The inscrutability of reference¹

1 I listened to Dewey on Art as Experience when I was a graduate student in the spring of 1931. Dewey was then at Harvard as the first William James Lecturer. I am proud now to be at Columbia as the first John Dewey Lecturer.

 

Philosophically I am bound to Dewey by the naturalism that dominated his last three decades. With Dewey I hold that knowledge, mind, and meaning are part of the same world that they have to do with, and that they are to be studied in the same empirical spirit that animates natural science. There is no place for a prior philosophy.

 

When a naturalistic philosopher addresses himself to the philosophy of mind, he is apt to talk of language. Meanings are, first and foremost, meanings of language. Language is a social art which we all acquire on the evidence solely of other people’s overt behavior under publicly recognizable circumstances. Meanings, therefore, those very models of mental entities, end up as grist for the behaviorist’s mill. Dewey was explicit on the point: ‘ Meaning... is not a psychic existence; it is primarily a property of behavior.'²

 

Once we appreciate the institution of language in these terms, we see that there cannot be, in any useful sense, a private language. This point was stressed by Dewey in the twenties. ‘ Soliloquy ’, he wrote, ‘ is the product and reflex of converse with others’ (170). Farther along he expanded the point thus: ‘Language is specifically a mode of interaction of at least two beings, a speaker and a hearer; it presupposes an organized group to which these creatures belong, and from whom they have acquired their habits of speech. It is therefore a relationship’ (185). Years later, Wittgenstein likewise rejected private language. When Dewey was writing in this naturalistic vein, Wittgenstein still held his copy theory of language.

 

The copy theory in its various forms stands closer to the main philosophical tradition, and to the attitude of common sense today. Uncritical semantics is the myth of a museum in which the exhibits are meanings and the words are labels. To switch languages is to change the labels. Now the naturalist’s primary objection to this view is not an objection to meanings on account of their being mental entities, though that could be objection enough. The primary objection persists even if we take the labeled exhibits not as mental ideas but as Platonic ideas or even as the denoted concrete objects. Semantics is vitiated by a pernicious mentalism as long as we regard a man’s semantics as somehow determinate in his mind beyond what might be implicit in his dispositions to overt behavior. It is the very facts about meaning, not the entities meant, that must be construed in terms of behavior.

 

There are two parts to knowing a word. One part is being familiar with the sound of it and being able to reproduce it. This part, the phonetic part, is achieved by observing and imitating other people’s behavior, and there are no important illusions about the process. The other part, the semantic part, is knowing how to use the word. This part, even in the paradigm case, is more complex than the phonetic part. The word refers, in the paradigm case, to some visible object. The learner has now not only to learn the word phonetically, by hearing it from another speaker; he also has to see the object; and in addition to this, in order to capture the relevance of the object to the word, he has to see that the speaker also sees the object. Dewey summed up the point thus: ‘The characteristic theory about B's understanding of A’s sounds is that he responds to the thing from the standpoint of A ’ (178). Each of us, as he learns his language, is a student of his neighbor’s behavior; and conversely, insofar as his tries are approved or corrected, he is a subject of his neighbor’s behavioral study.

 

The semantic part of learning a word is more complex than the phonetic part, therefore, even in simple cases: we have to see what is stimulating the other speaker. In the case of words not directly ascribing observable traits to things, the learning process is increasingly complex and obscure; and obscurity is the breeding place of mentalistic semantics. What the naturalist insists on is that, even in the complex and obscure parts of language learning, the learner has no data to work with but the overt behavior of other speakers.

 

When with Dewey we turn thus toward a naturalistic view of language and a behavioral view of meaning, what we give up is not just the museum figure of speech. We give up an assurance of determinacy. Seen according to the museum myth, the words and sentences of a language have their determinate meanings. To discover the meanings of the native’s words we may have to observe his behavior, but still the meanings of the words are supposed to be determinate in the native’s mind, his mental museum, even in cases where behavioral criteria are powerless to discover them for us. When on the other hand we recognize with Dewey that1 meaning... is primarily a property of behavior’, we recognize that there are no meanings, nor likenesses nor distinctions of meaning, beyond what are implicit in people’s dispositions to overt behavior. For naturalism the question whether two expressions are alike or unlike in meaning has no determinate answer, known or unknown, except insofar as the answer is settled in principle by people’s speech dispositions, known or unknown. If by these standards there are indeterminate cases, so much the worse for the terminology of meaning and likeness of meaning.

 

To see what such indeterminacy would be like, suppose there were an expression in a remote language that could be translated into English equally defensibly in either of two ways, unlike in meaning in English. I am not speaking of ambiguity within the native language. I am supposing that one and the same native use of the expression can be given either of the English translations, each being accommodated by compensating adjustments in the translation of other words. Suppose both translations, along with these accommodations in each case, accord equally well with all observable behavior on the part of speakers of the remote language and speakers of English. Suppose they accord perfectly not only with behavior actually observed, but with all dispositions to behavior on the part of all the speakers concerned. On these assumptions it would be forever impossible to know of one of these translations that it was the right one, and the other wrong. Still, if the museum myth were true, there would be a right and wrong of the matter; it is just that we would never know, not having access to the museum. See language naturalistically, on the other hand, and you have to see the notion of likeness of meaning in such a case simply as nonsense.

 

I have been keeping to the hypothetical. Turning now to examples, let me begin with a disappointing one and work up. In the French construction ‘ne.. .rien’ you can translate ‘rien’ into English as ‘anything’ or as ‘nothing’ at will, and then accommodate your choice by translating ‘ ne ’ as ‘ not ’ or by construing it as pleonastic. This example is disappointing because you can object that I have merely cut the French units too small. You can believe the mentalistic myth of the meaning museum and still grant that ‘ rien ’ of itself has no meaning, being no whole label; it is part of ‘ ne. . . rien ’, which has its meaning as a whole.

 

I began with this disappointing example because I think its conspicuous trait - its dependence on cutting language into segments too short to carry meanings - is the secret of the more serious cases as well. What makes other cases more serious is that the segments they involve are seriously long: long enough to be predicates and to be true of things and hence, you would think, to carry meanings.

 

An artificial example which I have used elsewhere³ depends on the fact that a whole rabbit is present when and only when an undetached part of a rabbit is present; also when and only when a temporal stage of a rabbit is present. If we are wondering whether to translate a native expression ‘gavagai’ as ‘rabbit’ or as ‘ undetached rabbit part ’ or as ‘ rabbit stage ’, we can never settle the matter simply by ostension - that is, simply by repeatedly querying the expression ‘ gavagai ’ for the native’s assent or dissent in the presence of assorted stimulations.

 

Before going on to urge that we cannot settle the matter by nonostensive means either, let me belabor this ostensive predicament a bit. I am not worrying, as Wittgenstein did, about simple cases of ostension. The color word ‘ sepia ’, to take one of his examples,⁴ can certainly be learned by an ordinary process of conditioning, or induction. One need not even be told that sepia is a color and not a shape or a material or an article. True, barring such hints, many lessons may be needed, so as to eliminate wrong generalizations based on shape, material, etc., rather than color, and so as to eliminate wrong notions as to the intended boundary of an indicated example, and so as to delimit the admissible variations of color itself. Like all conditioning, or induction, the process will depend ultimately also on one’s own inborn propensity to find one stimulation qualitatively more akin to a second stimulation than to a third; otherwise there can never be any selective reinforcement and extinction of responses.⁵ Still, in principle nothing more is needed in learning ‘ sepia ’ than in any conditioning or induction.

 

But the big difference between ‘rabbit’ and ‘sepia’ is that whereas ‘sepia’ is a mass term like ‘water’, ‘rabbit’ is a term of divided reference. As such it cannot be mastered without mastering its principle of individuation: where one rabbit leaves off and another begins. And this cannot be mastered by pure ostension, however persistent.

 

Such is the quandary over ‘ gavagai ’: where one gavagai leaves off and another begins. The only difference between rabbits, undetached rabbit parts, and rabbit stages is in their individuation. If you take the total scattered portion of the spatio-temporal world that is made up of rabbits, and that which is made up of undetached rabbit parts, and that which is made up of rabbit stages, you come out with the same scattered portion of the world each of the three times. The only difference is in how you slice it. And how to slice it is what ostension or simple conditioning, however persistently repeated, cannot teach.

 

Thus consider specifically the problem of deciding between ‘ rabbit ’ and ‘ undetached rabbit part ’ as translation of ‘gavagai ’. No word of the native language is known, except that we have settled on some working hypothesis as to what native words or gestures to construe as assent and dissent in response to our pointings and queryings. Now the trouble is that whenever we point to different parts of the rabbit, even sometimes screening the rest of the rabbit, we are pointing also each time to the rabbit. When, conversely, we indicate the whole rabbit with a sweeping gesture, we are still pointing to a multitude of rabbit parts. And note that we do not have even a native analogue of our plural ending to exploit, in asking ‘gavagai?’. It seems clear that no even tentative decision between ‘ rabbit ’ and ‘ undetached rabbit part ’ is to be sought at this level.

 

How would we finally decide? My passing mention of plural endings is part of the answer. Our individuating of terms of divided reference, in English, is bound up with a cluster of interrelated grammatical particles and constructions: plural endings, pronouns, numerals, the ‘ is ’ of identity, and its adaptations ‘ same ’ and ‘ other ’. It is the cluster of interrelated devices in which quantification becomes central when the regimentation of symbolic logic is imposed. If in his language we could ask the native ‘Is this gavagai the same as that one?’ while making appropriate multiple ostensions, then indeed we would be well on our way to deciding between ‘rabbit’, ‘ undetached rabbit part ’, and ‘ rabbit stage ’. And of course the linguist does at length reach the point where he can ask what purports to be that question. He develops a system for translating our pluralizations, pronouns, numerals, identity, and related devices contextually into the native idiom. He develops such a system by abstraction and hypothesis. He abstracts native particles and constructions from observed native sentences, and tries associating these variously with English particles and constructions. Insofar as the native sentences and the thus associated English ones seem to match up in respect of appropriate occasions of use, the linguist feels confirmed in these hypotheses of translation - what I call analytical hypotheses.⁶

 

But it seems that this method, though laudable in practice and the best we can hope for, does not in principle settle the indeterminacy between ‘rabbit’, ‘undetached rabbit part’, and ‘rabbit stage’. For if one workable over-all system of analytical hypotheses provides for translating a given native expression into ‘ is the same as’, perhaps another equally workable but systematically different system would translate that native expression rather into something like ‘belongs with’. Then when in the native language we try to ask ‘ Is this gavagai the same as that? ’, we could as well be asking ‘ Does this gavagai belong with that? ’. Insofar, the native’s assent is no objective evidence for translating ‘gavagai’ as ‘rabbit’ rather than ‘ undetached rabbit part ’ or ‘ rabbit stage ’.

 

This artificial example shares the structure of the trivial earlier example ‘ ne. . . rien’. We were able to translate Tien’ as ‘anything’ or as ‘nothing’, thanks to a compensatory adjustment in the handling of ‘ ne ’. And I suggest that we can translate ‘gavagai’ as ‘rabbit’ or ‘undetached rabbit part’ or ‘rabbit stage’, thanks to compensatory adjustments in the translation of accompanying native locutions. Other adjustments still might accommodate translation of ‘ gavagai ’ as ‘ rabbithood ’, or in further ways. I find this plausible because of the broadly structural and contextual character of any considerations that could guide us to native translations of the English cluster of interrelated devices of individuation. There seem bound to be systematically very different choices, all of which do justice to all dispositions to verbal behavior on the part of all concerned.

 

An actual field linguist would of course be sensible enough to equate ‘ gavagai ’ with ‘ rabbit ’, dismissing such perverse alternatives as ‘ undetached rabbit part ’ and ‘ rabbit stage ’ out of hand. This sensible choice and others like it would help in turn to determine his subsequent hypotheses as to what native locutions should answer to the English apparatus of individuation, and thus everything would come out all right. The implicit maxim guiding his choice of ‘rabbit’, and similar choices for other native words, is that an enduring and relatively homogeneous object, moving as a whole against a contrasting background is a likely reference for a short expression. If he were to become conscious of this maxim, he might celebrate it as one of the linguistic universals, or traits of all languages, and he would have no trouble pointing out its psychological plausibility. But he would be wrong; the maxim is his own imposition, toward settling what is objectively indeterminate. It is a very sensible imposition, and I would recommend no other. But I am making a philosophical point.

 

It is philosophically interesting, moreover, that what is indeterminate in this artificial example is not just meaning, but extension; reference. My remarks on indeterminacy began as a challenge to likeness of meaning. I had us imagining ‘ an expression that could be translated into English equally defensibly in either of two ways, unlike in meaning in English ’. Certainly likeness of meaning is a dim notion, repeatedly challenged. Of two predicates which are alike in extension, it has never been clear when to say that they are alike in meaning and when not; it is the old matter of featherless bipeds and rational animals, or of equiangular and equilateral triangles. Reference, extension, has been the firm thing; meaning, intension, the infirm. The indeterminacy of translation now confronting us, however, cuts across extension and intension alike. The terms ‘rabbit’, ‘undetached rabbit part’, and ‘ rabbit stage ’ differ not only in meaning; they are true of different things. Reference itself proves behaviorally inscrutable.

 

Within the parochial limits of our own language, we can continue as always to find extensiona! talk clearer than intensional. For the indeterminacy between ‘rabbit’, ‘ rabbit stage ’, and the rest depended only on a correlative indeterminacy of translation of the English apparatus of individuation - the apparatus of pronouns, pluralization, identity, numerals, and so on. No such indeterminacy obtrudes so long as we think of this apparatus as given and fixe*d. Given this apparatus, there is no mystery about extension; terms have the same extension when true of the same things. At the level of radical translation, on the other hand, extension itself goes inscrutable.

 

My example of rabbits and their parts and stages is a contrived example and a perverse one, with which, as I said, the practicing linguist would have no patience.

 

But there are also cases, less bizarre ones, that obtrude in practice. In Japanese there are certain particles, called ‘ classifiers ’, which may be explained in either of two ways. Commonly they are explained as attaching to numerals, to form compound numerals of distinctive styles. Thus take the numeral for 5. If you attach one classifier to it you get a style of ‘ 5 ’ suitable for counting animals; if you attach a different classifier, you get a style of ‘ 5 ’ suitable for counting slim things like pencils and chopsticks; and so on. But another way of viewing classifiers is to view them not as constituting part of the numeral, but as constituting part of the term-the term for ‘chopsticks’ or ‘ oxen ’ or whatever. On this view the classifier does the individuative job that is done in English by ‘sticks of’ as applied to the mass term ‘wood’, or ‘head of’ as applied to the mass term ‘cattle’.

 

What we have on either view is a Japanese phrase tantamount say to ‘five oxen’, but consisting of three words ;⁷ the first is in effect the neutral numeral ‘ 5 ’, the second is a classifier of the animal kind, and the last corresponds in some fashion to ‘ ox ’. On one view the neutral numeral and the classifier go together to constitute a declined numeral in the ‘animal gender’, which then modifies ‘ox’ to give, in effect, ‘five oxen’. On the other view the third Japanese word answers not to the individuative term ‘ ox ’ but to the mass term ‘ cattle ’; the classifier applies to this mass term to produce a composite individuative term, in effect ‘ head of cattle ’; and the neutral numeral applies directly to all this without benefit of gender, giving ‘ five head of cattle’, hence again in effect ‘five oxen’.

 

If so simple an example is to serve its expository purpose, its needs your connivance. You have to understand ‘cattle’ as a mass term covering only bovines, and ‘ ox ’ as applying to all bovines. That these usages are not the invariable usages is beside the point. The point is that the Japanese phrase comes out as ‘five bovines ’, as desired, when parsed in either of two ways. The one way treats the third Japanese word as an individuative term true of each bovine, and the other way treats that word rather as a mass term covering the unindividuated totality of beef on the hoof. These are two very different ways of treating the third Japanese word; and the three-word phrase as a whole turns out all right in both cases only because of compensatory differences in our account of the second word, the classifier.

 

This example is reminiscent in a way of our trivial initial example, ‘ ne. . . rien ’. We were able to represent ‘ rien ’ as ‘ anything ’ or as ‘ nothing ’, by compensatorily taking ‘ ne ’ as negative or as vacuous. We are able now to represent a Japanese word either as an individuative term for bovines or as a mass term for live beef, by compensatorily taking the classifier as declining the numeral or as individuating the mass term. However, the triviality of the one example does not quite carry over to the other. The early example was dismissed on the ground that we had cut too small: ‘rien’ was too short for significant translation on its own, and ‘ne. . .rien’ was the significant unit. But you cannot dismiss the Japanese example by saying that the third word was too short for significant translation on its own and that only the whole three-word phrase, tantamount to ‘five oxen’, was the significant unit. You cannot take this line unless you are prepared to call a word too short for significant translation even when it is long enough to be a term and carry denotation. For the third Japanese word is, on either approach, a term; on one approach a term of divided reference, and on the other a mass term. If you are indeed prepared thus to call a word too short for significant translation even when it is a denoting term, then in a back-handed way you are granting what I wanted to prove: the inscrutability of reference.

 

Between the two accounts of Japanese classifiers there is no question of right and wrong. The one account makes for more efficient translation into idiomatic English; the other makes for more of a feeling for the Japanese idiom. Both fit all verbal behavior equally well. All whole sentences, and even component phrases like ‘five oxen’, admit of the same net over-all English translations on either account. This much is invariant. But what is philosophically interesting is that the reference or extension of shorter terms can fail to be invariant. Whether that third Japanese word is itself true of each ox, or whether on the other hand it is a mass term which needs to be adjoined to the classifier to make a term which is true of each ox - here is a question that remains undecided by the totality of human dispositions to verbal behavior. It is indeterminate in principle; there is no fact of the matter. Either answer can be accommodated by an account of the classifier. Here again, then, is the inscrutability of reference - illustrated this time by a humdrum point of practical translation.

 

The inscrutability of reference can be brought closer to home by considering the word ‘alpha’, or again the word ‘green’. In our use of these words and others like them there is a systematic ambiguity. Sometimes we use such words as concrete general terms, as when we say the grass is green, or that some inscription begins with an alpha. Sometimes on the other hand we use them as abstract singular terms, as when we say that green is a color and alpha is a letter. Such ambiguity is encouraged by the fact that there is nothing in ostension to distinguish the two uses. The pointing that would be done in teaching the concrete general term ‘green’, or ‘ alpha ’, differs none from the pointing that would be done in teaching the abstract singular term ‘green’ or ‘alpha’. Yet the objects referred to by the word are very different under the two uses: under the one use the word is true of many concrete objects, and under the other use it names a single abstract object.

 

We can of course tell the two uses apart by seeing how the word turns up in sentences: whether it takes an indefinite article, whether it takes a plural ending, whether it stands as singular subject, whether it stands as modifier, as predicate complement, and so on. But these criteria appeal to our special English grammatical constructions and particles, our special English apparatus of individuation, which, I already urged, is itself subject to indeterminacy of translation. So, from the point of view of translation into a remote language, the distinction between a concrete general and an abstract singular term is in the same predicament as the distinction between ‘ rabbit ’, ‘ rabbit part ’, and ‘ rabbit stage ’. Here then is another example of the inscrutability of reference, since the difference between the concrete general and the abstract singular is a difference in the objects referred to.

 

Incidentally we can concede this much indeterminacy also to the ‘ sepia ’ example, after all. But this move is not evidently what was worrying Wittgenstein.

 

The ostensive indistinguishability of the abstract singular from the concrete general turns upon what may be called ‘ deferred ostension' as opposed to direct ostension. First let me define direct ostension. The ostended point, as I shall call it, is the point where the line of the pointing finger first meets an opaque surface. What characterizes direct ostension, then, is that the term which is being ostensively explained is true of something that contains the ostended point. Even such direct ostension has its uncertainties, of course, and these are familiar. There is the question how wide an environment of the ostended point is meant to be covered by the term that is being ostensively explained. There is the question how considerably an absent thing or substance might be allowed to differ from what is now ostended, and still be covered by the term that is now being ostensively explained. Both of these questions can in principle be settled as well as need be by induction from multiple ostensions. Also, if the term is a term of divided reference like ‘ apple ’, there is the question of individuation : the question where one of its objects leaves off and another begins. This can be settled by induction from multiple ostensions of a more elaborate kind, accompanied by expressions like ‘ same apple ’ and ‘ another ’, if an equivalent of this English apparatus of individuation has been settled on; otherwise the indeterminacy persists that was illustrated by ‘ rabbit ’, ‘ undetached rabbit part ’, and ‘ rabbit stage ’.

 

Such, then, is the way of direct ostension. Other ostension I call deferred. It occurs when we point at the gauge, and not the gasoline, to show that there is gasoline. Also it occurs when we explain the abstract singular term ‘ green ’ or ‘ alpha ’ by pointing at grass or a Greek inscription. Such pointing is direct ostension when used to explain the concrete general term ‘green’ or ‘alpha’, but it is deferred ostension when used to explain the abstract singular terms; for the abstract object which is the color green or the letter alpha does not contain the ostended point, nor any point.

 

Deferred ostension occurs very naturally when, as in the case of the gasoline gauge, we have a correspondence in mind. Another such example is afforded by the Gödel numbering of expressions. Thus if 7 has been assigned as Gödel number of the letter alpha, a man conscious of the Gödel numbering would not hesitate to say ‘ Seven ’ on pointing to an inscription of the Greek letter in question. This is, on the face of it, a doubly deferred ostension: one step of deferment carries us from the inscription to the letter as abstract object, and a second step carries us thence to the number.

 

By appeal to our apparatus of individuation, if it is available, we can distinguish between the concrete general and the abstract singular use of the word ‘ alpha ’; this we saw. By appeal again to that apparatus, and in particular to identity, we can evidently settle also whether the word ‘ alpha ’ in its abstract singular use is being used really to name the letter or whether, perversely, it is being used to name the Gödel number of the letter. At any rate we can distinguish these alternatives if also we have located the speaker’s equivalent of the numeral ‘ 7 ’ to our satisfaction; for we can ask him whether alpha is 7.

 

These considerations suggest that deferred ostension adds no essential problem to those presented by direct ostension. Once we have settled upon analytical hypotheses of translation covering identity and the other English particles relating to individuation, we can resolve not only the indecision between ‘ rabbit ’ and ‘ rabbit stage ’ and the rest, which came of direct ostension, but also any indecision between concrete general and abstract singular, and any indecision between expression and Gödel number, which come of deferred ostension.

 

However, this conclusion is too sanguine. The inscrutability of reference runs deep, and it persists in a subtle form even if we accept identity and the rest of the apparatus of individuation as fixed and settled; even, indeed, if we forsake radical translation and think only of English.

 

Consider the case of a thoughtful protosyntactician. He has a formalized system of first-order proof theory, or protosyntax, whose universe comprises just expressions, that is, strings of signs of a specified alphabet. Now just what sorts of things, more specifically, are these expressions? They are types, not tokens. So, one might suppose, each of them is the set of all its tokens. That is, each expression is a set of inscriptions which are variously situated in space-time but are classed together by virtue of a certain similarity in shape. The concatenate xy of two expressions x and y, in a given order, will be the set of all inscriptions each of which has two parts which are tokens respectively of x and y and follow one upon the other in that order. But xy may then be the null set, though x and y are not null; for it may be that inscriptions belonging to x and y happen to turn up head to tail nowhere, in the past, present, or future. This danger increases with the lengths of x and y. But it is easily seen to violate a law of protosyntax which says that x = z whenever xy = zy.

 

Thus it is that our thoughtful protosyntactician will not construe the things in his universe as sets of inscriptions. He can still take his atoms, the single signs, as sets of inscriptions, for there is no risk of nullity in these cases. And then, instead of taking his strings of signs as sets of inscriptions, he can invoke the mathematical notion of sequence and take them as sequences of signs. A familiar way of taking sequences, in turn, is as a mapping of things on numbers. On this approach an expression or string of signs becomes a finite set of pairs, each of which is the pair of a sign and a number.

 

This account of expressions is more artificial and more complex than one is apt to expect who simply says he is letting his variables range over the strings of such and such signs. Moreover, it is not the inevitable choice; the considerations that motivated it can be met also by alternative constructions. One of these constructions is Gödel numbering itself, and it is temptingly simple. It uses just natural numbers, whereas the foregoing construction used sets of one-letter inscriptions and also natural numbers and sets of pairs of these. How clear is it that at just this point we have dropped expressions in favor of numbers? What is clearer is merely that in both constructions we were artificially devising models to satisfy laws that expressions in an unexplicated sense had been meant to satisfy.

 

So much for expressions. Consider now the arithmetician himself, with his elementary number theory. His universe comprises the natural numbers outright. Is it clearer than the protosyntactician’s? What, after all, is a natural number? There are Frege’s version, Zermelo’s, and von Neumann’s, and countless further alter¬ natives, all mutually incompatible and equally correct. What we are doing in any one of these explications of natural number is to devise set-theoretic models to satisfy laws which the natural numbers in an unexplicated sense had been meant to satisfy. The case is quite like that of protosyntax.

 

It will perhaps be felt that any set-theoretic explication of natural number is at best a case of obscurumper 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 more desirable than the familiar opposite. On the other hand our impression of the charity even of the notion of natural number itself has suffered somewhat from Gödel’s proof of the impossibility of a complete proof procedure for elementary number theory, or, for that matter, from Skolem’s and Henkin’s observations that all laws of natural numbers admit nonstandard models.⁸

 

We are finding no clear difference between specifying a universe of discourse - the range of the variables of quantification - and reducing that universe to some other. We saw no significant difference between clarifying the notion of expression and supplanting it by that of number. And now to say more particularly what numbers themselves are is in no evident way different from just dropping numbers and assigning to arithmetic one or another new model, say in set theory.

 

Expressions are known only by their laws, the laws of concatenation theory, so that any constructs obeying those laws - Gödel numbers, for instance - are ipso facto eligible as explications of expression. Numbers in turn are known only by their laws, the laws of arithmetic, so that any constructs obeying those laws - certain sets, for instance - are eligible in turn as explications of number. Sets in turn are known only by their laws, the laws of set theory.

 

Russell pressed a contrary thesis, long ago. Writing of numbers, he argued that for an understanding of number the laws of arithmetic are not enough; we must know the applications, we must understand numerical discourse embedded in discourse of other matters. In applying number the key notion, he urged, is Anzahl: there are n so-and-sos. However, Russell can be answered. First take, specifically, Anzahl. We can define ‘ there are n so-and-sos ’ without ever deciding what numbers are, apart from their fulfillment of arithmetic. That there are n so-and-sos can be explained simply as meaning that the so-and-sos are in one-to-one correspondence with the numbers up to n.⁹

 

Russell’s more general point about application can be answered too. Always, if the structure is there, the applications will fall into place. As paradigm it is perhaps sufficient to recall again this reflection on expressions and Gödel numbers: that even the pointing out of an inscription is no final evidence that our talk is of expressions and not of Gödel numbers. We can always plead, deferred ostension.

 

It is in this sense true to say, as mathematicians often do, that arithmetic is all there is to number. But it would be a confusion to express this point by saying, as is sometimes said, that numbers are any things fulfilling arithmetic. This formulation is wrong because distinct domains of objects yield distinct models of arithmetic. Any progression can be made to serve; and to identify all progressions with one another, e.g., to identify the progression of odd numbers with the progression of evens, would contradict arithmetic after all.

 

So, though Russell was wrong in suggesting that numbers need more than their arithmetical properties, he was right in objecting to the definition of numbers as any things fulfilling arithmetic. The subtle point is that any progression will serve as a version of number so long and only so long as we stick to one and the same progression. Arithmetic is, in this sense, all there is to number: there is no saying absolutely what the numbers are; there is only arithmetic.¹⁰

 

2 I first urged the inscrutability of reference with the help of examples like the one about rabbits and rabbit parts. These used direct ostension, and the inscrutability of reference hinged on the indeterminacy of translation of identity and other individuative apparatus. The setting of these examples, accordingly, was radical translation : translation from a remote language on behavioral evidence, unaided by prior dictionaries. Moving then to deferred ostension and abstract objects, we found a certain dimness of reference pervading the home language itself.

 

Now it should be noted that even for the earlier examples the resort to a remote language was not really essential. On deeper reflection, radical translation begins at home. Must we equate our neighbor’s English words with the same strings of phonemes in our own mouths? Certainly not; for sometimes we do not thus equate them. Sometimes we find it to be in the interests of communication to recognize that our neighbor’s use of some word, such as ‘cool’ or ‘square’ or ‘hopefully’, differs from ours, and so we translate that word of his into a different string of phonemes in our idiolect. Our usual domestic rule of translation is indeed the homophonic one, which simply carries each string of phonemes into itself; but still we are always prepared to temper homophony with what Neil Wilson has called the ‘ principle of charity’.¹¹ We will construe a neighbor’s word heterophonically now and again if thereby we see our way to making his message less absurd.

 

The homophonic rule is a handy one on the whole. That it works so well is no accident, since imitation and feedback are what propagate a language. We acquired a great fund of basic words and phrases in this way, imitating our elders and encouraged by our elders amid external circumstances to which the phrases suitably apply. Homophonic translation is implicit in this social method of learning. Departure from homophonic translation in this quarter would only hinder communication. Then there are the relatively rare instances of opposite kind, due to divergence in dialect or confusion in an individual, where homophonic translation incurs negative feedback. But what tends to escape notice is that there is also a vast mid-region where the homophonic method is indifferent. Here, gratuitously, we can systematically reconstrue our neighbor’s apparent references to rabbits as really references to rabbit stages, and his apparent references to formulas as really references to Gödel numbers and vice versa. We can reconcile all this with our neighbor’s verbal behavior, by cunningly readjusting our translations of his various connecting predicates so as to compensate for the switch of ontology. In short, we can reproduce the inscrutability of reference at home. It is of no avail to check on this fanciful version of our neighbor’s meanings by asking him, say, whether he really means at a certain point to refer to formulas or to their Gödel numbers; for our question and his answer - ‘ By all means, the numbers ’ - have lost their title to homophonic translation. The problem at home differs none from radical translation ordinarily so called except in the willfulness of this suspension of homophonic translation.

 

I have urged in defense of the behavioral philosophy of language, Dewey’s, that the inscrutability of reference is not the inscrutability of a fact; there is no fact of the matter. But if there is really no fact of the matter, then the inscrutability of reference can be brought even closer to home than the neighbor’s case; we can apply it to ourselves. If it is to make sense to say even of oneself that one is referring to rabbits and formulas and not to rabbit stages and Gödel numbers, then it should make sense equally to say it of someone else. After all, as Dewey stressed, there is no private language.

 

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 Gödel, numbers. Surely this is absurd, for it would imply that there is no difference between the rabbit and each of its parts or stages, and no difference between a formula and its Gödel number. Reference would seem now to become nonsense not just in radical translation, but at home.

 

Toward resolving this quandary, begin by picturing us at home in our language, with all its predicates and auxiliary devices. This vocabulary includes ‘rabbit’, ‘ rabbit part ’, ‘ rabbit stage ’, ‘ formula ’, ‘ number ’, ‘ ox ’, ‘ cattle ’; 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 and that the same rabbit, and this and that different parts. In just those words. This network 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 distinctively of rabbits and parts, numbers and formulas. Next, as in recent paragraphs, we contemplate alternative denotations for our familiar terms. We begin to appreciate that a grand and ingenious permutation of these denotations, along with compensatory adjustments in the interpretations of the auxiliary particles, might still accommodate all existing speech dispositions. This was the inscrutability of reference, applied to ourselves; and it made nonsense of reference. Fair enough ; reference is nonsense except relative to a coordinate system. In this principle of relativity lies the resolution of our quandary.

 

It is meaningless to ask whether, in general, our terms ‘rabbit’, ‘rabbit part’, ‘number’, etc., really refer respectively to rabbits, rabbit parts, numbers, etc., rather than to some ingeniously permuted denotations. It is meaningless to ask this absolutely; we can meaningfully ask it only relative to some background language. When we ask, ‘ Does “ rabbit ” 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. The background language gives the query sense, if only relative sense; sense relative in turn to it, this background language. Querying reference in any more absolute way would be like asking absolute position, or absolute velocity, rather than position or velocity relative to a given frame of reference. Also it is very much like asking whether our neighbor may not systematically see everything upside down, or in complementary color, forever undetectably.

 

We need a background language, I said, to regress into. Are we involved now in an infinite regress? If questions of reference of the sort we are considering make sense only relative to a background language, then evidently questions of reference of the background language make sense in turn only relative to a further background language. In these terms the situation sounds desperate, but in fact it is little different from questions of position and velocity. When we are given position and velocity relative to a given coordinate system, we can always ask in turn about the placing of origin and orientation of axes of that system of coordinates; and there is no end to the succession of further coordinate systems that could be adduced in answering the successive questions thus generated.

 

In practice of course we end the regress of coordinate systems by something like pointing. And in practice we end the regress of background languages, in discussions of reference, by acquiescing in our mother tongue and taking its words at face value.

 

Very well; in the case of position and velocity, in practice, pointing breaks the regress. But what of position and velocity apart from practice? what of the regress then? The answer, of course, is the relational doctrine of space; there is no absolute position or velocity; there are just the relations of coordinate systems to one another, and ultimately of things to one another. And I think that the parallel question regarding denotation calls for a parallel answer, a relational theory of what the objects of theories are. What makes sense is to say not what the objects of a theory are, absolutely speaking, but how one theory of objects is interpretable or reinterpretable in another.

 

The point is not that bare matter is inscrutable: that things are indistinguishable except by their properties. That point does not need making. The present point is reflected better in the riddle about seeing things upside down, or in complementary colors; for it is that things can be inscrutably switched even while carrying their properties with them. Rabbits differ from rabbit parts and rabbit stages not just as bare matter, after all, but in respect of properties; and formulas differ from numbers in respect of properties.

 

¹ This is part of the first of two Dewey lectures delivered at Columbia University on 26 March 1968 and published under its original title, ‘Ontological Relativity’ in The Journal of Philosophy, volume LXV, no. 7, 4 April 1968.

² Experience and Nature (La Salle, Ill.: Open Court, 1925, 1958), p. 179.

³ Word and Object (Cambridge, Mass.: MIT Press, 1960), §12.

⁴ Philosophical Investigations (New York: Macmillan, 1953), p. 14.

⁵ Cf. Word and Object, §17.

⁶ Word and Object, §15. For a summary of the general point of view see also §1 of my ‘ Speaking of Objects’, Proceedings and Addresses of American Philosophical Association, XXXI (1958): 5 ff; reprinted in Y. Krikorian and A. Edel, eds., Contemporary Philosophical Problems (New York: Macmillan, 1959), and in J. Fodor and J. Katz, eds., The Structure of Language (Englewood Cliffs, N.J.: Prentice-Hall, 1964), and in P. Kurtz, ed., American Philosophy in the Twentieth Century (New York: Macmillan, 1966).

⁷ To keep my account graphic I am counting a certain postpositive particle as a suffix rather than a word.

⁸ See Leon Henkin, ‘Completeness in the Theory of Types’, Journal of Symbolic Logic, xv, 2 (June 1950): 81-91, and references therein.

⁹ For more on this theme see my Set Theory and Its Logic (Cambridge, Mass.: Harvard 1963, 1968), §11.

¹⁰ Paul Benacerraf, ‘What Numbers Cannot Be’, Philosophical Review, LXXIV, 1 (January 1965): 47-73, develops this point. His conclusions differ in some ways from those I shall come to.

¹¹ N. L. Wilson, ‘ Substances without Substrata’, Review of Metaphysics, XII, 4 (June 1959): 521-39, p 532.

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