ON NUMBERS

by LINDA ELIZABETH WETZEL

B.A. City College of New York (1975)

Submitted to the Department of Linguistics and Philosophy in Partial Fulfillment of the Requirements of the Degree of

DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 1984 @ Linda E. Wetzel

The author hereby grants to M.I.T. permission to reproduce and to distribute copies of this thesis document in whole or in part,

Signature of Author: Department of Linguistics and Philosophy January 13, 1984

Certified by : .-_ - Richard L, Cartwright Thesis Supervisor

Accepted by: v Richard L. Cartwright Chairman, Departmental Graduate Committee ON NUMBERS

Linda E, Wetzel

Submitted to the Department of Linguistics and Philosophy of October 21, 1983 in partial fulfillment of the requirements for the degree of Doctor of Philosophy

ABSTRACT

We talk as though there are numbers. The view I defend, the "popularn view, has it that there -are numbers. However, since they clearly are not physical objects, we reason that they must be abstract ones. This suggests a realm of non-spatial non-temporal objects standing in numerical relations; arithmetic knowledge is then knowledge of this realm. But how do spatio~temporal creatures like ourselves come to have knowledge of this realm? The problem ("Benacerrafls probleni") can be avoided by arguing that there are no numbers. In "What Numbers Could Not Bew Benacerraf himself took such a route. In chapter one, I discuss three of Benacerrafls arguments, showing that the first is circular, that the second involves a consideration that can be explained by less drastic means than supposing there are no numbers, and that the third would, if successful, show that neither sets nor expressions exist either. Yet despite the lack of success of arguments purporting to show that there are no numbers, accounts of arithmetic which make no reference to numbers might be thought preferable to the popular view on the ground that they manage to avoid the epistemological problem. Accordingly, in chapter two I examine three quasi-formalist accounts that llreducen number talk to talk about other sorts of objects, to see whether this is so. I show that each of then1 involves commitments to other familiar mathematical objects, and hence makes no headway on the epistemological problem. In view of the failure of quasi-formalist positions, the next logical step for someone anxious to avoid the epistemological problem by denying there are numbers is to retreat t; full-blooded formalism, maintaining that there are no mathematical entities, only linguistic ones. The presupposition here seems to be that unlike mathematical entities, expressions are epistemologically "accessible" (and therefore not subject to the epistemological problem.) By way of justifying the presupposition, I believe that a formalist would adva-nce the fcllowlng grounds: (1) Expressions are concrete whereas mathematical entities, including numbers, are not,

(2) Even if expressions are abstract objects, we can know about them on the basis of their tokens, which -are concrete objects; but since numbers have no tokens, this sort of explanation is not available. In section one of chapter three I examine (I), siiowing that the claim that expressions are concrete is just not tenable, Then I examine (2), arguing that it, too, is false, because in whatever sense we can know about expressions on the basis of interacting with their tokens, we can know about numbers, too, on the basis of interacting with pairs, trios, quadruples, etc.

Thesis Supervisor: Richard L, Cartwright Title: Professor of Philosophy A plurality is not an instance of num- ber, but of some particular number. A trio of men, for example, is aninstanoe of the number 3.. .. This point may seem elementary and scarcely worth mention- ing; yet it has proved too subtle for the philosophers, with few exceptions. --Bertrand Russell ACKNOWLEDGEMENTS

I owe an enormous debt fo Richard Cartwright and to George Boolos. In the course of many discussions, Richard Cartwright patiently guided me through fhe jungle of ideas that beeame this thesis. His advice and detailed criticisms were invaluable. No less a debt do I owe George Boolos. Besides providing much needed guidance and encour- agement, he first sensitized me to,the difficulties with nominalism in philosophy of mathematics--albeit through his devastating but elegant criticisms of an early paper I wrote. My intellectual\ development was greatly aided by exposure to the rigorous ways and keen insight of these two people. To Sylvain Bromberger I am.also deeply beholden. He not only played a substantial role in my education, especially as regards philosophy of language, but by generously bringing his philosophical acumen to bear upon chapter 111, greatly improved.it. My heartfelt gratitude goes to my friend Lon Berk. Always supportive, he discussed with me each argument, and read every word I wrote. Not only would this thesis have suffered without his valuable criticisms, but it might not have been at all. It was his tips on how-to- write-a-thesis that eventually carried the day. For general discussions in aid of the present project, not to mention the accompanying good times, I thank Jerrold Katz and Raymond Smullyan. My debt to them extends beyond that, however, for it was Jerrold Katz who intro- duced me to philosophy of language, and Raymond Smullyan who ffrsh taught me.to relcfsh logic and paradoxes. For specific discussions in aid of the present project, a special thanks goes to Paul Benacerraf. Along with the members of his spring 1981 seminar in philosophy of mathe- matics at Princeton, he endured a presentation of many of the ideas contained in chapters I and 11. His insightful response to those ideas proved most useful. Thanks also to Harold Hodes, whose cheerful espousal of nominalism in part prompted chapter 111. Others whose conversations were helpful and encouragement timely include Ken Albert, John Bacon, (belatedly) Joseph Bevando, Ned Block, Jack Cobetto, Judi DeCew, James Higginbotham and Joseph Ullian. Finally, I would like to thank my parents, William and Elizabeth Wetzel, brothers Stephen and John, and good friends Deborah Bowen-Weil, Eleanor Druckman and Valerie Johnson, for the support, good humor and kind words they lavished on me throughout the duration. TABLE OF CONTENTS

ACKNOWLEDGEMENTS ...... 5

INTRODUCTION ...... 7

CHAPTER IIBENACERRAF ...... 18 1 . First Argument ...... 18 2 . Second Argument ...... 36 3 . Third Argument ...... 49

CHAPTER II/SOME RIVALS OF THE POPULAR VIEW .... 78 1 . Introduction ...... 78 2 . Harman's Account ...... 84 3 . White's Account ...... 92 4 . Benacerrafvs Account ...... 99 5 . A Final Word ...... 117

CHAPTER III/FORMALISM...... 126 1 . Introduction ...... 126 2 . First Formalist Claim ...... 127 3 . Second Formalist Claim ...... 162

SELECTED BIBLIOGRAPHY ...... 214 7

INTRODUCTION

My concern is whether there are numbers, especially natural numbers. I believe th~tpopular opinion, not only with the public (if introductory philosophy students are any guide), but with mathematicians and probably scientists as well, has it that there are numbers. Monk,' for example, has estimated that the mathematical world is populated with 65% platonists, 30% formalists, and 5% intuitionists.' There seems little doubt that we talk as though there are numbers. Why then am I concerned with whether there are numbers? Why not just conclude that there are? Because the popular view may run, into insuperable epistemological difficulties, difficulties 2 detailed by Benacerraf in "Mathematical Trutht'. Here is how those difficulties might be seen to arise. To the philosophically uninitiated it is natural to suppose not only that there are numbers, but also that we are quite familiar with at least some of them. Every- one can add and divide them (although not always too well); quantities having nuaber are all around us, The more mathematically sophisticated among us can prove

1. J. Donald Monk, Mathematical Logic (New York: Springer- Verlag, 1976) p. 3.

2, Journal of Philosophy 70 (1973): 661-679. esoteric truths about them. But when asked: what -is a number? the philosophical novice is likely to point to numerals. Ah, but numbers cannot be identical to numerals, it will be said, because although 5 + 7 = 12, for example, it is not true that '5 + 7' = '12'. So numerals must refer -to numbers. This routine initiation into philosophy leads to what I will call the popular view. According to it, the following sort of procedure is unproblematical. Suppose a bare uninterpreted number calculus is given, using the symbols lo', 'Sf, etc. where the usual concatenation rules are employed, so we can get the sequence lo1, 'SO1, lSSO1, ...

The semantics can then be given in the fo1,lowing. , terms:

'0' refers to 0, 'SO1 refers to 1, lSSO1 refers to 2, and in general '0' preceded by -n lSWsrefers to the number -n. If it is asked: what does '2' refer to, though, or ltwol, or 'seventeen1? the answer would be: why, the numbers two, two, and seventeen, respectively, of course. The basic idea behind this view is that the mathe- matical terms in question are already a part of English, which is an interpreted language. Hence these terms already refer. To what? To numbers. What's the problem? So far there is no problem, When queried as to the ontological status of numbers, an advocate of the view I am considering will say: numbers are abstract objects. Having traipsed this far into realist territory, when pressed for further details he or she may say: numbers are non-spatial and non-temporal; they are members of a platonic 9 realm of mathematical objects that stand in mathematical relations to each other. It is a realm of mathematical facts, as it were, whereby every mathematical statement is either truo or false. The mathematician's task is to go out there and, not invent, but discover the principles, or laws, we could even say, governing this realm. So numbers are to the arithmetician what animals are to the zoologist, after all. Now given this picture, whic.h is really a picture of what it is for a mathematical proposition to be true, Benacerraf argues that "It will be impossible to account for how anyone knows any properly number-theoretical proposition^.^' because llIf, for example, numbers are the kinds of entities they are normally taken to be, then the connection between the truth corlditions for the statements of number theory and any relevant events connected with the people who are supposed to have mathematical knowledge cannot be made outew3 In short, since we are spatio-temporal objects, and numbers are not, we can never causally interact with them or an-{ part of the abstract realm of mathematical entities whereby arithmetical propositions are true -- but interaction is necessary for us to be able to know such propositions non-innately. Where and how does the mathematician do his/her field work? (G6del attempted to

3. -Ibid, p. 673. answer this question, but his answer is generally thought to be unsatisfactory.).

Thus it appears that in the absence of a plausible account of how mathematical knowledge is obtained on this view, we must reject the popular view, on the ground that it leads to insurmountable epistemological difficulties. The main theses of what I am calling "the pop~larview1' are that (I) There are numbers. and (11) Numbers are abstract objects. These entail, respectively, (111) There are mathematical entities. ', and (IV) There are abstract objects. Of course the view presupposes that mathematical statements are, in the main, either true or false: in particular^, that (V) Arithmetic statements are true or false. So in rejecting the popular view, one must deny sorne or all of the above theses. But before we consider any rejections, let us first take note of the fact, as I see it, that were it not for the epistemological difficulties it seems to involve, the popular view would win hands down; it is popular with good reason. The good reason is that it best accords with usage. That is, ---we talk as though there -are numbers, so of course the thesis that there are numbers accords very well with this fact. While it is hard to make precise the sense in which the italicized claim is true, the following points may help to amplify the claim, In "Ontological Relativity1' Quine argues that, as ''there is no fact of the matterf1, inscrutability of reference can be applied not only to remote bushmen, but to our neighbors, and even to ourselves. But then

We seem to be maneuvering ourselves into the absurd position that there is no difference on any terms, interlinguistic or intralinguistic, objective or subjective, between referring to rabbits and referring to rabbit parts or stages; or between referring to formulas and referring to their GBdel numbers. Surely this is absurd... 5 I

He defends himself against this charge by saying:

Toward resolving this quandary, begin by picturing us at home in our language, with all its predicates and auxiliary devices. This vocabulary includes 'rabbit1, 'rabbit part1, 'rabbit stage1, 'formula1, 'number1, ,..; also the two-place predicates of identity and difference, and other logical particles. In these terms we can say in so many words that this is a formula and that a number, this a rabbit and that a rabbit part. ... This netwollk of terms and predicates and auxiliary devices is, in relativity jargon, our frame of reference, or coordinate system. Relative to it we can ---and do talk meaningfully -and disflncticlx?f - rabbits -and parts, numbers -and formulas .6

4. Ontological Relativity And Other Essays (New York: Columbia University Press, 1969) p. 47. 5. -Idem. 6. bid ., p. 48, my italics. Not only do we begin at home in the coordinate system of our own language, for Quine, -- where 'numbert refers to numbers just as surely as 'rabbit1 refers to rabbitst7 --- but we wind 9 there after having tried to scrutinize reference for a while:

It is meaningless to ask whether, in general, our terms 'rabbit', 'rabbit part1, 'number1, etc., really refer respectively to rabbits, rabbit parts, numbers, etc., rather than to some ingeniously premuted denotations. It is meaningless to ask this absolutely; we can meaningfully ask it only relative to some back- ground language. When we ask, "Does 'rabbitT really refer to rabbits?" someone can counter with the question: "Refer to rabbits in what sense of 'rabbits'?" thus launching a regress; and we need the background language to regress into. ... Are we involved now in an infinite regress? ... [I]n practice we end the regress of background languages, in discussions of reference, & acquiescing our mo her tongue and taking its words face value.4- 7 at

I think it would be fair to say that Quine thinks that our mother tongue or the theory embedded therein commits us to the existence of numbers (at least so long as we are speaking it). And we know what ontological commitment amounts to for Quine. So, apparently he thinks that (VI) We quantify over numbers,

No doubt we do. A nice example is the following, extracted

7, This emerged during a lecture he gave- at Princeton in February 1981: Cf. ~heoriesand Things (Cambridge, Mass : Belknap Press of Harvard University Press, 1981) p. 20.

8. Ontol gical Relativiby And Other Essays, p. 49, my italics. from the programfbipleyfs Believe It Or NO^:

For some number 2, there is a famous person who was n years old the year the King Jamesf version of tEe Bible was writteri, and, some believe, helped write that Bible; because if you concate- nate the nth word of the nth Psalm of the King Jamesf version of the ~ibiewith the nth-from- the-last word (of the last sen ence) of that same Psalm, you have his name. b

Besides quantifying over them, there seem to be other respects in which we talk as though there are numbers. In statements such as (S1) Two is a prime number. and

'twof and '2' appear to be uniquely referring singular terms, not predicates, functors, or relational expressions. On the popular view, they -are names -- names of abstract objects. Thus several fu2ther theses might be stated, as evidence for the view:

(VII) Numerals like '2' and number words like 'twof in

such statements as (S2) and (S1) respectively are singul erms .. (VIII) The surface structures on (S1) and (S2) are identical to their logical structures,'

(IX) The appropriate semantics for (Sl) and (S2) are referential semantics.

9. ;is 46. The writer in question is Shakespeare. Since the popular view involves theses (I) - (IX), in rejecting it, one must deny one or more of these theses. And indeed, denials have abounded. In the last century, Mill denied (II), arguing that numbers are properties of objects. lo While neatly sidestepping the epistemological problems of the popular view, it is less successful in other respects, respects that Frege was quick to point out. 11 Though Frege is usually taken to have ltburiedl' Mill, recent efforts have done much to resurrect his outlook. Charles Lambr4os, for example, has argued that Fregels attack on Mill's "predicative view of number words,'1 -- which proceeded partly via (VII) -- is far from conclusive. la And Glenn Kessler has presented a theory wherein numbers are relations I. between aggregates and properties. 13

Of course Frege himself, and Russell and Quine after him hold versions of the popular view. They subscribe to (I) and (11) through, the thesis that numbells are sets, or classes. In "What Numbers Could Not Be'' Paul Benacerraf

10. John Stuart Mill, A System of Logic (New York: Longman, Green & Co., 1936) Book 11, Chapter VI, Sec. 2. 11. Gottlob Frege, The Foundations of Arithmetic, reprinted and translated by J.L. Austin (Evanston, 11: Northwestern University press, 1980) sections 7, 8, 9 and 25. 12. "Are Nmbers Properties of Objects?, Philosophical Studies 29 (1976):381-389. 13. "Frege, Mill, and the Foundations of Mathematics," Journal of Philosophy 77 (1980):65-79. mounts a vigorous attack on this latter thesis. l4 In the process he does not merely deny (I) and (II), but argues for the very strong result that numbers could --not be objects --at all. For this reason, chapter one of my dissertation (which begins shortly below) is an examination of how successful his attack is. Interestingly, Benacerraf seems to think that although numbers do not exist,15 other sorts of mathematical entities, like sets, do. He appears to subscribe to (111), (IV) and (V). This leaves him open to his own criticism, mentioned earlier, that such a position is epistemologically untenable. 16 Hence some version of formalism, Curry's or Hilbertls

14. Philosophical Review 74 (1965) : 47-73. When this article is referred to in chapter one, below, page numbers will be included parenthetically in the text; in later chapters lWNCNBt will be appended. 15. So far as I am aware, Benacerraf does not claim to have shown "numbers do not existv per se. Usually his claim is that l1numbers are not objects at alln. Although the two might be thought equivalent, they are nct. Because the former does not entail the latter unless the following is true: (S3) If numbers exist, they are objects. And there is nothing immediate about (S ); numbers might exist but as properties or relations, as the ii illians claim. How- ever I shall use "numbers do not existu and llnumbers are not objects at all1' interchangeably when discussing Benacerrafls article, because he gives no evidence that he thinks numbers might be properties or relations, In fact, he spends a good deal of time criticizing the view that number words are predicates (pp, 5c-61), and concludes the article by saying ''...if the truth be known, there are no such things as numbers..." (p. 73). 16. Haskell Curry, Outlines of a Formalist Philosophy of Mathematics ,(Amsterdam: North-Holland, 1950). For Hilbertls view see, e.g., "On the Infinite," in Benacerraf and Putnamls Philosophy of Mathematics (Englewood Cliffs, N.J.: Prentice Hall, 1964) 134-151. for example, might be thought preferable, involving as they do denials of (111). Yet neithzr of the formalists cited reject (IV); Curry is committed to formal systems, and Hilbert, to simple signs (at least). Even this commitment is too un-nominalistic for Hartry Field, who wants to deny every one of (I) - (v).17 However, unlike most of the other writers whose positions involve not only rejections of one or more of (I) - (V) but also one or more of (VI) - (1x)18, it is not clear that Field would dispute the claim that we talk as though there are numbers -- i.e., deny any of (VI) - (IX). Nor is he suggesting that we revamp our speech habits to accord with nominalism. His point appears to be that 'lone can always reaxiomatize scientific'< theories so that there is no reference to or quantification over mathematical entities in the reaxiornatization... 1119 But whether he successfully does away with all abstract objects and the consequent epistemological problems, as he seems to want to, is another matter. As can be seen, there is quite a range of views that reject the popular view, from Mill's rejection of (11)

17. Science Without Numbers (Princeton, N.J.: Princeton University Press, 1980). 18. Usually not (IX). Even the intuitionists are probably better understood as denying (VIII),the thesis that the logical structure of an arithmetical statement is identical to'its surface structure, rather than denying referential semantics. to Field's rejection of (I) through (V). In later chapters,

I deal with a number of these possibilities, mainly with an eye towards seeing whether they solve the epistemological problem that appears to be a consequence of the popular view. But first, I shall concern myself at some length with the arguments given by Benacerraf in !?What Numbers Could Not Be." That is the subject of chapter one, which comprises the remainder of what follows. CHAPTER I : BENACERRAF

1. Benacerraf's First Argument

Judging from the flurry of articles it generated, What Numbers Could Not Be" is one of the most important pieces written in recent years on the question of whether there are numbers. So a critical examination of the arguments contained therein seems called for. Benacerraf addresses the question of whether there are numbers head on. No, he says, there are no numbers. His -main argument has two conclusions. The first is that numbers are not sets, and the second is that numbers are L_- - not- objects --at all. An abbreviati~nof the argument that numbers are not sets is the folldwing: (1) Peano Arithmetic (P.A.) has infinitely many models in set theory. For example, all the theorems of P.A. come out true if we let 0 = 0, and n + 1 = [n], or alternatively, if we let n + 1 = n U [n]. Moreover, infinitely many of these accounts satisfy whatever constraints (like recur- siveness) we find necessary to impose with regard to "numberhoodl' (based, perhaps, on considerations involving counting and measurement). This is illustrated by the cases of Ernie and Johnny, one of whom is taught that the Zermelo numbers are -the numbers, and the other of whom is taught that the Von Neumann numbers are -the numbers. (2) Therefore, either (a) None of the accounts are correct; or, (b) Exactly one of them is correct; or (c) More than one, perhaps all, are correct. (3) (c) is defeated by the following argument. If, say, Ernie's and Johnny's accounts are -both correct, then it

follows that [O, [@I, [@, [@]]I = 3 = [[[@111. But this is clearly false. 20 (4) Benacerrafls argument against (b) is that if exactly one account is correct, then we should be able to give rea- sons for preferring one account over all the others. The

position that it is an unknowable truth that, e.g., 3 = *--[[[@]]I (rather than [,,] is untenable. Because We do not know what a proof of that could look like. The notion of "correct account" is breakin-Toosefromits moorin s ii we a-dmit dome~stence-+o un~ustifiable but correct answers to questions such as this. To take seriously the question "is 3 = [[[O]]]?" -tout court (and not elliptically

20. I think this argument is perfectly straightforward, that one need not appeal to such statements asl'If numbers are sets, then they are particular sets'' the way Kitcher does in "The Plight of the Platonist" Nous 12 (1978): 119- 136, p. 120, when discussing ~enacerra-article. Kitcher seems to think that the assumption that if numbers are sets then they are particular sets, is required, and that Quine thinks it false. If so, then it is up to Quine to show how numbers are sets, but not particular sets. And I don't think that Quine would be willing to want to show this. He probably doesn't think the claim makes sense, without further qualifications. He usually says things like: any progression (of a certain sort) will do as the numbers. That is very different from claiming~t~uchprogressions -are the numbers. for Ifin Ernie's account?"),. - in the absence of any way of settling it, is to lose onels bearings completely. (p. 58, my italics.)

However we cannot give reasons for preferring one account over all the others. Although, for example, Fregels identification of the number 3 with the class of all classes equivalent with a given (3-membered) class is I1...an appealing notion, there seems little to recommend it over, say, Ernie'st1 (p* 58). So that

Relative to our purposes in giving an account of these matters, one will do as well as another, stylistic preferences aside, There is no way connected with the reference of number words that will allow us to choose among them, -for -the accounts differ g& places where there is -no connection whatever between features of the accounts and our uses of words question*m)- -

(5) Therefore we are left with (a), that none of the set- theoretic accounts is the correct one, and so numbers are not sets at all, I am inclined to agree with Benacerrafls argument so far, i.e., up to the conclusion that numbers are not sets, in spite of the vagueness associated with the metaphors "lose one's bearings completely" and "break loose from its moorings". It does seem that it would be completely arbitrary and hence unjustified to identify the numbers with one set progression rather than another, And this is a telling objection to the view that numbers are sets. It might be worthwhile at this point to discuss what the claim that numbers are not sets, or are not to be identified with sets, amounts to, so as to preve~tany misunderstanding. It is not meant that numbers cannot be identified with sets -for certain purposes. But, rather, that they ought not to be identified with sets once and for all, as it were -- i.e., in a more fundamental, ontological sense. Benacerraf discusses the difference as one between explication and reduction:

In putting forth an explication of number, a philosopher may have as part of his explication the statement that 3 = [[[fl]]]. Does it follow that he is making the kind of mistake of which I accused Frege? I think not. For there is a difference between asserting that 3 is the set of all triplets and identifyin 3 ath that set, which last is what, might b: done in the context of some explication. I certainly do not wish what I am arguing in this paper to militate against identifying 3 with anything you like. The difference lies in that, normally, one who identifies 3 with some particular set does so for the purpose of presenting some theory and does not claim that he has discovered which object 3 really is, We might want to know whether some set (and relations and so forth) would do as number surrogates. In investigating this it would be entirely legitimate to state that making such an identification, we can do with that set (and those relations) what we now do with the numbers...Under our analysis, any system of objects, sets or not, that forms a recursive progression must be adequate. It is therefore obvious that to discover that a system will do cannot be to discover which objects the numbers are. (pp. 67-68)

Like Benacerraf, I am interested in the answers to the questions !'Are there numbers?" and "If so, what are they?" as these questions are understood in the reductive sense described above. Unlike him, however, I shall speak of whether we are to identify the numbers with this or that progression (when the issue is that of what the numbers really are, as opposed to what will do as an explication.21) Another point that should be made explicitly before we continue is this. Throughout I assume that, whatever numbers are, they are all the same sort of thing -- that if some are sets, say, then all are. And so I move comfortably back and forth between such statements as: Numbers are not sets. and : The natural number progression is not a set-theoretic progression. This assumption seems innocuous enough to me. It might seem to need defending, though, in view of the fact that Benacerraf appears not to make it sometimes. For example, in (4) on p. 12 above, Benacerraf is quoted as saying:

To take seriously the question "Is 3 = [[[g]]]?" -tout court (and not elliptically for ''in Ernie's accou'nt?"r, in the absence of any way of settling it, is to lose onels bearings completely. (p. 58)

But this is misleading. The assumption in question seems to be operative in his concluding remarks about 3:

21. Although this is at odds with Benacerrafls use of the term Ifidentify" ("you can identify 3 with anything you likeu) I think my usage is justified because we want to discover which progression out of all the possible progressions that "will dou as the numbers is identical to the numbe~s. Then we can assert that the natural numbers are, or are identical to, that progression -- and not relative to scme purpose or other. ...any feature of an account that identifies 3 with a set is a superfluous one -- and that therefore 3, and its fellow numbers, could not be sets at all. (p., my italics)

I believe that he just wants to emphasize the lack of rationale for identifying numbers as, say, sets, and that this is more readily apparent when individual numbers and sets are involved than whole progressions. The same remarks apply to the second conclusion of what I have been calling l1the main argument1' -- that numbers are not objects at all. Benacerraf claims that this is arrived at by an "extension" of the argument which I have summarized by (1) to (5) above. His reason for thinking so is that "...there is no more reason to identify any individual number with any one particular object than with any other (not already known to be a number)''. (p. 69) Unfortunately, he does not explicitly give the extension. Using (1) to (5) as a model, perhaps it would look like this: (1') P.A. has many models even outside of set theory. (2I) Therefore either none of the models is the right one; exactly one of them is; or, more than one is. (3') Not more than one can be correct because they are not extensionally equivalent. (In one, for example, the number 3 might be Julius Caesar, while in another, 3 might be Marc Antony. Clearly Ca~saris not Ant ony . ) (4') If one account is the correct one, then we should be able to specify why this is so. But we cannot. One will do as well as another. (5') Therefore, numbers are not objects at all.

If the preceding argument worked, then the conclusion that Benacerraf urges would (of course) be forthcoming. But it seems full of gaps, at least as it stands. For starters, we can ask about (1'). Everyone knows that there are models of P.A. in set theory. But (I1)claims there are models of P.A. outside of set theory. Where? we may ask. It might be suggested that the natural numbers themselves provide many examples. For example, the odd numbers, with successor, addition, and multiplication properly defin,ed, constitute a model for P.A. So do the numbers without 0, the numbers without 0 and 1, the numbers without 0, 1, and 2, and etc. In short, any recursive progression composed of natural numbers constitutes a model for P.A. There are infinitely many models of P.A. to be ootained from the natural numbers themselves. So it might be thought that a successful extension of (1) to (5) -- unlike the unsuccessful (1') to (5') -- might look like this: (1") P.A. has infinitely many models within the natural numbers themselves. For example, there is a model whose domain is just the odd numbers, and another whose domain is just the evens. (2") Theiefore either more than one model is the correct one, one of them is, or none are.

(3'') Not more than one can be correct, because they are not

extensionally equivalent. In the odd model, 2 Is 5,

so to speak, while in the even model 2 is 4. But as we all know, 4 # 5. (4") There is no reason for identifying one model as the correct one over all the others; one will do as well as another. (5") Therefore numbers are not objects at all. Clearly the above argument is fallacious. (4") is just false, Because if it is natural numbers we are talking about when we say loddsl and 'evensf then of course one I model to be preferred over all the others -- viz,, the standard model. It is easily enough specified, given the terms at hand; it is simply the one whose domain consists of odds and evens together (all the natural numbers) with the successor function, addition, and multiplication as they are usually understood. Since (411) is false, we do not get (5") at all. Consider anyhow if we had obtained (5n). (5") implies that there are no natural numbers. If that were so, then there would be no models of number theory consisting of natural numbers, and so (1")would be false. Thus we cannot use the fact that there are multiple models of P.A. within the natural numbers themselves to discredit the hypothesis that numbers exist. Perhaps it should be mentioned that there is a variant of the last argument that eludes the objection to (4") mentioned above. Let (2"), (311), and (4") remain as they are, but replace (1") by:

(1) There are infinitely many proper subsets of the natural numbers that provide models of P.A. Now (4") will be true, because there will be no reason for identifying the natural numbers with any proper subset of them. But we cannot conclude (5") from this, i.e., that numbers are not objects at all. For just as (1) to (4) only show (5), that numbers are not sets, so this variant of (1") to (411) only shows (5111) The natural number series is not to be identified with a proper subset of itself. So it appears that for an extension of (1) to (5) to work (in such fashion as will not rule out the conclusion) we must find models of P.A. whose domains contain neither numbers nor sets. In view of the fact that numbers and sets are abstract objects, one might be tempted to look to the realm of physical objects for a model. Benacerraf discusses the status of the statement vJulius Caesar is the number 43.' (although he later comes to the position that the statement is non-semantical, or, at best, false), But there is no wholly satisfact,ory way to ensure ourselves of an infinite domain of (discrete) physical objects. Of course, it depends upon just what constitutes a physical obdect, but suppose that there are no physical objects smaller than a quark (or a gluon, or whatever sort of particle you like). Clearly at any given time there are only a finite number of such things. However, if we don't mind having our numbers spread out in time, then (since the universe had a beginning) if we assume that the universe has no end, perhaps we might obtain an infinite number of quark-sized things. But assuming the universe has no end is a strong, and probably false, physical assumption. Perhaps a physical model of a progression consists in the edge of my desk, divided into segments like so: the first segment is half the edge; the second segment is the quarter after that; the third is the eighth after that;and so on. But this seems to presuppose that the edge of my desk is infinitely divisible -- which is another strong empirical assumption. I do not wish to pursue further the question of whether there -is a physical object model of a progression, at least not here, Because it seems likely to me that if there is one, there is bound to be more than one. If the edge of my desk is infinitely divisible (thus providing us withaphysical model of a progression) so is the edge of your desk, not to mention a light ray from here to Alpha Centauri, and so forth, And if there is more than one, then in the absence of any good argument showing of one of them that it comprises the natural numbers, we can construct the following Benacerrafean argument, and agree with the conclusion :

(14 ) There are many physical models of number theory. 4 (2 ) Either more than one is correct, exactly one is, or none are. 4 (3 ) Not more than one can be correct, since they are not extensionally equivalent. 4 (4 ) There is no reason to identify one model over all the others as the natural numbers. 4 (5 ) Therefore, numbers are not physical objects at all. In casting about for models of arithmetic that are not composed of numbers, sets, or physical objects, we come upon expressions. Benacerraf himself ,favors expressions:.

...the sequence of number words is just that -- a sequence of words or expressions with certain properties. There are not two kinds of things, numbers and number words, but just one, the words themselves. Most languages contain such a sequence, and any such sequence (of words or terms) will serve the purposes for which we have ours, provided it is recursive in the relevant respect. (p. 71)

Appropriate progressions of expressions constitute models for P.A. Examples include the following progressions : lone, 'two, ' 'three,! ...

lone,' 'three,' ... 'two, 'four, l 'six, ... However one may individuate expressions, surely 'two1 is different from 'zweil, and from 'threet, and hence the procgressions mentioned above are just a few of infinitely many different ones. So we know that

(15 ) There are many different models of P.A. consisting of expressions.

(25 ) So either more than one is correct, exactly one is, or none are. 5 (3 ) . Not more than one can be correct, since they are not extensionally equivalent.

(45 ) If exactly one progression is natural numbers, then we should be able to give reasons why this is so, But we cannot.

(55 ) Therefore, numbers are not expressions at all, Benacerraf himself urges that ws dispense with numbers in favor of number words. So he might object to (45) on the ground that we -can identify the natural number progression with some one particular progression of expressions. (Another wzy of reading Benacerraf does not have him identifying numbers with expressions. This will be investigated later. For the next few pages, I shall only be considering the proposal that the numbers are to be identified with expressions. ) The following remarks suggest that he thinks there is one particular progression that may be singled out:

Although any sequence of expressions with the proper structure would do the job for which we employ our present number words, there is still some reason for having one, relatively uniform notation: ordinary communication. (pp. 71f, my italics.).

Which sequence is the preferred one? There appears to be some legitimacy to preferring the progression 'one,' 'two,' 'three,' ... over: 'two,' 'four,' 'six,' ... or: 'one,' 'three,' 'five,' ... But it seems patently unfair to the Germans to prefer it over: 'eins,' 'zweiY1 'drei,' ... Suppose instead that we identify the numbers with the numerals: '1,' '2,' '3,' ... thereby cutting out the previous cultural bias, as both groqs recognize this progression. ** Note that we cannot identify the progression: 'one,' 'two,' 'three,' ... with the progression: '1,' l2,' '3,' ... on the grounds that they are indistinguishable when read aloud by us, because the latter is indistinguishable from the progression: 'einsl, 'zweil, 'drew', ... when read aloud by Germans, and, as we have already noted, 'one', 'twot, 'three', ,.. is a different progression from 'einsl, 'zwei',. 'dreil, ... But then it appears that we have just replaced one cultural bias by another, this time against those groups who do not employ arabic numerals, but instead use binary notation, or Roman numerals, to name a few of the

-- -p 22. But is it a progression of expressions? Only if numerals are expressions. For those who reckon not, skip to the next paragraph. possibilities. The following passages of Benacerrafls, however, suggest a different view, one which does not insist upon the identification of one sequence ,f expressions as -the numbers: Most languages contain such a sequence. (p, 71)

The usual objection to [the account that there are no numbers, only number words] -- that there is a distinction between numbers and number words which it fails to make will, I think, not do. It is made on the grounds that 'two1, 'zwei', 'deuxT, '2', are all supposed to "stand for" the same number but yet are different words (one of them not a word at all). One can mark the differences among the expressions in question, and the similarities as well, without conjuring up some extralinguistic objects for them to name. One need only point to the similarity of function: within any numbering system, what will be important will be what' place in the system any particular expression is used to mark. (p. 72).

Perhaps, then, the implication to be drawn from this is not that there is one preferred sequence of expressions, but -one per language. And that we ought to acquiesce in the fact that most languages have their own recursive sequence of expressions. Of course this suggestion immediately runs into trouble in the case of languages with more than one such sequence and no reason to single out one as l'preferredll. But besides that it seems to me that the above suggestion violates the following principle: If x = y and x = z, then y = z. That is, if we identify the sequence of natural numbers as the sequence: 'onet, 'two1, 'three1, ... and we identify it with the sequence: leinst, 'zweil, dre. then we would have to identify the former sequence with the latter. But this cannot be done, as we have already agreed that they are different sequences OF expressions. It might be objected that, as speakers of English, we cannot identify the natural numbers with the sequence 'einsl, 'zwei', 'dreil, ... because they are not terms of English; they are unavailable to us for use. Only the Germans can so identify the numbers. But why? What is to stop me from saying: . , (S) The sequence one, two, three, ... is identical to the sequence 'einst, 'zweil, Idreit, ,.. Nothing, might be the response, except that (S) is false, because the English number words refer to English words, not German words. 'Onet, for example, refers to a certain three-lettered word of English spelled '0' - In1 - 'el, and this word is not identical with leins.' Of course, according to the suggestion we are entertaining, the following is a true German sentence: (Sf) Die Folge eins, zwei, drei, ... ist mit der Folge 'einsl, 'zwei', 'dreil, ... identisch. This maneuver has the unpalatable consequence of rendering (St)not a translation of (S). (However queasy one feels about meaning, it should be noted that even Quine insists that a proper translation preserve truth value.). The Germans would mean one thing by "numbert1 and we would mean something different. There would be German arithmetic, English arithmetic, -- in fact, as many different arithmetics as there are natural languages possessing a recursive number word progression. But for mathematics to retain its status as a universal science, and not become so many different provincial sciences, we would have to formulate what it is that German arithmetic and English arithmetic and so on have in common. This formulation, I suspect,would bear a striking resemblance to arithmetic as we now know it. But then the question of the ontology of this latter theory would arise, and we would be back where we started, on page one above.

It appears then that (45 ) stands unrefuted, although not unchallenged. And if (45 ) stands, so does (5~~1,the conclusion that numbers could not be expressions at all. At this point, let us take stock of what we were trying to do, and what we have done, We were trying to see how to extend (1) to (5), (which showed that numbers are not sets) so as to conclude that numbers are not objects at all; i,e., that no progression of objects is to be identified as the natural numbers. From (1"') to (4'") we concluded that the natural number progression is not a progression composed of less than all of the numbers. From 4 4 (1 ) to (4 ) we concluded that the natural number progression is not a progression of physical objects. And from (15 ) to (45 ) we concluded that, Benacerraf (possibly) to the contrary, the natural numbers are not expressions. What this suggests is the following. The only way of llextendingll (1) to (5) so as to show that numbers are not objects at all would be to produce a collection of arguments. First we would have to make an exhaustive list of all the kinds of things there are -- a grand partition of the whole universe of entities. 23 For example, one such list might be : 'sets, expressions, physical objects1. Then for each sort of object 0 on our list, we must give an argument that shows that numbers are not O1s, i,e., are not to be identified as ols. Taken all together, these arguments show that numbers are not objects at all. Of course, the individual arguments might vary, one to the next. They might parallel (1) to (5) the way the argument consisting of (1") to (5") does. That is, an argument showing that numbers are not 0 l s might be of the form: i (1 ) There are (at least) several 0-progressions that are models for P.A., and could serve as the basis

23. Not every sort of partition will do. Someone who thought that there were only numbers and physical objects could exhaustively specify his or her universe with the list: 'even numbers, odd numbers, ph sical ohjectsl, and then, using arguments of the form (1I ) to (5i) perhaps show that numbers don't exist. This obviously won't do. There- fore, the list must be subject to certain sorts of restrictions, but I will not try to specify them, as it would be very difficult, and I do not think anything important hinges on it for present purposes. for an account of natural number. i (2 ) Either more than one is co??ect, exactly one is, or none are, i (3 ) It cannot be the case that more than one is correct. i (4 ) There is no reason to identify one o-progression as -the natural numbers over all the others, i (5 ) Therefore, numbers are not O1s at all.

But that is not the only type of argument that would show that numbers are not 0's. If, for example, one could show that there are no mgdels of P.A. composed only of 0 ls -- say, because there are only a finite number of O1s -- then that too would suffice to show that numbers are not 0's. Taken all together, then, such a collection of arguments would, if sound, justify the conclusion that numbers are not objects at all, that they do not exist -- since there is no kind of object 0 that they are. However, the success of the overall argument is contingent upon the initial list. For example, if that list is: 'sets, physical objects, and expressions1, then (1) to (5), (14 ) to (54 ), and (15) to (55 ) together show that numbers do not exist. But if the initial list includes 'numbers1, then the above sort of argument will not work. Because there --is no argument that can show that numbers are not numbers. The absurdity of --- -7 i i any such argument of the form (1 ) to (5 ) was indicated above, in connection with (1") to (5"). So to show that numbers are not objects, when we list the sorts of things that we think there are, we could not put 'numbers' on that list. The upshot, then, is this. If we assume there are no numbers, we can prove as much, by employing a coLlection of arguments in the manner dzscribed above, and in the process make it seem as thouga we have genuinely proved there are no numbers. And I believe this is what Benacerraf has, in effect, done. So his main argument that there are no numbers fails, since it begs the question. But he does buttress the main argument with subsidiary arguments and considerations, which appear to discredit the thesis. Sections 2 and 3 below will be concerned with these.

2. Benacerraf's Second Argumemt

. I It is natural to suppose that if numbers are objects, then in our non-philosophical capacity we can readily render a verdict on the truth or falsity of t43 = Julius Caesar' (or, if you like, 'Forty three is identical to Julius Caesart>and '3 = [[[@]I]'. Yet we are reluctant to. This is puzzling because it is not as though we lack historical information about Julius Caesar, or mathematical information about 43, so that when the facts are available we will unhesitatingly render a verdict of 'true' or 'false'. All the relevant facts are before us. Only someone ignorant of the facts would seriously ask, in a non-philosophical context, whether 43 is Julius Caesar. Also, as Benacerraf notes, "we do not know what a proof of [3 = [[[m]]]could look like," (p.58) nor of 43 = Julius Caesar. "...[N]ormally, one who identifies 3 with some particular set does so for the purpose of presenting some theory and does not claim that he has discovered which object 3 really is" (p. 68). Indeed, it would be very odd for someone to claim that 43 # Julius Caesar. How can we account for these facts? And do they disconfirm the thesis that numbers are objects? First, we should note that '43 = Julius Caesar1 is either true, false, or neither. And secondly, that it

cannot be true, because, as we concluded earlier, ((54 )), numbers are not physical objects. Benacerrafls own account has it that '43 = Julius Caesar1 is unsemantical, and therefore (presumably) neither true nor false. One might think that he would give as his grounds for this that numbers are not objects at all, and hence number words (and expressions) do not refer, and so statements containing them are neither true nor false. But no, not at all, Because he does want to hold that many identity statements containing number expressions -are semantical. So the question is: how does he distinguish the semantical from the unsemantical? (We might also ask 'IWhy?" but I won't, because as the reader will see, we will not get to that, since the classification itself tends to undermine the claim that some identity statements \ containing number expressions are unsemantical.). The classification is this.

There are three kinds of identity statements [of the form tn = sfrwhere n is a number expression] corresponding to the three kinds of expressions that can appear on the right: (a) with some arithmetic expression on the right as well as on the left (for example, 'I2 = 4,892," and so forth; (b) with an expression designating a number, but not in a standard arithmetical way, as "the number of apples in the pot,ll or 'Ithe number of F1sll(for example, 7 = the number of the dwarfs); (c) with a referring expression on the right which is of neither of the above sorts, such as "Julius Caesar," ll[[B]]fl (for example, 17 = [[C%I11). (pa 63).

Benacerraf proposes discarding all identities of type (c), as they are "senseless and unsemanticall'. (p. 64)

'# Following Richard L. Cartwright, we may simplify Benacerrafls classification as follows:

Identities of the form ln = sf,where n is an arithmetical expression, may be divided into one of three types : (a) those wheresis an arithmetjcal expression; (b) those where s is a non-arithmetical expression, but designates a number;.and (c) those where s is neither an arithmetical expression, nor an expression that designates a number. more those where -s --does not designate -a number. So Benacerrafls claim boils down to this: identities containing only expressions which designate numbers are semantical, but those where the expression on one side of the identity sign design-tes a num?er, and that on the other side does not, are unsemantical. Now, Cartwright would argue, if -n designates -a number, and g does not, then surely there is no question but that the things designated (if s designates something) are not identical. And so the identity statement to the effect that they are must be false. '43 = Julius Caesart for example, is an instance of type (c); '43' designates a number and 'Julius Caesart does not. Therefore they are not identical -- 43 and Julius Caesar, that is -- and hence the statement that they are is simply false.

Although his classification does notL . support his claim that '43 = Julius Caesar1 and '3 = CC[01llt are meaningless, instead of false, we can see more clearly what Benacerraf is intent upon from the following remark: (6) If an expression of the form 'x = y1 is to have a sense, it can be only in contexts where it is clear that both x and y are of some kind or category C, and that it is the conditions which individuate things ----as the same C which are operative and determine its truth value. (pp. 64-5) As stated, it is unclear whether the insistence here is that the terms flanking the identity sign be from the same category of expressions (both number-words, or set-referring expressions, or names of physical objects, to name a few such categories) or whether the things referred -to by the terms flanking the identity sign must be of the same category (numbers, sets, physical objects, say), or something else. It matters, because the second inter- ?retation seems to presuppose that the terms do indeed refer, in order for such identities to be meaningful. But we know from Russell that they need not; 'Scott = the present king of France1, for example, is sensical even though lScottl and 'the present king of France1 do not refer to things in the same category, because one of them does not refer at all. On the other hand, the first interpretation would suggest, mistakenly, that we are discussing individuating conditions for, say, number words, rather than numbers. Therefore, I think that he had the following sort of claim in mind, and I shall henceforth construe (6) this way:

If an expression of the form lx = y1 is to have a sense, it can be only in contexts where it is clear that both x and y are terms that ostensible (or, P they refer --at all) refer to objects of some kind or category C, and that it-is the conditions which individuate things as the same C which are operative and determine its truth value.

At first blush it appears that this fails to have the consequence that '43 = Julius Caesar1 is meaningless, since we can construe that as '43 is the same entity, or object, as Julius Caesar1 thereby bringing both 43 and Julius Caesar under the umbrella of the category llentityrl. But this is no good according to Benacerraf; lllEntityl is too broad". He also maintains that I1C must be a 'well- entrenched1 predicater1 (p. 65). Presumably this last condition rules out statements employing imaginatively invented terms, like lnumero-physical object', which are meant to apply to both numbers -and physical objects, thus rendering '43 = Julius Caesar1 meaningful. (6) is a consequence of Benacerraf's overall view of identity and objecthood. He "...agrees with Frege that identity is unambiguous, always meaning sameness of object, but that (contra-Frege now) the notion of an object varies from theory to theory, category to category..," (p. 66). This is not the place to attempt an in-depth appraisal of Benacerrafls metaphysics over those of, say, Frege's. But I believe a number of criticisms and remarks are in order, some directed towards Benacerraf's arguments for (6), and others questioning the compatibility of (6) with other claims he makes in the article. First, Benacerraf cites as evidence that llcontexts of the form 'the same GI abound. . . l1 whereas

Very rare in the language are contexts open to (satisfiable by) any kind of "thingv what- soever. There are some -- for example, llSam referred to.. . ," "Helen thought of.. .I1 -- and it seems perfectly all right to ask if what Sam referred to on some occasion was what Helen thought of. But these are very few, and'they all seem to be intensional, which casts a referentially opaque shadow over the role that identity plays in them. (p, 66)

Michael Resnik has pointed out in connection with this claim that the predicates 'is the third member of some series' and 'belongs to a set1 are non-intensional and "satisfiable by any kind of thing whatsoeverll. That is, applied set theory and number theory presuppose the universality of identity. 24 And there is reason to believe that Benacerraf would find these predicates unobjectionable; he says that "...any object can be the third element; in some progression1' (p. TO), and in conversation, has said that he thinks there are sets (an inference surely justified by the article, anyway). Secondly, it is not as though Benacerrafls approach is the only one to account for the oddity (among other things) of either claiming that 43 = Julius Caesar, or claiming that 43 # Julius Caesar. He can account for their oddity on the grounds that they are meaningless -- so of course no one would be tempted to ;hem. A rival. account, proposed to me by George Boolos, has it that identities of the sort under consjleration (like '43 = Julius Caesar1 and '3 = [[[0]]]1) make- sense but are false; that their negations are true; and that the oddity of making such claims is best explained via pragmatic considerations. A first approximation of an account might run as follows. They zre odd, because there is virtually no circumstance in the ordinary (and non-philosophical) course of things in which a person might want to make such claims. No one would claim '43 = Julius Caesar1 because, like a contradiction, it is so patently false. The

24. Frege and the Philosophy of Mathematics, (Ithaca: Cornell University Press, 1980), p. 196. inclination to pronounce it meaningless (in the absence of theoretical considerations) can be seen as the same one that prompts introductory philosophy students to pronounce contradictions meaningless. (No doubt this latter fact is attributable to the introductory student's lack of long +,raining in discriminating among absurdities . ) . On the other hand, 43 # Julius Caesar, while true, is never claimed because to do so would suggest that one thought one's listener (or reader) needed informing as to this obvlous fact -- a suggestion that would be warranted by the a~sumptionthat you are not violating the Gricean maxim to "make your contribution as informative as is requiredffa5. Anybody who understood the sentence would know it to be true, so it couldn't be informative for any one, Now it might be thought that this argument also Jubtifies the conclusion that tautologies (and such statements as 'All bachelors are unmarried1) are never used, either, because anybody who understood them would know them to be true. And this would refute the argument, because of course tautologies -are used. But, first, this is not the case for complex tautologies. And second, with respect to simple and obvious tautologies for which it -is the case (that anybody who understood them would know them to be true), we can make a rough distinction between

25. "Logic and Conversationn, The Logic of Grammar edited by Gilbert Harman and Donald Davidson (Encino, Cal.: Dickenson Publishing Co., 1975) p. 67. two sorts of uses of such sentences. (i)They can be used to draw the listener's attention to some particular fact that the sentence is prima facie In the case

of 'It is not the case that Adam begat Seth and Adam did not beget Seth', for example, it is a fact "aboutu, among other things, Adam and Seth. In the case of '/F= ;/m, for some natural numbers -n and m_, or it is not the case that there are 11 and g such that &- = ;/fi it is a fact about, among other things, c.(ii) They can be used to draw the listener's attention to some other point, generally by a seeming-violation of the Gricean maxim cited above. For example, 'It is not the case that Adam begat Seth and Adam did not beget Seth1 might be used to 4raw the listener's attention to the Law of Contradiction, by way of giving an example -- because the empirical claim itself, about Adam and Seth, is too trivial to be informative, so the listener would be likely to latch onto something else as the point of the remark, on the assumption that the speaker is not violating s Gricean maxim. In these cases it is precisely because these sentences are trivially true and hence uninformative that, by means of the Gricean maxim, they can be used to convey information. I may be suffering from a lack of imagination, but it seems to me that simple and obvious tautologies used in the style of (1) above only occur in mathematical contexts and then not as the conclusion, but only as steps in the proof. As for '43 # Julius Caesar1, it is hard to see how the fact that 43 # Julius Caesar would need appealing to in the course of proving an interesting mathematical theorem, This would seem to indicate that the bare-faced claim that 43 # Julius Caesar is useless. And it is scarcely easier to imagine a type-(ii) use for '43 Z Julius Caesar1. If the speaker wanted to alert the listener to the fact that numbers are not physical objects, he or she would probably not do so by means of the instance '43 # Julius Caesar1 but by uttering the equally short and more to the point: 'Numbers are not physical objects1. It could happen that 'Is the number 43 different from Julius Caesar?' comes to be used for the same purpose as '1s the,Pope Catholic?l is used now, But I doubt it will.

None of which is to say that there can be no use whatever for '43 # Julius Caesart, but only that if there is, it is likely to be an "indirectr1, i.e., type-(ii) sort of use (and even this seems a remote possibility). The sentence cannot be used in a type-(i) sort of way because anyone who understands '43 # Julius Caesar1 is thoroughly enough acquainted with the conceptual scheme that underpins our language to know that 43 is a number and Julius Caesar is a person, and as they are not even the same sort of thing, they are of course different things. And that everyone knows this, and hence no one needs to be told it. Thus, as I see it, it is the peculiar inability of '43 # Julius Caesar1 to convey useful information, especially in its type-(i) usage, that explains why we think a claim to that effect so odd, and, for similar reasons, why we are (at first) reluctant to assign truth values to it, or to its negation. The main point to be derived from this, however, is that we can explain, among other things, that

43 # Julius Caesar is an odd claim to make, and yet maintain that such sentences as '43 # Julius Caesar1 and '3 = [[[fll]]' make- sense, which the,y certainly seem to. This is one important advantage the present approach has over Benacerraf's. Another advantage is that if Benacerraf thinks 1, that 'Ronald Reagan = the Great Spiral Galaxy1 makes sense, presumabl it is because it is clear that both terms, if they refer, refer to physical objects. But it seems to be equally clear that both '3' and f[[CB]]]l, if they refer, refer to abstract objects. And therefore that '3 = [[[B]]]' has just as good grounds to be considered sensiaal (although of course false) as 'Ronald Reagan = the Great Spiral Galaxyt does. Benacerraf might reply that 'Ronald Reagan = the Great Spiral Galaxy' can be known to be false, whereas "it will be just as hard to explain how one knows thatL143 =

Julius Caesarf and '3 = [[[8]]111 are false as it would be to explain how one knows that they are sen~eless...~~(p. 67) ---Not at all. Benacerrafls own argument, (1) to (5) above, shows that numbers are not sets, which entails that 3 # [[[BII], and hence that I3 = [[[B]]]' is false. Benacerrsf seems to think it is confirmation for the nonsensicality of '3 = [[[@II]' that from the vantage point of one who seriously entertains the possibility that 3 is identical to [[[B]]], there appears to be no method of determining whether it is true or false, No doubt there isn't. But this need not be seen as reflecting on the sensicality of '3 = [[[B]]]'. In going so far as to entertain this possibility, one has "lost one's bearingsff, to use a

phrase of Benacerraf's, just as surely as if one were to seriously entertain the possibility that Ronald Reagan is

identical to the Great Spiral Galaxy, or that Socrates both was Greek and wasn't Greek. Such musings are outside the constraints of the conceptual s heme that we operate within. Benacerraf raises skeptical doubts about the reference of number words, then tries to resolve the difficulty with a brand of referential positivism. But this usolutionw (that '43 = Julius Caesart, etc., are meaningless) is at variance with other claims he makes in the article -- e*g*, that numbers are not sets. Presumably this claim makes sense to him. Employing the obvious symbols, it might be represented as:

(7) (x) (Nx + -Sx) or, what is logically equivalent, as: (8) -ExEy (Nx & Sy & x = y). But (8) does not make sense, by Benacerrafls standards, because surely if l3 = [[[B]]]l does not, on the grounds that there is no category C which it is clear that the referents of the terms (if such there be) belong to, then neither can (8). Lest there be any doubt as to this objection, let me recast it as an objection to Benacerrafls clalm that what constitutes an object is theory dependent, and that, therefore, '43 = Julius Caesar1, for example, is senseless. This is just to assume that no (reasonable) theory has both numbers an.d physical objects in its universe of discourse. But doesnlt scientific theory have both? At least on the face of it. All things'considered, I think the view I have been urging, that sentences like '43 = Julius Caesar1 and '3 = [[[B]]]' are false, fares better than Benacerraffs, at least with respect to the considerations he adduces for it (not to mention being more consistent with the ocher claims he makes in the article.). It might not seem to matter, any- way, since Benacerraf says he is u.,.certainly happy with the conclusion that all identities of type (c) are either senseless -or false.ll (p. 67, my italics). But I hope the preceding discussion has (i) disposed of whatever doubt the reader might have had concerning the claim that numbers are objects, based on the facts cited on pp. 22-23 above, e.g,, that it would be very odd for someone to claim that 43 f Julius Caesar. This was mainly accomplished, if it was, by refuting the alternative presented by Benacerraf. And (ii), that it has thrown some light on Benacerraf's peculiar attitude towards numbers as abstract objects, which I shall now attend to more specifically.

3. Benacerraf's Third Argument

As we have seen, Benacerraf's main argument does not work. He summarized it as: ...numbers could not be objects at all; for there is no more reason to identify any indiv- iduak number with any one particular object than with any other (not already known to be a number). (p.69) To see at a glance that the argument is fallacious, follow Richard Cartwright's suggestion: substitute !peopleg throughout for 'number'. It is true that "there is no more reason to identify any individual person with any one par- ticular object than with any other (not already known to be a person),'I Yet one would not conclude from this that people are not objects. Benacerraf certainly would not have advanced the "people'l argument. He must believe that there are independent reasons for thinking that numbers don't exist, reasons that do not apply to people. What might those reasons be? I think the answer may be gleaned from the following passage; (I have taken the liberty of numbering the sentences for future reference): (9) For arithmetical purposes the properties of Tnumbers which do not stem from the relations they bear to one another in virtue of being arranged in a progression are of no consequence whatsoever. (10) But it would be only these properties that would single out a number as this object or that. (11) Therefore numbers are not objects at all, because in giving the properties (that is, necessary and sufficient) of numbers you merely characterize an abstract structure -- and the distinction lies in the fact that the "elementsf1 of the structure have no properties other than those relating them to other "elements1' of the same structure. (12) If we identify an abstract structure with a system of relations (in intension, of course, or else with the set of all relations in extension isomorphic to a given system of relations), we get arithmetic elaborating the properties of the "less- thanff relation, or of all systems of objects (that is, concrete structures) exhibiting that abstract structure. (13) That a system of objects exhibits the structure of the integers implies that the elements of that system have some properties not dependent on structure. (14) It must be possible to individuate those objects independently of the role they play in that structure. (15) But this is precisely what cannot be done with the numbers, ,,.(16) To be the number 4 is no more and no less than to be preceded by 3, 2, 1, and possibly 0, and to be followed by [5, 6, and so forth 1. ... (17) Arithmetic is therefore the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions. It is not a science concerned with particul-ar objects -- the numbers. (pp. 69-70).

Sentences (9) and (10) entail: (18) Only the properties of numbers which stem from the relations they bear to one another in virtue of being arranged in a progression could single out a number as this or that object. Sen,tences (13) through (16) entail: (19) It is possible to individuate numbers indepen- dently of the role they play in the "abstract structurev'. This notion of an abstract structure is not without difficulties, and I shall attend to them shortly. (181, though, and (19) as I understand it, are both false. 1 I can individuate 4 as: the number of Jupiter's Gali- lean satellites. Similarly sentence (11) might be read as implying: (20) The "elements" of the abstract structure (here I, the numbers, presumably) have no properties . :-r other than those relating them. to other numbers. Again, false. 4 has the property of being the number of Jupiter's Galilean satellites. The fact that (18), (19) and (20) can readily be seen to be false suggests that something else is involved. And in fact, viewed from the perspective of pure, abstract, unapplied number theory the passage makes much more sense. Each of (181, (19) and (20) look (to me, at least) not implausible, when viewed from that perspective. To the number theorist, 4 just -is, as Benacerraf puts it at (161, no more and no less than to be preceded by 3, 2, 1 and possibly 0, and to be followed by 5, 6, and so forth. It seems as though it is not possible to individuate numbers independently of the role they play in relation to one another. Quine was writing from that vantage point when he said: "It is in this sense true to say, as mathematicians often do, that arithmetic is all there is to number..,there is no saying absolutely what the numbers are; there is only arithmetic. 11 2 6 But one cannot conclude from this, as Benacerraf does in (ll), that numbers do not exist. One could just as reasonably conclude that there is no id, no ego, and no superego. Because from the perspective of pure psycho- analytic theory these elements of the mind have no properties not dependent on structure, on relations they bear to one another. Benacerraf might reject the analogy, perhaps on the grounds that the "right" (or, the l1preferredU, or, the ;only "legitimatet1) perspective on the numbers is from that of abstract number theory; but the Itright" psychoanalytic perspective on the mind is from that of applied psychoanalytic theory. And maybe these claims (or others rejecting the analogy) could be made out more adequately and defended. But an analogy that cannot be rejected is one with set theory. Anyone who bought (9)

1

26, In fact, Quine refers to Benacerrafls artlcle as Itdeveloping this pointtt. p. 45, llOntological Relativityu. through (17) would be obliged to buy the set-theoretical version, (9') through (17') (which I write out for the convenience of the reader):

(9') For set-theoretical purposes the properties of sets which do not stem from the relations they bear to one another in virtue of being arranged in a set-theoretical hierarchy are of no consequence whatsoever. (10') But it would be only these properties that would single out a set as this object or that. (11') Therefore sets are nct objects at all, because in giving the properties (that is, necessary and sufficient) of sets you merely characterize an abstract structure -- and the distinction lies in the fact that the "elements" of the structure have no properties other than those relating them to other "elements" of the same structure. (12') If we identify an abstract structure with a system of relations (in intension, of course, or else with the set of all relations in extension isomorphic to a given syptem of relations), we get set theory elaborating the properties of the !'is a member of1' relation, or of all systems of objects (that is, concrete structures) exhibiting that abstract structure. (13') That a system of objects exhibits the structure of the set-theoretical hierarchy implies that the elements of that system have some properties not dependent on structure. (14') It must be possible to individuate those objects independently of the role they play in that structure. (15') But this is precisely what cannot be done with sets. (16') To be [[B]] is no more and no less than to be the set whose only member is [a]. (17') Set theory is therefore the science that elaborates the abstract structure that all set-theoretic hierarchies have in common. It is not a science concerned with particular objects -- the sets.

Just as (9) to (17) entail (18) to (20), so (9') to (17') entail (18') to (20'): (18') Only the properties of sets which stem from the relations they bear to one another in virtue of being arranged in a set-theoretic hierarchy could single out a set as this or that object. (19') It is not possible to individuate sets independently of the role they in the "abstract structurev. (20') The llelementslTof the abstract structure (here the sets) have no properties other than those relating them to other sets. And, in similar fashion (18'), c19t), and (20') appear false from the viewpoint of applied set theory. Sets can be individuated independently of the role they play in the abstract structure. [[B]], for example, is the set whose only member is the set whose only member is the set of moons of Venus. [Russell] is the set which contains a man surely in a class by himself. Like its numerical counterpart, the long indented passage above makes more sense from the viewpoint of pure, abstract, theory -- in this case, pure set theory. But is it -true when viewed from that perspective? More importantly, is what is claimed in the original (numerical) passage true? And even if some of the claims are true from that perspective, like (18) to (20), we have to consider whether this warrants the conclusion that numbers are not objects at all. In a way, it would be very odd if it did. Imagine the nl~mbertheorist busily characterizing the abstract structure of arithmetic and investigating the relations between the (numberg) of the structure. He or she seems like the last person to volunteer the opinion that numbers do not exist, engrossed as he or she is with the structure and its elements, rather than with applications of the theory. Nonetheless, perhaps the passage can be justified from a "mixedn perspective. In any event (18) through (20), if true, are true from the perspective of pure theory. Are they and their set-theoretic counterparts (18') through (20') true? It mlght be objected that there is an asymmetry here. Consider (20'). It might be thought to be false, on the grounds that fl has the property of being memberless and this property does not "reiate it to other setsu. But then (20) is also false, since the property of 0 of being predecessorless does not "relate it to other ', numbersf'. Similarly, It might be thought that (19') is false because B can be individuated as:

(~x)(y) (y is a member of x * y # y) And this does not mention "the role 6 plays in the abstract structureff. But if so, then (19) too is false, since 0 can be Individuated as:

(7~)(y) - (X is the successor of y) And so far as mentioning "the role 0 plays in the abstract structureu it doesn't, if the earlier individuation of 0 doesn't err in that regard. Of course the problem here is with what counts, for Benacerraf, as a property that relates x to other lfelementsn of the llabstract structure", and how to interpret "individuate x independently of the role x plays in the 'abstract structurefv. These difficulties in turn seem to rest on what an "abstract structureu is. So let us look at what Benacerraf has to say about abstract structures. Benacerraf tells us in (12) that an abstract structure can be identified with either (1) a system of relations

in intension; or (ii)the set of all relations in extension isomorphic to a given system of relations. Further, we can infer from the scare quotes in (11) that either abstract structures do not have elements, or else that their elements are second-class citizens, object-wise. "The distinctionu mentioned in (ll),presumably between abstract and concrete structures "lies in the fact that the !elementsf of the abstract structure have no properties other than those relating them to other 'elementsf of the abstract structureu, whereas the elements of the concrete structure, I take it, -do have properties other than those relating them to their fellow elements. Also, concrete structures are "systems of objectstf, and they "exhibitff abstract structures (from (12)). A typical model-theoretical account of a structure is that a structure consists of a nonempty set of elements (the domain), a family of relations that the elements bear to one another, a family of functions, and a set of distinguished elements. 27 So a natural way to view what

27. This f'typical accountu is taken from Gerald Sack's Saturated Model Theory (Reading, Mass.: W.A.Benjamin, Inc., 1976) pp Ilf. an abstract structure is is as a structure in the above sense, but one whose domain consists of abstract rather than concrete objects. Since in one use of 'abstract1 numbers -- and all mathematical objects for that matter -- are abstract objects, the abstract structure of the natural numbers is

simply the so-called standard model of arithmetic! The members of its donain, 0, 1, 2, etc., would exist just as surely as the structure does. Concrete structures can be

said to "exhibitn it in thbt they are isomorphic with it but consist of concrete objects. While this notion of structure may be applicable to Benacerraf's concrete structures, it is not appropriate for his notion of abstract structure, for it is patent: y incompatible with his claim that numbers are not objects at all. In another use of Iabstract1, some models (structures)

of set theory are said :o be abstract, while others are

I said to be concrete. Although it may be that the domains of the abstract models are not proper sets, and hence the abstract structures themselves are not mathematical obJects, nonetheless (most of) che elements of the abstract structures -- sets -- exist just as surely ae do those of the concrete structures -- also sets. And having the elements of both abstract and ccncrete structures exist on a par is one thing Benacerraf clearly wants to avoid. A uniform account of structure is not particularly to be expected anyway. Since concrete structures are systems of objects, whereas Benacerraf characterizes abstract structures as systems of relations, in intension -- a difficult notion that surely requires elaboration. He gives as an alternate account of abstract structure that it is "the set of relations in extension isomorphic to a given system of rel~tions~~(at (12)). That phrase, 'system of relations1, again. I do not know what a system is supposed

to be, but I imagine that what he means by a 'relation in extension1 is a set, or possibly class, of ordered pairs. If so, he has to face the charge that there are no ordered pairs -- cn the basis of an argument he is particularly susceptible to. Here is the argument. 6 (1 ) We have many set-theoretic accounts of what it is to be an ordered pair, all of which satisfy the

conditions of containing a pair such that if

= , then x = z, and y = w. For example, i = {{XI, {x,y)) (Kuratowski); or cx,y> = {Ex), ifl,y)) (Weiner); or, cx,y> = {{XI, {{%I, yll; or etc. 6 (2 ) Therefore either more than one account is correct, exactly one is, or none are. 6 (3 ) Not more than one can be correct, because they are not extensionally equivalent. 6 (4 ) There is no reason for identifying one account as the correct one, over all the others; one will do as well as another. 6 (5 ) Therefore ordered pairs are not sets. b (6 ) And by an extension of the above argument, they are not objects at all.

Of course, Benacerraf might well have repudiated this sort i i of (1 ) to (6 ) argument by now, even for the numbers (in fact, he has), Hence an ap2eal to the notion of a relation as a set of ordered pairs is one that, perhaps, he can legitimately make. Yet under this construal a rzlation is a sort of object; so a system of relations would then just be a system of objects, and the distinction between abstract and concrete structures collapses. 28 Perhaps the matter can be cleared up by looking instead at the notion of a "concrete structurev, and ' , reflecting on what a concrete structure could "exhibit1'. As systems of objects concrete structures seem somehow less problematical than abstract ones. In the least, each consists of a domain of objects which are related, maybe even ordered, in certain ways. And presumably these objects are concrete. But what does "concrete objectu amount to? Benacerraf seems to be assuming throughout the article that sets exist. Are sets concrete objects? In mathematics, it is common to see sets categorized as concrete -- unlike groups, for example, which are said to be abstract. But Benacerraf cannot be relying upon this distinction, because it places numbers squarely in the "concrete1' category also, Perhaps he has other reasons for regarding sets as concrete.

-- - 28. Lon Berk pointed this out to me. But I wonder how principled they could be. Because what- ever consequences (9) through (17) have for numbers, (9')

through (17') have for sets. In particular, if numbers are

not concrete objects, neither are sets. If numbers do not exist, neither do sets. Physical objects might be thought good candidates for concrete objects, as it is hard to imagine how they could be categorized as "abstractf1 (at least with sense data nowhere in sight). But are there enough of them to constitute the donlain of a concrete structure, particularly one that exhibits the abstract structure of arithmetic? That is, are there Infinitely many physical objects? (It goes

without say that any structure which is to. modela arithmetic must have a domain containing infinitely many objects, in

order to satisfy the axioms: 0 is a number; the successor of any number is a number; no two numbers have the same successor; and, 0 is not the successor of any number.). Without infinitely many physical objects (and, in the absence of other sorts of concrete objects) arithmetic will have no concrete structures to exhibit its abstract structure, Unfortunately, as I indicated on pages 15-16 above, some pretty strong empirical assumptions appear to be needed in order to conclude that there are infinitely many physical objects. The reason is -- to recap the main argument presented above -- that if space-time is finite, then given any positive real value v, no matter how tiny, if we regard the whole physical universe as being divided up into space-time volumes of size v (quadrubic inch-seconds, or whatever the unit would be), there still will not be infinitely many physical objects (i.e., space-time volumes). So it looks as though we have no good reason to believe that there are enough physical objects to constitute a concrete structure. There would seem to be enough expressions however. But (I hope to show) not for Benacerraf, What are expressions? Quine has an illuminating discussion of the matter in connection with his thesis that ontology is theory-relative. He asks us to consider the case of the thoughtful protosyntsct ician, working with a formalized system of first-order proof theory. The universe of this theory of protosyntax is meant to be expressions, of course, Not expression tokens, since there are only finitely many of these (hence not enough), but expression types. What are expression types? Identifying a type with the set of its tokens involves a serious danger of violating the law of protosyntax which says that x = z whenever x concatenated with y = z concatenated with y. It may be that the expressions x, y, and z have tokens, and x # z, but the expressions x concatenated with y, and z concatenated with y have no tokens, and hence are both identical to the null set. The next thing (s)he might try is to identify expression types with sequences of single signs. (A single sign can be identified with the set of its tokens, since these primitives will no doubt have tokens.). But what are sequences? A familiar construal of them is as mappings of things on numbers. So expressions turn out to be finite sets of pairs of signs and numbers. Alternatively, (s)he could have started by Ggdel-numbering the expressions and gotten by on a universe of just numbers. Quine concludes from this that ll...in both constructions we were artificially devising models to satisfy laws that expressions in an unexplicated sense had been meant to satisfyv1,29 So it appears that there are several different accounts of what an expression could be. No two of which are extensionally equivalent. Oh dear. I guess expressions

llcould not be objects at all. l1 GI won't bore the reader i by writing out a (1 ) to (6i)-tyye argwnept.). But, aa 1 mentioned earlier, Benacerraf has repudiated this sort of i i (1 ) to (6 )-argument, so let us ignore it and just concentrate on which account of expressions Benacerraf could choose. The possibilities are to take expressions (i) as sequences of single signs; or, (ii) as sets of pairs of signs and numbers; or, (iii) as numbers; or, (iv) in an unexplicated sense, i.e., as primitives.

Obviously (ii)and (iii)are not available to someone who is unwilling to countenance numbers, and unable to

29. W.V. Quine: vOntological Relativitytt, Ontological Relativity and Other Essays, pp. 41-43. countenance sets. Then (i)perhaps? Needless to say, a 6 6 trivial modification of (1 ) to (6 ) -- the argument that ordered pairs could not be objects at all -- would have the conclusion that sequences could not be objects either. (Again, I won't bore the reader by producing it.). But that aside, taking expressions as sequences of signs necessitates countenancing sequences in some unexplicated sense -- or at least, not explicated by means of numbers and/or sets. If numbers and sets have been axed, why should sequences be spared? I apologize for going on to anyone for whom the point is obvious. I shall merely try to enunciate my reasons for finding it obvious. For one thing, the notion of a sequence fairly screams thataf number. And traditionally, sequences have been reduced to numbers, or to sets. For another, sets have proved very useful, more useful than sequences, at least in providing a "foundation for mathematics1'. Considerations of mathematical utility therefore would have us dispense witth sequences (in an unexplicated sense) sooner than with numbers or sets. Someone might conceivably urge, however, that sequences are to be preferred on other grounds, They might emphasize the epistemological accessibility of certain sorts of sequences. After all, so many aspects of our daily lives are ordered. First we turn off the alarm, then we get out of bed, then we put the coffee on, then ,.. The things we do before getting to work are done in a i sequence. Every line -- of people in a supermarket, or waiting for a bus -- is a sort of finite sequence. Starting nearest to the sun, the first planet in the sequence of planets in our solar system is Mercury, and the ninth and possibly last is Pluto. I think these observations -do support the claim that some sequences at least are epistemologically accessible. But no more than they do for numbers and sets, Because we can also count the number of people in line, planets going around the sun, or things we did before getting to work. And we can form sets out of them, too -- the set of planets in our solar system, fpr example -- as easily as regard them as sequentially arranged, Perhaps these points do not suffice to show, as I believe they do, that it would be a very thin ontological story indeed that countenanced sequences but neither numbers nor sets. Yet they seem sufficient to enable us to conclude that any reasonable division of objects into "concrete" and "abstract" that Benacerraf might employ would not place sequences in the concrete category while placing numbers (and sets) in the abstract category. Since under our present construal an expression is just a sort of sequence, and since sequences are not concrete objects (because numbers are not), neither are expressions. And if expressions are not concrete objects , they cannot form concrete structures and thereby "exhibit the abstract structure of arithmeticu. It remains to be seen whether Benacerraf can afford to countenance expressions in an unexplicated sense, and not as sequences of signs. He certainly requires that there be expressions; he advocates that number wordd Itdo the jobf1of the numbers. What this thesis amounts to and how it fares will be gone into later. Note for the moment 4 its emphasis on linguistic entities standing in for mathematical ones. In keeping with this formalistic spirit let us return then to our protosyntactician and his/her formalized system of first-order proof theory, whose universe we shall take to be expressions. Here, an expression is just a fqnitely longUstringl1 of single'. signs. (In a sense this is an explication,,as it construes some expressions in terms of others; but these latter are primitively adopted. Also, the notion of a "string1,,is problematical since it is usually explicated as merely a sequence, and we have already seen that Benacerraf cannot countenance sequences. But let us assume, for the moment at least, that the notion of a string is one that is available to him.). The single signs will typically include variables, and logical and non-logical constants. These expressions of length one can be concatenated to form longer expressiqns. Depending upon the arrangement of their single signs, some longer strings will be formulae, and some formulae will be sentences. The proof theorist will be concerned with some of the relations that occur among these expressions, or sets of them -- relations like I1x is derivable from yI1 -- and functions on the expressions -- like "the result of substituting x for all occurrences of y in zI1. But (s)he will not be concerned with any properties expressions might have beyond those relating them to other expressions (or sets thereof). The vexpressions" of length one, the "single signsf1, could be anythinq, and, subject to certain restrictions, flconcatenationu could be any operation. The resultant longer expressions would be infected with the same sort of indeterminateness, since they would merely be the result of the vconcatenationn operation applied to the llsingle signsv1. In fact, it is very popular to take expression^^^ to be numbers. One just picks some numbers arbitrarily to be the "single signs". lfConcatenationll of the "single signs" nl, ..., nk may be defined so as to yield a Iflonger expre~sion~~m as follows :

(where pl, ..., pk are the first k primes). Bearing these facts in mind, consider the following passage :

(911) For proof-theoretical purposes the properties of expressions which do not stem from the relations they bear to one another in virtue of being concatenated with each other ' and derivable from one another are of no consequence whatsoever. (1011) But it would be only these properties that would single out an expression as this object or that. (1111) There- fore expressions are not objects at all, because in giving the properties (that is, necessaw and sufficient) of expressions you merely characterize an abstract structure -- and the distinction lies in the fact that the "elements1I of the structure have no properties other than those relating them to other llelements" of the same structure. (13") That a system of objects exhibits the structure elaborated by proof theory implies that the elements of the system have some properties not dependent on structure. (14") It must be possible to individuat.e those objects independently of the role they play in that structure. (15") But this is precisely what cannot be done with expressions. (16") To be the expression '(x) (FX)~is no more and no less than to be the result of concatenating the following expressions in the order presented: '(l, Ix1, ll?l,lxl,l)l. (1711) Proof theory is therefore the science that elaborates the abstract structure that all models of proof theory have in common. It is not a science concerned with particular objects -- expressions.

I think it is pretty clear that this adulteration of '# Benacerrafls (9) to (17) shows that expressions do not exist, if (9) to (17) shows that numbers do not exist, Actually, to put the matter more carefully, (9") to (l7I1) show that proof-theoretical expressions do not exist, if (9) to (17) show that natural numbers do not exist. And for present purposes, we can also conclude that expressions, or at least proof-theoretical ones, are no more "concreteI1 than natural numbers or sets, and so are not available to comprise the domain of a concrete structure that would exhibit the abstract structure of arithmetic.

It may be argued that (i)through (iv) on p. 51 above do not exhaust the possibilities for what expressions can be.

But it would surprise me a great deal if there was some other account of expressions, and under this account expressions turn out to be concrete objects, and there are enough of them to exhibit the abstract structure of arithmetic. Consider, for example, the expressions of a natural language, of English, more specifically. There is, first of all, the problem of whether there are infinitely many of them. Most linguists think there are, but the matter is not entirely settled. 30 Second, it is doubtful whether linguists or philosophers will provide an account of expressions different from one of (i)- (iv). Emmon Bach, for example, in Syntactic Theory says:

...we will take a language to be 2 set of sentences and a grammar to be some explicit account of (among other things) the sentences of the language, We consider each sentence to be a string or sequence of zero or more elements put together by an operation of concatenation (literally "chaining together"). The elements might be anything whatsoever; in our discussions of natural languages we may thjnk of them as representing distinctive sounds (phcnemes) or other linguistic elements (words, morphemes), depending on the focus of our discussion. We call the set of basic elements the (terminal) alphabet or vocabulary of the 31 language (and its grammar). (my italics).

30. For arauments to the contrary, see Charles F. Hockettls The State OF the Art (The Hague, ig68), Peter Reichrs ''The Finiteness of Natural Language", in Language 45:831-843

(19691,. - -.- and D.L. Olmsted's "On Some Axioms about Sentence Lengt hn in Language 43:303-305 31. Syntactic Theory (New York: Holt, Rinehart and Winston, 1974) p. 26. This passage again raises the question: is a string a

sequence, and if not what is it? Perhaps arguments similar to those showing that by Benacerrafls lights sequences cannot exist would show that strings do not either.

A fuller comparison of expressions with numbers will appear in chapter three below. For now, it is enough to consider the following. Imagine a finished syntactic theory T of English expressions -- not of the English language with all its history, but merely a theory that gives the necessary and sufficient (syntactic) groperties of English expressions. (It is important to restrict the properties to just the necessary and sufficient ones, because Benacerraf restricted it tq just these in the case of numbers, in (ll).). The expressions themselves would just be sentences and words, and maybe letters and phonemes, Along with their structural descriptions they would be arranged in a sort of abstract structure. That is, the theory will describe the various relations these abstract objects bear to one another (such as "is composed ofu and "has the deep structure of1') and the properties

that stem from these relations, but it will not refer to any other, "externalt1, or unnecessary properties (such as ''was said by Nixon'', or "was commonly said in 1974"). To be a given sentence, for example, 'There was a child found

in the bullrushes', is no more and no less than to be the string of words: 'there1, 'was1, 'a1, 'child1, 'found1 'in1, 'the1, lbullrushesl. To be the word 'there1, in turn, is no more and no less than to be the string of letters: t,h, e, r, e. (Alternatively, it could be identified with a certain string of phonemes. But it cannot be identified with both a string of letters, -and a string of phonemes, because these strings will normally not be identical to each other. We could conclude from the fact of these multiple possible identifications, la

Benacerraf, that words are not objects at all. Or we could conclude that words are "composed ofw1letters, and words are "composed ofqwphonemes but the composition relation is more complex than one would have thought, In any event, for our immediate purposes, it is immaterial whether we identify a word as a string of letters, ? string of phonemes, or a string 'wcomposed ofv either. The point cernains that these things are abstract objects standing in certain relations. This goes for phonsmes, too. I have been told by James Harris that [a] current phonemic theory has it that phonemes are sets, or sequences, of features. To simplify matters then, I shall assume our theory identifies a word with a string of letters.). So according to our theory sentences are strings of words, and have deep structure, and words are strings of letters. But what are letters? They may well just be

given in a list, the "terminal alphabetf1, as: law,Ibf, c,. x,y, z. And this may be done, because further reduction might be deemed unnecessary (everyone knows what the letters of the alphabet are, just as everyone knows what the (natural) numbe~sare). Or, too, because fur8ther reduction might be deemed impossible. The letter lo' for example, cannot be icentifird with any one sort of physical object because the letter lo1 is neutral with respect to its modes of representation: in print, a circle; in Morse code, two dots; in Braile, a chara.cter of raised

I) dots arranged so . . ; In the code th~tAnne Sullivan taught Helen Keller, a certain hand con:iguration; nct to mention flag arrangements, smoke signals, and so on. Again, we could conclude Prom the fact of multiple possible reductions, la Benacerraf, that letters are not objects at all -- although of course we won't. But we cannot say about letters anything analogous to v;hat we said about words at this point because, ~compositions~aside, it will always be ~ossibleto come up with a new encoding of the alphabet. So it is not possible for our theory T to I specify a disjunction that captures all the modes of represent,tion of a given letter, th~1.sidentifying it in ternis of properties not stemming frm the reiations it bears to other letters, words, and sentences. Of course, it is possible to identify a leitcr by means of a contingent property it has -- such as "being the first letter of the second ward of the third line on this page", just as it is possible to identify a number as "being the number of the

Apostles". But our theory T xi11 not concern itsrlf with auch properties, H~ncewe may conclude that letters are primitively adopted elements of the theory, specified by a finite list. The important thing will be how the alphabet composes words and sentences, not the nature of the lettem themselves, Taking what Emmon Bach said about the elements that comprise sentence-strings a step further, the letter lo1 itself may be taken to be anything at all for the purposes of the theory, just so long as the relations between letters, words, sentences, deep structures, etc. (if there is any cetera) hold. Bearing all this in mind, then, consider the following passage.

(glll) For T-theory purposes, the properties of expressions (and other items of sxntax) which do not stem from the relations they 'bear to one another are of no consequences wfiatsoever. (lofi') Therefore, expressions are not objects at all, because in giving their properties (that is, necessary and sufficient) you merely characterize an abstract structure -- and the distinction lies In the fact that the uelements" of the structure have no properties other than those relating them to other llelernentsu of the same structure. (13"') That a system of objects exhibits the syntactic structure of sentences,words, deep structures, etc,, implies that the elements of the system have some properties not dependent on structure. (14" ) It must be poaslble to individuate those objects independently of the role they play in that structure. (15n1) But this is precisely what cannot be done with expressions. (16"') To -be the sentence 'There was a child found in the bullrushesl is no more and no less than to be the string of words: 'there1, 'was1, 'a1, 'chilzI1, found1, 'in1, 'the1, lbullrushesl. (17"l) T is the~eforethe theory that elaborates the abstract structure of English syntax. It is not cor~cernedwith particular objects -- expressions. 73

Again, the point is not that (9"') to (17"') are --true. Expressions -can be individuated by means of properties which do not stem from their relations to other items of syntax -- such as "being the first letter of the second word of the third line on this page1' -- just as numbers can be so individuated -- e.g., as "being the number of wqrds on this page" or "being the number of the apostlest1. The point to be drawn is the analogousness of the ,passage (gltl) to (117111) with the passage (9) to (17). Of course, it is true that T is not a theory currently being studied as arithmetic is; and it may never be more than a goal. So the parallel is not perfect, But I believe the analogy is strong enough to enable us to draw thy following conclusions: (i) if (9) to (17) show that natural numbers do not exist, (9"') to (17tt') show that English expressions do not exist, and, (the weaker conclusion) (ii)if natural numbers are not "concrete objects1' (whatever this may amount to) neither are the expressions of English. By now the reader may have lost track of the structure of the argument. So perhaps we should do some reviexing, and take stock of the conclusions reached so far. First we saw a presentation of Benacerrafts argument that numbers are not sets -- viz., (1) to <5)* In trying to extend the argument to show that numbers are not objects at all, as Benacerraf claimed could be done, we saw that indeed numbers are neither expressions nor physical objects. But, since it cannot be shown that numbers are not numbers, at best this shows that numbers cannot be ureducedtl to anything else, as it were, and not that they are not objects at all, Hence Benacerrafls first argument begs the question. We then discussed whether the oddness attached to claiming '43 # Julius Caesar1, for example, should be taken as evidence for the thesis that numbers are not objects. Because if numbers are objects, then it would appear that it is true that 43 # Julius Caesar, and there should be nothing queer about saying so. But clearly there is some- thing queer about such a claim. If it were unsemantical, as Benacerraf claims, then that would go a I~ngway towards accounting for the oddness. However this solution is ', strikingly at variance with other claims '~enacerrafmakes. Nor is it the only solution; we saw the sketch of a rival story that accounts equally well for the oddness factor, but according to which such statements have truth values. Another argument for the conclusion that numbers are not objects seemed to be present in the passage I numbered (9) to (IT), (ll),for example, begins: "There- fore numbers are not objects at all becau~e,,.~~.We noted of the claims made therein that some of the more important ones, like (la), (lg), and (20), were prima facie false, but seemed at least somewhat plausible from the perspective of the pure number8 theorist. In his/her capacity as a number theorist, (s)he might reasonably be concerned merely with the relations the numbers bear to each other, and properties derived from such relations. This, as opposed to ''accidental" properties of numbers that concern, for example, planetary scientists (like "being the number of moons of Saturnu), or even the number theorist acting in a different capacity (like ''being the percentage of one's incme that may be deposited in an IRA account tax-free1'). Instead of proceeding to an evaluation of (9) - (l7), though, we constructed (9') - (IT'), a set-theoretic version of (9) - (17). It looked as plausible as (9) - (17). In particular, the corresponding claims (18:), (lgl), and (20') were supported by the observation that the pure set theorist might be concerned solely with the relations the sets (s)he studies bear to each other,', and properties derived therefrom. Like the number theorist, (s)he is not concerned with "accidental1I properties of sets, like Ifbeing the set of planets in the solar system beyond the orbits of Pluto and Neptunev or "being- -.. khe same size as the set of Americans who don't own a TVn. Two questions then arose. First, are (9) - (17) and (9') - (17') definitely analogous, so that if the former show that numbers are not objects, the latter show that sets are riot objects? They appearsd to be, but the matter seemed contingent on what, precisely, an abstract as opposed to a concrete structure is; in particular, whether there is some principled distinction whereby the standard model of set theory is a concrete structure but the standard model of arithmetic is a mere abstract one. Secondly, do they, or does one of them, show that numbers and/or sets do not exist? This question also hinges, even more obviously, on what an abstract structure is. Benacerraf does not say much about abstract structures, and what he does say leads to problems, for him, with relations. This prompted

us to examine instead the notion of a concrete structure. One construal that looked promising was that or a structure whose domain consists of ccncrete objects. While physical objects seem like ideal candidatez for the job, such concrete structures composed of them cannot be what Benacerraf had in mind, since there are not enough of them to constitute a structure that exhibits the abstract structure of arithmetic. Expressions seemed like good 'I candidates too, especially in view of Benacerrafrs desire to let "the sequence of number words1' perform the function of thc natural numbers. But, even if Benacerraf had expressions in mind as conclete objects, we saw that they are not available to him for use as such. Because some of the things expressions might be, such as sequences, cannot plausibly be considered concrete objects if numbers 1 and sets cannot. And because if (9) - (17) show that numbers are not objects, then (9") - (17") and (g1I1) - (17lV1)together show that expressions (or important varieties thereof) are not objects, So therefore it cannot reasonably be urged that expressions are concrete objects. Hence neither numbers, sets, physical objects nor expressions are concrete objects that could comprise the domain for a ''concrete structure" that exhibits the abstract structure of arithmetic. This does not exhaust the ontological~posslbilities for what could comprise the domain of such a structure. But in ruling out some of the most obvious and desirable candidates, it suggests that, whether or not chere are any concrete structures at all, there are no concrete structures that exhibit the abstract structure of arithmetic, This in turn makes it seem likely that there could be no non-trivial distinction between concrete and abstract structures that would support (9) - (17) and the conclusion that numbers are not objects at all, But even if such a distinction could be found, it would probably also support (9') - (ITt),

(9") - (ITu), and (9lt1) - (l7"l), thereby showing that there are no sets or expressions either. I think it fair to conclude that (9) - (I?'), BenacerrafTs second and last explicit mgument in "What Number Could Not Be1' for the conclusion that numbers are not objects at all, fails to show that numberbs are not I objects at all. CHAPTER 11: SOME RIVALS OF THE POPULAR VIEW

After a short introduction that will, I hope, give us some perspective on the popular view, and in particular on a way of extending it in light of conclusions reached in the preceding chapter, I consider a number of the popular view's llrivalsll. The rivals considered were put forward by Gilbert Harmsn, Nicholas White, and Benacerraf himself, largely in response to the negative conclusions reached In "What Numbers Could Not Be". They will be presented, criticized and, most importantly, compared with the popular view, especially epistemologically.

Recall that the popular view consists of the following theses: (1) There3are .numbers . (11) Numbers are abstract objects. (111) There are r-athematical entities. (IV) *There are abstract objects.

(V> Arithmetic statements are tr~eor false. It is supported by the fact that we talk as though there are numbers, I claimed, and that whatever else this may involve, it involves at least the following theses: (VI) We quantify over numbers.

(VII) Numerals like '2' in '2 + 2 = 4' and number words like 'two1 in 'Two is a prime numberr are singular terms.

(VIII) The surface structures of '2 t 2 = 4l and 'Two is a prime numberr are identical to their logical structures. (IX) The appropriate semantics for statements such as '2 + 2 = 4' and 'Two is a prime number' are referential semantics. Frege took the nature of number to be problematic, and tried to reduce arithmetic to logic. In his account the number 5 turns out to be the class of all n-membered . 0 - classes. But his class theory was not consistent. Russellls account of natural numbar is muc? the same as Fregefs except that according to his type theory there turns out to be a different class of things that are the natural numbers for each type. Although apparently consistent, Russellls theory lacked the simplicity of

Zermelols theory, wherein 0 = 0, and n + 1 = En], or of Von Neumannls, wherein n t 1 = n u [n]. Each of these accounts is consistent with theses (I) - (V); sets and classes, after all, are abstract objects. Thus each represents a possible extension of the popular view, But as Benacerraf argues, this very profusion of nonextensionally equivalent accounts cases suspicion i upon the credibility of any particular account's claim to being the correct one. The Frege-Russell account has f a certain appeal, admittedly, but, as Benacerraf remarks, "there seems little to recommend it over, say, Ernie's1' ( WNCNB , p. 58). After all, "Erniels" account figures elegantly in the context of the more popular ZF set theory . So in showing, by (1) - (5), that numbers are not sets, Benacerraf terminated a long tradition, one that began with Frege and continued through to Quine. (Quinets position will be gone into in the chapter on Quine.) In trying to extend Benacerraf's (1) - (5) argument we saw 4 5 that (5 ) and (5 ) are 'true. if 4 (5 ) Numbers are not physical objects.

(55 ) Numbers are not expressions. These subconclusions do not justify the conclusion that Benacerraf wanted to reach, that "numbers 2ould not be objects at allf1, I argued, because ta reach that conclusion utilizing these arguments, one would have to show that for every type of object 0 there is, i (5 ) Numbers are not 0's (i.e., no number is an 0). But of course since it cannot be shown that numbers are not numbers, the argument was un~uccessful;to work, it had to presuppose that numbers do not constitute a type of object at all. Actually, in view of the philosophical tradition from Frege to Quine, the presupposition that Benacerraf makes does not seem to be as patently question-begging as I have mace it out to be. Stated more generously, the presupposition seems to be that (21) Numbers are not -sui generis. They do not constitute an ontologically irreducible sort of object . If numbers are objects (as opposed to properties or relations) then they belong to some "othert1 category -- like sets -- that prima facie does not include them. The real difficulty with Benacerrafls argument lay in the fact that (21) goes unargued for (as was indicated above). It is not my intention here to manufacture arguments Benacerraf might have produced to support (21) (particularly as I haven't a clue as to what they might be). My object is to defend the popular view against the arguments Benacerraf has actually produced. I bring up the matter so as to show in what direction the popclar view may be extended, ,in the light of Benacerrafls arguments in "What Numbers Could Not Be", and the preceding discussion of them. The direction is this. We can state a further thesis that we may regard as being associated with the popular view: (22) Numbers are neither sets, nor classes, nor physical objects, nor expressions. (22) may be extended, too -- e.g., by the words 'nor eventsv. (It is fairly obvious, I think, how that argument would go.). The point is to try to refine the popular view, even as it is being defended. And I am claiming that (22) represents a reasonable first extension. It is supported by the fact that Russell, himself an advocate of what

Benacerraf calls "the traditionally favored view1' (p. 62) (and hence of (21)), felt the need to defend his identifi- cation of numbers with classes. Witness the following:

To regard a number as a class of classes must appear, at first sight, a wholly indefensible paradox, Thus Peano [Formulaire & Mathdmat iques , 1901, sec. 321 remarks that Itwe cannot identify the number of [a class] a with the class of classes in question [Fee, the class of classes similar to a], for the.,se objects have different properties. 'l' He does not tell us what these properties are. , ,Probably it appeared to him immediately evident that a number is not a class of ~lasses.3~

Having further refined the popular view, or, more strictly speaking a philosophical position based upon it, and noted some non-alternatives to it, let us proceed to consider the alternatives (or a number of them). To prevent any misunderstanding, let me reiterate the basis upon which rival accounts will be evaluated. The charge that initiated this dissertation was that the

32. Bertrand Russell, Principles of Mathematics (New York: W.W. Norton & Co., 1938) p. 115. popular view was untenable, because it made it impossible to account for our knowledge of aritlmetic. The basic

idea was that if numbers are non-spatial and non-temporal (i.e. abstract) objects, we cannot interact with them to

learn about them. So rival accounts will be compared with an eye towards seeing how successfully they handle this epistemological problem. Other bases of comparison are possible; e,g., ones involving considerations of ontological economy. That is, one might be concerned to find an account of number, perhaps as part of a larger theory, that involves quantification over either: a) the smallest number of entities, or b) the smallest number of sorts of entities; whilst conforming to: c) phenomenological experience, or d) physics, or e) religious dogma,

or f) ordinary language usage among other things. Ontological economy, per E, means nothing to us in the present context -- unless it cures, or eases, the epistemological problem we are confronting. This may lend an air of unfairness to the proceedings,

since no doubt not all the authors whose accounts we will consider were aiming to solve epistemological problems, Yet there is a point at which the issues blur. This can best be seen by looking at our first rival accourit --

Harman I s . 2.Harmants Account

Harmants account, which will be quoted in its entirety, is given in a one-page note. 33 It begins:

To say that numbers can be identified with sets is not...to say either that numbers -are sets (which is too stzong) or that number theory has a model in set theory (which is to6 weak). It is to say, rather, that any w-sequence of different sets (or other things) can be used as the numbers.

What does Harmac mean by this? Surely not that I would be understood by the grocer if (s)he asked how many pounds of mushrooms I wanted, and I replied: ''the unit set of the unit set of the null setff, because the unit set of ', the unit set of the null set is the second member of some w-sequence. The claim made in the last sentence of the quoted paragraph is more than a little similar to Benacerrafts claim that "Under our analysis, any system of objects, sets or not, that forms a recursive progresston must be adequateu (WNCNB, p. 68). Discussion of just what it might mean will be postponed until later. Note for the,moment the reference to sequences of sets, Apparently Harman would affirn, (111) and (IV). Continuing with that Harman says:

With this in mind, we might consider the following hypothesis cxicerning the logical grammar of talk about numbers: numerals are best analyzed as function eymbols rather than names.

33. ftIdentifying Numbers," Analysis 35 (1974):12. It appears then that he would deny (1TIII); he therefore owes us an account of what !'the logical grammar1' of a statement such as '5 + 7 = 12' is. Not unaware of this debt, he writes:

Knowing that 5 t 7 = 12, we know that fo? any sequence s, the sum operation for that sequence applied to the fifth and seventh members of the sequence yields the twelfth membzr.

This suggests that Harman wants to claim that the logical grammar ~f '5 t 7 = 12' is given, in English, by (23): (23) For any 2, if -s is an w-sequence then the sum operation for -s applied to the 5th agd 7th members yields the 12th member. 1, (23) represents only one of several ways of interpreting Harman, as we will see shortly, although I believe it is the most straightforward and plausible way. Yet this analysis, wherein (23) is khe analysans, d.oes not work, as it subject -to the- -same --fatal cbjectims that . Philip Kitcher brinp:,- againsl Nicrlolas White's an~lysis.3 4 In 2onnection with this pofnt, the rsader may want to note £,>I- hidherself the cimilarity of (23) ~5th(30) on page 81 below, which is 'Xhite's acalysis of '5 + 7 = 12'. I AlLhough Kitcherls objections apply equally well to

34. Philip Kitcher. ''The Plight of the ?latonist,ll Nous 12. ( &978 ) : 119-136; and Nicholss White, "What N'lmSors the analysis represented by (23), since it is White he

brings them againstydiscussion of them will be postpaned until we get to White's account, which is next after Harman ' s . So let us conzider other ways of interpreting Harman, In the very next sentence (after the last one quoted abcve) he seems to suggest a different analysis:

We can make ttis more explicit by writing 5, tS 7, = 12,. Here '5', '7' and '12' are not names but function symbols. The letter 'st names a sequence and '5,' names the fifth member of that sequence...

This is strange. Clearly 'sf does not name a sequence. It is either a variable or a schematic letter. So our I second interpretatioh of Harman is that k.e is advocating tne following analysiz: (24) The logical grammar of '5 t 7 = 12' is, or is # I best represented by, '5, ts 7, = 12,'. - But '5 t 7 = 12' is a true sentence, whereas '5, +, 7, - 12,' has no truth valve;t it is a schema. Are we to under- stand that Harman thi~ks'5 t 7 = 12' is merely a schema then? I don't know what Harman means to be givins, in

giving an analysis, but presumably it is something that is equivalent, in sone sense, to the original, although more perspicuous from the point of view of grammatical

structure. So that if '5 t 7 = 12' is to be analyzed as a schema, it is in some sense equivalent to a schema, , and has no truth value. Does Harman mean to be denying (V)? I doubt it, because he claims that we know that I 5 + 7 = 12, which suggesLs that he thinks it is true. Besides, if he thought that '5 t 7 = 12' is not true, why would he bother to analyze it at all? There is no apparent problem, if it is not true. In that case, one need only account for our knowledge of so-called llapplications of arithmetic1' (which is precisely what Hartry Field tries to do. 35) Perhaps, then, (24) is not strictly speaking an / actual statement, but is a schema itself, to be understood on a par with this passage from The Theory of' Numbers 36.. Corollary 13.1 (Fermatls ~heorem):,Ifp is a prime, then aP E a(rnod p). of course 13.1 might be lriewed as an abbreviation for a bona fide statement beginning with the quantifier 'For any integer a,..'. But 13.1 can also be viewed as a schema that yields a truth when the name of an integer

1s substituted for la1. So understood, Harrnan1s schema (24) ought to yield only truths when names of sequences are substituted for Is1. The following is an example,

35, In Science Without Numbers (Princeton, N.J.: Princeton University press,.

36. By Anthony A. Gioia, (Chicago : Markham Publishing Co. , 1970) p. 40. where 'zt abbreciates 'the Zermelo sequence, 0, {@I,

UflIl, .me:

(25) The logical grammar of '5 t 7 = 12' is (or is

best represented by) '5, t, 7, = 12,l. (When read aloud it might sound like this: 'The logical grammar of '5 + 7 = 12' is (or is best represented by) 'the sum operation for the Zermelo sequence applied to the fifth and seventh members is identical to the twelfth member1.). (26), too, should be true, where lvnl abbreviates 'the Von Neumann sequence, 8, (01, {fl,{%I}, ,.. (26) The logical grammar of '5 + 7 = 12l is (or is best represented by) - '5,, tVn7vn - 12vn1. These disparate analyses suggest that Harman give us some '8 criteria showing their equivalence -- or say that there --is no unique logical grammar. The third and final way I shall consider of inter- preting Harman incorporates the schematic aspect of what he says, without the unattractive consequence that one

seemingly unambiguous sentence, viz., l t 7 = 12', has many seemingly different logical grammars. It is this:

/ (28) The logical grammar of '5 t 7 = 12' is (or is best represented by) (29).

(29) The result of replacing Is1 by the name of an w-sequence of sets (or other things) in the schema:

is always true. What appears to be a major difficulty with this analysis is that according to it '5 t 7 = 12' is really a meta- mathematical rather than a mathematical statement. I say

lappearsf because a formalist might argue that this in itself is no difficulty at all, but is to be expected if one forsakes traditional mathematical objects. But is Harman a formalist? The reference to w-sequences of sets suggests not. In fact, that takes us to the heart of the difficulty. Here is the remainder of Harman1s I article. t

...but we need not suppose that '5' by itself names anything. It is the 'fl in 'fxl. In particular, to say that numerals are function symbols would not be to,say that they are names of functions. They'would not be names at all. It follows that they would not be names of objects (which confirms Benacerrafls conclusion in "What Numbers Could Not Bef1.. . ) . It also follows that they would not be names of objects that have essential properties...

It is fairly clear, I think, that Harman would deny (I) and (11), but accept (1111, (IV) and (V). And perhaps it looks as though doing so successfully makes some headway on,Benacerraffs epistemological problem. After all, some measure of ontological economy would have been C achieved -- there would be no numbers to worry about anymore. There would be only uw-sequances of sets (or other things)", and (perhaps it will be said) we had to worry about these anyway. I I find such considerations unconvincing. Suppose it wese persuasively argued that angels are pure spirits and, as. such, are insensible. It would not lessen the problem, in my estim~tion,if it were proved that Thrones, Dominions, Prir,cipalities, Virtues and Powers do not exist; that claims about angels require only four of the nine choirs: Archaagles, Angels, Seraphim and Cherubim, A number of grounds might be advanced by someone anxious to deny that the comparison is apt. It might be claimed that while the nine choirs of angels may be equally epistemologically (in)accessible, that is not so for mathematical entities. But for which mathematical entities, and why not? Harman nppeals to w-sequences of --sets. So perha?s he thinks that sets are epistemologi- tally more accessible than numbers. But this requires an argument because even if true, it is hardly self- evident. Consider Quinefs remark:

It will perhaps be felt that any set-theoretic explication of natura.1 number is at best a cake of obscurum per obscurius; that all explications must assume something, and the natural numbers themselves are an admirable assumption to start with. I must agree that a construction of sets and set theory from natural numbers and arithmetic would be far mor desirable than the familiar opposite. 3 Li

Like Harman, Benacerraf also seems to think, in "What Numbers Could Not Bef1, that sets are epistemol ogically

37. Ontological Relativity and Other Essays (New York: Columbia University Press, lm)33. more accessible than numbers. But as we saw in chapter one, arguments like -his against numbers would work equally well against sets. So no support can be expected from that quarter. Another argument that might be given in Harman1s behalf is that the sets Harman refers to are sets of everyday objects. These are thought by some to be less abstract (and hence less problematical) than either pure sets or numbers, because at least we are familiar with the things they are sets of. For example, it might be urged that Snow White's acquaintance with each of Sleepy, Sneezy, Dopey, Doc, Yappy, Grumpy and Bashful makes it possible for her to 5e familiar with the -set of dwarves. But surely the same fact makes it possible for her to be familiar with the number of dwarves too: Besides, Harman seems to want there to be wLmang sets,

But if there are only finitely many everyday objects, as seems likely, then there will be anly finitely many sets of them (6ecauqe lf 2 is finite, the cardinality of l' l' the power set of q, 2-, is also finite.j. In response to this it might be pointed out that Harman does not need w-sequences of sets; w-seq~ences of anything will do. But the same sorts of difficulties seem t~ emerge here. If the w-sequences are supposed to be abstract but harmlessly so, because composed of concrete objects, the risk is run of there not being any w-sequences at all, because there may be only finitely many concrete objects, (In chapter one above we saw this problem arise in connection with Benacerrafls claim that there are only "concrete structures that exhibit tne abstract structure of arIthmeticl1.). If, on the other hand, w-sequences are full-fledged abstract entities of pure mathematics, then they would seem no more epistemologically accessible than numbers. So llanalyzing awayv1numbers in terms of' them p~esentsus with no epistemological gain. What thei~is Harman to do? As I see it, the only hope of consistently extending his position so as to procure something philosophically acceptable lies in analyzing the logical grammar of statements abaut w-sequences so that w-sequences, $00, get "analyzed away1'. But there is very good reason to think that this cannot be done, and we will now look at White's article and Kitcherla critique af it to see why this is so.

3. White1s Account

In "What Numbers ~rell' Nicholas P. White argues that

38, Synthese 27 (1974): 111-124. If the existence of a model of arithmetic in set theory was ever a reason for identifying numbers with sets, then I maintain, the existence of multiple set- theoretic models of arithmetic should prompt us, not to say with Benacerraf that numbers cannot be sets, but rather to scggsst that there are multiple full- blown series of natural numbers. Thus, for example, instezd of there being only one three, there are after all many threes, and many thirty-sevens, and so on... The first step of the suggestion is to suppose that we can, in our arithmetical discourse, replace our numerical singular terms, such as 'three1, by atomic general terms, such as 'is a threef, or 'threesf. (P. 112)

White is not denyi~(VII) so much as recommending that we revamp our arithmetical speech habits to reflect the facts (much the way, he says, a person who concluded that the Homeric ?oems were written by several people might claim that "there turn out to have been several HomersH (p, 113)). What does the revamping consist in?

The intuitive idea behind the suggesticn is that to be a three, e-g., is to stand in a certain position (viz., fourth, counting zero) in some progression or other. Plainly, what we therefore want is a relativized notion of what it is to be a three -- relativized, that is, to progressions. (p. 113)

So 'x- is a zero in p1 is explained as 'there are no elements of p- which precede -x in pl; 'x- is a 1 in p1 as 'there is a precursor, x, of x in p and there is no precursor of -x in p which is not identical with xl; and so forth. The Peano axioms can be rewritten as statements about progressions. For example, The sum of 5 and 0 is 0. would be rewritten as:

(x)- () (E) (if -x is a zero in and x is in Q, then the sum of x and in -p is identical with x). And The sum of n and the successor of m- is the successor of the slm of -n and m.- would be rewritten as: I

(5) (E) (if -X is in E and x is In E, en (the sum of -x and the successor of y in 2) in 2 is identlcal with the successor in g of the sum of q and x in E.). '5 + 7 = 12', then, would be explained as; (30) (E) (the sum in 2 of a 5 in and a 7 in p is a 12 in 2). So, like Harman, White would deny (VIII), and propose as an analysis of '5 + 7 = 12' something very like Harman's (23). True, Harman speaks of w-sequences instead of progressions, and 'the 5th member of1 instead of 'a 5 in1, But since all it means, for White, for -x to be a 5 in g is for & to be the 5th (actually, 6th since he starts with 0) member of g, the difference is trivial. So if White's analysis fails, so will Harrnan1s (23). Interestingly, although 'rJllitelsanalysis is very like Earman's, and of course both would agree to (III), (IV) and (V), they disagree about (I). Harman says there are no numbers, and White says there are plenty of numbers. Yet this does not rule out their being in complete agreement as to which individual objects exist! The reason for this is that Harman probably thinks that, if there were numbers, they would be abstract objects of a certain sort (i.e., (11) would be true). For White,

(11) is false; something is a number if it is a member of a progression. Anything that is the third member of some progression -- and wnat isn't? -- is a 3. In fact, every number is every other number- So the difference between Harman and White is a. bit like that between one who maintains that there are no witches, on the ground that witches consort with the de7:il and there is no devil to consort with, and one who maintains that there are witches but no devil, because to be a witch one need only think that one consorts with the devil, Actually, I find the inference from Ix is an element in a progression1 to 'x is therefore a number1 -- in fact, 'for every n, x is an n1 -- rather implausible and unmotivated. But there is more the matter with White's view than this superfluity. The main difficulty stems from the fact that for White, as for Harman, there have to be progressions. And as I hope was evident from our discussion of progressions in connection with Harman, whether and what progressions are for White is crucial for determining whether his analysis provides us with a net epistemological gain. Now it is clear from the first quote (above) from White's article that his reason for analyzing number statements (like '5 t 7 = 12') as being statements about progressions (like (30)) is that "there are multiple set-theoretic models of arithmetictt. It turns out, however, there are multiple set-theoretic models of what It is to be a progression. We can see this as follows. To explicate the notion of a progression, or w- sequence set-theoreticaily, we might start, as Ptlilip

Kitcher does in "The Plight of the ~latonist"~by analysing them in terms of sets and relations:

A set x is a quasi-n~!mber set wfth respect to a relation R just in case there is a unique element l(g,R) E x, such that R is a 1-1 correspondence from x to x\ {l(x, R)) [i.e. x without its "firstT memEer under R]. set x is a number -set with respect to R iff it-is a quasi-number set with respect to R and is the smallest quasi- number set with respect to R containing l(q,R). (p. 124).

A number set is, of course, a progression, or w-sequence. To complete the reduction, we need to say what a relation is. The usual txplication is that a relation is a set of ordered pairs. But now the fact stated in 6 (1 ) on p. 49 above poses a problem for White: There are nany set-theoretic accounts of what it is to be an ordered pair. It follows that there are many set-theoretic accounts of what a progression is. Were we to use Benacerrafean reasoning, we would conclude that there are no progressions. Kitcher, however, cleverly suggests that we reason like White -- i.e. that we try to solve the problem by employing the same trick he employed in connection with numbers: relativize. Just as something is a 3 only relative to a progression, something is an

ordered pair only relative to an ordered-pair (OP) class. (Perhaps we should say something like Ivy is

paired with z relative to the OP,class Cvv to make this maneuver palatable to Harman, who, as noted earlier, needs it, but probably would not want to speak of ordered pairs, even relatively, any more than he wanted to speak of numbers.). So, for example, the class of all Weiner-ordered-pairs is an OP class, as is the class of all Kuratowski-ordered-pairs.

But there is a fundamental problem with this approach, which Kitcher presents as follows:

There are no OP-classes simpliciter; a class is only an OP class relative to a correlation of its elements with pairs of sets. To see this, consider the class of' Kuratowski-ordered- pairs, {{{XI, Ix,y)): x,y E V). This is an OP-class with respect to the uniform correlation of with {{XI, {x,y)), but not with respect to the uniform correlation of cx,y> with {{x?). Moreover, consider any two arbitrary sets a,b. Let f be a function of two arguments such that f(x,y) = EExI, {x,y)) (x,y E V). We can now define a function g(x,y) thus:

The class of Kuratowski-ordered-pairs is an OP-class with respect to g as well as with respect to f. Since a,b are arbitrary, there is no saying absolutely which element of the class is a . Any member of the class can be a if we choose our correlation function correctly. (pp. 125-6)

The point is this. According to White, [[[0]]] is a 3 only relative to, for example, the Zermelo progression of wnumbersu -- that~is,the set ordered by the relation R = [ : y = [x]]. Yet what precisely is R? Is R identical to the class B = [[[XI, [x,[xll1: x vl or to the class

C = [[[XI, CB, Cx111: x E Vl or to something else? Kitcherls argument shows that R = B relative -to the uniform correlation of with [Ex], [x,y]]; but relative to the unizorm correlation of

wlth [[x], [%,y]], R = C. Thus cannot even be said to be "a Zermelo progression relative to the

OP-class [[[x], [x,:~]]: x,y c V]", Nor can we speak of its being "a Zermelo progression relative to the- uniform correlation of with [[XI, [x,y]lU because that would be tu presuppose that we had a non-aribtrary account of what an ordered pair is -- and the problem, precisely, is that we don't. Hence the explication problem that White was trying to solve reoccurs at the level of progressions. That is, (30) for example was supposed to eliminate arbitrariness and vagueness; but it fails to do so, because the same features reoccur at other levels. So White's analysis is not successful.

-4.Benacerraf1s Account

I would like now to examine Benacerraf's own positive suggestion. Although motivation for it nay have declined somewhat along with the success of his argument to the effect that numbers do not exist, there is something appealing in what he says; 50 a consideration of the merits and flaws of his account seems warranted. Here is what he says:

Slogans like "Arithmetic is about "Number words refer to numbers,l1 when properly urged, may be interpreted as pointing out two quite distinct things: (1) that number words are not names of special nonnumerical entities, like sets, tomatoes, or Gila monsters; and (2) that a purely formalistic view that fails to assign any meaning whatsoever to the statements of number theory is also wrong. They need not be incompatible with what I am urging here. This last formalism is too extreme. But there is a modified form of it, also denying that number vords are names, which constitutes a plausible and tempting extension of the view I have been arguing. Let me suggest it here. On this view the sequence of number words is just that -- a sequence of words or expressions with certain properties. There are not two kinds of things, numbers and number words, but just one, the words themselves. Most languages contain such a sequence, and any such sequence (of words or terms) will serve the pu7poses for which we have ours, provided it is recursive in the relevant respect. In counting, we do not correlate sets with initial segments of the numbers as extra- linguistic entities, but correlate sets with initial segments of the sequence of number words, The central idea is that this recursive sequence is a sort of yardstick which we use to measure sets. Questions of the identification of the referents of number words should be dismissed as misguided in just the way that a questlor, about the referents of the parts of a ruler would be seen as misguided. Although any sequence of expressions with the proper structure would do the job for which we employ our present number words, there is still some reason for having one, relatively uniform, notation: ordinary communication. Too many sequences in common use would make it necessary for us to learn too many different equfvalences. The usual objection to such an account -- that there is a distinction between numbers and number words which it fails to make, will, I think, not do, It is made on the grounds that "zwei l1 "deux-3 l1 112,l' are all supposed to "stand foru the same number but yet are different words (one of them not a word at all). One can mark the differences among the expressions in question, and the similarities as well, without conjuring up some extralinguistic objects for them to name, One need only point to the similarity of function: within any numbering system, what will be important will be what place in the system any particular expression is used to mark. (pp. 71-72, WNCNB).

The gist of Benacerrafls proposal seems to be that all we need in order to count and do number theory is a recursive progression. And we have one in the number words themselves. Numbers are therefore redundant. There are several attractive aspects to what he says. It makes sense to say that in counting objects we are correlating words with objects, because we usually do recite s sequence of words when we count. So, for example, instead of analyzing (31) There are five lions in the zoo. as (32) There Is a 1-1 correspondence between the numbers

from one to five and the lions in the zoo. I suppose we could analyze it instead as (33) There is a 1-1 correspondence between the number

words from 'one1 to 'five1 and lions in the zoo. And

(34) There are more lions than tigers in the zoo, ', instead of

(35) There are numbers II, m- such that there are m lions and -n tigers in the zoo and m_ > n. could be analyzed as: (36) There are number words g, m_ such that 'there are m- lions and -n tigers in the zoo7 is true, and m- comes before -n. Also, in comparison with Harman1s and White's proposals, Benacerrafls appears to have the merit of placing less reliance on other sorts of mathematical entities. There is no reference to sets, for example; nor does he quantify over progressions. Of course, if he were to embrace (33) as the analysis of (31) he wzuld need to give some account of what a correlation is. We would be anxious to learn, among other things, how it stacks up episBemologically against numbers themselves. He requires that there be number words, but so, it might be said, does everyone else. Hence at first blush it looks as though some serious headway may have been made on the eplsternological problem presented by abstract mathematical entities. Yet a serious difficulty confronts us immediately.

Benacerraf wants it to be a consequence of his proposal that statements of pure number theory, like these: (Sl) Two is a prime number. (S2) 2 t 2 = 4. are true, even though there are no numbers. This is evidenced by the remark he concludes !'What Numbers Could Not Be" with: vl...there are no such things as numbers; which is not to say that there are not at least two prime numbers between 15 and 20." (p. 73). But it is very hard to see what his proposal comes to for statements like (S1) and (S2), much less how it accomplishes this. However, let us try. Given that (S1) and (S2) are true, (VII), (VIII) and (IX) together seem to have the consequence that numbers exist. So Benacerraf has to deny one of (VII), (VIII) or (IX). From conversations with him, I doubt that he would want to give up referential semantics, i.e., deny (1x1. Therefore he has to deny (VII) and/or (VIII). Now one way of doing this is to provide a general analysis which entails, for example, that the

logical structures of (S1) and (S2) differ from their surface structures in not having the singular terms ltwol and '2' in them. This amounts to only denying (VIII) -- precisely what Harman1s proposal does. In fact, one way of looking at Harman1s much more explicit proposal is as -a development -of Benacerrafls -own position, especially of the claim, noted earlier, that "Arithmetic is the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions1' (WNCNB, p. 70). It may be that Harmanfs proposal, or Whitels perhaps, is much nearer to being an explicit statement of what Benacerraf is proposing that anything I shall come up with. But since we have seen that Harman's proposal fails, it seems a good idea to concentrate on any alternatives, and I can think of only one. Here it is. Benacerraf seems to view statements

like (S1) and (S2) as uquasi-interpretedu. They are true, he thinks, so they must have an interpretation, but somehow he sees the interpretation as arbitrary, so long as it meets certain conditions. For example,

consider his remark that ll., .any system of objects. , . that forms a recursive progression must be adequate [as number surrogate^]^^ (p. 68). What such a claim might really amount to will be discussed at some length in the next, and last, section of this chapter. But in conjunction with the long passage quoted above, especially the claim that "There are not two kinds of things, numbers and number words, but just one, the words them- selves," it suggests that Benacerraf thinks the number words ought to "do the job" of the numbers, "stand inf1 for them. The most straightforward way of accomplishing this is by analyzing (S1) and (S2), respectively, as: (37) 'Two1 is a prime number. (38) 'two1 plus 'two1 = 'four1. The problem is that (37) just looks plain false, and (38) does not even make sense. So let us try: (39) 'Two1 is a prime number word. (40) ;two1 ''addedu (relative to the number word

progression) to ltwol = 'four1. But contrary to what analyses are supposed to do, the truth of the analysanda, (39) and (40), seems to presuppose truth of (S1) and Because what does it mean for -x to be a prime number word? Presumably that -x designates a prime number! So (39) would be true because (S1) is true, rather than vice versa. Also, the concept of l1addingV words, required by (40) seems to presuppose that of adding numbers. But maybe this is wrong. Maybe "Expression Arithmetic", as I shall call it, can be understood without prior understanding of arithmetic. A good case might be made that understanding arithmetical predicates of, and operations on, number words in fact precedes, in childhood, understandipg of arithmetical predicates, and operations on numbers. Let us, then, consider what Expression Arithmetic would be about. What is -the sequence of number words? If we wanted to keep things nice and simple, we might say, in analogy to the Peano axioms, that the sequence of number words is 'zerot, 'the successor of zero,' 'the successor of the successor of zero,' etc. But this will nct do, because the purpose of having -one sequence, Benacerraf said, was ordinary communication, and these terms are far from ordinary. Consulting my dictionary, I learn that the name '< of the first cardinal number is 'naught,' or lzero,l or 'cipher,' and that that of the second cardinal number is 'one. 140 This fact alone defeats the construction of a straightforward analog to the Peano axiom: (41) No two numbers have the same successor. which presumably would be: (42) No two number words have the same successor. because 'zero1 and 'naught1 are two number words that have the same successor. Actually, what this shows is that there is more than one sequence of lumber words in English. Perhaps one sequence could just conventionally be adopted, and then (42) would be acceptable. My

40. Websterls Seventh New Collegiate Dictionary (Spring- field, MA: G. & C. Merriam Co.; 967) p. 579. dictionary shows no more ambiguous terms until 101, which would correspond to either lone hundred and onet or lone hundred one1. Again, either one of these could be conventionally adopted so that '101 t 1 = 102l for example, would have only one logical analysis (as seems desirable). Moving right along, we have, corresponding to 10' lone billion1, to 1@12Ione trillion1, to 1015 lone quadrillion1, . . . to 10333 lone de~illi&~,to lone undecillionl, to lone duodecillionl, . . . and to 'one vigintillion1. But what do we do thereafter? The only later entry my distionary has is 'one centillion1 for 10303. What we do do about 1066? I suppose we could say lone thousand vigintillions ; and for l one million.vigintillions , but it does not takz a Georg Cantor to see that we are going to need a convention that involves indefinite reiteration of at least one number word. Perhaps there is such a convention, and the dictionary and I are unaware of it. Nonetheless, there --could be such a convention. So we can recast Benacerrafls proposal as one about what the analyses of statements of pure number theory should be, relative to the assumption that there is one standard sequence of number words in English (or rather, American English, because the British have a radically different system after one million). That is, it makes sense to view Benacerraf as proposing that (i)we select one of the several w-sequences of number words available and make that the standard one; and (ii)we interpret statements of pure number theory as being Maboutll(in the appropriate sense) members of this sequence.

So interpreted, this might be thought to contradict two tenets of Benacerrafls. First, he explicitly denies that "number words are namesf1 (WNCNB, p. ?I), and on the construal we are considering number words do seem to function as names -- names of themselves. The second tenet that might appear to have been violated is Benacerrafls thesis that we cannot identify the numbers with this or that pr~gression-- of sets, or of anything (including, presumably, number words) because to do so would be perfectly arbitrary. (Indeed, such was the basis of the argument we saw on p. 22 above that led to the conclusion (55 ), that numbers are not expressions). Yet it might appear that our construal of Benacerraf has done just that: arbitrarily identified the numbers with a progression of number words. I believe neither consideration need detain us. First, because I have doubts as to whether either objection is on the mark. It is not readily apparent to me that even if (S1), for example, is to be analyzed as (39) that ltwo' names itself, or that we have identified the number two with the numSer word 'two'. It depends on what the relation ''is to be analyzed asf1amounts to, and this is a subject I would prefer not to go into, in part because neither Harman, White nor Benacerraf does. Secondly, even if it -is a consequence of the analysis that now number words are names -- i.e., names of themselves -.- and/or numbers have been identified with expressions in some sense,, I believe it can be reconciled with the two tenets of Benacerrafls mentioned in the preceding paragraph. We can view the claims embodied in those tenets as having been made in an earlier context -- before his proposal. So that, for example, when he says "number words are not names1', this is to be construed as "number words have not functioned as names up until now, although if it is a consequence of our adopting my proposal that they do so now, why, so be itr1. Similarly, Benacerraf is not claiming to have discovered that the numbers are really words; to the extent that his proposal involves Itidentifying numbers with wordst1 it only amounts to the suggestion that we endorse the ', identification as a matter of conventl.on. The difference here is like that between the claim (to use Benacerrafls own llyardstickll analogy) made in the absence of a standard meter bar, that such and such a bar has been discovered to be the standard one, and the proposal that we let such and such a bar be the standard meter, My own inclination is to view the proposal as involving the denial of (VIII) but not, or not necessarily, (VII). So that 'two1 or '2' may or may not be singular terms in (Sl) and (S2), respectively, but the issue of what they refer to does not arise because such terms do not appear in the corresponding analysanda, (39) and (40).

It is now time to look at objections to the proposal itself. The proposal should not be seen as being merely about English. Presumably, Benacerraf is advocating that each group of different language users select some expression progression as standard, and analyze numerical truths in terms of it. This would mean that there would -- be no standard model -of arithmetic, not even one composed of expressions. Each group of different language users could use a translation of the phrase 'the standard model1, but each would thereby be referring to a different progression of expressions. Benacerraf is aware of this counterintuitive consequence, it would appear. In response (this is from the long passage quoted above) he says: "One can mark the differences among the expressions in question, and the similarities as well, without conjuring up some extra- linguistic objects for them to name. One need only point to the similarity of function.ll (p. 72). This sounds not implausible, but I wonder whether the matter is cleared up so easily. I understand that (43) Two plus two is four. is a good translation for (44) Deux plus deux font quatre. and for (45) Zwei plus zwei ist vier. They all express the fact that 2 + 2 = 4, to use a some- what more neutral terminology.41 But now it turns out that (43) is true because its analysans, (46), is true. (46) 'Two1 "added" to Itwo1 is 'four?. And (44) is true because (47) is true. (47) 'Deuxl ajout6 2 Ideux1 est 6gal B 'quatrel. And (45) is true because (48) is true. (48) tZweil llhinzufiigt"zu 'zweil 1st 'vier1. Although (43), (44) and (45) express the same fact, (46), (47) and (48) express different facts. (46) and (47) are not translations of each other. The translation of (47) is: and that of (48) is: (50) 'Zweil "added" to 'zweil is lvierl. Insofar as we deem (49) true, it is because we understand it as meaning that 'deuxl l1addedl1to Ideux1, relative to the progression of French number words, is 'quatrel;whereas - 7 -- insofar as we deem (50) true, it is because we understand it as meaning that lzweil I1addedl1to 'zweil,relative to the progression of German number words, is 'vier1. Clearly, 7 7 (46), (49) and (50) express different, even if comparable,

CI1. Since it 1s more neutrai, why aiunlt Benacerraf choose lo1, '17 '2', '3l,. .. as the standard progression (or, why didn't we construe him as so choosing)? Because, as is evident from the long passage quoted above, the number surrogates he is advocating are number words, and '2', he also claims there, is "not a word at allu. Even had he so chosen, objections like those I am urging would still be frameable, as not all cultures that employ numerals employ arabic numerals. facts; they are llaboutl' different progressions. So it looks as though Benacerraf would have to deny that (43), (44) and (45) are translations of each other.

A second, related, objection is this. According to the proposal under consideration, '2 t 2 = 4' is true because 'two1 "addedf1 to 'two1 is ffourf. But a French propor,ent would say, and here I am translating, that

'2 t 2 = 4' is true because ldeuxl l1addedr1to ldeuxl is 'quatrel. And a German proponent would say it is true because 'zweil ''addedv to 'zweil is 'vier1. Yet, as was remarked above, (46), (49) and (50) express different facts. So they cannot all be right, unless they are discussing something different -- i.e., unless '2 t 2 = 4' is highly ambiguous, meaning something different in each language. But this seems counterintuitive, Yet another way of putting essentially the same objection is this. You would think that '2 t 2 = 4 ' is true by virtue of some -one fact, mathematical, linguistic or otherwise. However, the considerations advanced in the preceding paragraph show that this is not so. Or do they? Perhaps there -is some one statement that, according to this proposal, when inserted in the blank of the following schema, yields a truth that Is not radically language-rel.at ive :

'2 -b 2 = 4' is true because We know that neither (46), (49) nor (TO) works. But perhaps '2 t 2 = 4l is true by virtue of all such statements. That is, maybe it is true because of the fact stated in (51). (51) Given any standard number word progression, the second member of it when "added1' to the second member of it yields the fourth member of it. And Benacerraf need not deny that the translation of (51) into, say, French & a translation. Unfortunately, this idea founders on the rocks that capsized Harmants (23) and White's (30). This can be seen by noting the similarity between (51) and (23) (on p. 76 above). The only significant difference is that Harman quantifies over all progressions, whereas Benacerraf quantifies over all standard number word brbgressions. But I cannot see how this would make Benacerrafls job any easier, because it still remains the case that, as we saw on p, 88 above, there is no absolute notion of a set of ordored pairs, and it is hard to see how the notion of a relation, and hence of a progression, can be unpacked without it. So it looks as though it is a consequence of Benacerrafts proposal that there is no single fact by virtue of which '2 t 2 = 4' , for example, is true. Also, that '2 t 2 = 4' is many-ways ambiguous, and that (43), (44) and (45) are not translations of each other. While unattractive, these consequences do not render the proposal incoherent. Perhaps some story can be told that will make them palatable. ,Perhaps something can be gained by pointing to "the similarity of function1' that (43), (44) and (45) would play, for example. Nonetheless, the difference between numerical truths of different languages makes numerical truths radically language-dependent. In fact, what it might be said to come down to is that there are no strictly numerical truths. The only truths to be had, whatever their similarity of function, are those such as (46), (47) and (48), and these are about words. Each language that has a standard progression of expressions can have an expression arithmetic about such expressions, but each expression arithmetic will differ from the others; each will have different truths (as (49) and (50) are different) about different objects (as the words 'deux' and 'zwei'). There would even be no one arithmetic, Upon analysis, for example, the axiom of Peano Arithmetic: 0 is a number. would be, in American-English Expression Arithmetic: 'Zerof is a number word. But in German Expression Arithmetic is would be the very different : 'Null' ist ein Zahlwort. And similarly for French Expression Arithmetic. If expressions are the proper objects of the domain, then it would seem that one could add further axioms onto one's expression arithmetic. The following statements, for example, seem of interest and importance, Why shouldn't they be theorems of American-English Expression Arithmetic?

Among the first ten number words the letter let occurs nine times. IThreel is not only a prime number word, it is the only

number word in the first ten with a 'th' in it. There are only fifteen letters of the alphabet occurring among the first ten number words. Of course, were we to try to make our expression arithmetics completer, each would start looking quite different from the others. One would wonder what had become of plain old arithmetic. No doubt someone would come along (this consideration may sound familiar from chapter one) and try to formulate what it is all these expression arithmetics have in common -- and we would be back to wondering about Peano Arithmetic again. I doubt that Benacerraf would consider even trying to make the consequences of the preceding scenario seem palatable. On the other hand, I do not know how he would attempt to avoid them, while still remaining faithful to the proposal under consideration. It seems to me that the most plausible way of avoiding all the unattractive consequences cited so far would be to find a set of truth conditions, conditions like (51), that are not- relative to a particular number word progression, and under which statements of pure number theory come out true. But we saw how (51) was not available to Benacerraf, Besides, even if it were, there is something queer about having the truth of an arithmetical statement like '2 + 2 = 4' be contingent upon the truth of (51). Recall that ''a standard number word progression" refers to one that had been agreed upon as the standard one for a certain language, and in this sense it could turn out that there may not be, now or ever, any standard number word progressions. Then

'2 t 2 = 4' would be false. Also, '2 + 2 = 4' is supposed to be the sort of thing that is true in all possible worlds, even worlds where there are no words. (I assume such worlds are possible.). So it seems unwise to make the truth of it contingent upon facts about word progressions. ' < Another serious question arises. Suppose Benacerraf were able to avoid committment to mathematical entities, that he were able to make do with linguistic expressions. Would we be any better off? Certainly, some level of ontological economy would have been effected But why do away with numbers ii: iavor of expressions? There is no transparently obvious reason for preferring the reduction, as it were, of numbers to expressions rather than that of expressions to If the former is to be preferred, reasons should be given. Benacerraf gives no reasons.

42. At least, not to me. And not to any one who thinks that my hypothetical little friend Kurt, discussed on p. 113 below, is no more remarkable than, say, Ernie or Johnny. On top of this, expressions are usually regarded as being abstract objects. In the absence of a different account of what they are, or arguments showing how, although expressions are abstract objects, propositions concerning them are knowable, I think we ought to be wary of supposing that much epistemological progress has been made. There will be more discussion of these matters in the next chapter. Thus we have seen that Benacerrafls proposal, in spite of several attractive features, runs into serious difficulties when it comes to giving an account of statements of pure number theory, like (S1) and (d2). The unattractive consequences of one way of construing Benacerraf are that ' , such statements turn out to be not simply true and in accordance with some one mathematical fact, but highly ambiguous things that mean something different in each language, This would lead to a multitude of different (exp~ession)arithrnetics, and make the truth of numerical statements contingent upon facts about our number-names. Such implausibilities are circumvented by the alternative construal; but that route, although in many ways better, interprets Benacerraf's rather inexplicit suggestion as amounting to Harmants, or something very like it, and we have seen how Harman1s proposal fares, 5.A Final Word on Benacerraf, Harman and White

Both Harman and White took their cue from Benacerraf. That is, they all agreed that we can conclude from the fact of multiple set-theoretic explications of number that there is no unique object that is the number one, nor one that is the number two, etc. And they agreed that there are sets. Since their accounts do not work, I believe we are entitled to tentatively conclude that these claims do not make satisfactory bedfellows. A bit more generally, I think we may view with suspicion any position that denies (I) and (11) while maintaining (III), (IV) and (V), unless we are actually presented with a thorough- going and consistent reductioi~. But there is little likelihood of the latter, it seems to me, because it is no coincidence that Harmanfs, White's and especially Benacerrafls reasons for analyzing away numbers (as unique objects) apply as well to other mathematical objects; mathematical objects are all of a piece, Non-spatial and non-temporal, they are members of a platonic realm of objects standing in mathematical relations to each other. Benacerraffs problem, of how we come to know truths about them, applies to sets as well as to numbers. Indeed we say in chapter one how arguments akin to his own for analyzing away numbers would have done away with sets. 43

43. I do not mean to imply that Benacerraf is unaware that his 1968 position is subject to his 1973 objections. Before leaving this threesome of philosophers, I would like to discuss a claim that many may find appealing, which appears in various forms in their and others' writings. It is supposed to be a primary motivator for the conclusion that there are no numbers. I discuss it with the hope of changing the minds of any readers who still feel the view- point represented by Benacerraf, Harman and White is basically sound, and it is just their particular formulations which are defective. Harman makes the claim in perhaps its bluntest form: (52) Any w-sequence of different sets (or other things) can be used as the numbers.

White writes : j,

For even if we take the references of numerical expressions in our ordinary arithmetical discourse to be fixed, the presence of multiple models of arithmetic makes us realize that we could equally well have referred differently and Still have come out with arithmetical truths. (P* 117)

Benacerraf writes:

Under our analysis, any system of objects, sets or not, that forms a recursive progression must be adequate, (WNCNB, p. 68)

In a similar vein, Quine writes:

Any progression will serve as a version of number so long and only so long as we stick to one and the same progression ...( Ontological Relativity p, 45) Dedekind, much earlier, may have had a like thing in mind when he wrote:

If in the consideration of a simply infinite system N set in order by a transformation @ we enTirely neglect the special character of the elements; simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the order-setting transformation @, then are these elements called natural numbers or ordinal numbers or simply numbers, and the base-element 1 is called t o base-number of the number- series N. With reference to this freeing the elements from every other content (abstraction) we are justified in calling numbers a free creation of the human mind. The relations or laws which are derived entirely from the conditions a, 8, 8, 6 in (71) and therefore are always the same in all ordered simply infinite systems, whatever names may happen to be given to the individual elements..form the first object of t& science of numbers or arithmetic.

What do these claims mean? In particular, what does (52) mean? Surely not that a TV newscaster would even be understood, much less saying something true, if (s)he were to announge: "The number of planets in our solar system has now been definitively established to be the unit set of the unit set of the null set." So the claim is not that anyone can use any w-sequence (s)he wants for any purpose. Perhaps it is that the arithmetician can use any w-sequence (s)he wants while doing arithmetic, in the

44. Richard Dedekind, Essays & the Theor of Numbers (New York: Dover Publications, Inc. 19-d! 3 p.T8. sense that (s)he can obtain results true of all w-sequences by reasoning about any one of them. If so, the claim is true (provided that the arithmetician does not focus upon properties of the particular w-sequence which other w-sequences lack) but unremarkable. Any model of a theory, however un- intended, is still a model. And if -A and -B wemodels of the same theory, then they model each other in certain respects. This no more-shows that there is no intended model than (to take an exaggerated example) the fact that chemists can arrive at hypotheses true at the molecular level by reasoning about tinker-toy models shows that there are no molecules. The rewon why the newcasterfs remark is unintelligible ' 8 has to do with communication. As Benacerraf points out: "Too many sequences in common use would make it necessary for us to learn too many different equivalence^.'^ (WNCNB, p. 72). Taking our cue from White's version of the claim, then, perhaps (52) should be interpreted to mean that all arithmeticians, or better, all of us, might have spoken a different language, one in which particular set-theoretical expressions systematically occur wherever we actually now use numerical expressions. So that, for example, 'the unit

Set of the unit set of the null set' might be used instead of !the number tenf. Then the newscasterfs rernmk would certainly be intelligible, Assuming this is possible, does it provide us with a reason for concluding that there are no numbers? Suppose that in the circumstance described, we had not developed set theory. Then I think the only thing this possibility demonstrates is that the numbers might have had different names. Suppose, instead, that in the circumstance described we -had developed set theory (so that 'the unit set of the unit set of the null setf, for example, would refer to a set, viz., {{O)), as it does now.). We would count using set-theoretic expressions, We would, in effect, be akin to Benacerraffs friend Ernie -- or Ernie before he learned how to "speak with the vulgar", as Benacerraf put it. That is, we would have set theory, and have developed a subset of it around a particular w-sequence of sets -- say, the sequence O, {a), {O, (011, (0, {%I, (0, (0111, . . . as Ernie did -- which we use for counting, and relations among whose members we study. Maybe we even call this sequence 'the numbersf and use the abbreviations 'zero1, 'one1, 'two1, etc. to refer to the respective members. If so, we are just like Ernie; we are an entire society of Ernies. What does this possibility show? I think it is that we might have had a different intended model of arithmetic than the one we do have, and still have arrived at arith- metical truths. That seems to be the gist of White's remark, and one of the conclusions to be drawn from Benacerraffs fable of Ernie and Johnny. First of all, I do not know if the situation is possible. That is, I do not know if it is epistemologically possible for a society to develop set theory before it 4 5 develops arithmetic. But if it is possible, it does not

follow that as things are either (i)we have no intended model of arithmetic, or that (ii)there are no numbers.

With regard to (i),if anything, the possibility of our functioning with a different intended model tends to support the hypothesis that we need some intended model or other, rather than undermine it. (The model in the Ernie society -is different, because we do not now, but would in that society, maintain that, say, 5 is a member of 7.). With regard to (ii), we might describe the situation as one in which we had no numbers, but instead used sets; but then again, we might describe it as one in which the numbers -are sets (e.g,, 2 = {%, (011). At best this shows that arithmetic, pure and applied, is doable even if there were no numbers, not that there -are no numbers, or that the hypothesis that there are no numbers is consistent with actual practices. But, it might be urged the fact that both pure and applied number theory is doable without supposing that

45. Notice that this difficulty is not one that Benacerraf has to face with respect to his Ernie-Johnny situation, since he asks us to imagine only -one Ernie, not a society of Ernies, He does presuppose, however, that a person could be taught a full-fledged set theory without even knowing how to count, and this may involve difficulties. there are numbers shows that numbers are dispensable, and so, by Occamls Razor, ought to be dispensed with. No, it does not; because the same sort of argument could be

advanced for any particular model of arithmetic -- e*g.9 Ernie's model. Then since it would be true to say of any particular model that we ought to dispense with -9it we would have to dispense with -all models of arithmetic, (Having no models, arithmetic would be inconsistent, by the Completeness Theorem!) Not so, might be the rejoinder, because we should dispense with only those models whose existence we lack independent justification for. But now the problem is with what counts as "independent ju~tification~~. Until we are told, and probably even after we are, nothing prohibits us from proposing similar arguments that would have the effect of doing away with, say, expressions, This move should be familiar from chapte~one above, but in case it is not, consider the following possibility. We might have spoken a different language, one in which we use numerical expressions instead of expression e~p~essions,That is, imagine & la Benacerraf, a militant Pythagorean who teaches his child, little Kurt, arithmetic and all about numbers, but when the time comes for Kurt to learn his alphabet, he is told that la1 is just another name for the number one, 'bl for two, and so on for the single signs; that vconcatenation" of "the lettersu nl, ...,

"k to yield the longer wexpressionllm is just this operation: (where p pk are the first k primes). So it would 1) p2, ... turn out that all talk of expressions is just talk of numbers, 4 6 for Kurt. Now imagine that we are a society of Kurts.

This shows that syntax is doable even if there are no expressions. Hence, were we to reason in a fashion parallel to thzt being hypothetically urged, we would conclude that expressions are dispensable, and so, by Occamls Razor, that they ought to be dispensed with. But this is ridiculous. So much for (52), the claim that "any w-sequence of different sets (or other things) can be used as the numbers1'. While at first blush, perhaps, alluring, we have seen that upon inspection it proves false if it means: Anyone can use any w-sequence (s)he wants for any purpose. And if it means either The arithmetician can use any w-sequence (s)he wants while doing arithmetic (in the sense that (s)he can obtain results true of all w-sequences

by reasoning about any one of them). or We might have spoken a different language, one in which particular set-theoretical expressions occur wherever we actually now use numerical expressions. 46. This possibility was suggested to my by Richard Cartwright . then, although true, it does not justify the conclusion that there are no numbers, As support for that conclusion, at best it represents an uninhibited appeal to Occam's Razor -- uninhibited by any explanation as to why the numbers, in particular, ought to be sacrificed. CHAPTER 111: FORMALISM

1. Introduction

One of the most serious challenges faced by the position that numbers exist comes from formalists, like Hilbert, who claim that we do not need the traditional non-linguistic mathematical entities to do mathematics at all; linguistic objects will do. This claim has often been criticized, but I wish to challenge not the claim itself, but the presupposition that seems to lie behind the claim, namely, that linguistic objects, expressions, are epistemologically accessible but mathematical entities are not. There are two main grounds as I see it that the formalist wouldadvanceby way of justifying the presup- position. First, (s)he would say that expressions are concrete whereas mathematical entities, including numbers, are abstract. And second, (s)he would say that even if expressions are abstract objects, they can be known about on the basis of their tokens, which are concrete objects -- unlike the case with numbers, because numbers have no tokens . In the next section of this chapter, section 2, i examine the first ground and argue that it is false; and in the following section, section 3, I examine the second ground and argue that it too is false. 2. First Formalist Claim

Benacerraf urged us to dispense with num'.)ers in favor of number words. Yet he was not anxious to foreso all mathematical entities -- e.g., sets. This half-hearted formalism is one source of the difficulties his position faces, as we saw earlier, So one might think that a more thoroughgoing formalism would be less problematical. These passages of Hilbertts suggest that he would deny not only (I) and (11), but (111) and possibly (IV) as well:

No more than any other science can mathematics be founded by logic alone; rather, as a condition for the use of logical inferences and the performance of logical operations, something must already be given to us in our faculty of representation, certain extralogical~concrete objects that ar? intuitively present as immediate experience prior to all thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that neither can be reduced to anything else nor requires reduction. This is the basic philosophical position that I regard as requisite for mathematics and, in general, for all scientific thinking, under- standing, and communication. And in mathematics, in particular, what we consider concrete signs themselves, whose shape, according to the conception we have adopted, is immediately clear and recognizable,..All the propositions that constitute mathematics are converted into formulas, so that mathematicq becomes an inventory -of formulas. 7-

47. "The Foundations of Mathematics," From Frege to Gadel, edited by Jean van Heijenoort (cambridgnass. ~arvGdm7) pp. 464f (my italics). Among the llconcrete signs" are

In number theory ...the numerical symbols

where each numerical symbol is intuitively recognizable by the fact it cc-tains only 1's. These numerical symbols ~hichare themselves -our subject mattgg have no significance in themselves. Formulas [are] always used exclusively for communication in intuitive number theory. The letters [stand] for numerical symbols and an equation communicated the fact that the two symbols coincided. In algebra, on the other hand, we regard expressions containing letters as independent structures which formalize the material theorems of number theory. In place of statements about --numerical symbols', we haveformulas which -are themselves the concrete objects of intuitive (p. 145, "On the Infinitelf, my italics

So for Hilbert the objects of study in mathematics are numerical symbols and formulas -- expressions, in other words. Notice that he refers to them as "concrete objects1'. Hartry Field goes a step further than Hilbert does; he explicitly denies (IV) and even (V). (Hence he would deny every one of (I) - (V),). This is evident in the following passages from Science Without Numbers. 49

Nominalism is the doctrine that there are no abstract entities. The term 'abstract entity1 may not be entirely clear, but one thing that does seem clear is that such alleged entlties as numbers, functions, and sets are abstract -- that is, they would be abstract if they existed. In defending nominalism therefore, I am denying

48. "On the Infinite,ll in Philosophy of Mathematics edited by Benacerraf and Putnam, p. 143,my Itallcs,

49. Princeton, N.J.: Princeton University Press, 1980, that numbers, functions, sets or any similar entities exist. (p. 1) Toward that part of mathematics which does contain references to (or auantifications over) abstract entities -- and this includes virtually all of conventional mathematics -- I adopt a fictionalist attitude: that is, I see no readon to regard this part of mathematics as true. (p. 2)

In spite of his disavowal of abstract entities, he endorses a nominalistic theory that "...contains, besides the usual quantifiers 'v' and 'a', also quantifiers like ' ~87(meaning 'there are exactly 87') ..." (p. 21). Since quantifiers are expressions, this suggests that he thinks that expressions are not abstract objects. And indeed this is supported by his statementithat

...nominalistically inclined philosophers [usually] try to reinterpret mathematics -- reinterpret it so that its terms and quantifiers don't make reference to abstract entities (numbers, functions, etc.) but only to entities of other sorts, say physical objects, or linguistic expressions, or mental constructions. (p. 1)

Like Hilbert, Field thinks (i) the objects that mathematics is traditionally viewed as being about don't exist, but (ii) there are expressions. How would either respond to the charge that I wish to make, that such a position is not justified on epistemological grounds, because numbers, at least, are on an eplsternological par (roughly) with expressions? I think the answer is clear enough from the passages quoted above: both would deny that numbers are even on a rough epistemological par with expressions, saying that numbers are- abstract, wherers expressions -are concrete objects. Whatever the abstract-concrete distinction amounts to, nuii.bers are generally regarded as abstract objects par excellence -- and I for one have no desire to challenge that. Expressions, however, are another story. Hilbert and Field classify them as concrete, but their reasons for doing so are obscure. Field says nothing further about the concrete/ abstract status of expressions that I am aware of. Hilbert, as we saw in the first passage quoted above, characterized the concrete objects in question as being wintuitively present as immediate experience prior to all thought.ll This suggests that the expressions in question, or at least all of their parts, should be directly experiencable, and not known by virtue of an inference. He also stipulates, in the same passage, that

...it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that neither can be reduced to anything else nor requires reduction.

I shall refer to this set of conditions that Hilbertls concrete objects must meet as the llperceptibility requiirernentvl, although it amounts to a number of separate requirements. (The requirement that "it must be possible to survey these objects completely in all their partd'is sometimes rei'erred to as the flsurveyabilityflrequirement. ). I call it the ffperceptibility requirementu because, while it is less than totally clear, it seems to stipulate that, at the minimum, the concrete objects In question (expressions) and various things about them, be perceptlble (by means af the senses, or, perhaps, with the "mind's eyeff). Logicians who apparently share Hilbertfs opinion about expressions belng concrete include Joseph Shoenfield, for whom "a sentence...is a concrete ~bject.,.'~because it is an "object which appears on paper when we write down 50 an axiomv , and Benson Mates, for whom "a sentence, at least in its written form, is an object having a shape accessible to sensory perception, or, at yorst, it is a set of such objectsn.51 No doubt some sorts of expressions -are perceptible -- that is, are physical objects or events, But are the ones Hilbert, Field, Shoenfield and Mates are talking about perceptible physical objects or events? Quite frankly, no. They cannot be, Shoenfieldfs formation rules for formulas, for example, entail that there are infinitely many formulas (p. 15). Field's nominalistic theory, as we saw, "contains, besides the usual quantifiers fvl and

'af, also quantifiers like lag7 . . .If. I take it he means:

50. Mathematical Logic (Reading,Mass Addison-Wesley, 1967) p*2. 51. Elementary Logic (New York: Oxford, 1972) p. 10. for every E, his theory contains a quantifier with the numeral for -n as a subscrlpt. (He allows himself to present his ideas platonistically.). This requires infinitely many quantifiers. And Hilbert requires infinitely many numerical symbols (1, 11, 111, ...) for the objects of study of number theory. Each of these authors needs infinitely many expressions of one sort or another, and there simply aren't infinitely many perceptible physical objects. Let us pause to see why. First of all, there may r~oteven be infinitely many physical objects. As we remarked in chapter one, if space-time is finite (and there is reason to believe it is), then given any standard positive real value v, no matter how tiny, if we regard the whole physical universe as being divided up into space-time volumes of size (quadrubic inch-seconds or whatever the unit would be) there would still be only finitely many such physical volumes. In fact, there would still be only finitely many -sets of them. Secondly, the requirement that the objects be perceptible, no matter now broadly construed, rules our taking teeny-tiny objects as expressicns; so how small -v can be is limited. And thirdly, given this last fact, there certainly aren't infinitely many strings of 1's that can be llsurveyed completely in all their partsf1, as Hilbert requires, because most would just be too long to be surveyed, But it might be objected that the surveyability requirement is just plain unreasonable, and so should be dropped -- or at least construed as "surveyable by Godff, or something equally generous. More importantly, it might be objected that the preceding discussion, especially wherein the physical universe was viewed as being partitioned into space-time volumes, presupposed that the objects in question could not c~verlap. Roderick Chisholm suggested that since a blank piece of paper could be cut up so as to form (a token of) an expression, say, of !catr, and since the token would just a particular bit of paper, the token of 'catt already exists before the paper is cut up, since that particular bit of paper does. ' # Alternatively, it could be cut up to form a token of !dog1. So that exists too, And so on. Thus there being a finite amount of matter might be compatible with there being infinitely many expression (tokens). There are two respects in which this suggestion fails, First, even if we agree that many words exist as parts of the paper even though the paper will never be cut up, it seems to me that these expressions fail to meet the perceptibility requirement. We have not inquired too closely into what this requirement amounts to. In a sense the token of 'catf, the token of 'dogt and all the other tokens are perceptible because they are lfpartsf?of the paper, and the paper is perceptible. Yet it seems clear enough from Hilbertfs insistence that "...the fact that [the expressions] occur, that they differ from one another... [should be] immediately given intuitively together with the objectsv1("Foundations of Mathematics1I, p. 464) that the expressions in question ought to be distinguishable from one another even if there is some overlap, and this seems violated in the present case. The second respect in which Chisholmls suggestion fails is that even if the paper token of 'cat1 exists, because it is part of the paper, and similarly for 'dog1 and so on, there will not be infinitely many expressions that are "parts" of the paper. The land so on1 cannot be construed as lad infinitum1 -- or at least I cannot see how it can be so construed. For consider written English expression (types). There are only finitely many letters, words, sentences of length 11 for any particular n, and sentences of length less than or equal to 5. The fact that there are infinitely many English expressions is due to the fact that given a sentence of length 5, there is always a sentence of length greater than n. Now I do not think it is unreasonable to assume that there is some positive real value, call it -r, that is the lower bound on how many square inches, or Angstrom units, are required to instantiate the smallest word. (Chalk this assumption up to the perceptibility requirement, if it is thought that some justification is needed.). Then even if the piece of paper is infinitely divisible (a dubious supposition, but the paper is in space, and doubtless consists of some space, and it may be that space is infinitely divisible) the paper itself is only finitely divisible by ;. In particular, if the paper is of size 2, then the largest number of words that can be instantiated by means of it is less than or equal to s/r (or, more precisely, it is less than or equal to the smallest whole number less than or equal to s/r). So there is an upper bound to the length of the sentence that can be accomodated; and hence infinitely many English sentences cannot be "partst1 of the piece of paper, even in this very liberalized sense of l1part1!. There is no need, it might be urged, for the expression tokens to be all of different types. Because if the piece of paper is, say, 8'' x ll", then a 1" x 2" token of 'catt cut out of the paper would be a different token from a 5" x 10" token of 'cat1 cut out of the paper, even though their parts overlap. Now, the objection continues, if space is infinitely divisible, then, as was indicated above, it is plausible to think that the paper is too. Hence there are infinitely many different tokens of 'cat between 1" x 2" and the 5" x 1011 tokens, all of which exist concurrently as ltpartslt of the paper. The appropriate rejoinder to this, I think, is to point out that infinitely many such tokens would be partitionable into a finite number of equivalence classes, depending on the atoms of paper they consist of. There will be nc possible way of perceptually discriminating among the members of an equivalence class. They will be perceptually equivalent. To suppose, therefore, as this objection must, that token a- is not identical with token b- even though and -b do not differ perceptibly is to violate the perceptibility requirement. Once again, it has the consequence that a piece of paper can only support the existence of a finite number of expression tokens.

The upshot of all this is that if being "concreteI1 entails being vperceptiblem, then even though some sorts of expressions -are perceptible physical objects or events, the sorts that Hilbert, Field, Shoenfield and Mates are concerned with are not. What these writers have in common '8 is that in their anxiety to avoid committing themselves to certain abstract objects that they found objectionable -- mathematical ones, in the cases of Hilbert and Field, meanlngs or facts, for Shoenfield (p. 21, and statements, propositions, thcughts, and judgments for Mates (p. 10) -- they forgot to take heed of the type-token distinction, Peirce, who introduced the nomenclature, described it thus:

A common mode of estimating the amount of matter in a MS. or printed book is to count the number of words. There will ordinarily be about twenty the's on a page, and of course they count as twenty words. In another sense of the word 'word1, however, there is but one 'the1 in the Engl-ish language; and it is impossible that this word should lie visibly on a page or be heard in any voice, for the reason that it is not a Single thing or Single event. It does not exist; it only determines things that do exist. Such a definitely Significant Form, I propose to term a Type. A Single event which happens once and whose identity is limited to that one happening or a Single object or thing which is in some single place at any one instant of time, such event or thing being significant only as occurring just when and where it does, such as this or that word on a single line of a single page of a single copy of a book, I will venture to call a Token. ...In order that a Type may be used, it has to be embodied in a Token which shall be a sign of the Type, and thereby of the object the Type signifies. I propose to call such Token of a Type an Instance of the Type. @

Once that distinction is before it is fairly clear, I think, that Hilbert, Field, Shoenfield and Mates ', are, or ought to be, concerned with expression types, not tokens. This is so because each of them wants there to be infinitely many such things, but unless the universe itself is somehow infinite there are only finitely many expression tokens, as these are physical objects or events, Nor could it be merely -sets of tokens that they are concerned with, contrary to what Mates said, because there are only finitely many of these too (again, unless the universe itself is somehow infinite). So it must be expression types -- and these are abstract objects. These points might well have

52. Collected Papers of Charles Sanders Peirce, Vol. IV, edited by Hartshorne and Weiss (Cambridge: Harvard University Press, 1958) p. 423, from ltProlegomena to an Apology for Pragmatism". been apparent to the authors in question had they not, as was indicated above, other fish to fry. Goodman and Quine consider the question directly of whether one can renounce abstract entities and still assume there are infinitely many concrete objects -- especially expressions. 53 At the time, both were nominalists; ("We do not believe in abstract

entities." (p. 137)). Here is how they answered it, and why :

We decline to assume that there are infinitely many objects, Not only is our own experience finite, but there is no general agreement among physicists that there are more than finitely many physical objects in all space-time. If in fact the concrete world is finite, acceptance of any theory that presupposes infinity would require us to assume that in addit$on to the concrete objects, finite in number, there are also abstract entities. (p. 174) Classical syntax, like classical arithmetic, presupposes an infinite realm of objects; for it assumes that the expressions it treats of admit concatenation to form longer expressions without end. But if expressions must, like everything else, be found within the concrete world, then a limitless realm of expressions cannot be assumed. Indeed, expressions construed in the customary way as abstract typographical shapes do not exist at all in the concrete world; the language elements in the concrete world are rather inscriptlons or marks, the shaped objects rather than the shapes. (p. 175)

By adopting the strategy they do, they run the risk, for example, that some (of what they consider to be) formulas

53. "Stens Toward A Constructive Nominalismn in Problems ana Projects (Indianapoiis ana New Yorlk: Bobbs-Merrill, 1972). will not be theorems merely because no inscription dld or will exist to be a needed line in a proof. To decrease the risk of this sort of thing happening, they adopt a ploy that appears very much like that suggested by Chisholm, above. They include in the stock of inscriptions ll.. .not only those that have colors or sounds contrasting with the surroundings, but all appropriately shaped spatio-temporal regions even though they be indistinguishable from their surroundings in color, sound, texture, etc.lt (p. 175). It is important to note that they have no illusions that adoption of such a ploy gains them infinitely, or indefinitely, many expressions: "The number and length of inscriptions will still be limited insofar as the spatio- temporal world itself is limited," (p. 175) they caution. Besides, there is reason to believe that their ploy differ? substantially from the one suggested by Chisholm l.n other respects, too. It is not clear from the above passage that Goodman and Quine wish to allow overlapping inscriptions (so that, e,.g., a token of 'dog' could consist of much the same paper as a token of tcatt), That Goodman, at least, would not so approve is suggested by the fact that in Languages of -~rt~~ he requires of a notational scheme (and he holds English alphabetical notation to be one (pp. 140ff)) that any two characters of it be disjoint; that is,.that no mark may belong to more than one character (p. 133). The

5 4 e Indianapolis and New York: Bobbs-Xerrill , 1968. ''marks'' here just consist of (parts of) the paper itself; the characters are 'cat' and 'dog1. In the Chisholm scenario a majority cf the paper molecules comprising the !dog1 mark will comprise the 'catr mark too, and at the same time. It is not perfectly clear whether Goodman's requirement that "no mark belong to more than one character" rules out this possibility, but I bring up the matter because it looks suspiciously like it might. Lest the reader think that short and uncharitable shrift is being made of Hilbert, let us discuss the matter a bit more. It might be urged that Hilbert requires only finitely many numerals (which, recall, are strings of lrs for him). Unfortunately, this is not so, because in his ' , view of multiplication, for example, the product is a longer numeral than either multiplicand. Or, it might be urged that Hilbert requires, not an actual infinity, but merely a potential- infinity of numeral (tokens). Yet it is hard to see how this observation, if true, helps. Hilbert himself, at one point in "On the Infinite1' writes:

Someone who wished to characterize briefly the new conception of the infinite which Cantor introduced might say that in analysis we deal with the infinitely large and the infinitely small only as limiting concepts, as something becomlng, happening, i.e., with the potential infinite. (p. 139)

But "as regar~sthe infinitely small and the infinitely large," he says, ''the universe is finite in [these] two respectsqq. (p. 137). And not only in these respects: lq...the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for thought." (p. 151). It sounds to me like Hilbert is renouncing even potential infinities. Besides, if everything in nature is finite, then there are only finitely many numeral tokens, and so there is a longest one(s). Perhaps it is possible that there could have been a longer one. But since we are, talking about concrete objects, the notion of possibility that seems most relevant is that of physical possibility -- and perhaps it is not physically possible that there c~uld have been a longer one, much less infinitely many longer ones, Another way to try to exploit the notion of possibility is to argue that even though there may be only finitely many actual tokens, there are infinitely many possible ones, But it is hard to see how this idea would help either. An infinity of possible objects strikes me as more problematical in many respects than one of abstract objects. Perhaps this is the way someone like Field would care to go, though, In that case, he owes us an account of these infinitely many possible objects: in what sense they are possible, and under what conditions they are identical. This last requirement is notoriously difficult to fulfill. But even if he were to succeed in providing us with such an account, in order to defend the thesis that expressions are concrete objects whereas numbers are not he would have to satisfactorily explain how it is that nonexistent objects are concrete ones. (If they were to exist, they would be concrete? This would appear to be true of Godzilla, King Kong and the twenty fat men in the doorway, none of whom, fortunately, seem the least bit concrete.). Michael Resnik points out that "it is possible that Hilbert thought of written numerals as Kantian constructions in intuitionv, (Frege and the Philosophy -of Mathematics, p. 99). While not as readily subject to difficulties involving infinity, this perspective is also subject, Resnik argues, to a Fregean objection to ant^^ which Resnik poses as follows: "How do we see by direcp inspection, that a numeral of length 1,000,003 is shorter than a numeral of length 1,004,004?fl (p, 99). Hilbert could say: we can see, because all such numerals are surveyable. But in what senae are very long numerals ~s~rveyable~~?Resnik suggests that

...[Hilbert] might argue that our operations with short numerals reveal patterns that we I1see" must hold for larger numerals and permit us to llsurveyuthem as well. For example, in rearranging the terms of small sums we observe a pattern which we express by asserting the general associative and commutative laws of addition for all numerals...Finitary statements about small numerals would be directly evident, as would be the patterns recognizable through their aid, Finitary statements involving large numerals would then be 55. Frege gives it in The Foundations of Arithmetic (Evanston, Ill. Northwestern ~nlverzyPress, 1980)sections 5 and 89. justified by applying these patterns or general laws ...Hilbertts ontological problems could be straightened out & recognizing abstract symbol types whose properties are known through observing the patterns to which their inscribed instances conform. (p. 100, my italics).

Once again, as the italicized passage is meant to indicate, Hilbert is driven to expression types, which are, of course, abstract rather than concrete objects. There seems no avoiding the conclusion that if you want infinitely many objects -- or even, perhaps, finitely many some of which are very very large -- you must renounce pure nominalism. I conclude that Hilbert and Field (and Mates and Shoenfield) are just wrong in claiming that expressions of the sort they are concerned with are concrete objects whereas numbers are abstract. Basically, the reason is because they need, or assume there are, infinitely many such expressions, and this assumption is justified only with respect to expression types, not tokens, and these are abstract objects. Some, like Goodman, might just take this to show that one oughtn't to assume that tl-?re are infinitely many expressions. By not so assuming, one can hope to consistently maintain that expressions are concrete objects. So let us consider the question: Are there infinitely many expressions? According to classical proof theory, yes, clearly.

Because if p and q are sentences, then their conjunction disjunction, or alternative denial, etc.) will be too, and no sentence is identical with a conjunction of which it it a part. On the other hand, in Goodman and Quinets nominalistic theory of syntax, as presented in llSteps Toward A Constructive N~minalisrn~~this need not hold. How are we to choose between them? There is no simple way, There is no obvious fact to the matter; some would say that there is no fact to the matter at all. What we do in such a situation presumably is to compare theories along several dimensions -- simplicity, fruitfulness, degree of confirmation, among other things'. And of Goodman and Quinels construction of a nominalistic theory of syntax, I think Samuel Johnson's remark about a woman preaching and a dog's walking on his hind legs is applicable: "It is not done well; but you '# are surprised to find it done at all.f1 For the result of dealing with expression tokens rather than types is a rather cumbersome theory, and one with the consequence that, for example, some formulas may fail to be theorems because there happen to be no sequences of inscriptions constituting proofs of them, as was noted earlier. This just runs contrary to the needs of most proof theorists, who prefer to be able to assume that the set of formulas is closed under such operations &s conjunction and alternative denial, and that the set of theorems is closed under logical entailment, Nor does the situation improve for a 'Ifiniti~t'~like Goodman when we look to linguistics for an answer to the question: are there infinitely many expressions? -- contrary what one might suppose. For although, again, there appears to be no longest sentence of English -- one can always preface a sentence with 'Harry thinks that1, or conjoin two sentences -- one might suppose that there was a bit more motivation for having a theory that countenanced only finitely many expressions. Because there is good reason to believe that no one ever did or will inscribe or utter a sentence of, say, one billion words, and what the set of English sentencesis~ssupposedto have some intimate connection with speaker-hearers of English. Yet confronted with such considerations, most linguists today would, I think answer the question whether there are infinitely many sentences in the affirmative, Here, for example, is Emmon Bach's response:

...the set of English sentences is not some- thing that exists apart from theories about language and about English ...The only way in which we can decide the question is to ask about the kinds of theories that would lead to one or the other answer..,On the one hand, it seems that there is no longest English sentence; on the other hand, this seems to lead us to say that there could be an English sentence a billion words long. The way out of the impasse is to... [separate] out a theory of grammar from a theory of performance. The set of grammatical English sentences is infinite; there is no longest English sentence. But the set of actually usable English sentences is finite, not because of the grammar of English, but because of other factors, mostly completely nonlinguistic: limitations on attention span, facts about the length of human life, and so on. ...If we decide to treat the obvious limitations on length and complexity as a matter to be dealt with in theories of performance, we can account for the way in which sentences are used as fundamental building blocks for other sentences, and in addition we will be able to account for other facts. If limitations on length and complexity are a function of language use rather than knowledge about language, then changing the conditions of performance ought to result in changes in those limitations, and this indeed seems to be the case. If we allow people increasing lengths of time to prepare an utterance, or if we furnish them with external aids to memory, they can produce and understand longer and more complicated sentences. All of this points up to the fact that the set of English sentences is a theoretical construct rather than something directly observable or given. The set of acceptable utterances o English is another theoretical construct.. .$6

I think these two important points emerged quite nicely in the course of Bachls discussion: (i)whether or not there are infinitely many sentences, even what a sentence is or what English is, is a theoretical matter, rather than a datum or a straightforward matter of empLrical fact; and (ii) according to the best theory(ies) available, there are infinitely many English sentences. (i)and (ii)seem to me to be correct. For besides the reasons Bach gives for thinking that there are infinitely many sentences, Chomsky cites this one:

56. Syntactic Theory (New York: Holt, Rinehart and Winston, 1974) PP. 25f. In general, the assumption that languages are infinite is made in order to simplify the description of these languages. If a grammar does not have recursive devices... it will be prohibitively complex. If it does have recursive devices of some sort, it will produce infinitely many sentences. 57

Simplicity is a well-known desideratum for scientific theories. And especially so in the case of a grammar for a natural language, as there are good arguments for the claim that in order for speaker-hearers to speak/hear, they must have knowledge of such grammars. So one might plausibly argue that since knowledge of a grammar, if acquired, is generally acquired in a few years1 time, it probably is not the case that the grammar itself is "prohibitively complexff. Sketchy though this has been, I do not intend to pursue further the issue of why the favored theories in linguistics are/should be committed to infinitely many sentences -- i.e., the status of (ii), For this seems to me to be a fairly complicated technical question that is best not gone into at length here. The point is that it -is a theoretical matter what a sentence is, and how many there are. And for various theoretical reasons, linguists doing syntax decided that there are infinitely many sentences, and that the sorts of expressions they are studying are

57. Syntactic Structures (The Hague: Mouton & Co., 1964) PP* 23f. types. 58 So it looks as though we are on pretty firm ground in claiming that just as if there are any numbers there are infinitely many, and they are abstract objects, so too there are infinitely many expressions, and hence (for the reasons given on pages 117 to 113 above) they are abstract objects. Some might feel that what current linguistic theory has to say does not decide the issue of how many expressions 6 there are, The following line of reasoning might be urged. "Clearly, -if there are numbers there have to be infinitely many of them, But it is not so with expressions. The choice of infinitely rather than finitely many expressions is precisely that -- a choice. It is a fairly arbitrary one at that, based as it is largely on considerations of simplicity. Unlike the situation with number, theories in which there are only finitely many expressions are possible. Writers like Charles P. Hockett and D,L. Olmsted even urge that we adopt such theories. But theories in which there are some but only finitely many numbers are nonsense. The point is that the situation with expressions is quite

58. For example, Chomsky writes that "each sentence is representable as a finite sequence of...phonemes (or letters)If (p. 13 of Syntactic Structures). Phonemes, in their turn, are portrayed by Jakobson and Halle in Fundamentals of Lang~iageas types: Whether studying phonemes or contextual variants ("allophones"), it is always, as the logician would say, the usign-designfl and not the "sign-eventu that we define." ('S-Gravenhage: Mouton, 1956, p. 13). disanalogous to that of the numbers; whatever current linguistic theory says, there still might be only finitely many expressions -- it at least makes sense to suppose that there are only finitely many expressions -- and if that is the case, then expressions need not be abstract objects.ll

What I want to urge in response to this line of reasoning is that the situation with expressions is very similar to that involving numbers, contrary to the view enunciated in the preceding paragraph. So that if there is reason to believe that if there are any numbers, there are infinitely many of them, there is reason to believe that there are infinitely many expressions. For, first, it is true that in a sense there was, or is, a choice as to how many expressions to assume there are. But this is so with numbers too; there is a choice as to how many numbers to assume there are: finitely many or infinitely many. Yet the fact that there is a choice does not mean that the matter is completely aribtrary. Good reasons can be given for the choice of infinitely many numbers -- and expressions too, as was indicated above. And, second, it is true that linguistic syntactic theories @v-finite models are possible (that is, they are not intrinsically incoherent) and even espoused some writers. But again, a similar situation obtains in mathema4;ics. That is, there are theories about number that have finite models, and writers who espouse such theories. Let me amplify. Recall that the basis for there being a I1choicef1 about how many expressions of English -- sentences in parti- cular -- there are lay in the fact that we have conflicting intuitions about the matter. On the one hand, if g and q are sentences of English, then so is their conjunction; hence it seems there is no longest sentence of English. On the other hand, we observed in support of the hypothesis that there are only finitely many that we can be reasonably confident that no one ever did or will inscribe or utter a sentence of, say, a trillion words. Because even at five words per second it would take nearly 6,342 years to utter the sentence. In current mainstream linguistic theory, resolution is achieved by distinguishing competence from performance, and saying that while there are infinitely many sentences, factors affecting performance render the number of acceptable sentences finite. This situation has its counterpart in a pair of t conflicting intuitions that occur with respect to numbers. On the one hand, given a number we can always add 1 to it to obtain a greater number, so there seems to be no greatest number. On the other hand, we can be reasonably confident that no one ever did or will count (by 1's) up to, say, a trio Because, again, 1012 seconds is nearly 31,710 years. This would support the hypothesis that there are only finitely many numbers. In the case of these Intuitions, too, resolution can be achieved by pointing to a sort of competence-performance distinction, and saying that while there are infinitely many numbers, factors affecting perfor- mance limit how many zan feasibly be counted up to. Letls call them the 'lfeasible" numbers (for reasons that will soon become apparent). I am not claiming that everyone will feel this last- 12 mentioned intuition -- that no one can count up to 10 -- to be pressingly relevant to the issue of how many numbers there are. An opponent may be tempted to exclaim: ltThatls absurd. If -n is a number, so is n- + 1. So if there are any numbers, a trillion is one of them." But this is just to insist on the first of the two conflicting intuitions; it will not make the second go away, or provide any resolution for it. In response to the suggestion that there are only finitely many sentences, a linguist could exclaim in similar fashion: "That's absurd. If the result of replacing In1- by an arabic numeral in 'There are g guinea pigs in the field1 is a sentence, so is the result of replacing, In1- by the next arabic numeral. 59 Or, if p and 9 are sentences, them .so is their conjunction. "But ,I1 my opponent may continue, '!why think that in order for a number to exist we should be able to count up to it? Why, the set of so-called feasible numbers wonlt even be well-defined.I1 This, too,

59. This is basically one of Bachls examples. It is from -An Introduction to Transformational Grammars (New York: Holt, Rinehart and winston, 1964), p. 13. has its linguistic analog: "Why think that for a sentence to exist we should be able to utter it? After all, the class of acceptable sentences is not going to be well-defined either." The point I wish to make here is not that there are only finitely many numbers, for that is a claim I would not care to defend. The point is that claiming there are only finitely many expressions is analogous in important respects to claiming there are only finitely many numbers. Just as the latter claim relies upon some assumption as that a number is something that is counted up to, or -can be counted up to, the former claim can be maintained only by insisting that for something to be a sentence it must be uttered or inscribed, or be capable of being uttered or inscribed. So if the existence of some numbers implies there are infinitely many, then it seems to me that we have ample ground for concluding that there are infinitely many expressions, too, Still my opponent may not be satisfied. Continuing the line of criticism sketched earlier, he or she may say that infinity is part of the very notion, or definition, of number -- whether or not therelactually are any -- so that no theory of number is possible that has a finite model. By way of supporting this claim, he or she may point to the fact that the currently and widely accepted theory of nctural number consists of the Peano axioms, which taken together have no finite models. And that even though this particular theory of number did not exist until Dedekind enunciated it in the late nineteenth century, previous notions of number had, whatever their differences and flaws, incorporated the intuition that there is no largest number -- that if -n is a number, so is -n + 1 -- thus ruling out the possibility of there being finitely many numbers. No such agreement exists on -the number of expressions, though (it will be said). For example, Charles F. Hockett, in State --of the 60 -Art claims that "it is empirically absurd.,,that the millionth -- or even the thousandth -- term of [the series one. one and one. one and one and one. ...I is in fact.. .English, just as a million is not a possible

football score." (p. 60). Of course, he would be inconsistent if he also maintained that if the nth term of the series is English, so iI the -n + lSt,but he does not. He foresakes this intuition, giving as justification:

this does not mean that we can specify exactly which term of the series is the largest that -is good English, any more than we can specify the largest possible football score or write down the formula for the largest possible methane-series hydrocarbon molecule. A8 one attempts a larger and larger sentence of the kind shown, or of the kind defined by any other open-ended pattern, one encounters certain- flexible constraints, that are, in my opinion, part of the language, just as the time limits orafootball -game are Dart of football.

60, The Hague: Mouton, 1968. (pp. 60-61) Hockett is not the only one to argue for "finitismlf in linguistics. D.L. Olmsted compares the situations of linguists studying language in the field with that of linguists approaching it "from a background in symbolic logic and in contexts which include programming 61 computers to aid in analysis or tran~lation.'~ He cl.aims that "the assumption that English sentences'are not always less than a million words in length smells of the computer center rather than of the field situation. The field worker is quite ready to assume the contrary." (p. 304). Another reason he gives for viewing the assumption in question "with su~picion,'~he says, is that "sentences consist of more than words...there are also intonation patterns ...[I]t may ' I be stated as a theorem thst a sentence-terminal pattern will coincide with the end of a breath group at ].east once in every 11 breaths, where -n is a digit greater than zero.!' (p. 304). And presumably, a million word sentence violates this themem. "So," my opponent might conclude, "theorizs about r~umberhave to be ffinfinitistic," but theowies about expressions need not be, as is evidenced by the existence of responsible alternative views. Hence, there -is an important disanalogy Setween numbers and expression^,'^ Nevertheless, I still think the analogy obtains, Here's why. It is true that there has been a fair degree of consensus on the issue of whether there is a greatest

61. I1On Some Axioms About Sentence Lengthff, Language 43 (1967) 303-5s P* 303. number. But not, as the preceding sketch would have us infer, unanimity. Just as llfinitism" in syntax, as I shall call it, has its proponents, ''finitism" in mathematics does too. Disagreement over whether there are any actual, as opposed to potential infinities goes back at least to Aristotle. But in modern times A.S. Yessenin-Volpin goes a step further. 62 He asks:

why has such entity as lo1* to belong to a natural number series [sic]? Nobody has counted up to it (1012 seconds constituting more than 20,000 yef$s) and every attempt to construct the 10 -th memberlqf sequence 0, 01, Ow, . . .requires just 10 steps. But the expression ln steps1 presupposes that -n is a natural number i.e. a number of a natural number series. So this natural attempt to construct the number 10J2, in a natural number series involves a vicious circle. (pp. 4f)

As a result he claims not to ".. .really believe in the existence of a series containing 10''~ (p. 5), evidencing more interest in "the series F of feasible numbers, 1-e,, of those up to which it is possible to count.'' (p. 5). My opponent's first inclination upon hearing this

62. In "The Ultra-Intuitionistic Criticism and the Anti- traditional Program for Foundations of Mathematics,l1 Intuitionism and Proof Theory (Proc. Conf., Buffalo,N.Y.1968) (Amsterdam: ~zhHolland, 15701, pp. 3-45. And he claims that "many people,, e.g. Borel, Frechet, Mannoury, Riezer and van Dautzig doubted [the uniqueness (up to isomorphism) of the natural number series], or the lfinitenessl of very great numbers like 10 (van Dantzig) . p. 4n. 10 might be to say that any such theory would be inconsistent; that -of course lo1* is a number, and to maintain otherwise is to contradict some induction axiom. But Yessenin-Volpin rejects the principle of mathematical induction (and many others of what he calls the "traditional assumptions under- lying the body of modern mathematicsv1 (p. 4)). So however eccentric his view, it cannot be faulted for out and out inconsistency. His/her second inclination might be to say something to this effect: Yessenin-VolpinTs theory, or view, is not one about -the natural numbers; for by "natural numberv1I mean something that accords with some one of the usual inductive definitions and behaves in accordance with Peano . 0 Arithmetic. Yessenin-Volpinls may be a theory about the vfeasiblev numbers, as he calls them, but at best this constitutes only a subset -- and probably an ill-defined one -- of the natural numbers. Such a response would be understandable. After all, the inductive-Peano Arithmetic conception of number is by now very well-entrenched, --But the linguist's --view of expressions -as abstract objects -of which there -are infinitely many -is quite well-entrenched, -too. We have, corresponding to the historjcal facts claimed by my opponent, some linguistic facts -- to wit, that according to current syntactic theory there are infinitely many expressions. And that even before current theory was enunciated language theorists were committed to more than the finite i~ linguistics. (For example, Chomsky refers to Wilhelm von Humboldtfs 1836 view that a language "makes infinite use of finite meansff.63) So it would -also be understandable if, onf fronted with the existence of a fffinitistic" view like Hockettfs, a linguist responded in a fashion similar to that manifested by my opponent when confronted by Yessenin-Volpinls lffinitisticff view of number, That is, he or she might first accuse Hockett of inconsistency. To which the corresponding appropriate reply would be to point out that Hockett forsakes the intuition that if the result of replacing In1- by an arablc numeral in a sentence such as 'There are -n guinea pigs in the fieldf is a sentence, then so is the result of replacing -n by the next arabic numeral. Secondly, he or she might say something to the effect that Hockettfs and Olmstedts views are not about the sentences of a syntactic theory, but merely about the acceptable sentences, or even, perhaps, about actual utterances. In fact, Bach comes quite close to saying precisely that :

The disagreements about whether to consider limitations on length and complexity part of a grammar, as evidenced in such writings as Hockett, 1968; Reich, 1969; and Olmsted, 1967, compared to the more or less standard transformational view expressed here, seem to stem largely from differences in the meaning of such terms as lfEnglish languagen and ItEnglish sentencell. It is ffempirically absurdff to think that there could be an English sentence a million words long, as

63. As ects of the Theory -of Syntax (Cambridge, Mass. MIT Press5-T-- 19 5 p.v. Hockett writes, only if you mean by "English sentence" either actual utterance or acceptable sentence. (Syntactic Theory, p, 26).

Actually, by 'English sentence1 Olmsted may mean 'actual utterance' for he says:

Indeed, it may be stated as an empirical generalization that, using my definitions of lsentencel and 'word1 so far found useful in describing actual utterances, every sentence in every language so far studied Is less than one million words long. (p. 304, my italics).

It seems to me that the "empirical generalizationu warrants the description only if by 'sentence1 he means 'actual utterance1. Then again, he says, as we saw earlier, that "...sentences consist of more than words.:.: there are also intonation patterns.ll Are intonation patterns abstract objects? One would be inclined to think so, but then, since sentences consist of them, this would seem to make sentences abstract, too -- which actual utterances, presumably, are not. It should be mentioned here that besides an unwillingness to make type-token discriminations for the reader (which is, perhaps, not such a se~iousmatter), Olmsted writes as though he has never heard of the competence- performance distinction. For his whole article is based on the differing perspectives of the lllogicianll linguist and the field worker. And obviously, unlike the former, the latter approaches his/her work with a theory of performance in tow, or some facsimile thereof. It is this which enables the field worder to reasonably conclude that any utterance he/she is likely to hear will be less than a million words in length, and not the assumption that Olmsted imputes to hidher, that I1English sentences are always less than a million words in length.I1 The "theoremu quoted above about sentence-terminal patterns coinciding with the end of breath groups that he mentions is another illustration of this; it seems a paradigm of what one would expect from a theory of performance. Perhaps Olmsted is not unaware of the competence-performance distinction and the fact that by means of it one can resolve the issues he raises; perhaps he rejects that distinction. But if so, he should make this clear, and state his reasons why. ' No such accusation can be directed towards Hockett, who explicitly rejects the competence-performance distinction (pp. 63-66), although he, too, appears to mean by 'English sentence1 'utterance or inscription of English1, But then, he seems to think that the English language is a set, not of sentences, but of possible sentences, for he says that there is "no definite boundary to the 'set of all possible sentences1 of the language; and just for these reasons languages are ill-defined.ll (p. 61). Of course for such an account to be philosophically satisfactory it must spell out (as was mentioned earlier) the identity conditions for these possible obj ects. Therefore he might prefer instead to identify the language with the set of acceptable sentences. But then Hockettls theory will be of no use to anyone who maintains that expressions are concrete, since most acceptable sentences will not be instantiated in an uttersnce or

inscription, and so will have to be considered types, and hence abstract objects. Unless, that is, recourse is made

to Goodman and Quinels trick of counting a frisbee, for example, as containing an inscription of an unused sentence such as the conjunction of (53) and (54). (53) Pauline batted .900 for the Yankees while singing the role of Tosca in -Das Rheingold. (54) Her brother ate nineteen pounds of popcorn while firing his B-B gun at unsuspecting lycanthropes, But this sort of terribly artificial philosophical contrivance is not likely to appeal to a linguist, as its sole function is to safeguard nominalism -- at the cost of rendering it impossible to determine what inscription, if any, a given object is, or ucontainsll. Not to mention that linguists are concerned not only with inscriptions, but also with utterances. For suppose that a linguist were to try to describe in Goodman-Quine fashion a natural language that had no inscriptions but only utterances. Naturally, he/she would not want to be limited to the prima facie actual I utterances any more than Goodman and Quine wanted to be limited to the prima facie actual inscriptions. So the corresponding move would be to count all manner of noise, e,g., robbins1 songs, as utterances of e.g., the conjunction of (53) and (54). But I cannot see how such an identification will be consistent with the truths of phonology -- for example, that the first phone of the sentence in question is an unaspirated bilabial voiceless stop, a sound that a robin cannot produce. In conclusion then, I think it fair to say that the claim made by our formalists that expressions are concrete objects rather than abstract ones, has been shown to be unjustified. It is unjustified, I argued, because only formalists willing to do without infinities can hope to consistently maintain that expressions are concrete objects. But none of the formalists we discussed seemed to be so willing, This is not to say that the only argument for expressions being abstract is that therq are infinitely many of them. Other considerations can also be advanced which motivate that conclusion. For example, it is very hard to see how one would go about specifying the syntax for a particular nominalistic theory of expressions. One difficulty lies in saying what the tokens of the language are without making reference to types. One can point to a specific ink mark, for example, and say "anything physically similar in shape to that is the same sort of tokenu -- as Hugly and Sayward do; see pages 185-186 below -- but as we shall see in connection with their proposal one then needs a fancy theory of when two things are "physically siniilar in shape1', Similar remarks apply to Goodman and Quinets attempt to specify the characters of their language in "Steps Toward a Constructive Nominalismw, The characters are not supposed to consist of the shapes 'vl, I", '(I,

')I, )If, and 'E" , of course, but rather of Il1v1-shaped inscriptionsI1 and "'('-shaped inscriptions", etc. (p. 138). Not that Goodman is unaware that no simple straightforward contrual of llphysically similar in shapef1 will do. He seems to be aware of it in Languages -of -9Art but he does not give a theory that would solve the problem. The point of emphasizing our formalists1 difficulties with infinity, rather than other sorts of difficulties they face, was to show why principle they cannot maintain that expressions are concrete objects,

3, Second Formalist Claim

I advanced the thesis that (most) expressions are on an epistemological par with numbers, so that if we think expressions exist -- or, more conservatively, if we think that infinitely many expressions exist, which is what I will assume throughout this section of the chapter based on considerations advanced in section one -- we should have no more qualms about admitting the existence of numbers. Opposed to this thesis are the uformalistsM-- those who contend that ekpressions exist, whereas mathematical entities, including numbers, do not. We examined their first line ! of defense in section one of this chapter, which was that numbers are abstract;, whereas expressions are concrete, and found it wanting. In reply, a llformalist" might well concede that expressions, or most of them, are abstract objects, like numbers. But then (s)he would fall back on his/her second line of defense, saying something like this: 64 "Even though expressions (types, that is) are abstract objects, there is still a mighty disanalogy between expressions and numbers. For expressions (again, types) have tokens -- splotches of ink, air disturbances, and so forth -- that are readily perceptible. We can come to know about expression types by means of interaction with their tokens, and thereby circumvent the epistemological problems chronicled by Benacerraf. No such explanation is available in the case of numbers, however, for numbers have no tokens. So even were they to exist, we would be unable to know anything about them."

I want to challenge this argument. I do not wish to challenge the assumption that expressions have tokens and numbers do not. But I -do wish (a) to question the claim that "we can come to know about expression types by means

of interacting with their tokens"; and (b) to dispute the claim that "no such explanation is available in the case of numbers". With regard to (a): not only is no mention made

-- 64,. This argument, or something very like it, was employed by Harold Hodes in conversation with me one day, by way of defending a "modal logicistI1 position he took in his (as yet unpublished, as far as I know) paper, vLogicism And The Ontological Commitments of Mathematics'l. Of course, he is in no way responsible for whatever defects exist in my formula~ionof the argument. of -how "we can come to know about expression types" by means of interaction with their tokens -- surely an essential ingredient in any fully adequate epistemological account, although, admittedly, a hard one to come by -- but the very claim itself that "we can come to know about expression types1' is quite obscure. What could the formalist mean by this? Here are a few of the possibilities:

(i) We can come to have propositional knowledge of expression types -- i.e., we can come to know facts about expressions. (ii) We can come to have what Russell called "knowledge bg acquaintancem regarding expression types; (iii) We can come to possess the concepts 9f expression types (e.g., to have a concept of the word lcatastrophel). It is hard to evaluate the formalist's claims without knowing what is meant by "we can come to know about expression typesw. Nonetheless, I want to claim that to the extent that the formalist is justified in claiming that 'I7,ve can come to know about expression typesv -- whatever this amounts to -- "by means of interaction with their tokensu we can also, and on a similar basis, llcome to know aboutu numbers. That is, if we can come to have propositional knowledge of expressions, then we can come to have propositional knowledge of numbers. If we can llbecome acquainted withv expressions, then we can become acquainted

I with numbers. And if it is a matter of coming to have conceptual knowledge of expressions, then we can come to have concepts of numbers, like that of 2, too. But on the

I basis of what? the reader may well ask, since it has already been admitted that numbers have no tokens. Here we come to point that I wish dispute the claim that "no such explanation is available in the case of numbers". That is, I will argue that there -are readily perceptible phenomena that function as numerical analogs actual utterances and ~nscriptions. Although not tokens of numbers,

I believe they can just as readily provide us with "knowledge ofv numbers -- in whatever sense the formalist cares to use this expression -- as utterances and inscriptions can provide 65 us, if they do, with "knowledge ofvf express>ons. There is a prima facie insurmountable difficulty with this position that I would like to get out of the way now.

It is this. Even if there are numerical analogs of tokens, we can have experience of only finitely many of these, So it would appear that even if they enable us to have knowledge of numbers, it will be of only finitely many numbers. Yet what we want to account for is our knowledge of infinitely many numbers if we are to account for our knowledge of arlthmet ic. - 65. Henceforth I shall freely use phrases like lfknowledge ofu and l'knows aboutu; the reader should bear in mind the discussion we had above on the possibility of construing the formalist's claim as (i),(ii) or (iii),and understand that such phrases, when they appear below, are to be construed in whatever sense is intended by the formalist in the aforementioned claim. The problem with this argument is that it assumes that

(F) If a person has acquaintance with only finitely many tokens, then (s)he can have knowledge of only finitely many types. yet if the formalistls argument that we are considering is to even get off the ground, (Fj must be false. For recall that the formalist's argument involves assuming that "we can come to know about expression types by means of inter- action with their tokensu. Clearly we interact with only finitely many expression tokens. So if (F) were true, we could "have knowledge of1' only finitely many expression types, too -- contrary to what we have been-assuming: that there are infinitely many expression types, and hence infinitely many to have to "know aboutl1. (I suppose it is not inconsistent to maintain that although we have propositional knowledge that there are infinitely mr.ny expression types, we are only llacquainted withf1 or have "the concepts ofv1only finitely marly of them -- with the ones that have tokens. But this position is very odd, since it makes it totally mysterious how we ever corn2 to utter a sentence that we have never seen or heard an instance of. Beaides, any account of our grammatical competence regarding expressions must account for our llknowledge of1', In some sense, infinitely many of them.). To get back to the formalistls argument: (s)he says we can come to have knowledge of exprersion types by means of interaction with their tokens. The general ide~.seems to be that by seeing/hearing/(perhaps) feeling a certain number of tokens of a particular expression type, one can come to know certain important facts, at least, about the type. We can see though, based on the foregoing discussion, that without'a supplemental account of how we came to know expressions, this method by itself will only afford vs knowledge of those expression types that hkve tokzns we physically interact with. And of course no-- one who has thought about the matter believes that these are the only types that we know about. Even someone anxious to deny that there are infinitely many sentences would agree that eqch of us knows more se~ltencesthan we have come into contact with. So the first thing to note about the formalist's account of how we can come to know about expressions -- through their tokens -- is that at best it too only explains how we come to know some of the types we do know, Still, that at least is something. And the formalistls point, I take it, is that even that explanation, however partial, is lacking in the case of numbers and mathematical entities generally, because they have no tokens. This is presumably why, according to the formalist, although they are abstract, expressions escape the clutches of Benacerrafls epistemological problem. I agree that there are no number tokens, -se. But it seems to me that there are things which funztion analogously: units, pairs, trics, quadruples, quintuples, ...scores, ...g rosses, etc. Admittedly --what this function is unclear -- as with the formalist's original doctrine. [As noted earlier, no mention w?,s made of -how interactio~. with tokens yields knowledge cf types.). But just as with learning in children, some plausible sort of story -- what- even it may be -- can be told. Identical twins, happy couples, pairs of shoes or jacks, and doubles in baseball or tennis enable us to know about the number 2. Triplets, happy threesomes, Beethoven trios, major r1.nd minor triads, and triples in baseball or bowling illuminate us about 3. Quadruplets, happy foursomes, Beethoven quartets, Greek musical tetrachords, and the quadruple of one's allowance inform us about 4. There are perceptible particulars experience of which enables us to "know our nuhnbersVf,just as expression tokens enable us to "know our letters", Of course, I would not deny that there -are differences, even important ones, between the two instantiation- relations

(if I may be YO bold as to call, for example, the First Couple, Adam and Eve, an "instancefVof 2) ; It is just that I do not think the differences suf'fic~ent to justify the formalistfs claim that although we can come to 1;ni3w about expressicn types by experiencing their tokens, Ifnu such explanation is available in the case of ilumbe~su (so that even were numbers to exist, they would be epistemologically inaccessible), I think that what this formalistic claim amounts to is that pairs, trios, and so forth are not the numerical analogs of expression tokens, that there are no such analogs. In view of the success of Sesame Street in teaching "the numbers1' to tots by means of just such devices as showing pairs of shoes and jacks, and happy threesomes, one wonders what the formalist could have in mind. The rest of this chapter, therefore, will consist in a number of objections that might be raised, and responses to them. Due to the immensity of the topic and topics it impinges t upon, it will of necessity be sketchy and have more nearly the nature of a preliminary study than a full-blown one, and for this I apologize in advance. My first objection is not one thatla formalist would 66 necessarily raise, but it is an interesting one. Objection #1: This attempt to explain how it is that we come to have knowledge of numbers is circular. In order to recognize a pair of shoes, for example, as a pai~ of shoes, one must already have the concept of 2, because that is (partly) what recognition of a pair of shoes consists in: recognition that there are 2 shoes. Response to #1: In discussing this objection we risk becoming embroiled in such questions as whether transitive

counting (1 shoe, 2 shoes, 3 shoes, ,..) or intransitive

- 66. This was suggested to me by Lon Berk. 6 7 counting (1, 2, 3, + *) llcomes first'' (in possible learning sequences). It might seem, for example, that we are off

the hook if transitive counting "comes firstv1, While

interesting, I believe this sort of issue can be sidestepped here. My claim is that -if expression tokens can give us knowledge of expression types, then pairs, triples, etc., can give us knowledge of numbers. Now it may be that knowledge of numbers is innate, has-- to be innate, or we would not be able to see a pair -as a pair or a triple -as a triple, and s3 on. But it seems to me that ---one can make just -as good ---a case for claiminq, as some do, -that knowledge -of expressions Is innate too; otherwise how, it might be asked, could we recognize a token of 'cat: as a token af 7 'cat1, unless we already had the concept of the word 'cat!? That is, it could be argued that In order to recognize a particular spoken utterance as a token of 'cat1, one would have to be able to recognize other utterances as tokens of ---the same word, and this requires knowledge of that word, of 'oat1. Again, I do not want to become entangled in empiricist- rationalist disputes about what, if any, innate knowledge or dispositions we have to have in order to do what we do. The point is that the intuition behind objection #1 cuts both ways; to the extent that it undercuts the hypothesis that we learn about 2 and 3 e.g,, from pairs and triples,

67. Benacerraf distinguishes and discusses these modes of coufiting in "What Numbers Count Not Bef1. it also undercuts the hypothesis that we learn about expressions from tokens of them. As long as we are assuming, with the formalist, that knowledge of expression tokens does yield knowledge of expression types, we are tacitly agreeing to not discuss the above issue (although it may well be that we are taking a stand on it). So let us nct discuss it. Objection #2: It may well be true that we could obtain knowledge of small numbers this way; but it does not account for how we could have knowledge, not only of infinitel: many numbers (as we saw earlier), but even of large numbers. We never experience, for example, a trillion- tuple of concrete perceptible particulars, so that we could come to know about a trillion on the basis of it. Yet surely if we know about a hundred, we know about a trillion. Response to #2: Yes, except in connection with the federal budget, numbers like a trillion are not likely to have "instantLations that we can experience of concrete perceptible particulars -- which is perhaps why people tend to think of such numbers as abstract compared to the number of, say, one's toes, or the planets, and why persons on TV after making reference to a large number (like a trillion) always append some comment such as: "Thht many dollar bills laid end to end would go to the moon and back 19 1,/2 times1'. But again, I do not think this fact need detain us here, for the same reason as was put forth in response to objection #l. That is, it is certainly true that this account needs supplementing to explain how we obtain knowledge of numbers whose instances have never been experienced. This is just to concede once again that the present account is only meant to be partial. In exactly the same way, though, the formalist's token-to-type account of how we .come to have knowledge of expressions is also partial (as we ~otedearlier), because it needs supplementing, too, to explain how we come to have knowledge of expressions (like long sentences) instances of which have never been experienced. I suspect that were the formalist to provide the needed supplementation, similar principles might be applicable towards explaining how we come to have knowledge of large numbers, too. Objection #3: In connection with Mill's account of how we know that '2 t 1 = 3' is true, Frege questioned whether it is necessary to observe "that collections of objects exist, which while they impress the senses thus, 0 r, ,168 , may be separated into two parts, thus, 0 . He noted that "If it were, the number 0 would be a puzzle; for up to now no one, I take it, has ever seen or touched 0 pebbles.I1 (p. 11). The objection Frege raised in this note can be raised equally appropriately in the present context: it leaves knowledge of 0 unexplained. Yet knowledge of 0 is

68. Foundations of Arithrnetlc 57, 8, pp. 9, 11. basic to knowledge of number; nor should we be made to wait for an explanation until the formalist puts forth some solution, because the formalist has no similar problem.

Response to #3: It does leave 0 unexplained. And let us assume that the formalist has no parallel problem. Does this constitute a serious objection to the epistemo- logical account we are considering? I don't think so. It does show that this account does not account for our knowledge of 0, which at first blush might appear to be a serious liability, But if there is good independent evidence that knowledge of 0 is more difficult to obtain, that even belief in the existence of 0 is not automaticaily concomitant with belief in the existence pf 1, 2, 3, etc. then I think that instead of this being a liability, it may well be confirmation for the account under consideration. And indeed there is such evidence. In the Encjclopedia -of Philosophy under the heading Number, Stephen F. Barker states that

the numbers involved in counting -- 1, 2, 3, etc. -- are of course the simplest and most fundamental kind. At least since late classical times these have been called the I natural numbers, thereby being c~ntrasted~~ with the llartificialll kinds of numbers.. .

So it appears that 0 was not corisidered a natural number, at least not in classical times. But Barker hastens to add :

69. (New York: Macmillan, 1967) Vol. 5, p. 527. A trifling ambiguity in the term Itnatural numberI1 arises because some writers choose to call zero a natural number, whereas others do not. (Idem).

As a matter of fact, ~re~e~'and ~usse11~' classified

0 as a natural number; ~edekind~~and ~eanc~~ do not. But

these are modern writers. As to the existence of even a numeral corresponding to our 'Of, Morris Kline tells us:

At first the Babylonians had no symbol to indicate the absence of a number in any one position, and consequently their numbers were ambiguous. Thus p<( could mean 80 or 3620, depending up whether the first symbol meant 60 or 3600. 74

In time, Kline tells us, this deficit was corrected:

'a Greek papyri of the first part of the Alexandrian period (first thrr;e centuries B.C.) contain symbols for zero such as

The zero of the Greek Alexandrian period was used, as was the zero of the Seleucid Babylonian period, to indicate mlssing numbers. According to Byzantine manuscripts, which are all we have of Ptolemyls work, he used 0 for [the numeral] zero both in

70. Foundations -of Arithmetic 574. 71. Introduction-- to Mathematical Philosophy (New York: Simon and Schuster, 1971) p. 13. 72, "Letter to KefersteinH in From Frege -to Gzdel, p. 100. 73. "The Principles of Arithmetic1! in From Frege -to Ggdel, P* 94. 74, Mathematical Thought from Ancient to Modern Times (New York: Oxford University Press, mp.T. the middle and at the end of a number. (P. 132).

In treating the numeral '0' as a mere placeholder, it is clear that the Babylonians and Greeks did not consider zero a number. That this is so is borne out by the following passage from Plato's Parmenides:

Now suppose we take a selection of these terms, [say] 'being' and 'different1, or 'being1 and 'one1, or 'one1 and 'differentt. In each case we are selecting a pair which may be spoken of as 'bothv...And a pair that can properly be called 'bothg must be -two. And if a pair of things are two, each of them must be one. This applies to our terms. Since each set forms a couple, each term must be one. And if so, then, when any one is added to any pair, the sum will be three. And three is odd, two even. Now if there are two, thzre must also be -..twice times, if three, three times since two is twice times one and three is three times one, And if there are two and twice times, three and three t,imes, there must be twice times two and three til7,es three. And, if there are three ~hichoccur twice and tvo which occur three times, there must be twice times three and three times -two. Thus there will be even multiples of even sets, odd I multiples of odd sets, odd multiples of even sets, and even multiples of odd sets. That being so, there is no number left, which must not necessarily be.75

The last line of the passage in particular suggests that Plato did not tnink 0 a number.

Apparently, it was not until after the year 600 that

75. Translated by F.M. Cornford in Plato: The Collected Dialo ues, ed, by Edith Hamilton and Huntington Cairns : Princeton University Press, 1961) lines 143~- 144a, pp, 936f. zero came to be considered a number, by the Hindus. Kline tells us that

. . .the zero, which the Alexandrian Greeks had earlier used only to denote the absence of a number, was treated as a complete number. ~ZhavFrasays that multiplication of a number by 0 gives 0 and that subtracting 0 does not diminish a number. (p. 185)

Historically, then it appears that 0 was not considered a number until after 600; and its addition to the numbers seems to have been motivated at least partly by closure considerations, rather than the less abstract sorts of considerations involving collections and counting that Barker hints at, and Dedekind seems to have been motivated by:

I regard the whole of arithmetic as a necessary, or ac least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual i defined by the one immediately preceding ...79

It is no wonder then that Dedekind does not treat 0 as a natural number, claiming that the "creation of zerof1 invoLves "an extension of the number-conceptu (p. 35).

And even though Russell includes 0 in the natural nmbers, it is clear from his discussion (quoted directly below) that he not only thinks 0 is a more difficult number to discover than, say, 2, but he endorses the

76. Essays on the Theory of Numbers, p. 4, epistemological account of number-knowledge (albeit a partial one) that we are considering.

To the average educated person of the present day, the obvious starting-point of mathematics would be the series of whole numbers, 1,2,3,4, ... etc. Probably only a person with some mathematical knowledge would think of beginning with 0 instead of with 1, but we will presume this degree of knowledge; we will take as our starting-point the series:

and it is this series that we shall mean when we speak of the "series of natural numbersu. It is only at a high stage of civilization that we could take this series as our starting-point . It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number 2: the degree of abstractlon involved is far from easy. And the discovery that one is a number must have been difficult. As far as 0, it is a very recent addition. pe Greeks and Romans had no such digit.

In view of the historical facts, 0 seems to require a different explanation from that of even the other non- large natural numbers about how we come to know about it, So although it is true that the account under consideration does not entail such an explanation, this appears to be an asset rather than a liability. Objection #4: We are construing the formalist as saying that even though or most of them, are abstract objects, they are accessible to us because we

'(1. Introduction to Mathematical Philosophy, p. 3, experience their tokens which, as physical objects (e.g. splotches of ink) or events (e.g. air disturbances) are concrete, are perceptible particulars. And you claim that units, pairs, twos, etc. -- pluralities let's call them -- are the numerical analogs of expression tokens, by means of which numbers are accessible to us. But these pluralities cannot be mere heaps, or aggregates, for the same sorts of reasons that Frege advanced to show that numbers are not properties. He wrote:

I am able to think of the Iliad either as one poem, or as 24 Books, or as some large Number of verses. Is it not in totally different senses that we speak of a tree as having 1000 leaves and again as having green leaves? The green colour we ascribe to each single leaf, but not the number 1000. -----If we call'all the leaves ---of a tree taken together -its foliage, then the foliage too is green, but it is not --71000, To what thendoes the property OF really belong? It almost looks as though it belongs neither to any single one of the leaves nor to the totality of them all; is it possible that it does not really belang to things in the external world at all? If I give someone a stone with the words: Find the weight of this, I have given him precisely the object he is to investigate. But if I place a pile of playing cards in his hands with the words: Find the number of these, this does not tell him whether I wish to know the number of cards, or of complate packs of cards, or even say of points in the game of skat. To have given bin the pile in his hands is not yet to have given him completely the object he is to investigate: I must add some further words -- cards, or packs, or points. Nor can we say that in this case the different numbers exist in the same thing aide by side, as different colo~rs do. I can point to the patch of ea:h individual colour without saying a word, but I cannot in the same way point to the individual numbers. If I can call the sane object red and green with equal right, it is a sure sign that the object named Is not what really has the green colour; for that we must first get a surface which is green only. ~imilari~,an object to which --f can ascribe different numbers with eaual riaht is --not what really has 5 number.10

Since the pluralities in question are not mere heaps, they must be sets or classes. But sets and classes are not concrete objects, they are abstract. In support of this point we may look to Quine, for example, who writes:

Continental United States is an extensive physical body (of arbitrary depth) having the several states as parts; at the same time it is a physical body having the several counties as parts. It is the same concrete object, regardless of the conceptual dissections imposed; the heap of states and the heap of counties are identical. The class of states, however, cannot be identified with the class of counties; for there is much that we want to affirm of the one class and deny of the other. We want to say emg,that the one class has exactly 48 members, while the other has 3075, We want to say that Delaware is a member of the first class and not of the second, and that Nantucket is a member of the second class and not of the first. These classes, unlike the single concrete heaps which their members compose, must be accepted as two entities of a non-spatial and abstract kind. 79

Since they are abstract objects, the pluralities in question -- pairs, trios, etc. -- are hardly on an epistemological par with tokens of expressions, which are concrete.

78. Foundations of Arikhmetic, 522 pp. 28ff (italics added), 79. Mathematical Logic (Cambridge, Mass.: Harvard University Press, 1940) p. 120. Response to #4: 1 do not think the preceding argument, at least as I will construe it, even begins to show that the pluralities in question are abstract rather than concrete. The skeleton of the argument is this: (55) Pluralities such as pairs, trios, etc., are either aggregates (in the philos.ophica1 sense of' tlheaplt) or sets. (56) Pluralities are not aggregates. (57) So pluralities are classes. (58) Classes are abstract objects. (59) So the pluralities in question are abstract objects. Of course I agree that (57) and (59) are consequences of preceding statements; I object to (55), (56) and (58) though. To begin with, no argument is given for (55). Why can't the pluralities in question be neither aggregates, in the full-blown philosophical sense (actu~lly,there is more than one) nor classes? Why aren't pairs just, well, pairs? Perhaps they are like aggregates in some ways, classes in other ways, but do not conform to either set theory or the calculus of individuals. But even if (55) is true, it does not seem to me that Fregels argument shows (56). Here is what I take to be the essence of that argument, applied to pluralities: (60) Every plurality consists of some one definite number of things. For example, a trio consists of exactly 3 things. (61) Aggregates do not have some one definite number of things. To use Quine's example, the physical. body that is the United States is both 50 states and 3075 counties.

(62) Therefore, aggregates are not pluralities. (Or,

what comes to the same thing, (56).). (61) may be right; but (60), although prima facie convincing, is dubious. For one can construct a plausible argument whose conclusion contradicts (60) (under what I think its most straightforward reading is): Pairs consist of exactly 2 things of the same sort, trios of 3, and in general -n-tuples of exactly -n. (I have in mind here vcountable" sorts like apples, states and counties, not water, or quantities of water.). Now the physical body that is the United States consists of, at one and the same time, 50 states and 3075 counties. So the United States is both a 50-tuple and a 3075-tuple at the same time. I take it th;t this last statement contradicts (60),

(60) derives its credibility, I believe from the fact that it resembles (63), a statement that is perhaps vague, but true.

(63) A plarality consists of scrne one definite nurnber of things --of a certain -sort. So even though a plurality (of pie pieces, say) may consist of some one definite number (2) 3f things of a certain sart (half-pies), it may also consist of some one definite ncmber (4) of things of a different sort (quarter-pies). (A pair of half-pies can comprise the same pie as a quadruple of quarte?-pies), What this suggests is that being a pair, say, is like being a father, in certain important respects. If x is a father then x is the father of someone.- If x is a pair, then x is a p2j.r oP things ----of the same sort So Fre~ewas right insofar as being -a pair is not like being green because, vagueness aside, notning can be both green and a different color liLke 17ed; whereas something can be, say, both a pair and a quadruple at the same time, ju~tas someone can be a father and 2. grandfather at the same time. But he was wrong insofar as he cc:ncluded that therelore being -a pair is nct an objective propl-rty of external things, lot a phyalcal property, A number of writers have criticized Frege on this and relstcd points. Charles Lambros, for example, argues that, 0 in the Fsqsage quoted frao~n 3rege above, Frege is concluding that '1000' is not a pro~ertyword cn $he grourds that its behavior is distinct fronl ordinary predicates like 'green1, but that this is a mistake in view of Itthe now familiar 80 distinction hetwe,en distributive and collective predicatestt. t 'Is 11)001 m3y be s collective .-edicate, applying to the c~llectionof' leaves (which Lambros construes as a class)

80. '!Are Numbers Properties 3f Objects?", Philosophical --Stud.ies 29 (1976): 381-389. (pp. 231-282). He also construes Frege as arguing there that properties of objects objectively 'reside1 in their objects, independently of how we think of or regard such objects, but numbers cannot be properties (nor number words predicates, therefore) because they do not (p. 384). (I will call this "Fregels relativity argumentu.). Lambros points out that number words may be syncategorematic predicates of material objects (p. 385). Glenn Kessler also comes to Mill's defense, in "Frege, Mill, and the Formdation* of ~rithmetic".~'~There he criticizes Fregels conclusion in the relativity argument, that statements about numbers are not statements about aggregates or objects. He shows how numbers -can be construed as applying to aggregates, although only with respect to some individuating property (the reference of a sort'al predicate), Under his coristrual., numbers turn out to be reiations- between ap~reqatesand individuating properties. I It is very likely that, by leaning on Kessl.erls account we could analyze & 3 pair -of as in lx is is pair of A's1 as a relation between an aggregate, x, and an individuating property, being an- A. It should not be thought however that therefore a pair turns out to be an obscure object, anymore than a father is, Frege criticized Millls construal of number as "the characteristic nannerl' in which an agglomeration can be separated into pots, on +ne grounds - 81. Journal. -of Philosophy 77 (1980): 65-79. that llthere are very various manners in which an agglomeration can be separated into parts1' 523, p. 30). This is correct, and in the present context amounts to the observation that there is often or always more than one individuating property that applies to a given aggregate. Yet its importance should not be exaggerated. It may well be that there is very often only one individuating property that seems natural and appropriate to apply (like being -a rabbit, e.g., as opposed to being -a -one --cubic -inch undetached rabbit part.) ~esides,/ if Frege s relativity argument showed that being -a pair is no; s property of objects or aggregates, then a similar argument would show that being -an expression is this token not a property of objects, eithe~., Foil I1readl1 phonetic sequence aloud: Ah 'key ess 'oon ah 'may sah, If my stumbling attempt at phonetization was successful, a Spanish listener would understand you to have said that here is a table, but a speaker-hearer of Yiddish would understand you to have said that a cow eats without a knife. This shows, I think, that the same perceptible particular may be a / token of type T relative to language L, aod a token of type T1 relative to language L1. And hence that an object is /i token of a type only relative to a language. Actually, Philip Hugly and Charles Sayward argue that [Tlhis isnlt enough. Note that it is only because of the Morse Code that objects made up of dots and dashes are tokens of English. A perceptible particular is a token of an expression of a language L only relative to some tokening system. Being a token of is a 4-place relation between perceptibie particular, expression, language and tokening system, rwhere]A tokening system is a set of instructions that tell how, for any given expression, a speaker of the language could construct a perceptible particular that tokens that expression given that the spea.er had enough physical and mental resources... If2

The point is, that whether a 3- or a 4-place relation, being of, -a token like our Kesslerian analysis of being a- pair of-9 is relative to one or more other things.83 But if Fregels argument were correct, this very fact would indicate that being -a token -of 'cat', say, was not a property that "residedff in the object in question, not a physical'property, or property of externzl things. The "relational aspect1' to -bein& -a token can be obscured by the fact that context often enables the listener to infer what the lai~guageand tokening system are; but similarly, the individuating I property that makes being pair, for example, relative is # often equally obvious.

82. ffExpressions and Tokens", Analysis 41, (1981): 181-187, p, 186. 83. Nor can the example be dismissed as so bizarre as not to warrant consideration. There arc many like it, particularly at the level of individual letters and phones -- as anyone who has ever struggled to decode a handwriting sample or utterance in a rather different dialect can attest to. I do not wish to insist that the pluralities in question -are aggregates. It is just that (60) is tantamount to the claim that they are classes and not aggregates. This may or may not be true, but it isn't obviously true -- as evidenced by the fact that the alternative of construing pluralities as aggregates seems to have a fair degree of plausibility. Of course, if they -are aggregates, pluralities (or, ratqer, the ones that are aggregates) are concrete objects.

But what if the most plausible way to construe pluralities is as classes; i.e., what if (57) is true? Should we conclude forthwith that (59) is true, i.e., that pluralities are abstract objects? If (58) is true, yes. But (58) seems to me false. Some classes are undoubtedly abstract -- e.g., the class of all ordinals, or the von Neumann t'numberstt. Other classes seem patently concrete &tough -- e.g., tb~ third grade class of P.S. 62 in New York City in June 1983. There is a gap in Quinels argument. He states, and argues for (64), then concludes (65) without further ado. (64) The class of states of the United States is not identica1,to the class of counties of the United States, I (65) These classes must be accepted as two entities of a non-spatial and abstract kind. But I do not think that he intends to be assuming something as strang as (58); else why would he spend so much time arguing/ for (64) before concluding (65)? I believe that the following statements more adequately represent Quinels suppressed assumptions. (66) The only spatial, non-abstract object that the class of states could be, if it is a concrete object, is the physical body that is the United States. (67) Similarly, the only spatial, non-abstract object that the class of counties could be, if it is a concrete object, is the physical body th~tis the United States, But they cannot both be the physical body that is the TJnited States, because then they would be identical to each other (since identity is transitive), contrary to (64). Presumably by the, or a, principle of sufficient reason, then, it follows that neither is the United States. And according to (66) and (67) there is no other concrete spatial object they could be; hence Quine concludes that (65). It is the suppressed assumptions that bother me. Of course, I am only surmising as to what QuineTssuppressed assumptions are, and what the intuitions arc that support (66) and (67). Quine might tell a different story. But I think anything he said that had the consequence that the sorts of pluralities in question (the sorts thht enable us, I am claiming, to have knowledge of numbers) are not- concrete would 84 be false, The notion of llconcreteu is vague at best. 84. Wetsterls Seventh Collegiate Dictionary- is .lo help, although its second definition is interesting. Here is .what it says: "1, formee by coalition of particles into one solid mass; 2: naming a real thing or class of things; 3a: charac- terized by or belonging to immediate experience of actual things or events b: SPECIFIC, PARTICULAR c: REAL TANGIBLE; 4: relating to or made of concreteH. (Springfield,kss. G & C Merriam Co; 1969) p. 172. And perhaps being concrete or being abstract is a matter of degree. But I should think that anything that can be made, bought, worn, worn out and thrown away, and that comes in various colors, sizes, and styles -- --like a pair -of' shoes -- is concrete, A brace of Old World pheasants may be more attractive, or more delicious, or weigh more, than a brace of bobwhites; such possibilities entail that the braces are concrete, I believe. According to the rules of poker, Mary's four of a kind, the very one she is holding in her hand, beats John's three of a kind. So three and four of a kinds are concrete. Minor triads can be heard, so they are too. And so forth. So if may be that pairs, trios, etc, -are classes (or are best analyzed as classes). If so, then it seems to me that we have good evidence for claiming that there -d1.e concrete class2s -- and hence that (58) and (59) are false. Soneone may balk at the notion of concrete classes, on the grounds that tken all but the purest classes (those ultimately composed of the null set) turn out to be concrete objects, For if any class of concrete objects is itself a concrete object, then it follows that a class of classes i of concrete objects is concrete, and so one. But this is not a necessary consequence of admitting that scrne classes are concrete. We could consider concrete only those; classes all of whose members are both concrete and non-classes. (Or, if we like, we could extend it one level up to classes of such classes -- or beyond.). Another reason that might be given for rejecting concrete classes is that then a pair of shoes would be a concrete object; so that in a room containing nothing but

a right shoe and a matching left shoe, there would be, not 2, but -3 concrete objects! However, this is not much of an objection, because anyone, even an ardent nominalist like Goodman, has to swallow this consequence already in virtue of admitting "fused individualsff -- such as the right shoe llfusedll, as it were, to the left shoe -- into his/her ontology. (Besides, even if one did not admit such nfusim~nsv, one would still haye to deal with the fact that there are four concrete objects in the room: two shoes and two shoelaces. ) . Much more could be said, pro and con, abouc concrete classes. Much more could also be said about what it is that pairs, trios, etc. are: aggrega.Les, classes, not fully

either, or something else entirely. But I hope enough has been said to render plausible my response to objection #4, which charged that the pluralities in question are abstract objects'themselves. In summary, my response was to argue that the objection did not show that pairs, trios, etc.

were abstract, because there is reason to doubt (ij whether the pluralities in question must be either aggregates or

classes; (ii)that the argument (which was basically Fregefs) that they cannot be aggregates is sound; and (iii)that,

(since ihey so patently are concrete), if they are classes, then they are not concrete objects.

LObjection #5: The reason why we can corne to know about expression types on the basis of their tokens is because types -are just -like their tokens, exoep.: that they -are --abstract whereas --tokens -a~~e concrete objects. For example, the word 'catt, like its tokens, Is composed of three letters, is spelled tcl-tat-ttl, and is pronounced ['kat 1. So empirical investigation --can give us knowledge of the word type 'cat1. Numbers on the other hand (assuming they I -do exist) are not just like what you are calling their llinstances". For example, pairs are composed of exactly two things -- but numbers are not. Therefore numbers and pluralities do not exhibit anything sufficiently akin to the type-token relationship to enable us to employ the same sort of explanation with respect to "our knowledge of numberstt. Response to #5: I must concede that, as the example above suggests, there are important d.ifferences between the type-token relationship and the number-plurality relation- ship -- which is why, after all, I did not try to subsume the latter under the headiKg of the former. But I do not think the type-token relationship is anywhere near 3s cozy, nor the number-plurality relationship as distant, as the objection seems to suppose. That is, the reason stated above as to how we can come to know about expression types on the basis of their tokens is not precisely correct. When spelled out in more detail, we will be able to see tha,t although not that of type to token, the relationship between a number and a plurality having .that number is sufficiently similar to begin to explain how we can have knowledge of numbers on the basis of our experience with pluralities (concrete ones, at least). In order to discuss the issue in a definitive sort of way, we would need an account of the type-token relation- ship that provides correct and precise answers to the following questions, among others: (68) What is a token, especially an expression token? (69) What is a type, especially an expression type? I (70) How do types, especially expressions, differ from other ugenerlclt entities, like universals? (71) Can something, especially an utterance or inscription, be a token of more than one type?

(72) Precisely what sorts of properties do ty7es have, especially expressions, and what sorts do tokens halre? (73) What is the relationship between the propertles the types have and the properties the tokens have, especially as regards expressior~s? The trouble is that there is no one account that I know of that pravides answers to all of the above questions, much less one that 1s generally regarded as being correct. Different authors say different and often conflicting things (as we shall soon see). Hence it would take a discussion of book length to do justice to the issue, and this I do not intend to provide (at least not in this work). So the discussion that f~llowswill be sketchy and prelirrdnary in many respects, although I hope not altogether without value. My only justification is that I believe enough can be said in a short space to at least make the suppositions behind objection #5 dubious, and render more plausible the analogy between type-tokens, and number-pluralities. The logical place to begin searching for a ccherent answer to the above questionz is in Peircels writings. One author refers to his account of what the termltypel and 'token1 mean as "the only adequate theory. ..and the only account most authors cite1'. 85 Nct unly is his theoxay of signs very much built around the distinction, but (in the passage on p. 126 above from his writings) Peirce apparently coined the terms 'type1 and 'token1 to denote the distinction. What he says there about there being two senses of 'word', such that in one sense there are likely to be about twenty "thelsl' on this page and in another that there is only one "thell in the English language, is enough to give anyone versed in English a rough feel for the distinction, Fie is fairly clear, too, about what tokens are: one time happenings (events) or spatio-temporal objects. 8 6 - 85. Randal R. Dipert, llTypes and Tokens: A Reply to Sharpelf, -Mind 89 (1980): 587-588, p. 588. 86. Collected Papers of Charles Sanderb Peirce, Vol. 4, p. 423. Future references to Peirce will' contain merely the volume number followed by the page number, Elsewhere he characterizes them (under :,he label Isinsign') as "individual objects or events" (8.334). This seems relatively unproblematical, and concurs nicely with Baruch Brodyts definition of a token in the Encyclopedia -of Philosophy as I1a specified utterance of a given linguistic expression or a written occurrence of itv1(Vol. 5, p. 6). What Peirce has to say about types, however, is quite another matter: types Ifdo not existv, yet they are "defin- itely Significant Formsr1 that Ifdetermine things that do existu (4.423). A type, or ulegisigriu as he also calls it, Ifhas a definite identity, though usually admitting a great variety of appearances. Thus, &, -and and the sound are all one wordf1 (8.334). He also tells us that a type is "a general law that is a sign. This law is usually established by men. Evel-y convention31 sign is a legisign. It is not a single object, but a general type which...shall be significant. ..every Legisign requires Sinsignstt (2.246)

I find these statements confusing at best, for I do not kncw how something which has a definite identity, is, in fact, a general law, could at the same time not exist. Perhaps all Peirce meant by saying they do not exist was that they are "not j.ndividua1 thingst1 (8.334), that they are, instead, I1generalsn. Yet what could he mean by calling them general laws? In Peircevs Concept -of Sign87 , D. Greenlee

-- 87. The Hague: Mouton, 1973. gives as a gloss on the matter that

Every sign is either qualified by a power [as tokens] ...or consists in a typical power [as types] ...because it is either associated with or consists in a habit of interpretation. The habit, of course, belongs to the interpreter, the power, let us say to the sign... (pp. 115-116).

Which sign? Surely not the type (unless something can be a power of itself), so perhaps the type is a power of some sort possessed by its tokens. On the other hard, it could be a "habit of interpretation1' because Greenlee says that a type is a "general that is a signu; that ''generals which are signs are called by Peirce 'habits1, in an extension of the notion of a habitu (P. 49) and, in the case of some types, "a habit controlling a specific way of responding interpretacively" (p. 137). The suggestion here seems to be that a type is a habit of llinterpretersll to respond In a certain way when expased to tokens of the type, These two construals of types (as powers of tokens, or interpreterst habits of responding to tokens) cannot both be right, it seems to me. But either way of construing types entails, as Peirco was quoted above as saying, that "every [type] requires (2.246) (which is why a type is a general rather than a universal for Peirce; the latter does not require instances). Either construal then is Incompatible with our requirement that, whatever types are, --there -are infinitely many --of them. So we cannot use Peirce's notion(s) of a type as ours. Besides, it is very hard to see how, under either construal there will be any sense in which types will be just like their tokens, as the formalist claims; how can a power or a habit be three- lettered, for example? Also, under the construal as a "habit of interpretersf1 types seen in danger of being psychological, contrary to our other proviso that they be abstract objects. (True, something can be an abstract psychological object, like a type of habit, but then that would seem to make tokens habits t~o.) What elsc csuld a type be? A class? I think that many think so, in spite of the fact, noted earlier, that it cannot be that all types are simply classes of their tokens (since many different ones will then be identical. in virtue of having no tokens). Of course, this is not the only class construal types can be given, For example, in the article by them cited above, Hugly and Sayward construe types as classes in a way that ensures that there are infinitely many. Starting with the ink marks:

(a) x (b) P (c) ' (d) N (el a they let the primitive sign (types) of the language be the five sets: A = {x: x is physically similar in shape to the ink mark at (a)1 B = {x: x is physically similar in shape to the Ink mark at (b)), and in the same way, sets C, D and E are respectively defined from the ink marks at (c), (d) and (e). Variables are defined as follows: is a variable; if a is a variable so is ; nothing else is a variable. Predicates, formulas, and sentences are similarly defined, as ordered 88 n-tuples of the primitive signs and each other. SO, for example, the type that (74) ax' Px' is a token of is , c, >>. There are several problems with construing types as classes in this way. First of all, while not an unattractive set-theoretic way of construing types, it is fairly arbitrary. For example, why should the primitive signs be A, B, C, D and E? Why not let them be: A1 = {x: x = the ink mark at (a)); B1 = {x: x = the ink mark at (b)), and so forth? Or, if these are too llconcretel', then let them be A" = ; Bfl = etc. Perhaps, might be the response, because these construals sever all ties between types and tokens, e.g., the seven ink marks that compose (74) are actually members

*cit 9 of E, A, C, B, C, A and C, respectively, by dint of being physically similar in shape to (el, (a), (c), (b), (c), (a) and (c), respectively. They are (some of) the ur-elements out of which the types are constructed. This response seems acceptable, but the question then arises: why construe variables in that way? Why not do the following: first, define a prime tuple: C is a prime tuple; if a is a prime tuple, so is ; nothing else is a prime tuple. Then define a variable as: any ordered pair where B is a prime tuple. It seems an equally attractive way. Perhaps Hugly and Sayward could justify their choice as being the most natural and attractive compared to this and all the other construals that exist. But I doubt it. The problem is that we, and they, purport to be answering the question "what kind of an abstract entity is [an expres~ion]?~(p. 184). If there is more than one equally attractive non-extensionally equivalent account -- which seems to be the case -- then ---none of them can be -the correct one. (I have in mind here the sort of argument so common in chapter one, above, the details of which I will skip.). So the first problem with the account Hugly and Sayward provide is that it cannot be --the correct one, as there are equally attractive rival set-theoretic accounts. The second problem is that the account is of very limited value, as it cannot serve as a model for what expressions of a natural language are -- at least, not in the obvious way, where we merely start off with many more primitive signs defined in terms of their physical similarity in shape to certain given signs, and then employ the phrase structure rules, transformational rules, --et a1 that linguists recognize. This is not just because by defining the primitive signs as sets of objects physically similar in shape to (some given sign) oral tokens are ruled out. That could be remedied (if- the technique worked) by taking the primitive signs to be sets of objects that are physically similar in sound to for example, ['eks]. (Or, if this could not work because the sound units -- phones -- do not correspond to the written units -- letters -- we could give up the idea of defining something that can be either spoken or written, and just try to separately define acoustical and shaped types.) The real problem lies in the fact alluded to on pages 151-152 above, that many inscriptions, particularly of letters, that ---we want to count -as being tokens of--- the same type ---are in fact more- physically similar -in shape -to tokens -of other types -than they --are to tokens -of their -own type. TO name just one of many possible examples: on the original copy of the Declaration of Independence, it seems to me that the first letter token of Samuel Adams1 signature is much more 'Iphysically similar in shape1' (in any remotely straight- forward sense) to the first letter token of John Hart's signature than to that of Samuel Chase's. Of course, Hugly and Sayward could just accept the consequence that therefore such Ifaberrations", although there are many of them, will in fact be members of primitive signs (which are sets, recall) the other members of which are predominantly tokens of a different type. They might say that it does not follow from this that the first letter token of Samuel Adams1 signature -is a token of 'J1 just because it is a member of the type lJ1. And that is true. But then even the weak connection between token and type postulated two pages earlier (that a token's atomic constituents would be (some of) the ur-elements out of which the type was constituted) has been severed, So that what makes an object a token of one rather than another type becomes a totally mysterious matter. Perhaps this problem could be eased. Perhaps one could start, for example, with sets like {x: x is the same phoneme as (some particular [PI)), and depend upon the phonologists to develop adequate criteria of identity for phonemes (i.e., for when two things are "physically similar in soundu). (The case for letters seems to me no easier; given the way medical doctors write, it may even be harder.) Then at least there would be some specifiable connection between types and their tokens. ----But is it the right --sort of connection? This brings us te the third problem that comes of construing expressions as classes in anything like the Hugly-Sayward manner. Let 3 be the set of tokens of the letter lcl. Let a be the set of tokens of the letter 'a1.

Let r be the set of tokens of the letter It'. It seems to me obvious that although (75) >is an ordered pair. and (76) >has a member that is an ordered pair, are true,

(77) > is pronounced [lkat]. (78) >is composed of 3 letters. and (79) > is spelled lcl-lal-ttl. are false. (Perhaps '>' has a pronunciation, though nat [lkat], but it doesn't help with (77).). It also seems to me obvious that most English speaker-hearers are familiar with some expression types -- e.g. the word 'cat1. We may not be confident of precisely what it is (which is why we are embarked upon this discussion) but we -do know some things about it, some things that need explaining in view of the fact that as an abst:?act object, a word type like 'cat1 cannot be perceived. For example, as the formalist assumed in his/her objection #5, we know that the following are true: (80) The word 'cat ' is pronounced [ lkat 1. (81) The word 'cat1 is composed of 3 letters. (82) The word 'cat1 is spelled '~~-~al-lt'. The formalistls explanation for -how we know that (80)) (81) and (82) are true, is that we experience tokens of 'catf; we hear thii2gs pronounced [Ikat], and see things composed of the 3 letters 'cf, 'af and Itf. Still, why does this enable us to infer that (80), (81) and (82) are true? As we saw, the formalistfs answer is that types are jl~stlike their tokens, except that they are abstract, whereas tokens are concrete objects. (How- we know this is another story, one that shall be ignored until we get to Sylvain Brombergerfs account; for present purposes it is enough that we know it or something like it.) Now we saw that in the Hugly-Sayward construal, the expression at (74) was supposed to be a token of "the expressionu , <, >>. So the word 'cat1 would probably turn out, under one of their construals, to be some set like >. Letfs suppose it does, Then it would follow that (77), (78) and (79) are true. But as we remarked earlier, they are false. Of course an --ad hoc redefining of "is pronounced," "is composed of," and "is spelled" might gratuitously have the consequence that (77), (78) and (79) are true. But I do not think we should be fooled by such possibilities into thinking that (77), (78) and (79) might really be true -- anymore than a dog would have five legs, if we redefined 'legf to include tails, as the old joke goes. What this point comes to is that we may be ignorant about precisely what it is that expressions are; but we are not so ignorant as to believe that they are classes, Besides, -a fornialist would never go --for a class -construal anyway, for two reasons: first, it violates the common sense basis for his/her objection #5 (the one we are considering); and second, (s)he is not likely to be charmed by the commitment to sets and n-tuples that it entails. Not that we ought to construe Hugly and Sayward as merely having made. some kind of fundamental metaphysical mistake. There are contexts in which it could be very useful to treat expressions as if they were classes, and Hugly and Sayward showed just what sort of classes they could be treated as so as to preserve certain inportant properties of expressions (like their being abstract and infinite in number). It is just that in doing so they sacrificed certain other properties of expressions (like their having the same shape and spelling as their tokens), properties that are important in the epistemological arena. There is one more difficulty that I would like to mention with Hugly and Sayward1s construal. One of the most useful applications of the type-token distinction has been in aesthetics. Richard Wollheim, for example, whose views on types we shall get to shortly, construes many works of art as types, whose tokens are the various copies of it, or its performances, or what have you. 89 I think it is desirable that any account of what an expression type is

89. ---Art and Its Objects. (New York: Harper and Row, 1968). should be part of a general account of types. But it is very hard to see how to generalize Hugly and Sayward1a account so as to explain what Handelts Messiah is, for example. Is it the set of its tokens? Again, this seems an in- appropriate construal because whereas the Messiah is notable for its many ~Hallelujahfstfthe set of its tokens (real and imagined performances, copies of the score, perhaps) contains -no ~Hallelujahtsff,only its members do. So let us put aside Hugly and Sayward1s account, and all accounts that construe types as classes, because most of the difficulties their account faces stem from this identification. Richard Wollheim, whose account we shall now consider, labels a type a non-particular, and constrasts it with other kinds of "generic entitiestf, or non-particulars like universals (which are said to have instances) but especially with classes (which are said to have members). "An example of a class would be the class of red things: an example of a universal would be redness: and examples of a type would be the word 'redl and the Red Flag.!' (p. 75). Although sketchy (we do not know, e.g., where numbers fit into this account on the basis of the above) this at least begins to constitute an answer to (70). Wollhelm discusses the relations between generic entities and their "element stf, as he calls them -- the things that 'Ifall underff them -- thereby further distinguishing the generic entities from each other: The various generic entities can be distinguished according to the ways or relationships in which they stand to their elements. These relationships can be erranged on a scale of intimacy or intrinsicality. At one end of the scale we find classes, where the relationship is at its most external or extrinsic: for a class is merely a mode of, or constituted by, its members which are extensionally con.joined to form it..,In the case of universals the relation is more intimate: in that a universal is present in all its instances. Redness is in all red things. With types we find the relationship between the generic entity and its elements at its most intimate: for not merely is the type present in all its tokens like the universal in all its instances, but for much of the time we think and talk of the type as though it were itself a kind of token, though a peculiarly important or pre-eminent one. (PP. 75-76).

So far, Wollheimls view may be too platonistic for the formalist, but perhaps the formalist can agree with what Wollheim says about types. The question is: how do numbers fit into this picture? The picture is really too vague and metaphorical for a definitive answer to be given, but my own impression is that the relation between 2, say, and pairs is not at all "extrinsi~~~,like that of classes to their members; that insofar as redness is "present" in red things and the word 'catr is present in this token. cat the number 2 can be said to be present in pairs; that we certainly -do treat 2 as a kind of particular, even a peculiarly important one; and that therefore, although it is not a type, 2 has to be rated, so far, as being rather similar to types. Wollheim then proceeds to give very general answers to questions (72) ,and (73), regarding the properties had by types and tokens, as well as those had by other generic entities and their elements. An important distinction emerges, between sharing properties (when A and B are both f, f is shared) and properties being transmitted (when A is f because B is f, or vice versa, f is transmitted between A and B). (p. 76). Wollheim contends that (83) Classes rarely share properties with their members, and no properties are transmitted. (84) (i)There are transmitted properties between universals and their instances, and between types and their tokens, but (ii)there are fewer transmissions for universals than for types, and (iii)no property that an instance of a universal has in virtue of being an instance (like being -red) can be transmitted to the universal (redness) -- unlike the situation with types (like the Red Flag), which do- have the properties their instances have in virtue of being instances (like their being red). Now bt might appear that the formalist's point has been provzd. If (84) is true, then since, as the formalist charges, being composed of exactly things is certainly a property that all pairs have in virtue of being pairs, but 2 lacks it, then by (84) it follows that the number- plurality relationship is not like the type-token one -- at best it is like that of a universal to its instances -- because there is at least one non-transmitted property that all the "instances" of 2 necessarily share. So if (84) were true, the formalist's objection would have received theoretical underpinning from Wollheim. However, I think that (84) is false -- or rather, that (84) (iii) is false when the types concerned are expressions. Earlier, we did not disagree with the formalist that types are just like their tokens, except that they are abstract whereas tokens are concrete objects; and that we know (80), (81), and (82) are true because we hear things pronourlced ['kat] and see things composed of the 3 letters 'c', 'a' and ltl. All of this vaguely suggests Wollheim is right and (84) is true. But what, precisely -are the transmitted properties in the case of 'cat1? A particular utterance of 'cat1 is a sound event, is a pronunciation of 'cat1, but it is not the sort of thing that, strictly speaking, -has a pronunciation. Yet the word type 'cat1 has a pronunciation, It is also composed of 3 letter types: the letter types lcl, lar and t. It is not and cannot be composed of letter tokens. Tokens of 'cat1, on the other hand, are composed of 3 letter tokens, tokens of the letters 'c', 'af and It'. Of course, both 'cat' and its tokens can be spoken of, as we did earlier, as "being composed or 3 letters", suggesting that there is a common transmitted property. But to treat being composed of 3 letter types as the same property as being composed --of 3 letter tokens Is to just ignore the type-token distinction -- hardly appropriate in the present context, when we are trying to understand what ea~his, and how they relate to each other. If "what we sayu were the only criterion for when properties are shared, even in the face of reasonable distinctions, then a trio would share the inability to be halved with 3, because we could say either (85) A trio cannot be halved. or (86) 3 cannot be halved.

For that matter, numbers would share many properties with numerals, because, like Kline (above) "what we sayf1 is often at odds with the use-mention distinction.

Therefore it seems to me that (84) (iii) is false as regards expressions. This may not matter to Wollheim, since he was probably thinkirg less of expressions than of works of art. But I believe his interest in works of art led him to over-emphasize the .importance of "a set of circumstances in which we postulate typest1, namely, those "where we can correlate a class of particulars with a piece of human invention: these particulars may then be regarded as tokens of a certain typeff. (F. 78). This is supposed to comprehend a spectrum of cases: "At the one end we have the case where the particular is produced, and is then copied; at the other end, we have the case where a set of instructions is drawn up which, if followed, gives rise to an indefinite number of particularsI1 (p. 78). While useful for works of art, it seems to be a little less useful to consider all expressions "inventionsrr, and completely wrong to consider types like The Tiger lrinventionsu, because its tokens are members of a very natural kind. Hence while it has many attractive features, Wollheirnfs account of types is too restrictive. (Note, however, that what he did say that was right dj.d not intro- duce any serious disanalogies between expressions and numbers.). Let us move on then to our last account of expression types, that of Sylvain Brombergerls, to see if it supplies the formalist with the theoretical means for claiming that types are just like their tokens, or for otherwise undermining the analogy between expressions and numbers. Brombergerls account has several advantages, for our purposes, over that of Wollheimls. Whereas Wollheim was primarily concerned with works of art, Bromberger is primarily concerned with expression,,types. And besides not having the restrictive view of types as "human inventions" (that rules out The Tiger) Brombergerfs account goes further towards explaining how we have knowledge of expression types, by relating such knowledge to our knowledge of natural kinds generally. The account in question is contained in an unpublished paper called "Mind, Language, and Knowledge: On Some Platonistic Relationships From An Erotetic Position" that was first presented at the APA meetings in December 1981. Page references will be to a more recent version of the paper. While all of it is interesting and 1rnportant;in what follows only the main points of the paper that bear upon our tcpic will be discussed. And they will be discussed, not critically (because that would take us a bit far afield) but only with an eye towards seeing how numbers stack up epistemologically against expressions in this, the best theory of what expressions are that I know of. Bromberger specifically asks: "what is the connection between tokens and their types by virtue of which inferences from tokens to types are possible?" (p. 6). His answer is sumqarized in these six conjectures (which I haverenum- bered). (87) "Toh.ens of a type make up a natural kind.ll (88) "Tokens of a type make up a natural kind, that has an archetype." (89) llArchetypes of a natural kind are abstract entities.

(90) "(The Platonic Relationship Principle: ) If A is an archetype of a natural kind K, then the essential properties of A are systematically derivable from some of the properties shared by all the members of the natural kind." (91) "Types are the archetype of the natural kind made up

by their members. l1 (92) llInfesences from tokens to their types are based on the Platonic Relationship Prin~iple.~'(p. 6). In connection with (87), Bromberger stresses the fact that members of a natural kind model each other, because by finding out certain things about one member, one can also find out certain things about another (pp. 6f). (More

strictly, I'M is a model of 0 relative to a set of triples

[Q,, Qo, A] if and only if in each triple Qm is a question

about M, Qo is a question about 0, and A is an algorithm that translates any answer to into an answer to go, and

correct answers to Qm into correct answers to Qo.ll (p 7) So for example, finding an answer to ''What is its boiling point under standard conditions?ll for one sample of pure water gives an answer, the same answer, to the question asked of other samples; or for tigers, "What is its anatomical structure?" Bromberger actually distinguishes three kinds of questions that are associated with natural kinds: projectible, quasi-projectible, and individuating questions (p. 9). Projectible questions -- like the examples in the preceding paragraph -- (a) "can be asked of each member of the natural kind;" (b) have the same answer for each member; and (c) "when the answer,..gives rise to a why question, then that

why-question must itself be projectiblell (p, 9). Individuating qsestions -- like "Where was it on September 4, 1983 at 4 p.m. e.s.t?I1 for samples of water, or "On what day was it born?I1 for tigers -- (a) "can be asked of each member of the natural 1cfnd;'I but (b) not all members receive the same .answer; (c) "the answers need not give rise to why-questions; and (d) these "need not be projectiblel1. Also, (e) "answers to the why-questions can be grounded on different nomological principles" (p. 10). What quasi-projectible questions are will not matter to us here. Now of course it is not enough for a set to be a natural kind that there be proj ectible, quasi-proj ect ible and individuating questions that can be asked of its members. But "what determines whether one is justified- in thinking of a set of objects as constituting a natural kind relative to three groups of questions" is IITheoryI1, says Bromberger. "The kinds of theory that are relevant are the sort of theories that group objects into certain hierarchical systems1' (p. 11) Brornberg~rthen claims that lflinguistics procedes on the presumption that there is a set of questions relative to which tokens of a type constitute a natural kind" (p, 14). For example, here are some projectible questions for tokens of the word 'catastrophe': (93) How many syllable (tokens) compose it? (94) What is the phoneme that its onset is a token of? (95) What is its underlying phonological structure? And here are some individuating questions: (96) Who uttered it? (97) Where was it uttered? It seems to me that it could with almost equal plausibility be said that arithmetic (not necessarily of the pure variety) also procedes on the presumption that there is a set of questions relative to which pluralities of a certain size constitute a natural kind. There are projectible and individuating questions for pluralities. For example, for any pair of things one can ask, and get the same answer to, the following questions: (98) How many things is it composed of? (99) Can it be (non-fractionally) divided 3 ways? (100) Why can't it be divided 3 ways? Examples of individuating questions, at least for most uconcrete" pairs, are: (101) Where is it? (102) What sort of thing is it 2 of? Nor does the analogy break down when it comes to the business of archetypes. An archetype of a natural kind (that has an archetype) is, I think, an object for which (a) there is an algorithm that indicates in what relatively slight but systematic way every projectible question associated with the kind, can be altered so as to be asked of it; (b) the correct answer to the question is, except for similar alterations given by the algorithm, the same as for aach member of the natural kind; and (c) none of the other questions (quasi-projectible or individuating) have a right answer (because "its status as archetype violates their presupposition" (p. 121.). Bromberger argues that -an expression type 2 an archetype the natural kind its tokens comprise. One can ask of the word type 'catastrophef, e.g., corresponding to (931, (94) and (95): (93') How many syllables compose it? (94') What is the phoneme that comprises its onset? (95') What is its underlying phonological structure? And I think it is obvious how the answers to (93), (94) and

(95), whatever they are, will correspond to those of (93'), (94') and (95'). On the other hand, (98), (99) and (100) suitably and slightly altered can be asked of -the number -2: (98') How many 1's is it composed of, i.e., add up to it? (99') Can it be (non-fractionally) divided by 3? (100') Why can't it be divided by 3? Much more could be said by way of exploring Bromberger's conception of expressions as archetypes of natural kinds, and to what extent numbers can analogously be construed as archetypes of pluralities. I admit that it seems to be to a lesser extent, But this suggests to me merely that mathematics gets more abstract, more quickly, than linguistics does, not that it does not have any empirical basis (epistemologically) at all. However I hope enough has been said to meet the formalist's objection #5. That objection has not been well supported by sny theory of expression types that we have looked at. If anything, it seems to me at least, that Bromberger's theory, the best of them all, supports the anti-formalist claim that our knowledge of the abstract objects of linguistics and mathematics Reems partly dependent, in both cases, on our experience of concrete things that model each other in certain respects, and which enable us to "grasp" the essential features of the abstract object. SELECTED BIBLIOGRAPHY

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