by Scott Soames
One wonders, in light of this, what to make of Frege’s methodological principle that numbers are whatever they have to be in order to explain our knowledge of them. Now that logic has been distinguished from set theory, there are two reasons to think that natural numbers aren’t sets. The first is that the problem of explaining our knowledge of sets—which are themselves mathematical objects—seems as daunting as that of explaining our knowledge of numbers. The second is that even if we could explain how we arrive at set-theoretic knowledge, there is no evident way of choosing among the many different ways of identifying natural numbers with sets. Although each reduction identifies individual numbers—1, 2, 3, etc.—with different sets than those provided by other reductions, the different systems do an equally good job of preserving all arithmetical truths. If this is the only criterion for justifying a reduction, we have no reason for thinking that any such reduction is uniquely correct. In fact, we may have reason to doubt that any reduction is; for surely, if the number 3 is identical with some set, there should be a reason for thinking it is one particular set rather than any other. No existing reduction of arithmetic to set theory has supplied it.13
Nevertheless, one should not reject Frege’s maxim—Numbers are whatever they have to be to explain our knowledge of them. In The Foundations of Arithmetic, he rightly says that nothing we can picture or imagine seems to be an apt candidate for being the number 4. But he isn’t deterred. Although we can, in his words, form no idea of the content of a number term, we can investigate which assignment of meanings and referents to number terms best enables us to explain our knowledge of arithmetical statements.
By “our knowledge,” I mean everyone’s knowledge—children who know only a little, adults who know more, and mathematicians who know much more. The first challenge is to explain how we achieve any of the knowledge of numbers that is common to us all. The second is to explain how instruction enables us to acquire more such knowledge. If we can meet these challenges, we may have a realistic starting point for explaining the rest of our mathematical knowledge.
One component of the needed 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 number of books now on my desk. So, the number of universities at which I have served is the same number as the number of books in front of me. Both the books and the universities are three (in number). What is this property, being three (in number), true of? It’s not true of any of my past faculty homes; neither Yale, Princeton, nor USC is three (in number). It is also not true of the set that contains them (and only them); since the set is a single thing, it’s not three either. Like the property being scattered, the property being three (in number) is irreducibly plural; it applies, not to any single instance of any type of thing, but to multiple things taken together. My former Ph.D. students are scattered around the world, even though no one of them is scattered, and the set containing them, which isn’t in space and time, isn’t scattered 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.14 Zero is a property that doesn’t apply to anything. Each of these properties is, like other properties, a way that a thing or things can be. 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. When counting, 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. 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 in section 1 of Ludwig Wittgenstein’s Philosophical Investigations. The book begins with a quotation from Augustine.
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.15
Wittgenstein uses the passage to illustrate a conception of language he rejects—one according to which 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 our awareness of things in the world always comes first, followed our decision to introduce certain words to talk about them. In language learning, we first focus on candidates for what our elders use 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 word we are trying to learn.
Having set up the picture he wants to reject, he immediately jumps to a use of language that, he thinks, doesn’t conform to it. 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.16
This emphasis on the use of the numeral ‘five’, rather than its referent, is 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 than our antecedent nonlinguistic 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 is in establishing an epistemic foothold on a vast domain that none of us, individually or collectively, will ever actually count. Most of us know what we would have to do to count 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 ninety-nine billion, nine hundred ninety-nine million, nine hundred ninety-nine thousand, nine hundred and ninety-nine. Fortunately, most people have mastered the system of Arabic numerals, in which each of the infinitely many natural numbers has a name.
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. That might seem problematic, 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 ca
n also count multiples consisting in whole, or in part, of properties, including cardinality properties (numbers) we have already encountered. Since this ensures there will be no end to larger and larger multiples, it also ensures the existence of infinitely many distinct cardinality properties.
As the philosopher Mario Gomez-Torrente has 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. In time, counting won’t always be necessary, because she will recognize at a glance when she is perceiving trios of familiar types. At that point she has the concept, being a trio of things, 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 perceptual recognition, even though counting will remain the fallback method when in doubt, or when the multiples increase in size. In this way we come to have perceptual and other beliefs about numbers. Much of this counts as knowledge. In short, 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.
In 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 of 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 recognizes numbers and uses 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, “There are four of these things but only three of those,” one uses the numerals to attribute cardinality properties of multiples. The properties one attributes are the numbers, which exist independently of us and of our language, but which we cognize only in virtue of the linguistic (or other symbolic) routines we have mastered.17
At this point, a word about grammar may be helpful. One might think of the word ‘three’ on analogy with the word ‘blue’ as capable of performing three grammatical functions. First, both can be used to designate certain properties of which other properties are predicated, as in 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. Here, 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 if and only if each is a singer on the stage.
The idea that properties, which are true of objects, might also be true of properties may cause worry about paradoxes analogous to those that plagued Frege and Russell. But it’s not clear that paradoxes must result from plural cardinality properties. No such property, except the degenerate case of the number 1, is true of itself, for the simple reason that no natural number other than 1 is true of any single thing. That’s not paradoxical. Is any N true of some things one of which is the property being N in number? Yes, each N (other than zero) is. But that’s not obviously paradoxical either. Is there any plural cardinality property N such that for all pluralities F, N is true of all and only those Fs no one of which is the property being N in number? No, that condition isn’t met by any plural cardinality property. Thus, we still have no paradox. There is, of course, no property that is true of any property p if and only if p isn’t true of itself. That would be problematic if we needed properties satisfying every intelligible condition to arrive at our knowledge of basic arithmetic, but no such derivation is contemplated. Though one must be careful, even in our limited project encompassing only natural numbers, no clear threat is apparent.
What about another worry? While looking at Peter, Paul and Mary (a singing group) standing on the stage next to the Rolling Stones (another group), I may say, equally, the number of singing groups on the stage is 2 or the number of singers on the stage is 8. Indeed, I do see eight singers and I do see two singing groups. But in saying this, I am not saying that any of the things I see are both two and eight in number. You might suppose otherwise if you thought that each singing group was somehow identical with—i.e., the very same as—its members. But the groups and the members that make them up aren’t the same. The singers were all much older than the groups, even though the groups weren’t older than the groups. Since they had different properties, they weren’t identical. In general, to count items—singers or groups—the items must already be 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. We can’t begin counting until we have sorted the items being counted into distinct items of some common type. Nothing more is needed to predicate the plural properties.
So far we have examined only very early stages of our acquisition of knowledge of natural numbers. Some of this knowledge is perceptual. If one’s knowledge of each of two things, x and y, that it is a finger, is perceptual, 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 has 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 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.
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 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 the 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.18
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 to know that no natural number M has two successors by observing that otherwise there would be properties F1 and F2 (where the F1s are N1 in number and the F2s are N2 in number, N1 ≠ N2) such that the F1s can’t be exhaustively paired off with the F2s without remainder, even though there are objects oF1 and oF2 such that the F1s excluding oF1 and the F2s excluding oF2 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 grade school 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 philosophical perspective advocated here. Thus, it may turn out that plural cardinality properties satisfy Frege’s methodological maxim by providing us with the best explanation of our arithmetical knowledge, and so should be taken to be natural numbers.19
That said, we should not go away with the idea that this way of reconceptualizing and following through on the Frege-Russell attempt to explain our knowledge of arithmetic has been definitively established. On the contrary, it remains a work in progress. However, one who has followed the line of thought presented here should go away with some idea of what philosophers working on the foundations of logic and mathematics are, and have been, up to, and how they attempt to advance our understanding of aspects of cognitive lives that are, although simple and central to who we are, nevertheless not easy to comprehend.