The Arithmetic of Intention

American Philosophical Quarterly Volume 52, Number 2, April 2015 THE ARITHMETIC OF INTENTION Anton Ford N 1. Two Methods ear the end of her book, Intention, G. E. M. Anscombe distinguishes two different ways of trying to explain what an intentional action is: If one simply attends to the fact that many actions can be either intentional or unintentional, it can be quite natural to think that events which are characterisable as intentional or unintentional are a certain natural class, ‘intentional’ being an extra property which a philosopher must try to describe. In fact the term ‘intentional’ has reference to a form of description of events. What is essential to this form is displayed by the results of our enquiries into the question ‘Why?’1 Anscombe here repudiates any account that attempts to describe an “extra property” in virtue of possessing which an event of someone’s doing something is an intentional action. It attests to the seeming naturalness of this kind of account that so much action theory in the last half-century has attempted to describe just such a property. According to what is now aptly called “the standard story of action,” an intentional action is an event of bodily movement distinguished by the property of its having been caused by the agent’s mental states. Though Anscombe’s opposition to the standard story of action is widely recognized, the ©2015 by the Board of Trustees of the University of Illinois grounds of her opposition are, I believe, commonly misunderstood. Her dissent is often thought to be “rooted in strong behavioristic assumptions,” which, it is said, lead her to reject the claim that what makes an action intentional is its having a certain psychological cause.2 However, the passage quoted above suggests that the roots of Anscombe’s objection to the standard story of action are quite different and much deeper than is commonly supposed. If, as she asserts, intentional action cannot be explained by describing an extra property that makes an action intentional, then, trivially, it cannot be explained by describing the extra property mentioned in the standard story of action. The latter is not a consequence of a doctrine about beliefs, or desires, or reasons, or causes; and it is not grounded in any particular theory of the mind: it is a trivial implication of rejecting a certain method, a method that purports to explain what intentional action is by appealing to a property that makes an action intentional. The passage quoted above is not unique in suggesting that Anscombe means to oppose herself to something much more general than a doctrine about the relation between intentional action and mental states.3 In fact, one only needs to read the table of contents to ind the following entry for §19: “We do not mention any extra feature attaching to an action at the time it is done by calling it 130 / AMERICAN PHILOSOPHICAL QUARTERLY intentional. Proof of this by supposing there is such a feature.”4 The reductio that Anscombe provides in § 19 proceeds from the simple assumption that there is some feature in virtue of which an action is intentional: “Let us call it ‘I,’” she writes.5 The argument that follows is dificult to understand; but at least this much is clear: it does not assume that the putative feature, I, pertains to either an action’s causal etiology or an agent’s mental states. On the contrary, the target assumption is that I might be “any” feature whatsoever. My aim in what follows is not to explain why Anscombe rejects the method exempliied by the standard story of action.6 I hope rather to clarify what may seem even more obscure, the alternative that Anscombe recommends. Anscombe says the alternative is to display “a form of description of events.” But even the most sympathetic reader may struggle to see what she means by this. It is helpful to be told, as she subsequently tells us, that “what is essential to this form is displayed in the results of [her] enquiries into the question ‘Why?’” However, this is only helpful if one knows what Anscombe takes to be her most important results. And even knowing that, it may remain obscure what kind of account she proposes to give if not the kind that identiies a property, or feature, in virtue of possessing which an action is intentional. For what other kind of account could there be? 2. “Why?” Anscombe begins her inquiry into the concept of intentional action as follows: What distinguishes actions which are intentional from those which are not? The answer I shall suggest is that they are the actions to which a certain sense of the question ‘Why?’ is given application; the sense is of course that in which the answer, if positive, gives a reason for acting.7 These remarks are often discussed as though they contained a thesis, or as though they presented, in outline, Anscombe’s account of intentional action. It can look as though what she does is this: irst she poses a question, the one her account will address; and then she announces her answer to it. Part of the reason it looks this way, insofar as it does, is that contemporary philosophers are accustomed to thinking that a theory of action is a theory of some distinction—typically, a distinction between an action and a “mere event” or between an intentional action and one that is unintentional.8 According to Harry Frankfurt, “the problem of action is to explicate the contrast between what an agent does and what merely happens to him, or between the bodily movements that he makes and those that occur without his making them.”9 And according to Arthur Danto, “An action [is] a movement of the body plus x . . . and the problem . . . is to solve in some philosophically interesting way for x.” 10 These are programmatic claims about the character of “the” problem addressed by a theory of action; and their authors claim, with some justiication, to represent the perspective of the discipline as a whole. From that perspective, Anscombe’s opening question will naturally appear to be a way of posing the so-called problem of action. If that is how one understands the point of her opening question, the sentence that follows it will lead one to think that Anscombe’s account of intentional action is, in outline, this: what makes an action intentional is its being done for a reason. In that case, the question “Why?” serves Anscombe as a way of isolating that which distinguishes an intentional action from one that is unintentional. And if that is how one understands the point of her invocation of a reason for acting, then, since a reason is often said to be a pair of mental states—namely a belief and a desire—it will be natural to conclude that Anscombe’s account is roughly the one that Michael Bratman attributes to her: “[I]t THE ARITHMETIC OF INTENTION / is standing in an appropriate relation to [the agent’s beliefs and desires] that makes an action intentional.”11 According to Bratman, Anscombe agrees on this point with Donald Davidson and Alvin Goldman. As Bratman sees it, the principal disagreement between Anscombe and her opponents is about “just what relation between desires and beliefs, on the one hand, and action, on the other hand, makes for intentional action: Davidson and Goldman insist, while Anscombe emphatically denies, that the appropriate relation is in some signiicant sense a causal relation.”12 Bratman’s view of the matter is not idiosyncratic. On the contrary, it is the textbook account of what Anscombe thinks and of how her thought is related to the standard story of action. Of course, the received view is based on more than Anscombe’s opening remarks: it is based partly on what she says elsewhere in Intention, partly on what she says in her other writings, and partly, one suspects, on what Donald Davidson says in the famous irst footnote and inal paragraphs of “Actions, Reasons and Causes.”13 But whatever its basis may be, a fresh look at Anscombe’s opening remarks may help to correct this common misconception.14 3. “Why?” and “How Many?” Compare what Anscombe says in her opening remarks to what Frege says at the beginning of his inquiry into the concept of number: In what follows, therefore, unless special notice is given, the only “numbers” under discussion are the positive whole numbers, which give the answer to the question “How many?”15 Just as there are “actions” (unintentional ones) that lie outside the purview of Anscombe’s investigation, so, also, there are “numbers” (negative ones, fractions, irrational ones, etc.) that lie outside of Frege’s.16 In drawing his reader’s attention to this, Frege is not advancing a thesis; rather, he is telling us 131 what his theses will be about. When he says that positive whole numbers give an answer to the question “How many?,” he is not offering an account of number, or the outline of an account, or a summary of his results, or even a faint suggestion about what his account will be; rather, he is telling us what his account will be an account of. If one does not approach Intention with the expectation that Anscombe’s aim is to explicate a contrast between intentional action and something else, then it is possible to see Anscombe as doing in her opening remarks what Frege is doing in the passage quoted above. In fact, I think, there is no reason that Anscombe could not have begun by saying: “In what follows the only ‘actions’ under discussion are those which give application to a certain sense of the question ‘Why?’” In that case, the point of her introductory question—“What distinguishes actions which are intentional from those which are not?”—is not to raise the problem of action, but simply to ask: “What distinguishes the actions that are, from those that are not, my topic?” Like Frege, Anscombe identiies her topic by appeal to a certain triangular nexus (Fig, 1). The nexus is between: (1) a theme, about which, and about which alone, (2) a certain kind of question can be asked, a question to which, and to which alone, (3) a certain kind of answer can be given. For Frege, the theme is a countable; the question is “How many?”; and the answer is a number.17 For Anscombe, the theme is an intentional action; the question is “Why?” (in its special sense); and the answer is a reason for acting. Their explananda lie on different points of the triangle: while Frege’s explanandum is the answer to a question, Anscombe’s explanandum is the theme of a question. But each lays hold of the explanandum by isolating a question to which it is internally related. In Frege’s case, it is clear that the questionanswer-theme nexus is a single intelligible 132 / AMERICAN PHILOSOPHICAL QUARTERLY Figure 1. Anscombe’s triangular nexus. unit. On the one hand, to understand the question “How many?” is already to understand what can answer it—that it can only be answered by a number.18 And since that is the only possible answer, the question “How many?” is, in fact, a mere notational variant of “What is the number of?” (“How many leaves are on the tree?” = “What is the number of leaves on the tree?”) On the other hand, to understand this question is already to understand what it applies to—that it applies to, say, the birds overhead, or the stars overhead, but not to the air, or the mist, or the cold. Anscombe thinks that her question is related in the same way to its answer and to its theme. Thus, she does not merely say that intentional actions are “the actions to which a certain sense of the question ‘Why?’ is given application; the sense is of course that in which the answer, if positive, gives a reason for acting.” This statement, which is often quoted, is immediately followed by a rarely quoted addendum: But this is not a suficient statement, because the question “What is the relevant sense of the question ‘Why?’” and “What is meant by a ‘reason for acting’?” are one and the same.19 Anscombe holds that explaining the relevant sense of the question “Why?” is equivalent to explaining the kind of answer it takes. Because only a “reason for acting” can be a (positive) answer to the relevant sense of the question “Why?,” she might just as well have appealed to a certain sense of the question “For what reason?” (“Why did you go upstairs?” = “For what reason did you go upstairs?”). This would have made it obvious that there is no difference between explaining the sense of the question and explaining the kind of reason that is a reason for acting. In the next paragraph, the second of her inquiry, Anscombe makes the additional point that there is no non-circular explanation of the theme of her question (intentional action) by appeal to its answer (a reason for acting): Why is giving a start or gasp not an ‘action,’ while sending for a taxi, or crossing the road, is one? The answer cannot be “Because the answer to the question ‘why?’ may give a reason in the latter cases,” for the answer may ‘give a reason’ in the former cases too; and we cannot say “Ah, but not a reason for action”; we should be going round in circles.20 Anscombe here is marking the familiar distinction between two different kinds of events: she contrasts those events that are intentional actions, like sending for a taxi, and those that are (as one says) “mere events,” or “mere happenings,” like giving a sudden start. With regard to an event of either kind, one may ask for the “reason” “why” it “happened.” On the one hand, there is the question “Why did you give a start?,” which might receive the answer “I thought I saw a face in the window.” On the other hand, there is the question “Why did you call for a taxi?,” where the reason might be “I am going to the airport.” Anscombe thinks that one cannot deine intentional action as that which is done for (or caused by) a “reason” because the “reason” mentioned in the would-be account must be of one of two determinate kinds: it must be either a reason-for-acting or a reason-for-merely-happening. But the latter is not what one wants; and one cannot appeal to the former except by presupposing what needed to be explained—the nature, namely, of acting.21 In sum, Anscombe introduces her topic by identifying a nexus between a question, its theme, and its answer; and this nexus, like Frege’s, is one whose three moments are explained and understood, either all together, or THE ARITHMETIC OF INTENTION / not at all. The received view of Anscombe’s account takes it to be her central thesis that what makes an action intentional is its being done for a reason. But for Anscombe, saying that an intentional action can be done for a reason is like saying that a countable can be counted: it is a tautology, not a thesis; it does not explain anything. The fact that it does not explain anything does not mean it is useless. On the contrary, its being a tautology is precisely what makes it useful for isolating a formal concept as the topic of one’s inquiry.22 4. The Null-Answer Anscombe is like Frege, not only in that she ixes her topic by reference to a question, but also in that she believes this question occasionally, but graciously, accepts a null-answer. The importance of this point is suggested by the fact that it is marked in Anscombe’s opening statement. What she says is, recall, that the relevant sense of the question “Why?” is one whose answer may give a reason for acting, but only if the answer is “positive.” The point of this qualiication is initially obscure. But it soon comes to light that, on Anscombe’s view, not every intentional action is done for a reason. Interestingly, she explains her position by comparing the question “Why?” to the question “How much?”: Of course a possible answer to the question ‘Why?’ is one like ‘I just thought I would’ or ‘It was an impulse’ or ‘For no particular reason’ or ‘It was an idle action—I was just doodling.’ The question is not refused application because the answer to it says that there is no reason, any more than the question how much money I have in my pocket is refused application by the answer ‘None.’23 It is clear that Anscombe’s point here does not depend on the difference between “How many?” and “How much?” She might just as well have written that the answer “Zero” is not a way of refusing application to the question how many coins I have in my pocket. So compare, again, Frege: 133 What answers the question ‘How many?’ is a number, and if we ask, for example, ‘How many moons has this planet?,’ we are quite as much prepared for the answer 0 . . . as 2 or 3, and that without having to understand the question differently. . . . What will not work with 0 . . . cannot be essential to the concept of number.24 Throughout the Grundlagen, Frege treats 0 as the limit case of his explanandum: though no more important than other numbers, it is on all fours with 5, 6, and 7. This is the attitude Anscombe takes to the null-answer to her question. The philosophical interest of reasonless intentional action is, she says, “slight.”25 But a reasonless intentional action is not, for being reasonless, any less intentional. In case more proof was needed, this shows deinitively that the textbook reading of Anscombe is not only mistaken, but mistaken about what it takes to be her most important thesis. Anscombe says, as clearly as possible, and with no sign of anxiety, that an action can be intentional even though it is done for “no reason.” In that case, she cannot think that what “makes” an action intentional is its being done for a reason, or that an action is intentional in virtue of being done for a reason, or that its being done for a reason is what distinguishes an action that is intentional from one that is unintentional. It is completely irrelevant to the present point how one conceives of a reason, whether as a pair of mental states, or as the content of those mental states, or as a fact, or as a wider pattern of behavior. It is also irrelevant how one conceives of the relation between a reason and an action, whether as an eficientcausal relation, or as a formal-causal relation, or even as some sort of non-causal relation. After all, a relation requires the existence of at least two relata; and according to Anscombe, one of the putative relata—a reason—need not exist. Thus, Anscombe does not think an action is intentional in virtue of being done for a reason on any understanding of what a 134 / AMERICAN PHILOSOPHICAL QUARTERLY reason is, or on any understanding of how a reason is related to an action. All that matters to her is whether the action is one to which the relevant question applies: if so, it is “intentional”; and if not, not. Anscombe’s nonchalance about reasonless intentional action makes for a bold contrast with Davidson’s insistence that there must always be some reason for which an intentional action is done: When we [treat wanting as a genus including all pro attitudes as species] and when we know some action is intentional, it is easy to answer the question, “Why did you do it?” with, “For no reason,” meaning not that there is no reason but that there is no further reason, no reason that cannot be inferred from the fact that the action was done intentionally; no reason, in other words, besides wanting to do it. This last point . . . is of interest because it defends the possibility of deining an intentional action as one done for a reason.26 Unlike Anscombe, Davidson does deine an intentional action as one that is done for a reason—that is, as one that is caused by a reason; that is, as one caused (in the right way) by a suitably related belief and desire. So, unlike Anscombe, Davidson is forced to deny the commonsense appearance that, at least occasionally, an intentional action, like balancing a penny on its edge, is done for no reason. Moreover, Davidson knows what is at stake: he knows that if this appearance were taken at face value, it would falsify his theory. It will clarify the contrast between the kind of account Anscombe provides and one like Davidson’s, which seeks to describe an “extra feature,” to develop Anscombe’s analogy between the question “Why?” and a question like “How many?” or “How much?” 5. “How Many?” and “How Much?” Philosophers sometimes distinguish between two different forms of quantity, between multitude, on the one hand, and Figure 2. Multitude and magnitude. magnitude, on the other.27 A multitude is countable, while a magnitude is measurable. And while the countable is discrete, the measurable is continuous. There are multitudes of “things” and magnitudes of “stuff.” Examples of “things” include a river, a statue, a horse, and a planet. Examples of “stuff” include earth, air, ire, and water. The distinction between multitude and magnitude is conveniently marked in the English language by two different questions: “How many?” and “How much?” (Fig. 2). The answer to the question “How many?” is, as we have already seen, a number: for instance, 12. By contrast, the answer to the question “How much?” is an amount: for instance, 12 gallons. There are many palpably formal differences between magnitude and multitude. 28 For example, the determination of a magnitude is always doubly relative—both in that it speciies a dimension of quantity, and also in that, given some dimension, it employs a conventional system of measure. Thus, for example, the question how much milk you have may be answered in terms of volume, or, again, in terms of weight; and in either case, its quantity may be given either in terms of the metric system, or, say, the British one. But the determination of multitude is not in this way relative. There is one, and only one, answer to the question how many eggs you have. Just as, on Anscombe’s view, the question “Why?” is answered by mentioning a special kind of reason, so, also, the questions “How many?” and “How much?” are each answered by mentioning a special kind of quantity. We could therefore redeploy Anscombe’s peculiar jargon and say that a thing “gives THE ARITHMETIC OF INTENTION / application” to the question “How many?,” and stuff to the question “How much?” To represent the theme, or substance, of one’s thought as a thing is to represent it as having a certain deinite logical shape, thanks to which the question “How many?” is sensibly asked and answered. Meanwhile, to represent it as stuff is to represent it as having a very different shape—a shape that is itted by the question “How much?” If there can be a logical “match” between a question and answer, on the one hand, and the associated theme, on the other, we might expect to ind the possibility of a “mismatch.” And that is what we do ind. Of tobacco in a pouch, you cannot ask how many there are. Of cigarettes in a packet, you cannot ask how much there is. The thought “There are 12 . . . s” can only possibly be of things: the concept of “a thing,” as introduced above, refers to the theme of a predication of multitude. Likewise, the thought “There are 12 gallons of . . .” can only be of stuff: “stuff” refers to the theme of a predication of magnitude.29 The questions “How many?” and “How much?” may both have application even where they have no “positive” answer. If the number of cigarettes in a packet is zero, there is no positive answer to the question how many cigarettes there are. But since a cigarette is a countable thing, the question still its; what is quantiiable—here, a cigarette—is, if you like, in the present circumstances “unquantiied.” Likewise, there is no positive answer to the question how much tobacco there is in the pouch, if the amount in the pouch is none. But the question “How much?” is not in that case “refused application.” Thus, the formal contrast between stuff and things is marked even in their absence. The distinction we saw between “12” and “12 gallons” is preserved in that between “0” and “0 gallons.” Remarkably enough, there are two formally distinctive ways of having no quantity whatsoever—two different ways of being absolutely nothing. 135 This explains why Frege (and by analogy, Anscombe) is undisturbed by the nullanswer: a formal distinction remains, even where the question through which this form is grasped receives no positive answer. If Anscombe (and by analogy, Frege) believed that her explanandum was distinguished by a property—“an extra feature”—she could not maintain her tranquility in the face of the void. A property or feature either does, or does not, attach; and if it does not, it simply does not: there are no two ways about it. Reasonless intentional action is a threat to Davidson’s account because an event either is, or is not, caused (in the right way) by a pair of mental states; and if it is not, it simply is not. 6. A “Certain Sense” of the Question We are now perhaps in a better position to understand what Anscombe says in the passage with which I began this paper: “the term ‘intentional’ has reference to a form of description of events.” In view of the phenomena noted in the previous section, it might also be said that the term “number” has reference to a form of description of quantity. Supposing, as I have, that there are two different forms of quantity—that of things and that of stuff—one of them, but not the other, is Frege’s topic in the Grundlagen. My interpretative suggestion is that Anscombe believes her topic to be of the same kind. That is, when Anscombe isolates her topic by appeal to a sense of the question “Why?,” she takes herself to be singling out a concept that has just as much formal integrity—and just as much claim to the epithet “logical”—as the concept of a natural number. It is striking, though, that, unlike Frege, Anscombe is compelled to speak of a “certain sense” of her question, and this points to an important difference between their two investigations. Frege has no need to speak of a “certain sense” of the question “How many?” for the 136 / AMERICAN PHILOSOPHICAL QUARTERLY obvious reason that German, like English, already marks the required distinction; but his prose would have been more cumbersome if his book had been written in French. Unlike its English and German counterparts, the French interrogative adverb “Combien?” is ambidextrous: it is used, on the one hand, in situations where an English-speaker would ask “How many?” and where a Germanspeaker would ask “Wie viele?,” and, on the other hand, in situations where an Englishspeaker would ask “How much?” and where a German-speaker would ask “Wie veil?” Unlike “How many?” and “Wie viele?,” the French “Combien?” is not a question that only takes a natural number as its answer. One can ask, either, how much time (combien de temps) it takes to do something, or how many times (combien de fois) it has been done. The answer to the irst question is, for example, 5 minutes; the answer to the second question is, for example, 5. The second answer is a natural number; the irst answer is not. Thus, had he written the Grundlagen in French, rather than German, Frege could not have said— simply—that his topic is that which gives an answer to the question “Combien?” In order to be strict, he would have needed to say, for example, that his topic is that which gives an answer to a “certain sense” of “Combien?”30 In fact, a Francophone philosopher writing a treatise on quantity might feel the need to guard against certain possible misunderstandings parallel to those that Anscombe thinks we face in giving a theory of intentional action. For instance, just as Anscombe warns of an explanatory circle relating to “action” and “reason,” her French-speaking counterpart might point out a similar circle arising in connection with “thing” and “quantity.” For someone might ask: Why is ire or water not a “thing,” while a horse or a planet is one? The answer cannot be “Because the answer to the question ‘Combien?’ may give a quantity in the latter cases,” for the answer may “give a quantity” in the former cases too; and we cannot say “Ah, but not a quantity of things”; we should be going round in circles.31 The need for such a warning could only arise in a language that did not have a question dedicated to asking for a number. But this is so far supericial. The difference between Anscombe’s predicament and Frege’s is not simply that Frege happens to ind his topic demarcated in German—as though he were just lucky. For, of course, it is not just a coincidence that so many natural languages already have a question perfectly suited to Frege’s purpose. The existence of arithmetic, an ancient science, which has, over millennia, formalized the relation between all natural numbers, and the nearuniversal spread of its elementary principles (even among children, and even among those who do not go to school) everywhere on Earth that human life depends, day-to-day, on the regular use of money—all of this makes it quite unsurprising that so many natural languages have developed a question whose only answer is a counting number. If anything should surprise us, it is how many languages are like French and do not have such a question. On the other hand, the ones that do not have such a question do not even need one because although it is far from obvious how to explain what the number one is, or what the symbol “1” means, in a way that does not run afoul of Frege’s withering criticism, still, it is perfectly obvious, at least to Frege’s contemporaries, which things are, and which are not, numbers, and so, which things are, and which are not, among the things that Frege means to discuss: it is obvious, for example, that “¼ teaspoon” is not one of them.32 This is not to deny that if Frege had walked to the nearest park, clipboard in hand, to survey passersby about what is, and what is not, a counting number, he might have got- THE ARITHMETIC OF INTENTION / ten results that suggested a different starting point from his own. He might have found, for instance, that a very high percentage of ordinary people, who, between surveys, work for a living and are mired in debt, have the “intuition” that negative numbers are used for counting. But if Frege then returned to his ofice and wrote the same Grundlagen, everyone, including the people he’d met that day, would understand immediately what his topic was. Anscombe’s plight is different. On the one hand, she knows that she cannot rely on the English words “intent,” “intention,” “intentional,” and “intentionally” to demarcate her topic: as she says on page 1 of her book, “we are in fact pretty much in the dark” about these words. Even if they will serve for gesturing in the direction of her topic and are useful in a discussion of it, there is no reason to expect these English words, as we ind them on the sidewalk, to be any different from “number” and “counting”: on the contrary, there is every reason to expect that they will have a range of ordinary use that overlaps with, but is not identical to, the contour of any formal concept, much less the one she wants to discuss; and as far as that goes, her situation is just like Frege’s. On the other hand, though, unlike Frege, Anscombe cannot rely on a pre-given question, whether in English or in any other language, with the desired range of application. And worse, she cannot fall back on anything like the science of arithmetic, or a familiar system of numerals, to delineate her topic. This might make one doubt that Anscombe’s topic is, like Frege’s, a strictly formal one. But such evidence is inconclusive. After all, arithmetic itself was a hard-won intellectual achievement. We—we humans—did not always have it. Perhaps it is true, at some level of abstraction, that “thought is in essentials the same everywhere.”33 But this much is certain: before, at the earliest, a mere few 137 thousand years ago, no human being had ever had a thought like the one that Frege used to embarrass Kant: 135664 + 37863 = 173527. A simple calculation like this is possible for us only thanks to ages of prior intellectual labor, which were required to devise an articulate system of numbers, and then, to formalize the relations between such numbers. Before this was accomplished, nine-tenths of human history had already elapsed. The irst gropings of the human mind in the direction of arithmetic, precisely because they were irst, had to have been a matter of going out on a limb. And sometimes that is the right thing to do. Sometimes and sometimes not. Obviously, reminders about the historical development of purely formal thought do not show that Anscombe is right to go out on her limb; it only shows that she is not necessarily wrong to do so. Since what I am offering is an interpretation, not a defense, of Anscombe’s account of intentional action, my point here is only that it is her being out on such a limb that explains why she proceeds in the peculiar way she does. Before she can even begin her account, she needs to induce her reader to see the rough perimeter of that of which it is an account, and that is a task that occupies Anscombe for twenty of the ninety-four pages of her book. The task is not complete until the end of §18, where she writes: “With this we have roughly outlined the area of intentional actions.”34 Having at that point isolated her explanandum, Anscombe proceeds, in §19, immediately to say how it cannot be explained: that is, by appeal to an extra feature. And that is exactly the right thing to say if the relevant sense of the question “Why?” is like the question “How many?” in being formally related to its answer and its theme. Thus, everything we come across in the irst third of Intention—apart from a brief “preamble” 138 / AMERICAN PHILOSOPHICAL QUARTERLY (§§1–5) and some important methodological remarks (§§19–20)—serves to demarcate the topic of an account that is still to come. 7. The Arithmetic of Intention Anscombe’s account of intentional action, which begins in §21, includes her famous discussion of a man who is pumping water in order to ill a cistern. At a pivotal point in that discussion, Anscombe turns from the question “Why?” to a corresponding question “How?,” and she takes up the subject of practical reasoning.35 She seems to think that in doing so she is not changing the subject, but approaching it from another direction; for she says that the importance of an account of practical reasoning is that “it describes an order which is there whenever actions are done with intentions; the same order as [she] arrived at in discussing . . . the man . . . pumping water.”36 It is somewhere in, or around, this discussion that we ind the “results” of Anscombe’s inquiries, the ones that are said to display “what is essential” to the relevant “form of description of events.”37 The material is familiar enough to anyone who has read Intention. What is needed is neither a summary or a list of Anscombe’s theses, but a way of framing the whole discussion so as to make it clear, both that it provides an account and what kind of account it provides. If the science of arithmetic articulates the basic relations holding among numbers, then Anscombe may be seen as attempting an arithmetic of intention. That is, her account may be seen as a ledgling attempt to articulate the basic relations holding among intentional actions.38 Arithmetic embodies a relective understanding of what a number is. Operators like “+” and “−” encode the most general and fundamental truths about numbers, and in doing so, they exhibit something essential about what it is to be a number. This can go unnoticed because we add and subtract on a daily basis without needing to stop and relect on abstruse questions of number theory. However, our ability to do so presupposes a colossal act of relection, distributed over millennia, whose results are crystallized in a mathematical notation that enables us to calculate unrelectively. As Frege puts it: It is possible, of course, to operate with igures mechanically, just as it is possible to speak like a parrot: but that hardly deserves the name of thought. It only becomes possible at all after the mathematical notation has, as a result of genuine thought, been so developed that it does the thinking for us, so to speak.39 What Frege refers to as “genuine thought” is the kind of thought that was required to discern, articulate, and systematize the relation between numbers. Intellectual labor of that sort is not a matter of deining or analyzing something, or of reducing something to something else, or of distinguishing one thing from another. And yet, it is an achievement of the highest order: Often it is only after immense intellectual effort, which may have continued over centuries, that humanity at last succeeds in achieving knowledge of a concept in its pure form, in stripping off the irrelevant accretions which veil it from the eyes of the mind.40 If an explanation is that which provides for an understanding of something, then what Frege describes here—the unveiling of a concept to the eyes of the mind—ought to be recognizable as a kind of explanation, and perhaps as the most fundamental kind. In any event, that is the kind of explanation that Anscombe tries to provide. Her questions “Why?” and “How?” are comparable to functions that operate on all and only intentional actions, functions that display the basic relations in which intentional actions stand to one another. Just as the function “+ 1”, which operates on all and only numbers, takes us from one number to another number, and just as the inverse function “− 1” returns THE ARITHMETIC OF INTENTION / us from the second number to the irst, so, also, the question “Why?” takes us from one intentional action to another intentional action, and the question “How?” performs the opposite operation.41 To see how Anscombe’s questions work and how they relate to each other, suppose that what I am doing is illing a cistern with water. Someone passing by might ask “What are you doing?,” and I might tell him matterof-factly: “I am illing the cistern.” However, once it is known to the passerby what I am doing, he might begin to wonder why. And so he might ask me “Why are you illing the cistern?,” and I might say, truly, “I am replenishing the house water-supply.” The answer to the question why I am doing what I am doing is, in that case, another description of what I am doing. What I am doing is two things at once: it is illing the cistern and replenishing the water supply—and the one for the sake of the other. Replenishing the water supply is that in the pursuit of which I am doing something else: it is, in a word, my end. It is an end, moreover, to which illing the cistern with water is a means. If what I am doing is two things at once, then in response to the question “What are you doing?,” I might have answered straightaway: “I am replenishing the water supply.” If the passerby had then asked how, I could have explained: “I am illing the cistern.” And he might have continued: “How are you illing the cistern?” And I: “I am operating the water pump” And he: “But how are you doing that?” And I: “I am moving the handle up and down.” The answer to the question how I am doing what I am doing, like the answer to the question why, is another description of what I am doing. What I am doing is not just two, but many things at once. It is replenishing the water supply, and illing the cistern, and operating the pump, and moving the handle up and down. The many things done in the course of acting intentionally are bound together as the 139 elements of a system. The question “How?” displays them as an ordered series of means. Given that I am doing D intentionally, the question “How?” brings out the fact that I am also doing C intentionally, and that I am doing D by means of doing C. Further applications of the question reveal that I am doing C by doing B, and doing B by doing A. D. I am replenishing the house water supply. C. I am illing the cistern. B. I am operating the pump. A. I am moving the handle up and down. Meanwhile, the question “Why?” presents the very same material in the reverse order, as a series of ends. Given that I am doing A intentionally, the question “Why?” brings out the fact that I am also doing B intentionally, and that I am doing A for the sake of doing B, and so on: A. I am moving the handle up and down. B. I am operating the pump. C. I am illing the cistern. D. I am replenishing the house water supply. Just as the road from Athens to Thebes is the same as the road from Thebes to Athens, the order of ends (A—D) is the same as the order of means (D—A). This series of action-descriptions, or of things intentionally done, might also be compared to a fragment of the series of natural numbers: 1, 2, 3, 4. The order can be looked at in two different ways. We can read from left to right or from right to left: either as a series of addends (1, 2, 3, 4), where each number yields the next through the addition of 1, or else as a series of minuends (4, 3, 2, 1), where each yields the next through the subtraction of 1. Addition and subtraction—two different, and, indeed, opposite operations—deliver what is manifestly a single order of numbers. That there is a single order of actiondescriptions in the case we have imagined 140 / AMERICAN PHILOSOPHICAL QUARTERLY is guaranteed by the fact that the two series, A—D and D—A, both describe the same unfolding action. We have not imagined that on Monday in St. Louis, I was moving a handle, and on Tuesday in Phoenix, I was operating a water pump, and on Wednesday in Pittsburgh, I was illing a cistern, while on Thursday in Seattle, I was replenishing a water supply. Instead, we have considered four things that I am doing here and now. As Anscombe puts it: “There is one action with four descriptions, each dependent on wider circumstances, and each related to the next as a description of means to end; which means that we can speak equally well of four corresponding intentions, or of one intention.”42 But what is true of the intention is true, also, of what is actually done: we can say that I am doing four things, or that I am doing one—and equally well. “For moving [my] arm up and down with [my] ingers round the pump handle is, in these circumstances, operating the pump; and, in these circumstances, it is replenishing the house water-supply.”43 In doing any one of the things I am doing intentionally, I am doing all the rest. I am moving the handle “in” replenishing the water supply. And, conversely, I am replenishing the water supply “in” moving the handle. Each of them is “in” the other: the one as end, and the other as means. The latter phenomenon, too, has an arithmetic correlate. Except in special circumstances, if someone asks me whether I have two dollars, and I have four, the answer is not no, but yes: for I do have two dollars. If a second person asks me whether I have three dollars, and a third, four, I will answer yes to both. That I answer all three questions in the afirmative does not imply that I have two dollars, and in addition, three dollars, and in addition, four dollars: I have four, not nine, dollars. I give an afirmative answer to all three questions because, in the circumstances, my having four dollars is my having three dollars (and vice versa), and it is my having two dollars (and vice versa). So again, it is “in” having two dollars that I have four, and “in” having four that I have two. Though they do represent a single order, the two different series, A—D and D—A, nevertheless correspond to two different practical-intellectual operations. On the one hand, the A—D series, unleashed by the question “Why?,” presents itself as an order of what Davidson called “rationalization.”44 Each item in the series looks to the next one for its reason, purpose, aim, or rationale. I am moving the handle up and down “in order to” operate the pump; I am operating the pump “for the sake of” illing the cistern; and so on. On the other hand, the D—A series, heralded by the question “How?,” presents itself as an order of deliberation. If my employer has ordered me to replenish the house water supply, I might need to relect on how to do that, and I might conclude that what I should do is to ill the cistern. But at this point, the deliberative question might reassert itself and lead me to inquire how to ill the cistern, and so on. As addition is to subtraction, rationalization is to deliberation. The latter are, on Anscombe’s view, rudimentary functions of the practical intellect, and they are, she thinks, as little apt for a psychological account as the functions of arithmetic. 8. Conclusion I have offered the sketch of an interpretation of Anscombe’s account of intentional action. On this interpretation, Anscombe denies that an action is intentional in virtue of being caused by a reason because she denies that an action is intentional in virtue of standing in any relation to a reason, or in virtue of standing in any relation to anything, or in virtue of having any property whatsoever. She denies this because she believes that the concept of intentional action is, like that of number, formal, and because such a concept requires a formal account. University of Chicago THE ARITHMETIC OF INTENTION / 141 NOTES For comments and discussion, I am indebted to Silver Bronzo, John Brunero, James Conant, Adrian Haddock, Irad Kimhi, Eric Marcus, Will Small, and Eric Wiland. 1. G. E. M. Anscombe, Intention (1957; Repr., Cambridge, MA: Harvard University Press, 2000), p. 84. 2. Michael Bratman, “Davidson’s Theory of Intention,” in Action and Events: Perspectives on the Philosophy of Donald Davidson, ed. Ernest Lepore and Brian McLaughlin (Oxford, UK: Blackwell, 1985), p. 14. 3. Anscombe, Intention, pp. 27–30, 87, 88. 4. Ibid., p. v; emphasis added. 5. Ibid., p. 28. 6. Thus, I will not discuss the argument of §19. For a discussion of the method Anscombe rejects, and of reasons for rejecting it, see Anton Ford, “Action and Generality,” in Essays on Anscombe’s Intention, ed. Anton Ford, Jennifer Hornsby, and Frederick Stoutland (Cambridge, MA: Harvard University Press, 2011), pp. 76–104; see also Candace Vogler, Reasonably Vicious (Cambridge, MA: Harvard University Press, 2002), pp. 213–229. For a discussion of § 19, see Rosalind Hursthouse, “Intention,” in Logic, Cause and Action, ed. Roger Teichmann (Cambridge, UK: Cambridge University Press, 2000), pp. 83–105. 7. Anscombe, Intention, p. 9. 8. See J. David Velleman, Introduction to The Possibility of Practical Reason (Princeton, NJ: Princeton University Press, 2000), pp. 1–31. 9. Harry Frankfurt, “The Problem of Action,” American Philosophical Quarterly, vol. 15, no. 2 (1978), p. 157. 10. Arthur Danto, The Transiguration of the Commonplace (Cambridge, MA: Harvard University Press, 1981), p. 5. 11. Michael E. Bratman, Intention, Plans, and Practical Reason (Cambridge, MA: Harvard University Press, 1987), p. 6. 12. Ibid., p. 6. 13. Donald Davidson, “Actions, Reasons and Causes,” in Essays on Actions and Events (Oxford, UK: Oxford University Press, 1980), pp. 3–19. 14. It may also help to clarify what is at stake in some recent work inspired by Anscombe’s Intention. Sarah Paul discusses, and raises a problem for, what she calls “the neo-Anscombean theory.” Sarah Paul, “Deviant Formal Causation,” Journal of Ethics and Social Philosophy, vol. 5, no. 3 (2011), pp. 1–23. The theory in question is very close to the one that Bratman attributes to Davidson and Goldman, holding both (1) that an action is intentional in virtue of standing in an appropriate relation to the agent’s mental states, and (2) that the appropriate relation is, in some signiicant sense, a causal relation. What distinguishes the “neo-Anscombean” is a thesis about just what causal relation between the agent’s mental states, on the one hand, and action, on the other, makes for intentional action: the neo-Anscombean holds that the appropriate causal relation is one of formal, not eficient, causation. Most of the authors whom Paul associates with this account would, I suspect, repudiate it (though not all for the same reasons). But Paul is careful not to attribute any such view to Anscombe. 15. Gottlob Frege, The Foundations of Arithmetic, trans. J. L. Austin (Evanston, IL: Northwestern University Press, 1950), p. 5n1. 142 / AMERICAN PHILOSOPHICAL QUARTERLY 16. Though he begins with the numbers used for counting, Frege’s aim is ultimately to extend his account of positive whole numbers so as to account for both negative numbers and fractions. 17. A countable (ein Zahlbar) is that to which singular reference can be made. Frege holds that “the truths of arithmetic govern the realm of the countables (das Gebiet des Zahlbaren).” Frege, Foundations of Arithmetic, p. 21. 18. Or by a surrogate for such, like “a couple” or “a few” or “several” or ‘‘many.” Interestingly, “a number” is a surrogate for a number. If asked how many boats one had seen on the lake this morning, one might give a number, or one might simply say “A number of them.” The latter is an informative answer: it normally means “at least two” (and not merely “not none”). It can be surprising to learn that the ancient Greeks thought of two as the smallest number—“How strange!,” one thinks—but, as Michael Kremer pointed out to me in conversation, if someone says that she has a number of errands to run today, or that there are a number of problems with the committee’s proposal, it is deinitely implied that the number is neither zero nor one. 19. Anscombe, Intention, p. 9. 20. Ibid., p. 10. 21. This way of putting the point is misleading insofar as it suggests, falsely, that a “mere event” is a certain kind of event, corresponding to certain sense of a question “Why?” with a certain sort of answer. In fact, the “mere” in “mere event” serves to negate something: it signiies that what we are talking about is not a certain kind of event (namely, an intentional action). But to say what something is not is not to say what it is. For example, to say of an animal that it is not a giraffe is to say what kind of animal it is not, but not what kind of animal it is: “not a giraffe” is not a kind of animal. Or again, to say of my keys that they are not in my pocket is to say where they are not, but not where they are: “not in my pocket” is not a place—otherwise, I would always know where my keys were just by knowing that they are not in my pocket, and I would never have to look for them. 22. Though Frege does not regard the concept of number as a logically primitive notion, he does think it is one whose ine structure can be completely explicated in terms of such notions. As Anscombe remarks in her Introduction to Wittgenstein’s Tractatus, one difference between Wittgenstein, on the one hand, and Frege and Russell, on the other, is that “for Frege and Russell, (natural) number was not a formal concept, but a genuine concept that applied to some but not all objects (Frege) or to some but not all classes of classes (Russell).” G. E. M. Anscombe, Introduction to Wittgenstein’s Tractatus (London: Hutchinson, 1959), p. 126. If Anscombe sides with Wittgenstein on this question, then the analogy between her and Frege is imperfect, but this will not matter for my purposes. Whatever their disagreements, Frege, Russell, and the early Wittgenstein are of one mind in thinking that the concept of number requires a formal, or logical, account, and not a psychological one. 23. Anscombe, Intention, p. 25. 24. Frege, Foundations, p. 57. 25. Anscombe, Intention, p. 31. 26. Davidson, “Actions, Reasons and Causes,” p. 6. 27. Aristotle, Metaphysics, in The Complete Works of Aristotle, ed. Jonathan Barnes (Princeton, NJ: Princeton University Press, 1984), 20a7–14. 28. See Henry Laycock, Words without Objects (Oxford, UK: Oxford University Press, 2006). 29. I have avoided speaking of a “formal object” or of an “object of thought” or of the “object of a question” because the term “object” is not neutral between the two forms under discussion: “object” is a count-noun, which takes an indeinite article; thus, if it were used, it would constantly suggest that we were concerned, speciically, with countable particulars—that is, with things—and this would erase THE ARITHMETIC OF INTENTION / 143 the formal distinctness of stuff. I speak instead of the “theme” or “substance” of thought because these terms are neutral between stuff and things. 30. Alternatively, Frege might have done like his contemporary, the French logician Louis Couturat, who regimented the relevant sense of the question “Combien?” as follows: “A première vue, l’emploi du nombre à la mesure des grandeurs parait tout différent de l’emploi du nombre au dénombrement des collections: les unes sont homogènes et continues, les autres sont discrétes et hétérogènes; dans un cas le nombre répond à la question: Combien? et représente une quotité; dans l’autre, il répond à la question: Combien Grand? et représente une quantité.” Louis Couturat, De l’inini mathematique, ed. Feliz Alcan (Paris: Ancienne Librairie Germer Bailliere, 1896), p. 523. 31. This block of text, though not a quotation, mimics the last block quotation of §3, above. 32. This is why the standard French translation of the Grundlagen can render Frege’s “Wie viele?” as “Combien?” without editorial comment and without any danger of misunderstanding, in spite of the fact that, strictly speaking, “Combien?” fails to isolate Frege’s topic. See Gottlob Frege, Les Fondements de l’arithmetique, trans. Claude Imbert (Paris: Édition du Seuil, 1969), p. 128n2. 33. Frege, Foundations, p. iii. 34. Anscombe, Intention, p. 28. 35. The question “How?” is introduced on p. 46 of Intention; the discussion of practical reasoning begins on p. 57. 36. Anscombe, Intention, p. 80. 37. Ibid., p. 84. 38. In that case, Anscombe’s account of intentional action is very different from Frege’s account of number. Frege is not developing arithmetic, but taking an existent science and arguing that its truths are analytic and a priori. Frege, Foundations, pp. 118–119. Whether or not Frege’s project is properly described as a “reduction,” it is at any rate the kind of project that is only possible once arithmetic is already up on its feet. 39. Frege, Foundations, p. iv. 40. Ibid., p. vii. 41. There is, of course, a striking disanalogy: though every application of “+ 1” takes us from one number to the next ad ininitum, not every application of the question “Why?” takes us from one intentional action to another intentional action. It came out in section 4, above, that a null-answer is possible. Anscombe’s position, argued in §§21–22, is that “the concept of voluntary or intentional action would not exist, if the question ‘Why?,’ with answers that give reasons for acting, did not. Given that it does exist, the cases where the answer is ‘For no particular reason,’ etc. can occur” (Anscombe, Intention, p. 34). Besides, if there were an ininite progression of answers to the question “Why?” in terms of other actions, this would make our desire “empty and vain.” Aristotle, Nicomachean Ethics, 94a20. Anscombe identiies a certain point—a “break”—beyond which answers to the question “Why?” do not provide further descriptions of intentional action: beyond the break, answers name the kind of end that an agent has “never completely attained . . . unless by the termination of the time for which he wants it (which might be the term of his life)” (Anscombe, Intention, p. 63). Anscombe thus acknowledges that while one can do one thing for the sake of doing another, one can also do something for the sake of justice. Anscombe’s topic in Intention is limited to the former kind of means-end relation. 42. Anscombe, Intention, p. 46. 43. Ibid. 44. See, again, Davidson, “Actions, Reasons, and Causes.”