Almog, Joseph - Nothing, Something, Infinity

Journal of Philosophy, Inc.

Nothing, Something, InfinityAuthor(s): Joseph AlmogReviewed work(s):Source: The Journal of Philosophy, Vol. 96, No. 9 (Sep., 1999), pp. 462-478Published by: Journal of Philosophy, Inc.Stable URL: http://www.jstor.org/stable/2564708 .Accessed: 10/04/2012 19:54

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462 THEJOURNAL OF PHILOSOPHY

NOTHING, SOMETHING, INFINITY 8

In a memorable passage, Bertrand Russelll instructs us that: Tllere is an argument, suggested by a passage in Plato's Parmenides, to the effect that, if there is such a number as 1, then 1 has being; but 1 is not identical with being, and therefore 1 and being are two, and therefore there is sucll a number as 2, and 2 together with 1 and being gives us a class of three terms, and so on. This argument is fallacious (ibid., p. 138).

In spite of its alleged fallaciousness, I shall try to justify a form of the Parmenides argument. SpeciElcally, I am after a proof of the con- ditional:

(Illf) If there exists at least one thing in the world, there exist in the world infinitely many things

Read contrapositively, (Inf ) asserts that, if the world does not have inElnitely many things in it, there is nothing in it. We are left with a stark choice: there cannot be in the world only a tnere something; it is either inElnity or nothing.

I. THE ONE-REALM PRINCIPLE The formulation of (Inf) speaks of 'the world' and 'things' in it. This calls for some discussion of what I intend by these phrases. Traditional metaphysical analyses distinguish, among beings, between reals and ideals. Reals (for example, orcas and diamonds) belong in the material world, often alluded to as the first realm. Ideals belong elsewhere, in "other realms." How many other such realms? 8 In the last few years, I published a trio of interrelated papers: "Logic and the World," Journal of Philosothical Logic, XVIII ( 1989): 197-220, "The What and the How," this JOURNAL, LXXXVIII, 5 (May 1991): 225-44, and "The What and the How II," Nous, xxx,4 (1996): 413-33. The trio investigated the kinds of things there are in the world and which, if any, the world must have. The present paper is a sequel applying itself to the question of just how many things there must be in the world. The paper has been delivered in the last four years at various institutions. I am grateful for these discussions. Individually, I owe thanks to Torin Alter, Philip Bricker, Tyler Burge, Edmund Gettier, John Heintz, Ignacio Jane, David Kaplan, Ali Kazmi, Calvin Normore, Seana Shiffrin, Sten Lindstrom, and Byeong-uk Yi. Special thanks for objections discussed in section rv are due to Guy Rohrbaugh, Andrew Hsu, Tony Martin, and Dominik Sklenar (on whose work on kinds, see be-

low) . l Introduction to Mathematzcal PhilosofAzy (London: Unwin, 1919). By 'term', Rus- sell does not mean a linguistic item but an object (entity, thing). 0022-362X/99/9609/462-78 (C) 1999 TheJournal of Philosophy, Inc.

463 NOTHING, SOMETHING, INFINITY

Some (for example, George Berkeley and David Hume) opt for an exclusively mental ("psychological") realm, often referred to nowa- days as the second realm. Yet others (for example, Gottlob Frege) speak of an additional third realm of nonpsychological, reputedly objective, abstracta, such as thoughts (Sinne), functions, and "logical objects" (for example, numbers).

It is against the background of realm proliferation that we are to read our one ur-principle:

(One Realm) There is only one realm of being, the real world.

Our ur-principle propounds a uniElcation of all beings in one re- ceptacle the real universe generated by the "Big Bang" a few billion years ago. Whatever is be it material, mental, or abstract is a real historical being, one that came to be and existed in this one cosmic realm.2

One realm, then. The proof of (Inf) is about it: if there is in the real universe at least one thing (for example, Nastassja Kinski), then there are in it infinitely many real things. But from Aristotle to Thomas Aquinas, from G. W. Leibniz to Immanuel Kant, and all the way to our logically sophisticated end of Millennium, it has been re- peated that the concrete (sometimes: "the sensible") pluralities of the world are merely Elnite or, at any rate, not provably inElnite. As long as our focus is pluralities of concreta (actresses, orca whales, or diamond stones), there is no proving their inElnitude.

Let this much be granted: (Infc), 'There are inElnitely many concrete things', is unprovable. But what about our project, proving 'There are inElnitely many real things' (InfR)? It is presumed that the unprovability of (Infc) transfers to that of (InfR). What lies behind the exclusion of a category of nonconcrete ("abstract") worldly things?

I.1. The worldly abstracta dilemma. For our paradigm of concrete subjects, consider a mischievous Puget Sound orca, Godiva, and the resplendent diamond on Liz's ring, Shine. Each such concrete object is both (i) objective and (ii) worldly (wirklich). Call the conjunc-

2 A more canonical gloss of the adjective 'worldly' ('real') shows up later in the paper (see 'the reality quintet'). At this point, I follow this 'do-not-define-it-use-it' procedure because I find the appeal of the notion worldly, real independent of my own specific annotations. For example, Frege's gloss of 'real' ('zviaklich') suits us very well: "The world of the real is a world in which this acts on that, changes it and again experiences reactions itself and is changed by them. All this is a process in time. We will hardly recognize what is timeless and unchangeable as real"- "The Thought," in P. F. Strawson, ed., Philosothical Logi.c (New York: Oxford, 1967) p. 37.


tion of traits the mundane conjunction, with both meanings of 'mundane' in mind: the in-the-world and the run-of-the-mill character of the conjunction of traits. But mundane as the combination is for Shine and Godiva, it appears intrinsically impossible for the abstracta manifested by this pair.

With these abstracta, it is either/or. Make it (the number Two, the species Orca, the substance Diamond, the colors Black and Blue) objective, as Platonists have, and it is banished away to another, oth- erworldly and acosmic, third realm. Alternatively, we may follow the nominalist and (like, for example, John Locke) make each such abstractum into an individual mental abstraction. Less "individualisti- cally," we may view such abstracta as essentially dependent on our linguistic use of the classificatory predicates '...are two' and '...is an orca' and '...is blue'. Either way, we have brought the abstractum back into the world the psycholinguistic second realm but at the cost of losing its objectivity.

This then is our problem: the abstractum (say, the species Orca) cannot co-exist in the first cosmic realm with the concreta (Godiva, Gieronimo, and the like) that realize it, make it real. The problem is metaphysical and is not to be confused with Paul Benacerraf's dilemma, which is essentially epistemic. Our problem does not con- cern sensory access and forms of knowledge; it arises for an omni- scient being like God: even He cannot make abstract objects bear the mundane conjunction, be both objective and in the cosmos. Even for Him it is either/or, what I shall call the worldly abstracta dilemma.

I.2. The worldly infinity dilemma. The worldly abstracta dilemma is our root problem. It leads to a derivative quandary: no plurality of things may be both of worldly constituents and yet contain infinitely many items. It is again either/or. The choice we are forced to make may be impressed on us by considering two classical arguments for (Inf), one due to Frege, the other to Richard Dedekind.

Frege's argument first. He begins with an ideal, off-world item, the predicative concept (Frege speaking here) 'not identical to itself', Fo We now assign Fo a yet higher (second) level predicative concept, a number concept, ZERO. Still on logical grounds, we now "take" the extension of ZERO. This is by Frege's lights a pure "logical object," the number 0. We existentially generalize: there exists at least one object and 0 is it. Frege now ascends back to the realm of predicates and considers this time the concept F1, 'be identical to 0'. Correlated with it is a number concept (namely, ONE) the number of things falling under F1. The extension of ONE is a new object, the


number 1. By now, Frege can prove the existence of two logical objects, 0 and 1. And so it goes: the process yields infinitely many items when Frege proves by induction on n that the extension of 'number of items smaller or equal to n' is always n+ 1.

What has Frege proved? The infinitude of what he calls logzcal objects, predicative concept derivatives, items that have it built into their very identity that they are extensions of concepts. Therein lies Frege's version of our dilemma: insist on wirklich subjects, and no proof of (Inf ) is forthcoming; shift to off-world predicative abstracta, and a proof becomes available.

A similar message is imparted by the main other classical proof I am familiar with, Dedekind's. In section 66 of his Was Sind und Was Sollen die Zahlen?3 Dedekind tries to prove that there exists at least one infinite "systemS' of objects. On his way, he argues that there is an infinite plurality of objects.

Dedekind locates the infinite plurality in his Gedankenwelt. His ur- object is his Ego. He then goes on to apply to this realm of thought a one-one map a that is not onto. The informal gloss offis 'can be the object of my thought'. We are told without further ado that a is one-one. And obviously it is not onto: Ego is not got as a result of the map. This much makes the realm of these Gedanken-objects Dedekind-infinite, hence infinite simpliciter. But there remains the question: What precisely is proved by Dedekind?

Russell's commentary says it all:

We are then to suppose that starting say with Socrates, there is the idea of Socrates, and so on ad inf: Now it is plain that this is not the case ill the sense that all these ideas have actual empirical existence in people's minds. Beyond the third or fourth stage they become mythical. If the argument is to be upheld, the "ideas" intended must be Platonic ideas laid up ill heaven, for certainly they are not on earth (op. cit., p. 139).

The worldly infinity dilemma, all over again. II. DEFUSING THE WORLDLY ABSTRACTA DILEMMA 4

Consider again the map used in Dedekind's proof. Instead of his original 'can be the object of my thought', substitute 'set of'. This operation is one-one: if xWy, then {x}${y}; and it is not onto; whatever it exactly is, 'meine eigenes Ich' is surely not a set. We thus have on hand a (Dedekind)-infinite plurality of real items. Was the above allusion to dilemmas much ado about nothing?

I shall not rest lny case on the categoly of sets. Granted, sets, as characterized by Dedekind, Georg Cantor, and eventually Ernest

3 Braunschweig: Viewey, 1960, eighth edition. 4 I am grateful in this section to Andrew Hsu and Guy Rohrbaugh.


Zermelo, are not mere predicative projections, like Frege's extensions or Russell's propositional functions. Nonetheless, sets are not what I here target by 'real things'. It is thus time to set out in detail what I intend by real (worldly, cosmological, wirklich) items, why sets will not do and which abstracta will.

Five marks of the real suggest themselves, what I shall call hence- forth the reality quintet:

(1) No prefabricated individuation (2) No reduced identity (3) Historicity I: constitutional challge (4) Historicity II: predicative change (5) Historicity III: counterfactual developments

Consider first the principle of no prefabricated individuation criteria. With real itetns (for example, Godiva), it is the actual existence of the entity, its coming into history and development in it that Elxes whatever is distinctive about it. No set of antecedent conditions prior to the historical item may define or dictate what it is to be this specific orca whale Godiva.

A very different reading of individuation principles applies to sets and Goodmanian sums. Here, we do have axioms of extensionality that regulate in advance what it is to be this specific construction (the set Xor the sum Y) in terms of members or parts. The fact that such a principle is formulable for sets and G-sums (as for other "constructions" like aggregates, tropes, and the like) is, for me, an indication that we do not have on hand a genuinely historical subject, pro- duced by the autonomous forces of cosmic evolution.

Let us now turn to a third category of items: for example, the species Orca and the Manchester United soccer team. Do we get the profile of Shine and Godiva, or that of sets and G-sums? The species and the soccer team are historically real phenomena: it is only their actual emergence in history that fixed the distinct identity of the item that emerged. We date the emergence, namely, the origination of the species and the founding of the team: the species did not exist five billion years ago and the team did not exist in Rene Descartes's time. No prefabricated set of criteria could legislate into existence this species or this soccer team.

The second mark no reduced identity is related. A set (Good- nzanian sum, aggregate of parts, trope) just is (is identical to) an ensemble of "atoms" making it up. As such, the identity of the "higher" entity is reducible to a constructional operation of "composition" (set of, aggregate of, and so on) on more basic ingredients. In con-


trast, Godiva and Shine are not so reducible to "constructions" out of ingredients. We find the species and the soccer team sharing this profile. Even though both the species and the soccer team have a multiplicity of tnaterial "members," neither is identical to a logical construction out of the membership.

This logical point regarding identity is manifested over the tempo- ral dimension, when we consider the manner in which the pertinent entities endure in history. The set and G- sum would not exist, if it were not for their "ingredients" (for example, {Godiva, Shine} did not exist before planet earth did). But once in history, the continuing existence of the assembled entity is tied intrinsically to this specific constitution; there is no way for this set or G- sum to persist except through the persistence of this specific constitution.

Not so for Godiva and Shine. Their survival through history depends on constitutional change: the higher entity goes on existing, as its constituting matter changes, only because it is realized in a different material basis. Again, the species and the soccer team follow this mould: the evolution in time of this single species and soccer team presupposes changes in membership; it is only because the higher entity can sustain itself by having other members that it en- dures.

Constitutional change leads us to another form of change: predicative change. Real subjects like Godiva and Shine constantly change the properties they bear. Such is also the profile of the species and the soccer team: each ages, evolves, scatters, becomes en- dangered all properties borne by the higher subject and not reducible to properties borne individually by its present members. Not so with constructions. Doubtless, the set-theoretic pair may change: it had a member (Godiva) that was hungry and now it has a member that is bellyful. But such predicative changes are essentially by proxy, analyzable via changes in the intrinsic ingredients.

The availability of an actual history leads us to a final characteristic mark of matters real the potential for alternative developments of that history. Such counterfactual developments are surely envisage- able for Godiva and Shine and, in turn, for the species and the team. No such counterfactual developments are open to abstracta of the constructional kind. Just as they lack a genuine actual history, they are not subject to alternative evolutions of that history.

Our reality quintet sets apart "blueprint" constructed abstracta from those which are historically real. Two observations are suggested by this separation. First, we have seen examples (the species Orca and the Manchester United soccer team) of genuine worldly abstracta,


bearers of the mundane conjunction; each is both objective and cos-

mic. Our root dilemma thus turns to be a false one. Second, regard-

ing the worldly infinity dilemma, we know what to look for: items

which are in the world like soccer teams and species but of which

there are infinitely many. III. INFINITY IN THE WORLD

What is proved below is a form of the Parmenides argument.5 We are

given the existence of one (purportedly the first) concrete item: the

orca Godiva.6 By its very primal existence, Godiva generates a variety

of real kinds: the color-kind kBlackk, the shape-kind kOval*, the bio-

logical species kOrcak, the kind of bodily-organ kBraink. One further

kind generated by Godiva is the arithmetical kind kOne*. She is most

definitely a black thing, an orca-thing, and so on, but she is also one

thing; at that, the first among all ones.

So now we have on hand an ordered two-some: Godiva and an

item she generated, kOnek. Both Godiva and kOnek exist two real

things. Thus, the first real instance of kTwok exists; in turn, kTwok is

generated. We now have on hand three ordered, real things: Go-

diva, kOne*, and kTwok. The kind kThreek is generated.

The foregoing generation of the finite ordinals produces a plural-

ity of the following form: Godiva, kOne*, kTwo&,... The plurality has

-many members. This generates the first infinite ordinal, the kind

k(,)k. Once it exists, we have on hand the plurality: kOnek, kTwok,....

*(,)k. Here lies the generative basis for the ordinal +1. Past the gen-

5 The idea of "proving" any variant of (Inf) may seem passe to many a reader; we

should rather state it as an "axiom" of infinity. Let me clarify what is here meant

by "proof." First, I am not out to prove that there exists at least one infinite system (set,

proper class), and so on. Second, neither Dedekind nor the present account aims

to prove (Inf) inside a formal (be it generalized first- or second-order) logic deduc-

tive system. It is a familiar fact that such formal logic systems (where (Inf) is ex-

pressible) afford models with finite domains. What is offered here is an informal proof that (Inf) at that (InfR) is actually

true. This much is not more ambitious than what is pursued by standard set-theo-

retic axiomatizations. Consider such an axiomatization, say, standard ZF. The au-

thor of the axiomatizatioll (textbook) explains to the student why the axioms are

taque. This much applies also to the axioms of pairing, union, and so on. Such an ex-

planation must also be given for (I) there exists an infinite set. After all, set theo-

ries explicitly denying (I) are available. So, do we or do we not give a "proof" of (I)?

The answer is that most careful authors back up their assertion of (I) with the dis-

play of an infinite set usually the one representing the natural numbers in the Zer-

melo or von Neumann systems. This is the textbook de facto proof. Indeed, such

proofs seek a more ambitious result than ours. For it is argued that the aforemen-

tioned infinite plurality of finite sets forms in turn a single infinite set. Our argument

does not take the last step (though I do not believe there is anything wrong with it).

6 In what follows, I use the notation kOrcah to make clear I speak of the entity,

the zoological kind of Orcas.


eration of kOne*, we use two further generative principles the very two used by Cantor (before he gave up on ordinals as primitive and reduced them to sets): the addition of one item to a given ordinal plurality and the "taking of a limit" of an unending plurality of ordinals, as in the case of k(])k.

III.1. Annotations to the argument. (i) Generation versus instantiation. We should separate the item(s) that bring a kind into being from item(s) that, once the kind exists, count as its instances. Suppose Adam is the f1rst being in the world. His existence generates that of kOnek. Later, Eve shows up. She counts as an instance of kOnek, not as its generator. And so it goes. When Adam generates kOnek, he and kOnek, the two things that they are, generate kTwok. Later, Eve shows up. Adam and Eve, the two-some plurality they make, provide an instance of kTwok. Still, they are not the generators of kTwok, a role reserved for Adam and kOnek.

On the present account, any n-some plurality, whether of numbers, cabbages, or kings, may serve as an instantiation Of knk. But only the preceding sequence of ordinal numbers, up to knk, can generate knk. In all, it is suggested both that (i) only a non-number, a prenumber concrete object, can generate kOnek and (ii) from then on, only numbers can generate their successors.7

(ii) Ordinals versus cardinals. The procedure described above generates first the (finite) ordinals. The ordinal is the prior existent. The equivalence class that it is, cardinal Card(kTwok) rests on the subsequent truth of the predication: 'plurality having as many items as the one that generated this existent, the ordinal (kTwok)'.

In this respect, Card(kTwok) is rather like the biological kind-pred- icate 'is an orca' which I read as 'is an instance of the kind kOrcak'. In turn, the objectual ordinal kind kTwok is the analog of the objectual kind kOrcak. In both cases, the kind-as-object precedes the pred- icate 'being a member of that kind'.8

7 On both (i) and (ii), see more in the annotations that follow. 8 In the context of infinite llumbers, the difference between generation and in-

stantiation, ordinals and cardinals, is quite clear. The ordinal *(Jjk iS generated first, then *(,+l*. By quite a different existence principle, *(JJk gives rise to the cardinal number btov Now, the instances of *(JJk and *(,+lk are quite different: all pluralities of the order type X are instances of the former; one more item in the plurality or an order change will be required to make a plurality that instantiates the second ordinal kind. But now when it comes to instances of the first infinite cardinal, all such instances, regardless of their order type, would do as instances.

The ordinal/cardinal distinction shows up clearly for infinite numbers; it also shows up with Zero. On the present account, Zero is not an ordinal. What "de- fines" Zero into existence is predication: it is the number of thillgs falling under empty predicates. With, for example, Card(*Two*), we test for one-one correspon-


(iii) The reality quintet. (iii. 1) Identity craterza. First on our list of marks was the subject's affording no prefabracated individuation. Classical con- ceptions of number supply such individuation criteria in, for example, the Frege-Russell vein: the number of Fs and the number of Gs are one and the same if and only if the classes (concepts) of Fs and Gs are in one-one correspondence (lately known, under the influ- ence of the late George Boolos, as "Hume's principle" and universally viewed as "logical," or at the very least "analytic" and "defining").

On the present account, it may turn out of such a (biological, chemical, arithmetic, and the like) kind that some fundamental feature of its generation process is uniquely true of it; thus, different persons (rivers, species, chemical elements, numbers), different generative histories. But these are ex post facto historacalfacts, not prefabricated regulative conditions. It may thus well turn out that different ordinal numbers have different generative histories (different numbers, different ancestors) but this much is simply a descrip- tion of how each came to be, not a stipulated convention.

(iii.2) No reduced identity. The reduction would amount to the number's being just a certain ensemble of more basic ingredients (parts, sets, concepts). Our number-kinds are not reducible to their members. The existence of members is critical for the generation and ongoing existence of the kind but the kind itself is never ex- hausted by this or that membership.

(iii.3) Historicity I: constitution change. Consider the difference between our kind kTwok and, for example, von Neumann's set sTwos= {Godiva, One} (s-quotes are to remind us the object is a set). First, regarding identity: von Neumann's sTwos is a specific two-membered set. Benacerraf's other problem "What numbers could not be"- strikes: Why is the number Two identified with this set and not any other two-membered set? In contrast, our kind kTwok is at the right level of generality: it comprehends all two-item pluralities as instances (the instances of the kind are two-some pluralities (for exam-

dence with the ur-two-some plurality that generated the ordinal kTwo*. In the case of Zero, we cannot so test, for there exists no such ur-exemplar and no generated ordinal. All there is is the one - one correspondence between all empty predicates. We may feel here, with Frege, that Zero is an adjective that has grown capitals (for example, ' there are no (zero) unicorns' ) .

The foregoing relates to a referee's question whether we could use our real- number-kinds (for example, kTwo*) to number third-realm items (for example, two forms in Plato's heaven or two Fregean extensions). The answer tracks back to my above (one realm) principle: all beings exist in this one real cosmological realm. What is not is not zvirklich cannot be numbered. Only reals can enter real relations be numbered by with a real like the number kTwo*. On which, see more in annotation (iii.4).


ple, me and you), not the subsequent unity (the two-membered set {me, you}) ) .

Second, regarding existence. von Neumann's sTwos is perilously dependent on the existence of Godiva: no Godiva, no sTwos. In contrast, the kind kTwok, like kOrcak, goes on existing while its members change. The number-kinds (again, like biological ones) were generated by a specific ur-item the historically first concretum. But to go on existing, all they need is to be instantiated by some any old- instances.9

(iii.4) Historacity II: predicative change. kOrcak and kDiamondk change through history. But do numbers ever change? At first blush, the difference between the cases seems striking. But let us have a closer look. Granted, 9 is fixedly odd and greater than 7. But this fixidity, just as with the temporally stable features of kOrcak, concerns features fixed by (and at the time of) the very generation of the item. On the other hand, 9 also numbers the coins in my pocket. This is a feature gained by 9 after its generation and thus subject to change. When we say the next day '9 used to number the coins in my pocket but it no longer does', we make a claim of de re change: some specific res was one way, now it is not. The question is: What res is that?

The natural answer seems to me: the number of the coins. Of course, at first blush, we think the change is, to speak loosely, "in the coins," not in the number. But what does it mean the change is "in the coins"? It could not be a de re change in the nine membered set (cl, c2, ..., c9}. This set cannot survive the addition or subtraction of a coin. Nor is it a change in the aggregate or the Goodmanian sum. I submit that the change in the coins induces a change in the number but it is not identical to it the change in the number may have taken place in other ways (instead of adding c10, I may have added

9 What if we added to the reality quintet a sixth condition: all reals must at one time or another cease to be? Would this not make the existence of an infinity of our kind of numbers ephemeral?

My answer is twofold. First, quite apart of numbers (and other real abstracta), I would not add the purported sixth mark of being real. Generation as a mark of the real is not "symmetric" with corruption. Many generated abstracta (including individuals, not just kinds) may go on existing by replenishing the materials that realize them. This gives such real subjects as opposed to sets of such material ingredients a long lease on life.

Second, regarding numbers. Once there was one concrete thing, there were numbers. Amd if per impossibile there were nothing concrete, there would be nothing at all. As will become clear in the next annotation, I believe that once there existed something concrete, there always (and of necessity) continued to be some concrete item. In time, the infinite generating sequence, with the specific Go- diva at its basis, will be gone, because Godiva will corrupt. But the numbers (the second place in this sequence, kOnek, the third, kTwok,...) are forever.


cll). Likewise, there may have been, in the concrete coins, the following change: the addition of one blue coin c10 (the original nine are red). But it would be a mistake to identit the change thus in- duced in the number of coins with the change in the color. Although actually realized via the addition of c10, the change in number and color might have occurred separately.

One may question the foregoing emphasis on the intuition of de re change. Thus runs what I call the no-change objection:

Consider "The president used to be George Bush and now it is Bill Clin- ton." Rather than a change in the state of a single underlying entity, we have replacement of one entity by another: Bush was the president and another individual, Clillton, is now the president. In a similar vein, we may suggest for the above claims that no number, color, and so on has really changed. We have instead replacement: one color, green, was the leaves' in the spring, now it is another, yellow; one number, 9, was re- placed by another 10, as the number of coins in my pocket.l°

In response to the no-change objection, I shall say this. First, we should separate in this discussion syntactic and metaphysical issues. Grammat- ically, we have on hand a uniformity between 'Bert's girlfriend (the U. S. president) is changing', 'The color(number) of the coins is changing'. Metaphysically, going by the present paper's outlook, species, temperatures, colors, and numbers are real kinds, coming in and out of cosmic history independently of our verbal activities. In contrast, the "roles" of being the American president and Bert's girlfriend are in- duced by predications of ordinary individuals. I shall attend here to real kinds and leave to the aforementioned linguistic companion piece the proper analysis of 'Bert's girlfriend (the president) is changing'.

Second, let us ask now: When I speak of "the number (color) of coins is changing" as involving de re change, in which res is the change to be located?

'° The no-change objection has been made to me by a variety of commentators in the University of California/Los Angeles Language Workshop dedicated to the Partee-Montague puzzle regarding the sentence 'The temperature of LA is ninety and it is rising'. It was also made by Francaois Recanati.

I view Montague's original analysis of 'The temperature is rising' as in the no- change objection vein. He lets the prime semantic value of 'the temperature' be a function from times to numbers. He then speaks of that function as "changing" ("rising"). In truth, the function does not change (rise) at all. It is as fixed as any thing could be, the set of ordered time-number pairs that it is. What is actually encoded by the function is replacement: the replacement of one extension (90 degrees) by another (91 degrees) as the extension of 'the temperature'. I discuss in detail Montague's analysis, the no-change objection, and the syntax-se- mantics of such constructions in my "The Subject-intransitive Verb Class" (in press) .


The question cannot be answered without understanding the prechange kind of fact reported in the likes of 'The species of Godiva is orca' or 'The color of leaves is green'. Some (for example, Richard Montague) regard the claims as having the form of identity statements. With this, I disagree. We have identity claims in the likes of 'Furze is Gorse' (species identity), the bilingual 'Green is Verde' (color identity), '34=XXXIV' (number identity).

In contrast to an identity, 'The species of Godiva is Orca' has the two-sorted relational form:

R (c,A)

where c indicates the concrete relatum, A the abstract relatum. Idem for 'The number of coins is 9', whose form I view as the relational: Has-number (coins in my pocket, 9).

I read much importance into the genitive form of our constructions, namely, the abstractum mentioned is of- the-concretum (a): its color, temperature, species, or, in the plural, their number. Gener- alizing, I would like to call such assignments of a concretum-to-a- kind kind-membership reports.

Now, when we report change in such membership relations, we report change in a relation. The primary change is in the concrete relatum. But its (their) failure to stay related on a given dimension (color, temperature, number, and so on) to the kind (red, 90 degrees Fahrenheit, 9, and so on) induces a change in this second relatum the kind. Here, the change regards what the color, temperature, and number are oyS 11

(iii.S) Historicity III: counterfactual developments. Nine actually numbers the coins in my pocket but it might have failed to number them, just as Nastassja Kinski was actually the leading actress of Tess but she might have failed to be if she resisted Roman Polanski's pressures. What makes these 'might have' claims true?

' x might have failed to F ' is true at a moment t in the actual history of the world if and only if there is in the actual history of x an earlier moment tt at which x could have gone on to be not E There are such moments pre-1976 for Nastassja and prior to my filling my pocket with these nine coins for the parking meter. For concrete subjects like her, the necessity of bearing a trait would amount to

11 It has been sometimes objected to me that this is a mere "Cambridge change" in the species, color, and so on. Not on the present view of kinds. We saw earlier that, for example, species identity and existence depend on (i) what animals the species was generated by, and on (ii) there beingsuch animals to generate it. In contrast, when I Cambridge-change because you had a hamburger for dinner, nothing relating to my very existence and identity depends on your feeding behavior.


there being no way herhistory could have gone otherwise. And in the same vein, this is our grounding of arithmetic necessities: although 9 might not have numbered the coins in my pocket, it could not have failed at any point in its history to be odd or greater than 7. Not only do we unify Nastassja and Nine in the same cosmic realm of existents; we also apply a unitary notion of de re necessity cosmic necessity to both Nastassja's being lauman and 9's being odd.

(iv) The necessary existence of infinitely many things. The idea of cosmological necessity assumes through and through that there is only one world the real world. For a trait Fto be necessary of the real world, we look for all the possible developments of the world past the moment of its actual origination. Now, looking at Nastassja, we do not consider among her alternative developments, a consistent story about some individual looking like Nastassja and having the same genetic endowment as she does but living, to begin with, in Descartes's tinae. In the same vein, we do not consider among the alternative developments of the history of this cosmos, consistent sto- ries about some universe that would look (by now) like ours. A story that makes the world have a radically different history to begin with (for example, a Big Bang 500 billion years ago, a substantially different rate of expansion, or, following Aristotelians, no beginning moment at all) is just that-a story ("theory") .

It is important to separate the conception of cosmic necessity from the more familiar "parallel worlds" conception of necessity. A sense of the difference in the case of numbers may be gleaned from the analogous and much rehearsed example of God.

On the "parallel worlds" conception, God's necessary existence is secured by his antecedent acosmic essence (definition). A plurality of possible worlds is given of which this historically real world is just one (one among many of a kind). It now turns out that in any arbitrary world zv, something bears the divine essence at w and that something exists at w; furthermore, what bears it in some world w, bears it in every other world (shades of the ontological argument).

In contrast, the cosmological necessity of God's existence does not proceed from essence to transworld existence. We rather seek to es- tablish why, among actual existents, His is special: whereas other real existents may be subtracted from the history of the real world, He stands in a special position in the actual cosmic history and may not be subtracted. His necessity lies in His insubtractibility from the history of this one and only world of ours.

So it is for God and so it is for the numbers. On the "parallel worlds" conception, our Parmenidean argument secures it that in


any arbitrary world w, if there is one thing, there are in w infinitely many things. But, alas, I do not know how to argue for "parallel worlds" (given disjointly and point-like) that, to begin with, each must have something in it.

On the cosmological conception, we start with the actual fact that the world itself is. This world is not a "point" or an index of evalua- tion or a set of propositions; it is simply this real evolving universe. So, something is. But is its existence necessary? There is a point in the history of the world in which it could have gone on to develop without Nastassja Kinski, the Nanga Parbat, or planet earth. But is there a point in the history of the world at which it could have gone to be such that it did not exist in the first place? With the necessity of some real thing thus secured, our Parmenidean argument carries us all the way to the necessity of infinitely many reals.

(v) The generation of kOne*. Some readers will want to grant that the sheer emergence of Godiva generates certain kinds (for example, kOrcak) but not the purported kind kOnek, simply because this last is not a genuine kind. There is no genuine "common nature" or "unifying essence" behind all ones.

The present account of real kinds (for example, of biology and chemistry) is this. A process in history produces items of the pertinent kind and it is thus and only thus that the species and the substance come into history. Such is the worldly picture of kOrcak and kDiamondk and such is the intended account of kOne*.

First, a concrete existent brings into the world this other, more abstract, kind of existent, the ordinal kOne1£. Eventually, all numbers are so generated, not "from above," by a third or second realm, defining predication but "from below," by in-history, preceding real existents. There is an intrinsic difference between (i) the first number, kOnek, which can only be generated by a non-number, and (ii) all remaining numbers which can be generated only by other numbers, each by its predecessor. We should note that in (i)-(ii), we have a nice metaphysical reflection of the familiar (Peano- Dedekind) axioms: the ur-number (for Dedekind and for us) 1 is intrinsically not a successor; all others are intrinsically successors.

In the present framework, predecessor numbers actually precede the next number in the historical order of existence. Thus, the familiar fact that 16 is "after" (succeeds) 12 is not a truth got "from above," as if after 16 and 12 have been separately defined by predication, it also turned out that a plurality with the cardinal number 16 has "more items" than one with 12 items. When it comes to our generated ordinals, we have here a "from below" fact about the histori-


cal order of generation: 12 is among the ancestors of 16. This is meant not as a "metaphor" or "logical analog" of real ancestry relations. Not at all. Without the existence of 12 in history, 16 would not emerge.

One final repartee and a forceful one at that the separate begzn- nings objection. Take to heart the foregoing claim that kOnek is as generable as ordinary natural kinds, and it is from the frying pan into the fire:

Consider the spatiotemporally separate Dobbin and Bertrand, the first horse-like (in looks and DNA) animals. By your ( JA's) own lights, the two generate two distinct biological kinds. But you say that number- kinds are like biological kinds. So, either you predict that (i) the world must have at most one primal concrete item or else (ii) if there were two (or more) simultaneous primal objects (for example, Electra (E) and Fidelia (F)), the two would breed two distinct kinds kOne*. This is an odd result.l2

The source of the separate beginnings objection lies in misinterpret- ing the analogy between arithmetic and biological kinds. I would like to explain in what way they are similar and in what way they are not.

Biological kinds (for example, kHorses*) make prominent two separate features:

(Generability) For a kind to exist, it has to have concrete generators. (Membership connectedness). To be a (late) member of kind K (for

example, a contemporary horse), x has to come from Ks (for example, from horses).

The point to notice is that (generability) and (membership connectedness) are logically independent. I definitely asserted above (generability) for any real kind whatsoever. But is (membership connectedness) also true of kinds in general or is it something pecu- liar (even if essential to) biological species?

One might be led to think that I submit (membership connectedness) for all kinds simply by confusing it with (generability). But

12 The objection that follows has been made to me early on by Martin, then by Normore, Shiffrin, and Dominik Sklenar at UCLA andJane in Barcelona. The discussion of kinds that follows owes much to conversations with Sklenar about his 1996 UCLA Ph.D. "Being of a Kind"; a published version of his interesting ideas is forthcoming.

I should like it noted that the objection has the form of a disjunction: either the world must have at most one primal item or else if there were more than one such, then.... I believe the first disjunct is true. The remarks in the previous annotation about the necessary existence of the world itself give a hint of the argument I would submit. But in what follows, I shall assume with my opponent that there could be more than one primal item.


even without such a confusion, one may reason as follows regarding the separate question of kind-membership.

The critical question to consider is this: What unifies two items x,y in a given kind K? Either the unification is "from below" or "from above." If it is "from below" due to a cosmological fact- the model must come from the unification operative in biological species, namely, via historical connectedness between members. If not, the only alternative unifier is an idea (Sinn, form) in another "realm" (for example, Locke's second, Frege's third, or Plato's "heaven"). This either/or I shall call the unification dilemma.

Driven by the dilemma, our objector may well reason thusly: the present approach would not allow unification by a second-or-third realm template. Therefore, the one intended must be unification as in the biological case by historical connectedness. By hypothesis, there is no connectedness between Electra and Fidelia. Thus, we are led to two number-kinds kOne*.

My answer is that the unification dilemma is a false dilemma. There are other "from below" ways (aside of historical connectedness) of unifying things in the cosmos. A pertinent example is pro- vided by chemical elements, to which I now turn.

What makes any old atom x a member of the kind kGermanium*? Let us assume that the emergence of this chemical element in the cosmos (so many (many) seconds after the Big Bang) is the result of two simultaneous but causally separate supernovas. Each super- nova produces from electrons and protons by a similar process an atom with the structure of atomic number 72. The one Germa- nium atom call g8, the other atom g88. In my view, both are of the kind kGermanium*. Yet surely, they are not historically connected.

What unifies gt and g88 in the kind is neither (i) historical connection nor (ii) satisfaction of an apriori third-(second-) realm template. What makes an atom a Germanium atom is the type of cosmic process by which it came to be: electrons/protons at the right temperatures (and so on) coming together to form a unity.

Let me draw from the foregoing a general conclusion. What matters for membership at the general level of all kinds is this:

(1bl) what makes x and y co-specific is that the processes by which they each came to be are fundamentally similar (in the pertinent respect: chemical, biological, and so on) and essential to x(y)'s very coming into being.


(M) is true of the historically bound tigers x,y; the historically dis- connected germanium atoms gt and g88; and lastly, the separate simultaneous primals Electra and Fidelia.

It is essential to each and every object's (including Electra and Fi- delia) process of coming into being that it would come to have unity, that it would become one thing. The coming together of the unity might involve the bonding of electrons and protons, DNA mole- cules, stars in a galaxy, and other more complicated modes of forma- tion, depending on the specific kind of item concerned. But over and above specificities due to varying kinds, each object's very coming into being has essentially this much in it: the coming into being of one thing. We might well say: no entity without numerical identity. Physically, Electra and Fidelia may or may not be of the same kind. But this much is given about the two as about any other emergent being: Electra and Fidelia are of a common arithmetic kind, the one and only kOne*, because in both we have the fundamentally similar process of a coming into being of a distinct new entity.

IV. EPILOGUE

I have argued that there can be no existence of a mere something, be it a single initial something or a plurality of such primals. Eacll such primal being (as any other object) cannot exist without being one thing, a number that itself could not come into being without having a generating instance, some one concrete thing. So: no number one without a concrete generator; but also, no such concrete generator without the number one. Very well then: coming into the world with each such primal entity is the number one; and with that two-some, the primal entity and one, comes into the world the number two; and so on. We can sum it all thusly: there is no real infinity without real being and no real being without real infinity.

JOSEPH ALMOG

University of California/Los Angeles

Documents

Almog, Joseph - Nothing, Something, Infinity