What Numbers Might Be Scott Soames John's anti-nominalism embraces numbers without, as far as I know, worrying very much about whether they fall under some other category like sets or properties. His strongest reason for embracing them is sound; our most advanced, reliable, and systematic knowledge-generating practices, mathematics and natural science, tell us that there are numbers, along with sets, functions, and so on. We also have reliable sources of knowledge testifying to the existence of other things -- expression types, languages, act and event types, properties, colors, shapes, species, and natural kinds in general -- about which some of the worries raised about numbers also arise. Still numbers have been the prominent targets of philosophers who are reluctant to grant the existence of anything beyond concrete particulars. John isn't sympathetic, nor am I. I sense that he also isn't too worried about trying to determine precisely what natural or other numbers are. If not, the reason may be that nothing in mathematics or natural science, requires, or even strongly favors, one proposed identification of natural numbers over another. So if none is needed, why bother? Certainly, none is needed to avoid skepticism. Still, we may want to know What explains our knowledge of numbers? -- all the way down to the knowledge children gain when they learn to count. Philosophical questions about how we know about all sorts of things can be puzzling, and ought to be pursued. But our knowledge of numbers seems to generate a special puzzlement not generated by our knowledge of rocks and trees. In part this is because our knowledge of the latter is largely perceptual whereas our knowledge of numbers is not generally thought of in this way. Outside of attempts to explain why there is no basis for radical skepticism about knowledge of the so-called external world,
philosophers haven't been overly perplexed by the idea that we know that there are mountains. But apart from Penny Maddy, who has argued that certain sets, the members of which can be seen, can also, themselves, be seen, most philosophers seem to take it for granted that we can't see sets or numbers. Even if we could see some sets, we surely can't see them all, and the usual constraints for evaluating different set-theoretic conceptions of natural numbers don't discriminate among them in that way. It is against this background that explaining our knowledge of natural numbers may worth special attention. Frege's idea, that we can find out what numbers are by finding out what best explains our knowledge of them, remains compelling. In sections 58 and 60 of The Foundations of Arithmetic, he says that nothing we can picture or imagine seems to be an apt candidate for being the number 4. But he is not deterred. Although we can, in his words, form no idea of the content of a number term, he insists that this no reason for denying that it does have content. Rather than considering the term in isolation, he thinks we should ask what it contributes to meaningful sentences in which it appears. He admonishes us "Always to keep before our eyes a complete proposition [he means 'sentence']. Only [he says] in a proposition have the words really a meaning." Although I like his emphasis on complete sentences, I would supplement his enigmatic "context principle" with a related, though perhaps equally enigmatic, principle: Natural numbers are whatever they have to be in order to explain our knowledge of them. There are lots of sentences containing numerical terms occurring as proper names, adjectives, or stand-alone predicates that express propositions we know. The strategy for finding out what natural numbers are is to investigate which assignments of meanings and referents to these terms best advances our ability to explain our knowledge of the truth of the propositions expressed by numerical sentences. 2
By 'our knowledge', I mean everyone's knowledge -- children who know only a little, adults who know more, and elementary number theorists who know much more. I presume that this vast population shares a fair bit of common knowledge, even though some know a great deal more than others. Assuming, as we must, that no one knows all arithmetical truths -- or even any set of truths from which all others could be validly derived -- the resulting assignment of true propositions to arithmetical sentences will outstrip all actual arithmetical knowledge. Still, we should be able to explain various possible extensions of the knowledge we now posses. I am most interested in explaining (i) how we achieve any knowledge of numbers at all, and (ii) how, with a little instruction, we acquire more knowledge by learning one or another set of arithmetical axioms, and/or convenient calculation procedures. Mainly, I am interested in what a realistic starting point for our arithmetical knowledge might look like. I suspect that one component of that starting point is what Frege called Hume s Principle, which specifies that the number of X s is the same as the number of Y s if and only if the X s and Y s can be exhaustively paired off (without remainder). As it happens, the number of universities at which I have been a regular faculty member -- Yale, Princeton, and USC -- can be exhaustively paired off (without remainder) with the fingers I am holding up now. So, the number of universities at which I have served is the same the number as the number of fingers I am holding up. Both the fingers and the universities are three (in number). What is this property, being three (in number), predicated of? It s not predicated of any of my past faculty homes; neither Yale, Princeton, nor USC, is three (in number). It is also not predicated of the set that contains them (and only them); since the set of those universities is a single thing, it s not three either. Like the property being scattered, the property being three (in number) is irreducibly plural. It is a property that applies, not to any 3
single instance of any type of thing (simple, composite, concrete, or abstract), but to multiple things of a given type considered together. My former Ph.D. students are scattered around the world, even though no one of them is scattered around the world, and the set containing them isn t scattered around the world either. With this in mind, consider the hypothesis that each natural number N greater than or equal to 2 is the plural property being N (in number), and that the number 1 is a property applying to each individual thing considered on its own. Zero is a property that doesn't 1 apply to anything, or to any things. Suppose then that natural numbers are cardinality properties of individuals and multiples of the kind just illustrated. How do we gain knowledge of them? In the beginning, we do so by counting. Imagine a child inferring that I am holding up three fingers from her perceptual knowledge that x, y, and z are different fingers. Having learned to count -- at first by memorizing a sequence of verbal numerals -- she concludes that the fingers are three in number, which is a property she has learned to recognize from other events of counting things of various types. In so doing, she pairs off, without remainder, the fingers I am holding up with the English words one, 'two, and three, thereby ensuring that the fingers and the numerals have the same number in Frege s sense. Even better, the number they share is designated by the numeral, three, that ends the count; it is the property being three (in number). With this, we have the germ of an idea that combines the best of the attempted Frege-Russell reductions with a striking, but flawed and incompletely developed insight gestured at in section 1 of Wittgenstein's Philosophical Investigations. The book begins with a quotation from Augustine. 1 This way of thinking of natural numbers grows out of two path-breaking articles one, Boolos (1984) -- by my former teacher, George Boolos, and the other -- Arabic Numerals and the Problem of Mathematical Sophistication, forthcoming by my former Ph.D. student Mario Gomez Torrente. 4
When they (my elders) named some object, and accordingly moved toward something, I saw this and I grasped that the thing was called by the sound they uttered when they meant to point it out. Their intention was shown by their bodily movements, as it were the natural language of all peoples: the expression of the face, the play of the eyes, the movement of other parts of the body, and the tone of voice, which expresses our state of mind in seeking, having, rejecting, or avoiding something. Thus, as I heard the words repeatedly used in their proper places in various sentences, I gradually learnt to understand what objects they signified. Wittgenstein uses the passage to illustrate a general conception of language he rejects -- a conception according to which the essence of all meaning is naming. One reason he rejects this conception involves an imagined priority in introducing words into a language, and in learning a language once the words have been introduced. The priority is one in which first comes our awareness of things in the world, then comes our decision to introduce certain words to talk about those things. In learning a language, we first focus on various candidates for what our elders are using a word to name, and then, having done so, we converge on the single candidate that best makes sense of the sentences they use containing the name we are trying to learn. Having set up the general picture he wants to reject, he immediately jumps to a use of language that, he thinks, doesn't conform to the picture. He says Now think of the following use of language: I send someone shopping. I give him a slip marked five red apples [On finding the apple drawer the shopkeeper] says the series of cardinal numbers--i assume that he knows them by heart--up to the word five and for each number he takes an apple of the same color as the sample out of the drawer. It is in this and similar ways that one operates with words But how does he know what he is to do with the word five? [W]hat is the meaning of the word five?--- No such thing was in question here, only how the word five is used. (PI section 1) This emphasis on the use of the numeral 'five', rather than its referent, is, I think, illuminating. But the lesson isn t that its meaning is its use; the meaning of the numeral 'five', which is also its referent, is the property being five in number, which isn t a use of anything. The proper lesson is that our use of the numeral in counting makes us aware of the property, which, as a result, becomes cognitively associated with the numeral, rather 5
than our antecedent non-linguistic recognition of the property making it available for naming. First the use, leading to the awareness of something to be named; not first the mystical awareness of number, and then the decision to name it. The importance of counting, which the passage from Wittgenstein emphasizes, is in establishing an epistemic foothold on a vast domain that none of us, individually or collectively, will ever actually count. Pretty much all of us know what we would have to do to count up to a trillion. But some of us don't know a verbal numeral in English for the number that comes after nine hundred ninety nine trillion, nine hundred and ninety nine billion, nine hundred and ninety nine million, nine hundred and ninety thousand, nine hundred and ninety nine. Fortunately, however, most people have mastered the system of Arabic numerals, in which each of the infinitely many natural numbers has a name, even though no one will ever use them all. In calling them 'names', I assume they are rigid designators the referents of which are fixed by descriptions that must be implicitly be mastered by those who understand them. Each of these distinct names can be taken as designating a distinct cardinality property, as long as we don t run out of multiples to bear those properties. This might seem to be a worry, since it seems likely that there are only finitely many electrons in the universe, and so only finitely many multiples of concrete things. But it s not a worry since we aren t restricted to counting only concrete things. We can also count multiples which have plural properties as constituents, including cardinality properties (numbers) we have already encountered. Since this ensures that there will be no end to larger and larger multiples, it also ensures the existence of infinitely many distinct cardinality properties. 6
As our former student Mario Gomez Torrente has recently pointed out, this picture gives us an opportunity to explain our knowledge of numbers. Consider the child inferring from her perceptual knowledge that the number of fingers I am holding up is 3. At first, she does this by counting -- saying the first three positive numerals -- pairing them off without remainder with the fingers I am holding up. In time, counting won t always be necessary, because she will be able to recognize at a glance when she is perceiving trios of familiar types. At this point the child has the concept, being a trio of things (of some common type or other), which is the plural property being three in number i.e. the number 3. The child learns a few other small numbers in the same way initially by counting, but eventually by ordinary perceptual recognition, and the formation of related perceptual beliefs. She will be able to perceptually recognize instances of these numbers, even though counting will remain the fallback method when in doubt, or when the multiples increase in size. Because of this, it is reasonable to think that much of our knowledge of natural numbers is knowledge of cardinality properties grounded initially in perception (visual and otherwise), in cognitive recognition of things being of various types, and in cognitive action e.g., counting the items falling under a given concept by reciting the relevant numerals while focusing one s attention on different individuals of a given type falling under the concept. When thinking of things in this way, it is important to realize that one doesn t first learn what numbers are, and then use them to count. On the contrary, one first learns the practice of articulating certain sequences sounds (i.e. sequences of numerals) and pairing them off with sequences of things. That is the origin of counting. The point at which one begins to recognize numbers and use numerals to refer to them is the point at which one has mastered this practice and integrated it into one s cognitive life. In saying things like, The number of blue books on the table is four, but the number of red ones is only three, one uses the 7
numerals to attribute cardinality properties of multiples. The properties one attributes are numbers, which exist independently of us and of our language, but which we come to cognize only in virtue of the linguistic, and other symbolic, routines we have mastered. Although that's the basic idea, various clarifications -- linguistic and otherwise -- are needed. First a bit of grammar. I am thinking of the English word 'three' on analogy with the word 'blue' as capable of performing three grammatical functions: First, these words can be used to designate certain properties of which other properties are predicated, as in and Blue is the color of a cloudless sky at noon and Three is the number of singers in the group. Second, they can combine with the copula to form predicates, as in The sky is blue and We are three, said by Peter, Paul, and Mary in answer to the question How many are you? Third, they can modify predicates, as in There is a blue shirt in the closet and There are three singers on stage. On the analysis I suggest, the numeral 'three' designates a plural property that applies to Peter, Paul, and Mary without applying to any one of them; the compound property being three singers on the stage applies to some individuals who are (collectively) three iff each is a singer on the stage. 2 In a formal language, these distinct uses of the word 'three' might be regimented into uniform uses of different words, but that needn't concern us here. The idea that properties, which we predicate of things in the domain of so-called individuals, are themselves members of that domain may open the door to paradox, unless restrictions of some sort are adopted. Though that is a matter of theoretical concern, that concern is not unique to this project which can, perhaps, be set aside for now. 2 To say that there are at least 3 singers on the stage is to say that some individuals who are three in number are each on the stage. To say that there are exactly 3 singers on the stage is to say that some individuals who are 3 in number are each singers on the stage, but no singers on the stage are collectively greater than 3. 8
What about the following observation, which updates a passage from section 46 of the Foundations of Arithmetic? While looking at Peter, Paul, and Mary standing on the stage next to Bill Halley and the Comets, I may say, equally, the number of singing groups on the stage is 2 or the number of singers on the stage is 10. Indeed, I can say both. I see 10 singers and I see 2 singing groups. In saying this, I am not saying that any of the things I see are both 2 and 10 in number. One might be driven to this if one thought that the 10 singers simply were the two groups. But they aren't. Although the 10 singers constituted the two singing groups there is no genuine identity in the offing. For example, the singers all had the property of being much older than the groups, though neither of the singing groups had the property of being much older than the groups. To count items, the items must already be cognitively individuated. The number 3 is the plural property applying to all and only those individuals x, y, and z none of which is identical with any other. In saying this, we presuppose the individuation we need in stipulating that the values of the variables are different. Nothing more is needed to predicate the plural property. Given this, we can accommodate what Frege was getting at in section 46. So far I have talked only about the very early stages of our knowledge of natural numbers. I claim some of this knowledge is perceptual belief that qualifies as knowledge. If one's knowledge of each of two things, x and y, that it is a finger is perceptual in this sense, and one's knowledge that x isn't y is also perceptual, then one's knowledge that x and y are two in number is also perceptual. If the fingers had been painted blue one could truly say, not only that one sees that those fingers are blue, but also that one sees that they are two in number. Indeed, if two people are standing at a distance from someone holding up two blue fingers, one of the observers, who is having trouble making out precisely what is being displayed, might ask Do you see the color of the fingers he is holding up? or Do you 9
see the number of fingers he is holding up? The one with better vision might reply, Yes, I see the color of those fingers; they are blue or Yes, I see the number of those fingers; they are two. So, it seems, there is a more or less ordinary sense of 'see' in which we can truly say that some color properties and some natural numbers, i.e. plural cardinality properties, can be seen. Don't wring your hands over the fact that this is surprising. If philosophy is worth doing, it should sometimes provide surprising, even shocking, knowledge. Here, it is knowledge about some of our knowledge of numbers. Systematic knowledge of arithmetic e.g. of the axioms and logical consequences of Peano Arithmetic is, of course, more complicated. This knowledge can t all be logical knowledge of the sort Frege imagined. If natural numbers are cardinality properties, logic alone can t guarantee that there are any individuals, multiples, or distinct cardinality properties of multiples, let alone infinitely many. But we can use logic plus updated versions of Frege s definitions of successor and natural number -- involving plural properties of multiples rather than sets to derive systematic knowledge of natural numbers. The definition of successor tells us that the plural property N is the successor of the plural property M if and only if there is some property F of individuals such that, some things that are F, of which a given object o is one, are N in number, while the Fs, excluding o, are M in number. Given definitions of zero and successor we can define natural numbers in the normal way as plural properties of which every property true of zero and of the successor of anything it is true of, is true. 3 From this plus our initial perceptually-based knowledge, we can derive arithmetical truths. We can come to know that zero isn't the successor of anything by observing that if it were, then some property true of nothing would have to be true of something. We can come 3 See pp. 435-436 of Boolos (1984) for a fuller discussion. 10
to know that no natural number M has two successors by observing that otherwise there would have to be properties N1 and N2 such that the N1s can't be exhaustively paired off with the N2s without remainder, even though there are objects o N1 and o N2 such that the N1s excluding o N1 and the N2s excluding o N2 are both M in number -- and so can be paired off. The impossibility of this is easy to see. Knowledge of the companion axiom, that different natural numbers N1 and N2 can't have the same successor, is explained in the same way. As for the axiom that every natural number has a successor, this can be seen to be true when we realize that the plural properties we arrive at by counting can themselves be included in later multiples we count. This ensures that we can always add one of them to any multiple that has given us a plural property M we have already reached. In this way, plural cardinality properties can allow us to explain not only the earliest knowledge of natural numbers we acquired as children, but also how systematic knowledge of elementary number theory can be acquired. There are, of course, other ways of expanding the meager knowledge of arithmetic acquired in kindergarten, or earlier. Most of us learned our arithmetic -- addition, subtraction, multiplication, division, and exponentiation -- in the early grades, without being exposed to the Peano axioms. No matter. The efficient, user-friendly computational routines we mastered are compatible with the epistemological and metaphysical perspective advocated here. That said, I don't claim to know that this perspective is correct. However, I take it to be more promising than any set-theoretic account of natural numbers. Paul Benacerraf's original problem alerted us the fact that any such reduction purporting to tell us what natural numbers really are -- as opposed to what, for one or another purpose, we might find it convenient to take them to be -- should provide a principled reason for selecting one and 11
only one set theoretic-reduction from among the many non-equivalent ways of identifying natural numbers with sets. Although each identifies individual numbers with sets that differ from those provided by other reductions, the different reductive systems do an equally good job of preserving all arithmetical truths, and the relationships among them. So, if this were the only criterion for justifying a reduction, we would have no reason for thinking that any of those reductions is uniquely correct. We might even have reason for doubting that any is correct. Surely, one may think, if the number three is identical with some set, there should be a reason it is one in particular, rather than any others. But, Benacerraf plausibly suggests, no set-theoretic reduction provides such a reason. If one believes, as I do, that there really are natural numbers, which all of us who have learned grade-school arithmetic know some things about, and which number theorists know much more about, then we should be looking for a metaphysical account that best facilitates an explanation of this knowledge. Although some set-theoretic accounts might be more promising than others, I think Paul was right in suggesting that set theoretical reductions can't pass the test. It's not that we don't have knowledge of sets; surely we must. But I doubt that it comes directly from anything as immediate as counting or perception. Rather, I suspect it arises from different sorts of collective activities, particularly those in which teams, committees, and groups of coordinated individuals collectively succeed in doing certain things than no one of the individuals does -- like winning a football game, pushing a stalled car to the side of the road, or carrying an object too heavy for any one of them. After we have admitted these, I suspect we find occasions to recognize collections of things that are noticeably similar in some way, even though they may not collectively do anything. Later, it occurs to us that we have no reason to exclude arbitrary collections of things. At this point the axioms of set theory can be considered and accepted. 12
Even then, however, it is hard to take seriously the idea that natural numbers are sets -- partly for the reasons that Paul stressed in the 1960s and 1970s, but also because by the time children and young adults have reached the required level to appreciate set-theoretic abstraction, the natural numbers have already been firmly cemented in their minds through a conceptual process in which the numbers are naturally taken to be plural properties. It is tempting to put it this way: just as the color blue is naturally understood to be the property commonly possessed by this, that, and the other individual blue thing, so the number three is naturally understood as the property commonly possessed by this, that, and the other multiple, each of which is three in number. But, this way of expressing the idea can be misleading. It makes it sound as if multiples were a kind of thing, which could, in principle, be counted, like anything else. That can't be. When we say that Peter, Paul, and Mary are three, nothing of any kind is said to be three. On the story I am telling, being three in number is a property that is never correctly predicated of anything of any type. Rather it is a property that is predicated of some things -- e.g. Peter, Paul, and Mary -- without being predicated either of them individually, or of any other single thing. The idea is coherent and, I think, intriguing. But it is one that was made explicit relatively recently and, for that reason perhaps, seems not yet to have penetrated very far into philosophical theorizing. This makes me uneasy. Can it really be as psychologically elementary as I have been making out, while being as philosophically unfamiliar as it seems to be? I'm not sure, which is the main reason I chose to talk about it here today. I can think of no better place than Princeton, and no more insightful and reliable experts than my former colleagues John, Gideon, and Paul -- among others here -- to help me think about it. 13