Does Arithmetic Rest on a Mistake?

 

How can the statement “1 + 1 = 2” be true? How can the operation of adding 1 to itself produce the number 2? There is only one number 1, so how could it by itself give rise to the distinct number 2? If you add the number 1 to itself, all you get is the number 1. It’s like adding Socrates to Socrates and hoping to get Plato (or “double Socrates”, whatever that may be). If anything we have the oddity “Socrates + Socrates = Socrates”. The Concise OED has an instructive definition of “add”: “to join to or put with something else”; the Shorter OED gives “join to or unite (one thing to another) as an increase or supplement”. Both stipulate that the added things must be distinct (“something else”, “another”): but 1 is not distinct from itself, so it can’t properly be added to itself. And how would doing that “increase” anything? In our initial statement we have two occurrences of the numeral “1” denoting the self-same number, asserting that this number added to itself gives 2 as sum. What is this strange kind of addition, and if it were to exist how could it yield the number 2? If “+” expresses a function, it would appear to have the same number occur in both argument places—yet we are told that this single number yields 2 as value of the function from 1 as argument. Notice that no one ever utters the sentence “1 added to itself equals 2”, because that makes the incoherence obvious—as with “Socrates added to himself equals Plato (or some other entity distinct from Socrates)”. On the face of it, then, arithmetic contains an absurdity—but one that escapes notice and goes unchallenged. What is going on?

            We must first observe that ordinary language contains two sorts of number word: adjectival and nominal. The arithmetical language we have been considering is nominal: nouns, singular terms and proper names that denote numbers conceived as objects. These terms form the subject of sentences to which predications are directed. But in much ordinary speech the adjectival use dominates: “five dogs”, “three cats”, “one car”. Here we are not using number words to denote objects but as components of predicates; they modify count nouns or sortals. In the adjectival use we can say things like, “One cat and one dog together add up to two animals”; or more formally, “One dog + one cat = two animals”. There is nothing puzzling here: there are many dogs and cats subject to counting and they can feature in equations (one dog and one cat are clearly distinct things). We are not trying to get two things out of one or engaging in peculiar acts of addition. We said nothing here about the object 1 and adding it to itself; we spoke only of the number of cats and dogs. I conjecture that people tend to hear the pure mathematical nominal statement as short for, or closely related to, the applied adjectival statement; and this leads them to overlook the peculiarities of the former kind of statement, logically speaking. Probably when we are drilled in early school years in academic arithmetic we are introduced to its formulas by means of adjectival paraphrases that lull the mind into a sense of familiarity, while actually changing the thought in fundamental ways. An ontology of cats and dogs is covertly replaced by an ontology of numbers denoted by proper names. Thus children don’t protest, “But you can’t produce 2 just by combining 1 with itself!”

            Abetting these adjectival uses in overlooking the logical problems inherent in “1 + 1 = 2”, we have sign-object confusion: we see two signs for 1 and conjure two number 1’s to go with them. This gives us the illusion that 1 can be converted into 2 by being added to itself. That clearly won’t work for “4 + 4 = 8” and infinitely many sentences like it, since there are not eight occurrences of “4” here; but anyway the fallacy is too blatant to bamboozle the mind for long. There is just the number 4 here, denoted twice, and it can’t be converted into 8 by being added to itself: 4 put together with itself gives just the same old number 4. In addition to this there is vagueness and uncertainty about what precisely these mathematical objects are, which allows the mind to imagine that they can increase in magnitude simply by self-adding. One has to focus on the logical character of the statements in question to see how peculiar they are, as standardly understood. In any case there are several factors that induce us to overlook the actual intended content of these sentences, the main one being the availability of adjectival counterparts to them, which are perfectly kosher.

            The problem I have indicated infects certain attempts to define the natural numbers. Leibniz’s approach, endorsed by Frege, has it that each number is composed of a series of 1’s (apart from zero). Thus “1 + 1 + 1 = 3”: we can define 3 in this manner, and so on for all numbers. But adding 1 to 1 is not a method for generating a new number; it is simply a way to remain stuck at the number 1. We can add 2 to 1 to get 3 because these are different numbers, but adding a number to itself can’t produce a new number. Non-identity is the essence of counting. It might be thought that there is a way out by exploiting the adjectival paraphrase as follows: the statement “one collection + another collection + one more collection = three collections” is perfectly meaningful, allowing us to identify these three entities with the number 3. That is not adding one thing to itself, but rather adding three distinct things together (as it might be, collections of dogs, cats, and mice). But really this says nothing like the original statement containing tokens of “1” that all denote the same number; it merely gives the false impression that such a statement makes sense by sounding similar to it.

            It might be said that we could save arithmetic by reformulating it adjectivally, ridding ourselves of nominal expressions and an ontology of numbers as objects. That sounds like a solid move in principle, but it won’t be able to save all of arithmetic as it now exists, because that subject has now taken on a life of its own. We would need to be able to restate all propositions about numbers in adjectival terms—for example, propositions affirming primes, cubes, successors, etc. How can theorems about numbers as such be represented in a language that declines to refer to them? What is called “number theory” will find it difficult to reformulate itself using only numerical adjectives and count nouns—how can we even say that a certain number is even? Adjectival arithmetic is fine in the market place, but it won’t do to encompass nominalized academic arithmetic.

            Could we ban all equations of the form “n + n = m” but keep the rest of pure arithmetic? There will still be infinitely many true equations to play with, such “5 + 3 = 8”. This doesn’t add any number to itself. But unfortunately the problem persists under the surface: for implicit in such a statement is an addition of one number to itself, viz. 3 added to 3, since 3 is part of 5 and so gets added to 3 (with 2 added to the result to give 8). Hidden in “5 + 3” is the addition “3 + 3”—as also is “4 + 1”. So we can’t avoid commitment to such equations even when they don’t appear on the surface; they lurk beneath because they are built into the whole conception. Numbers can always be broken down so as to generate them. You can’t have arithmetic, as it now exists, without these kinds of equations, despite their manifest weirdness (they don’t even fit the dictionary definition of “add”). One might even say that they claim a metaphysical impossibility: adding an object to itself (itself an impossible operation) to produce a quite distinct object (impossible ontologically). This is what you get if you nominalize adjectives illicitly. Here is an analogy: talk of large and small objects is common, as in “Jumbo is a small elephant” and “Mickey is a large mouse” (attributive adjectives). There is no logical problem about such sortal-relative adjectives in their proper grammatical position, but if we try to abstract them away from this position in order to form nominal expressions we get ourselves into trouble. Thus we might elect to speak of an entity called “largeness” and regard it as self-subsistent, as if “large” had a meaning independently of the nouns to which it is usually linked. Then we would wonder how a single animal could have both largeness and smallness at the same time, given that Jumbo is a large mammal but a small elephant (small for an elephant). Similarly, number words originally belong with count nouns, in which position they are unproblematic; but if you abstract them from that context and nominalize them, you find that the ontology thus created produces monsters like “1 + 1”. You can certainly add one cat to one dog and get two animals, but if you try to add the object 1 to itself to get the object 2 you run into incoherencies. In effect, there has been an illicit reification—of attributive adjectives or of numerical adjectives. Singular terms have been introduced and objects assigned to them, along with certain operations (like addition): but the coherence of the whole structure has not been demonstrated. In fact, the structure is built on equations that have no clear sense—or else are demonstrably nonsensical.

            So what is the status of arithmetic as it is commonly understood? Is it simply nonsense? Are its propositions analogous to “Largeness is larger than smallness” or “Largeness added to largeness equals even larger largeness”? That is, does it consist of mangled adjectives forced to dress up as pseudo proper names? Should it therefore be dropped, eschewed, and ridiculed? That seems harsh. Perhaps a form of fictionalism will serve to save it: arithmetical facts in the shape of adjectival constructions have been converted into propositions about fictional entities, obeying fictional laws. Names have been introduced and formulas manufactured, so that we end up with the likes of “1 + 1 = 2”. We drill kids in this discourse, as we drill them in other fictional discourse masquerading as fact (e.g. religion) and they are forced to accept it at face value. People end up believing in the Holy Trinity, a piece of transparent nonsense; and they end up believing that there are objects that when added to themselves produce other greater objects, which is scarcely more credible than the Holy Trinity nonsense. So maybe the whole shebang is carefully curated fiction presented as sober truth. And there is no denying that the edifice rests on perfectly sensible foundations in the use of number words in adjectival form; it is not pure nonsense. Nor is nonsense always and necessarily pernicious; it may even be useful (“useful fictions”, e.g. the average man). Is it an accident that Charles Dodgson was both a mathematician and a creator of delightful nonsense? Arithmetic, as we have it, is a human construction, according to fictionalism, like the creatures of the Jaberwocky, and it does not need literal truth in order to captivate the human mind. And it’s sort of true, given its sterling adjectival origins. We can carry on cheerfully intoning such nonsense as “1 + 1 = 2” while accepting that we are engaging in metaphysical quackery. The whole history of mathematics is littered with controversy about the reality of this or that newly created mathematical entity (zero, the infinitesimal, the irrational, the negative, etc.): is it inconceivable that the arithmetic of positive whole numbers is also steeped in ontological mud?

 

Colin McGinn                

Ears

 

It began over a year ago when a small red patch appeared on the top of my right ear. At first it was diagnosed as inflammation of the cartilage probably brought on by wearing too tight headphones. The cure was not to put any pressure on it and wait for the inflammation to die down. After several months it had increased in size and painfulness, so I returned to the dermatologist. This time a biopsy revealed skin cancer (squamous cell). Now it was necessary to perform surgery. This was duly done: a five-hour MOHs procedure followed by a three-hour reconstruction (local anesthetic). I won’t specify the details but it was all pretty brutal (twenty odd stitches, serious pain). I lost half my ear. The cause: sun damage, possibly a result of playing tennis in the Florida sun.

            At the same time the ear was declared healed a lump appeared in my neck. This was duly scanned and biopsied: it was more cancer. The cancer in my ear had evidently spread to my neck and was now growing apace. A full-scale operation was quickly scheduled. This was performed two weeks ago. Again, I will spare you the gory details except to say that it was an eleven-hour operation requiring a hospital stay and subsequent daily home visits from a nurse. Full recovery is not guaranteed. It is presently difficult to eat and my right shoulder is compromised.

            I relay all this in order to encourage readers to protect their ears from the sun, especially if they live in a place with a lot of it. It is easy to neglect the ears while protecting the face. Sunscreen is not enough, especially if you spend time doing water sports (as I do). Wear a broad-brimmed hat (not a baseball cap). Ideally use a fabric covering such as a skullcap or full-face sun protection balaclava (as I have done for some years). Apparently my experiences are not uncommon and you don’t want to share them.

 

Co

 

 

Saying and Showing

 

Wittgenstein famously introduced the distinction between saying and showing in the Tractatus. I won’t be concerned with his treatment of the distinction, either by way of interpretation or evaluation; but I will be using the terminology. I want to say that every speech act includes an act of showing, as well as an act of saying, and also that showing is not a type of saying. It is not (pace Wittgenstein) that what is shown cannot be said, but only that the showing that occurs in speech acts is not a type of saying in that very speech act. The kind of showing I have in mind is perfectly familiar and non-mystifying: it is the mere utterance of a sentence with its characteristic form. The speaker displays or exhibits or presents a sentence to the hearer’s senses (generally vision and hearing), thus showing him that sentence. The speaker doesn’t say he is showing the hearer a sentence; he simply does it—as he might show the audience a coin in his hand. But the hearer is then in a position to know what the speaker is saying: the speaker says something to the hearer by showing him a sentence. He might, for example, produce a written sentence from behind his back that the hearer is now in a position to interpret and make an ascription of saying. Showing in this sense is an act of proffering an item to the senses, and this is what enables the speaker to communicate by acts of speech. So two acts are performed in a given speech act: an act of showing and an act of saying, where the former enables the latter. In other words, the act of uttering (saying in the oratio recta sense) is an act of showing (displaying, exhibiting, etc.) in the performance of which something is said in the oratio obliquasense. Utterance is not merely sounds issuing from the speaker’s mouth—that might be just involuntary babble—but an intentional presentation to someone’s senses of a sentence (conceived as such) for a specific purpose, viz. to say something to that person. The speaker is showing something to the hearer in roughly the sense in which a tour guide might show you the way to a cathedral. The purpose of the showing is to perform an act of saying and it is a sine qua non of that.

            In fact there is a bit more complexity here. First, there is not just the speech act of saying but also of commanding and questioning (and any other type of speech act you may believe in). The speaker shows the sentence “Shut the door!” in order to command the addressee to shut the door, or shows the sentence “What time is it?” in order to ask what the time is. Notice that the variety of speech acts performed is accompanied by uniformity in the act of showing: all speech acts involve showing, though not all involve saying. So there is something in common to all speech acts—they all involve an act of showing. Moreover, they all involve something not specifically linguistic, because showing occurs in a wide range of activities: we show sentences to each other in much the way we show things in general to each other (maybe the one capacity derives from the other). Second, there are two parts to the kind of showing that occurs in acts of communication: one part is the act of the speaker in showing a sentence to the hearer; the other is the sentence itself showing its form to the hearer (this is closer to Wittgenstein’s use of the concept). I show you a sentence S and S shows you its grammatical and logical form—without saying anything about this form. A conjunctive sentence doesn’t say it is a conjunction—it just is one. So strictly there are two acts of showing in any speech act: speaker showing and sentence showing. The speaker shows you a sentence and the sentence shows you its form (as well as its vocabulary). The speaker doesn’t say, “I am showing you this sentence” but simply does it, and the sentence doesn’t say, “I am an existentially quantified sentence” though it manifestly is one. This double showing is integral to the success of the speech act and is not to be viewed as mere acoustic production: it is part of speech as a rational purposive activity.

            I am tempted to suggest that this way of talking comes naturally because we are a theatrical species. Our social interactions have a theatrical character (think Shakespeare and Erving Goffman). We are always “putting on a show”. Thus the idea that speech involves performance is a theatrical idea (we perform speech acts as actors perform their lines). Our speech comprises a display that is designed to be interpreted by an audience as an act of saying (etc.). If all the world’s a stage and we are merely players, then our speech will involve acts of theatrical showing—skilled presentations that reveal states of mind. We show other people things in order to get things across to them: we wave our hands, point our fingers, make urgent sounds when in extremis, and produce grammatical strings.  We proffer things to other people’s senses in the hope that we will be understood. This requires skills akin to those of an actor; and isn’t speech often a kind of acting? We have to put in a goodperformance, a convincing verbal display (e.g. Winston Churchill giving a mesmerizing speech). So we naturally think of our speech performances in theatrical terms—as a type of show we put on. The great speaker or writer is exceptionally good at showing people sentences that produce the best audience effect. In any case, speaking involves showing—displaying, exhibiting. In speaking I reveal my thoughts, parade my desires, and exhibit my intentions—I show you sentences from which you make suitable inferences. Sign language is exemplary in this respect: here the speaker uses her hands to show the audience signs that communicate states of mind—and it lookstheatrical. The poet, too, shows you words and sentences that convey ideas and emotions (the love letter is similar). The language teacher may proceed by exhibiting sentences for the student to learn—while simultaneously saying something.

            The point of my saying all this is to supplement the usual conceptual apparatus of speech act theory with another layer of concepts: it is not just about saying, commanding, etc. but also a sophisticated type of action aptly captured by the showing terminology. Nothing like this exists in the work of the later Wittgenstein or Austin or Searle or Strawson or Grice: we just have an etiolated notion of “utterance”. But the speech act as it exists in humans is a more complex and subtle phenomenon than this terminology suggests, possibly because of lingering behaviorist assumptions. The general form of a speech act consists of a double act of showing combined with an illocutionary act of saying (commanding, questioning, etc.). It isn’t just people making noises that other people interpret this way and that but an act of putting words on display with all that that implies.[1]   

 

Colin McGinn

[1] Showing is not opposed to saying but the two form an indissoluble whole: in saying we show and in showing we say. I will also note that showing is a type of externalization: an inner process is externalized as we show sentences to interlocutors—thought is revealed in spoken words.

 

Action As Externalization

 

The causal theory of action says that actions consist of a causal link between an inner mental state (a reason, an intention, a willing) and a piece of behavior (an arm rising, flicking the switch). The mental state causes the behavior in the same way a tap on the knee can cause the lower leg to rise quickly (patellar reflex). Acts of speech, say, are composed of an internal state of intending or trying or simply desiring and the distinct event of vocal utterance, where the former causes the latter. But this omits an important aspect of action: the fact that behavior often externalizes states of mind—expresses them, manifests them, embodies them. Dance and song can externalize inner mental states, and speech behavior can externalize states of mind. Language can and does externalize thought (as well as desire and emotion). The body puts the mind into the public realm. It isn’t just that the mind operates to cause behavior, as measles causes a rash; behavior can be the outward expression of mind. This expression relation is not merely a causal relation; it is more intimate than that. The psychophysical system has the capacity to generate externalizations—bodying forth, as it were. No doubt this idea is difficult to articulate, but it appears to be a real phenomenon—which is why we often express ourselves in these terms. As perception internalizes its object, so action externalizes the inner states that lead to it. Ordinary human behavior is not a mental blank; it has mind inscribed on it. We may not understand how this can be but we readily talk this way. A theory of action needs to find room for it—and the usual causal theory falls short of achieving this.

            Spoken language externalizes internal language, the kind that deals with thought.[1] So it is natural to suppose that we possess two languages—a language of thought and a language of communication. Generalizing this point, we can say that non-linguistic action constitutes a kind of second mind—externalized mind. This mind is derivative on internal mind, as communicative language is derivative on cognitive language; but both can be described as mind made external. What we think of as external behavior is mentally imbued: for it is expressive of the inner, not merely causally connected to it. Thus the mind has the power to externalize itself in overt action, extending its reach beyond the private sphere (“the externalized mind”). Just as spoken words have meaning, in addition to words in inner speech, so actions in general are mentally endowed in virtue of the externalization exerted by the mind. Among the mind’s powers is its ability to transfer itself to the body. It isn’t that action is just mindless movement caused by a mental substratum; rather, it is the mind in action, so to speak. So the idea of a double mind, like the idea of a double language, comes to seem natural and attractive. This is another way of expressing the familiar point that the right conception of behavior is mentally defined. It is the process or act of externalization that enables the mind to spread itself to the body, which now becomes a site of mentality. The gap between mind and body is not as rigid as tradition has supposed. This is because behavior is not just caused by the mind but also the externalization of the mind—a much closer relation. Our paradigm of action should be singing and dancing not arms rising because of sudden urges (as if the movement is merely triggered). Behavior is not the non-mental effect of mind but a species of mindedness in its own right.

            As the mind externalizes itself in the body, so in turn it internalizes the result: we are aware of our own body in acts of perception (including proprioception). A singer is aware of herself singing and a dancer is aware of her own movement. Given that perception is a type of internalization, we can say that the mind internalizes its own externalizations. This may lead to further externalizations as the agent modulates her behavior according to what she perceives: for example, the singer might adjust her volume and pitch according to what she hears. So controlled action is rightly seen as a process of externalization and internalization: first the mind pushes itself outward, then it internalizes the result of this act, leading to further externalizations. It’s a feedback loop, to use cybernetic jargon; but the language of externalization and internalization better captures the nature of the process in question. There is no comparable process in non-mental cybernetic systems such as thermostats: these are devoid of the operations we are calling internalization and externalization. A psychology based on the latter concepts is more appropriate for systems that genuinely perceive and act. So we should drop talk of stimulus and response and replace it with talk of internalization and externalization; and the same for the terminology of cybernetics. Perception allows the organism to take in the world, and action allows the organism to project the mind into the body: this is the conceptual framework within which to view psychology, animal and human. It isn’t old-fashioned behaviorism, and it isn’t modern cognitive science; it’s a view of the behaving organism that emphasizes the twin powers of the mind to take in and give out. The mind is forever internalizing and externalizing, not being stimulated and responding.

[1] Here I am assuming, not defending, the perspective on language favored by Chomsky and Fodor.