets

This diagram accompanies a comment made at Beyond Necessity.

  

These two diagrams may help explain the difference between Ockham sets and mathematical sets, as I understand it. It's very much a case of reference, which is depicted by the arrows. Linguistic terms are shown on the left, in quotes to emphasise their symbolic nature. The Ockham term 'a and b' refers directly to the two objects a and b. In contrast the mathematical term '{a,b}' refers to a 'set-object', depicted as a quiver containing two arrows pointing to the objects a and b. Thus the mset {a,b} is distinct from its members, a and b, though it refers to them. The empty set, {}, is not nothing. It's a quiver with no arrows. It's this indirection that enables msets not so much to contain other msets but to refer to other msets that give the mset language it's referential power. My view is that msets are an artificial extension to the referencing capabilities of natural language. This may explain why we feel that they have to be 'constructed'. Contrast with the Ockham world in which such intermediaries are absent.