Semiotica 72, vol 1/2, 1-35
As the sign system whose grammar has determined the shape of Western culture’s techno-scientific discourse since its inception, mathematics is implicated, at a deeply linguistic level, in any form of distinctively intellectual activity; indeed, the norms and guidelines of the ‘rational’ – valid argument, definitional clarity, coherent thought, lucid explication, unambiguous expression, logical transparency, objective reasoning – are located in their most extreme, focused, and highly cultivated form in mathematics. The question this essay addresses – what is the nature of mathematical language? — should therefore be of interest to sernioticians and philosophers as well as mathematicians.
There are, however, certain difficulties. inherent in trying to address such disparate types of readers at the same time which it would he disingenuous not to acknowledge at the outset.
Consider the mathematical reader. On the one hand it is no accident that Peirce, whose writings created the possibility of the present essay, was a mathematician; nor one that I have practiced as a mathematician; nor that Hilbert, Brouwer, and Frcge – the authors of the accounts of mathematics I shall dispute – were mathematicians. Mathematics is cognitively difficult, technical, abstract, and (for many) defeatingly impersonal: one needs, it seems, to have been inside the dressing room in order to make much sense of the play. On the other hand, one cannot stay too long there if the play is not to disappear inside its own performance. In this respect mathematicians confronted with the nature of their subject arc no different from anybody else. The language that textual critics, for example, use to talk about criticism will be permeated by precisely those features – figures of ambiguity, polysemy, compression of meaning, subtlety and plurality of interpretation, rhetorical tropes, and so on – which these critics value in the texts they study; likewise mathematicians will create and respond to just those discussions of mathematics that ape what attracts them to their subject matter. Where textual critics literize their metalanguage, mathematicians mathcmatize theirs. And since for mathematicians the principal activity is proving new theorems, what they will ask of my description of their subject is: can it be the source of new mathematical material? Does it suggest new notational systems, definitions, assertions, proofs? Now it is certainly the case that the accounts offered by Frege. Brouwer, and Hilbert all satisfied this requirement: each put forward a program that engendered new matbematics: each acted in and wrote the play and, in doing so, gave a necessarily truncated and misleading account of mathematics. Thus, if a semiotic approach to mathematics can be made to yield theorems and be acceptable to mathematicians, it is unlikely to deliver the kind of exterior view of mathematics it promises; if it does not engender theorems, then mathematicians will be little interested in its project of re-describing their subject – ‘queen of the sciences’ – via an explanatory formalism that (for them) is in a pre-scientific stage of arguing about its own fundamental terms. Since the account I have given is not slanted toward the creation of new mathematics, the chances of interesting mathematicians – let alone making a significant impact on them – look slim.
With readers versed in semiotics the principal obstacle is getting sufficiently behind the mathematical spectacle to make sense of the project without losing them in the stage machinery. To this end I’ve kept the presence of technical discussion down to the absolute minimum. If I have been successful in this then a certain dissatisfaction presents itself: the sheer semiotic skimpiness of the picture I offer. Rarely do I go beyond identifying an issue, clearing the ground, proposing a solution, and drawing a consequence or two. Thus, to take a single example, readers familiar with recent theories of narrative are unlikely to feel more than titillated by being asked to discover that the persuasive force of proofs, of formal arguments within the mathematical Code, are to be found in stories situated in the metaCode. They would want to know what sort of stories, how they relate to each other, what their genres are, whether they are culturally and historically invariant, how what they tell depends on the telling, and so on. To have attempted to enter into these questions would have entailed the very technical mathematical discussions I was trying to avoid. Semioticians, then, might well feel they have been served too thin a gruel. To them I can only say that beginnings are difficult and that if what I offer has any substance, then others – they themselves, perhaps – will be prompted to cook it into a more satisfying sort of semiotic soup.
Finally, there are analytic philosophers. Here the difficulty is not that of unfamiliarity with the mathematical issues. On the contrary, no one is more familiar with them: the major thrust of twentieth-century analytic philosophy can be seen as a continuing response to the questions of reference, meaning, truth, naming, existence, and knowledge that emerged from work in mathematics, logic, and metamathematics at the end the last century. And indeed, all the leading figures in the modern analytic tradition – from Frege, Russell, and Carnap to Quine, Wittgenstein, and Kripke – have directly addressed the question of mathematics in some way or other. The problem is rather that of incompatibility, a lack of engagement between forms of enquiry. My purpose has been to describe mathematics as a practice, as an ongoing cultural endeavor: and while it is unavoidable that any description I come up with will be riddled with unresolved philosophical issues, these are not – they cannot be if I am to get off the ground – my concern. Confronted, for example, with the debates and counter-debates contained in the elaborate secondary and tertiary literature on Frege, my response has been to avoid them and regard Frege’s thought from a certain kind of semiotic scratch. So that if entering these debates is the only route to the attention of analytic philosophers, then the probability of engaging such readers seems not very great.
Obviously, I hope that these fears arc exaggerated and misplaced, and that there will be readers from each of these three academic specialties prepared to break through what are only, after all, disciplinary barriers.
My purpose here is to initiate the project of giving a semiotic analysis of mathematical signs; a project which, though implicit in the repeated references to mathematics in writings on signs, seems not so far to have been carried out. Why this is so – why mathematics, which is so obviously a candidate for semiotic attention, should have received so little of it – will, I hope, emerge. Let me begin by presenting a certain obstacle, a difficulty of method, in the way of beginning the enterprise.
It is possible to distinguish, without being at all subtle about it, three axes or aspects of any discourse that might serve as an external starting point for a semiotic investigation of the code that underlies it. There is the referential aspect, which concerns itself with the code’s secondarity, with the objects of discourse, the things that are supposedly talked about and referred to by the signs of the code; the formal aspect, whose focus is on the manner and the material means through which the discourse operates, its physical manifestation as a medium; and the psychological aspect, whose interest is primarily in those interior meanings which the signs of the code answer to or invoke. While all three of these axes can be drawn schematically through any given code, it is nonetheless the case that some codes seem to present themselves as more obviously biased toward just one of them. Thus, so-called representational codes such as perspectival painting or realistically conceived literature and film come clothed in a certain kind of secondarity; before all else they seem to he ‘about’ some world external to themselves. Then there are those signifying systems, such as that of non-representational painting for example, where secondarity seems not to be in evidence, but where there is a highly palpable sensory dimension – a concrete visual order of signifiers – whose formal material status has a first claim on any semiotic account of these codes. And again, there are codes such as those of music and dance where what is of principal semiotic interest is how the dynamics of performance, of enacted gestures in space or time, are seen to be in the service of some prior psychological meanings assumed or addressed by the code.
With mathematics each of these external entry points into a semiotic account seems to be highly problematic: mathematics is an art that is practiced, not performed; its signs seem to be constructed – as we shall see – so as to sever their signifieds, what they are to mean, from the real time and space within which their material signifiers occur; and the question of secondarity, of whether mathematics is ‘about’ anything, whether its signs have referents, whether they are signs of something outside themselves, is precisely what one would expect a semiotics of mathematics to be in the business of discussing. In short, mathematics can only offer one of these familiar semiotic handles on itself – the referential route through an external world, the formal route through material signifiers, the psychological route through prior meanings – at the risk of begging the very semiotic issues requiring investigation.
To clarify this last point and put these three routes in a wider perspective, let me anticipate a discussion that can only be given fully later in this essay, after a semiotic model of mathematics has been sketched. For a long time mathematicians, logicians, and philosophers who write on the foundations of mathematics have agreed that (to put things at their most basic) there are really only three serious responses – mutually antagonistic and incompatible – to the question ‘what is mathematics?’ The responses – formalism, intuitionism, and platonism – run very briefly as follows.
For the formalist, mathematics is a species of game, a determinate play of written marks that are transformed according to explicit unambiguous formal rules. Such marks are held to be without intention, mere physical inscriptions from which any attempt to signify, to mean, is absent: they operate like the pieces and moves in chess which, though they can be made to carry significance (representing strategies, for example), function independently of such – no doubt useful but inherently posterior, after the event – accretions of meaning. Formalism, in other words, reduces signs to material signifiers which are, in principle, without signifieds. In Hilbert’s classic statement of the formalist credo, mathematics consists of manipulating ‘meaningless marks on paper’.
Intuitionists, in many ways the natural dialectical antagonists of formalists, deny that signifiers – whether written, spoken, or indeed in any other medium – play any constitutive role in mathematical activity. For intuitionism mathematics is a species of purely mental construction, a form of internal cerebral labor, performed privately and in solitude within the individual – but cognitively universal – mind of the mathematician. If formalism characterizes mathematics as the manipulation of physical signifiers in the visible, intersubjective space of writing, intuitionism (in Brouwer’s formulation) sees it as the creation of immaterial signifieds within the – inner, a priori, intuited – category of time. And as the formalist reduces the signified to an inessential adjunct of the signifier, so the intuitionist privileges the signified and dismisses the signifier as a useful but theoretically unnecessary epiphenomenon. For Brouwer it was axiomatic that ‘mathematics is a languageless activity’.
Last, and most important, since it is the orthodox position representing the view of all but a small minority of mathematicians, there is platonism. For platonists mathematics is neither a formal and meaningless game nor some kind of languageless mental construction, but a science, a public discipline concerned to discover and validate objective or logical truths. According to this conception mathematical assertions are true or false propositions, statements of fact about some state of affairs, some objective reality, which exists independently of and prior to the mathematical act of investigating it. For Frege, whose logicist program is the principal source of twentieth-century platonism, mathematics seen in this way was nothing other than an extension of pure logic. For his successors there is a separation: mathematical assertions are facts – specifically, they describe the properties of abstract collections (sets) – while logic is merely a truth-preserving form of inference which provides the means of proving that these descriptions are ‘true’. Clearly, then, to the platonist mathematics is a realist science, its symbols are symbols of certain real – pre-scientific – things, its assertions are consequently assertions about some determinate, objective subject matter, and its epistemology is framed in terms of what can be proved true concerning this reality,
The relevance of these accounts of mathematics to a semiotic project is twofold. First, to have persisted so long each must encapsulate, however partially, an important facet of what is felt to be intrinsic to mathematical activity. Certainly, in some undeniable but obscure way, mathematics seems at the same time to be a meaningless game, a subjective construction, and a source of objective truth. The difficulty is to extract these part truths: the three accounts seem locked in an impasse which cannot be escaped from within the common terms that have allowed them to impinge on each other. As with the scholastic impasse created by nominalism, conceptualism, and realism – a parallel made long ago by Quine – the impasse has to be transcended. A semiotics of mathematics cannot, then, be expected to offer a synoptic reconciliation of these views; rather, it must attempt to explain – from a semiotic perspective alien to all of them – how each is inadequate, illusory, and undeniably attractive. Second, to return to our earlier difficulty of where to begin, each of these pictures of mathematics, though it is not posed as such, takes a particular theory-laden view about what mathematical signs are and are not; so that, to avoid a sell-fulfilling circularity, no one of them can legitimately serve as a starting point for a semiotic investigation of mathematics. Thus, what we called the route through material signifiers is precisely the formalist obsession with marks, the psychological route through prior meaning comprehends intuitionism and the route through an external world of referents is what all forms of mathematical platonism require.
A semiotic model of mathematics
Where then can one start? Mathematics is an activity, a practice. If one observes its participants it would be perverse not to infer that for large stretches of time they are engaged in a process of communicating with themselves and each other; an inference prompted by the constant presence of standardly presented formal written texts (notes, textbooks, blackboard lectures, articles, digests, reviews, and the like) being read, written, and exchanged, and of informal signifying activities that occur when they talk, gesticulate, expound, make guesses, disagree, draw pictures, and so on. (The relation between the formal and informal modes of communicating is an important and interesting one to which we shall return later; for the present I want to focus on the written mathematical text.)
Taking the participants’ word for it that such texts are indeed items in a communicative network, our first response would be to try to ‘read’ them, to try to decode what they arc about and what sorts of things they are saying. Pursuing this, what we observe at once is that any mathematical text is written in a mixture of words, phrases, and locutions drawn from some recognizable natural language together with mathematical marks, signs, symbols, diagrams, and figures that (we suppose) are being used in some systematic and previously agreed upon way. We will also notice that this mixture of natural and artificial signs is conventionally punctuated and divided up into what appear to he complete grammatical sentences; that is, syntactically self-contained units in which noun phrases (‘all points on X’, ‘the number y’, ‘the first and second derivatives of’, ‘the theorem alpha’, and so on) are systematically related to verbs (‘count’, ‘consider’, ‘can be evaluated’, ‘prove’, and so on) in what one takes to be the accepted sense of connecting an activity to an object.
Given the problem of ‘objects’. and of all the issues of ontology, reference, ‘truth’, and secondarity that surface as soon as one tries to identify what mathematical particulars and entities such as numbers, points, lines, functions, relations, spaces, orderings, groups, sets, limits, morphisms, functors, and operators ‘are’, it would be sensible to defer discussion of the interpretation of nouns and ask questions about the ‘activity’ that makes up the remaining part of the sentence; that is, still trusting to grammar, to ask about verbs.
Linguistics makes a separation between verbs functioning in different grammatical moods; that is, between modes of sign use which arise, in the case of speech, from different roles which a speaker can select for himself and his hearer. The primary such distinction is between the indicative and the imperative.
The indicative mood has to do with asking for (interrogative case) or conveying (declarative case) information – ‘the speaker of a clause which has selected the indicative plus declarative has selected for himself the role of informant and for his hearer the role of informed’ (Berry 1975: 166). For mathematics, the indicative governs all those questions, assumptions, and statements of information – assertions, propositions, posits, theorems, hypotheses, axioms, conjectures, and problems – which either ask for, grant, or deliver some piece of mathematical content, some putative mathematical fact such as ‘there are infinitely many prime numbers’, ‘all groups with 7 elements are abelian’, ‘5 + 11 + 3 = 11 + 3 + 5’, ‘there is a continuous curve with no tangent at any point’, ‘every even number can he written as the sum of two prime numbers’, or, less obviously, those that might be said to convey metalingual information such as ‘assertion A is provable’, ‘x is a counterexample to proposition P’, definition D is legitimate’, ‘notational system N is inconsistent’, and so on.
In normal parlance, the indicative bundles information, truth, and validity indiscriminately together; it being equivalent to say that an assertion is ‘true’, that it ‘holds’, that it is ‘valid’, that it is ‘the case’, that it is informationally ‘correct’, and so on. With mathematics it is necessary to be more discriminating: being ‘true’ (whatever that is ultimately to mean) is not the same attribute of an assertion as being valid (that is, capable of being proved): conversely, what is informationally correct is not always, even in principle, susceptible of mathematical proof. The indicative mood, it seems, is inextricably tied up with the notion of mathematical proof. But proof in turn involves the idea of an argument, a narrative structure of sentences, and sentences can be in the imperative rather than the indicative.
According to the standard grammatical description, ‘the speaker of a clause which has chosen the imperative has selected for himself the role of controller and for his hearer the role of controlled. The speaker expects more than a purely verbal response. He expects some form of action’ (Berry 1975: 166). Mathematics is so permeated by instructions for actions to be carried out, orders, commands, injunctions to be obeyed – ‘prove theorem T’, ‘subtract x from y’, ‘drop a perpendicular from point P onto line L’, ‘count the elements of set S’, ‘reverse the arrows in diagram D’, ‘consider an arbitrary polygon with k sides’, and similarly for the activities specified by the verbs add’, ‘multiply’, ‘exhibit’, ‘find’, ‘enumerate’, ‘show’, ‘compute’, ‘demonstrate’, ‘define’, ‘eliminate’, ‘list’, ‘draw’, ‘complete’, ‘connect’, ‘assign’, evaluate’, ‘integrate’, ‘specify’, ‘differentiate’, ‘adjoin’, ‘delete’, ‘iterate’, ‘order’, ‘complete’, ‘calculate’, ‘construct’, etc. – that mathematical texts seem at times to be little more than sequences of instructions written in an entirely operational, exhortatory language.
Of course, mathematics is highly diverse, and the actions indicated even in this very incomplete list of verbs differ very widely. Thus, depending on their context and their domain of application (algebra. calculus, arithmetic, topology, and so on), they display radical differences in scope, fruitfulness, complexity, and logical character: some (like ‘adjoin’) might be finitary, others (like ‘integrate’) depend essentially on an infinite process; some (like ‘count’) apply solely to collections, others solely to functions or relations or diagrams, whilst others (like ‘exhibit’) apply to any mathematical entity, some can be repeated on the states or entities they produce, others cannot. To pursue these differences would require technical mathematical knowledge which would be out of place in the present project. It would also be beside the point: our focus on these verbs has to do not with the particular mathematical character of the actions they denote, but with differences between them – of an epistemological and semiotic kind – which are reflected in their grammatical status, and specifically in their use in the imperative mood.
Corresponding to the linguist’s distinction between inclusive imperatives (‘Let’s go’) and exclusive imperatives (‘Go’), there seems to be a radical split between types of mathematical exhortation: inclusive commands marked by the verbs ‘consider’, ‘define’, ‘prove’ and their synonyms – demand that speaker and hearer institute and inhabit a common world or that they share some specific argued conviction about an item in such a world; and exclusive commands – essentially the mathematical actions denoted by all other verbs – dictate that certain operations meaningful in an already shared world be executed.
Thus, for example, the imperative ‘consider a Hausdorff space’ is an injunction to establish a shared domain of spaces: it commands its recipient to introduce a standard, mutually agreed upon ensemble of signs – symbolized notions, definitions, proofs, and particular cases that bring into play the ideas of topological neighborhood, limit point, a certain separability condition, in such a way as to determine what it is to dwell in the world of such spaces. By contrast, an imperative like ‘integrate the function f’ for example, is mechanical and exclusive: it takes for granted that a shared frame (a world within the domain of calculus) has already been set, and asks that a specific operation relevant to this world be carried out on the function f. Likewise, the imperative ‘define…’ (or equivalently, ‘let us define . . .’) dictates that certain sign uses be agreed upon as the shared givens for some particular universe of discourse. Again, an imperative of the type ‘prove (demonstrate or show) there are infinitely many prime numbers’ requires its recipient to construct a certain kind of argument, a narrative whose persuasive force establishes a commonality between speaker and hearer with respect to the world of integers. By contrast, an imperative like ‘multiply integer x by its successor’ is concerned not to establish commonality of any sort, but to effect a specific operation on numbers.
One can gloss the distinction between inclusive and exclusive commands by observing that the familiar natural language process of forming nouns from verbs – the gerund ‘going’ from the verb ‘go’ – is not available for verbs used in the mood of the inclusive imperative. Thus, while commands can always (with varying degrees of artificiality, to be sure) be made to yield legitimate mathematical objects – ‘add’ gives rise to an ‘adding’ in the sense of the binary operation of addition, ‘count’ yields a ‘counting’ in the sense of a well-ordered binary relation of enumeration, and so on – such is not the case with inclusive commands: normal mathematical practice does not allow a ‘defining’ or a ‘considering’ or a ‘proving’ to be legitimate objects of mathematical discourse. One cannot, in other words, prove results about or consider or define a ‘considering’, a ‘proving’, or a ‘defining’ in the way that one can for an ‘adding’, a ‘counting’, etc. (An apparent exception to this occurs in metamathematics, where certain sorts of definitions and proofs are themselves considered and defined, and have theorems proved about them; but metamathematics is still mathematics – it provides no violation of what is being elaborated here.)
The grammatical line we have been following formulates the imperative in terms of speakers who dictate and hearers who carry out actions. But what is to be understood by ‘action’ in relation to mathematical practice? What does the hearer-reader, recipient, addressee actually do in responding to an imperative? Mathematics can be an activity whose practice is silent and sedentary. The only things mathematicians can be supposed to do with any certainty are scribble and think; they read and write inscriptions which seem to be inescapably attached to systematically meaningful mental events. If this is so, then whatever actions they perform must be explicable in terms of a scribbling/thinking amalgam. It is conceivable, as we have seen, to deny any necessary amalgamation of these two terms and to construe mathematics purely as scribbling (as entirely physical and ‘real’: formalism’s ‘meaningless marks on paper’) or purely as thinking (as entirely mental and ‘imaginary’: intuitionism’s ‘languageless activity’); but to adopt either of these polemical extremes is to foreclose on any semiotic project whatsoever, since each excludes interpreting mathematics as a business of using those signifier/signified couples one calls signs. Ultimately then, our object has to be to articulate what mode of signifying, of scribbling/thinking, mathematical activity is, to explain how within this mode, mathematical imperatives are discharged; and to identify who or what semiotic agency issues and obeys these imperatives.
Leaving aside scribbling for the moment, let us focus on mathematical ‘thinking’. Consider the imperative ‘consider a Hausdorfl space’. ‘Consider’ means view attentively, survey, examine, reflect, etc.; the visual imagery here being part of’ a wider pattern of cognitive body metaphors such as understand, comprehend, defend, grasp, or get the feel of an idea or thesis. Therefore, any attempt to explicate mathematical thought is unlikely to escape the net of such metaphors; indeed, to speak (as we did) of ‘dwelling in a world of Hausdorff spaces’ is metaphorically to equate mathematical thinking with physical exploration. Clearly, such worlds are imagined, and the actions that take place within these worlds are imagined actions. Someone has to be imagining worlds and actions, and something else has to be performing these imaginary actions. In other words, someone – some subjective agency – is imagining itself to act. Seen in this way, mathematical thinking seems to have much in common with the making of self-reflective experiments. Such indeed was the conclusion Peirce arrived at:
It is a familiar experience to every human being to wish for something quite beyond his present means, and to follow that wish by the question, ‘Should I wish for that thing just the same, if I had ample means to gratify it?’. To answer that question, he searches his heart, and in so doing makes what I term an abstractive observation. He makes in his imagination a sort of skeleton diagram, or outline sketch: of himself, considers what modifications the hypothetical state of things would require to be made in that picture, and then examines it, that is, observes what he has imagined, to see whether the same ardent desire is there to be discerned. By such a process which is at bottom very much like mathematical reasoning, we can reach conclusions as to what would be true of signs in all cases… (Buehler 1940: 98)
Following the suggestion in Peirce’s formulation, we are led to distinguish between sorts of mathematical agency: the one who imagines (what Peirce simply calls the ‘self’ who conducts a reflective observation), which we shall simply call the Mathematician, and the one who is imagined (the skeleton diagram and surrogate of this self), which we shall call the Agent. In terms of the distinction between imperatives, it is the Mathematician who carries out inclusive demands to ‘consider’ and ‘define’ certain worlds and to ‘prove’ theorems in relation to these, and it is his Agent who executes the actions within such fabricated worlds, such as ‘count’, ‘integrate’, and so on, demanded by exclusive imperatives.
At first glance, the relation between Mathematician and Agent seems no more than a version of that which occurs in the reading of a road map, in which one propels one’s surrogate, a fingertip model of oneself around the world of roads imaged by the lines of the map. Unfortunately, the parallel is misleading, since the point of a road map is to represent real roads – real in the sense of being entities which exist prior to and independently of the map, so that an imagined journey by an agent is conceived to be (at least in principle) realizable. With mathematics the existence of such priorly occurring ‘real’ worlds is, from a semiotic point of view, problematic; if mathematical signs are to be likened to maps, they are maps of purely imaginary territory.
In what semiotic sense is the Agent a skeleton diagram of the Mathematician? Our picture of the Mathematician is of a conscious – intentional, imagining subject who creates a fictional self, the Agent, and fictional worlds within which this self acts. But such creation cannot, of course, be effected as pure thinking: signifieds are inseparable from signifiers: in order to create fictions, the Mathematician scribbles. Thus in response to the imperative ‘add the numbers in the list S’ for example, he invokes a certain imagined world and – inseparable from this invocation – writes down an organized sequence of marks ending with the mark which is to be interpreted as the sum of S. These marks are signifiers of signs by virtue of their interpretation within this world. Within this world ‘to add’ might typically involve an infinite process, a procedure which requires that an infinity of actions be performed. This would be the case if, for example, S were the list of fractions 1, 1/2, 1/4, 1/8, etc. obtained by repeated halving. Clearly, in such a case, if ‘add’ is to be interpreted as an action, it has to be an imagined action, one performed not by the Mathematician – who can only manipulate very small finite sequences of written signs – but by an actor imagined by the Mathematician. Such an actor is not himself required to imagine anything. Unlike the Mathematician, the Agent is not reflective and has no intentions: he is never called upon to ‘consider’, ‘define’, or ‘prove’ anything, or indeed to attribute any significance or meaning to what he does; he is simply required to behave according to a prior pattern – do this then this then . . . – imagined for him by the Mathematician. The Agent, then, is a skeleton diagram of the Mathematician in two senses: he lacks the Mathematician’s subjectivity in the face of signs; and he is free of the constraints of finitude and logical feasibility – he can perform infinite additions, make infinitely many choices, search through an infinite array, operate within nonexistent worlds – that accompany this subjectivity.
If the Agent is a truncated and idealized image of the Mathematician, then the latter is himself a reduced and abstracted version of the subject – let us call him the Person – who operates with the signs of natural language and can answer to the agency named by the ‘I’ of ordinary nonmathematical discourse. An examination of the signs addressed to the Mathematician reveals that nowhere is there any mention of his being immersed in public historical or private durational time, or of occupying any geographical or bodily space. or of possessing any social or individualizing attributes. The Mathematician’s psychology, in other words, is transcultural and disembodied. By writing its codes in a single tense of the constant present, within which addressees have no physical presence, mathematics dispenses entirely with the linguistic apparatus of dcixis: unlike the Person, for whom demonstrative and personal pronouns are available, the Mathematician is never called upon to interpret any sign or message whose meaning is inseparable from the physical circumstances – temporal, spatial, cultural – of its utterance. If the Mathematician’s subjectivity is ‘placed’ in any sense, if he can be said to be physically self-situated, his presence is located in and traced by the single point – the origin – which is required when any system of coordinates or process of counting is initiated: a replacement Hermann Weyl once described as ‘the necessary residue of’ the extinction of the ego’ (1949: 75).
I want now to being this trio of semiotic actors – Agent, Mathematician, and Person – together and to display them as agencies that operate in relation to each other on different levels of the same mathematical process: namely, the centrally important process of mathematical proof. In the extract quoted above, Peirce likened what he called ‘reflective observation’, in which a skeleton of the self takes part in a certain kind of thought experiment, to mathematical reasoning. For the Mathematician, reasoning is the process of giving and following proofs, of reading and writing certain highly specific and internally organized sequences of mathematical sentences – sequences intended to validate, test, demonstrate, show that some particular assertion holds or is ‘true’ or is ‘the case’. Proofs are tied to assertions, and the semiotic status of assertions, as we observed earlier, is inextricable from the nature of proof. So, if we want to give a semiotic picture of mathematical reasoning as a kind of Peircean thought experiment, an answer has to be given to the question we dodged before: how, as a business having to do with signs, are we to interpret the mathematical indicative?
The answer we propose runs as follows. A mathematical assertion is prediction, a foretelling of the result of performing certain actions upon signs. In making an assertion the Mathematician is claiming to know what would happen if the sign activities detailed in the assertion were to he carried out. Since the actions in question are ones that fall within the Mathematician’s domain of activity, the Mathematician is in effect laying claim to knowledge of his own future signifying states. In Peirce’s phrase, the sort of knowledge being claimed is ‘what would be true of signs in all cases’. Thus, for example, the assertion ‘2 + 3 =3 + 2’ predicts that if the Mathematician concatenates 11 with 111, the result will be identical to his concatenating 111 with 11. And more generally, ‘x + y = y + x’ predicts that his concatenating any number of strokcs with any other number will turn out to he independent of the order in which these actions are performed. Or, to take a different kind of example, the assertion, that the square root of 2 is irrational is the prediction that whatever particular integers x and y are to be taken to be, the result of calculating x2 – 2y2 will not be zero.
Obviously such claims to future knowledge need to he validated: the Mathematician has to persuade himself that if he performed the activities in question, the result would be as predicted. How is he to do this? In the physical sciences predictions are set against actualities: an experiment is carried and, depending on the result, the prediction (or rather the theory which gave rise to it) is either repudiated or receives some degree of confirmation. It would seem to he the case that in certain very simple cases such a direct procedure will work in mathematics. Assertions like ‘2 + 3 = 3 + 2’ or ‘101 is a prime’ appear to be directly verifiable: the Mathematician ascertains whether they correctly predict what he would experience by carrying out an experiment – surveying strokes or examining particular numbers – which delivers to him precisely that experience. But the situation is not, as we shall see below, that straightforward; and in any case assertions of this palpable sort, though undoubtedly important in any discussion of the epistemological status of mathematical ‘truths’, are not the norm. (The Mathematician certainly cannot by direct experiment validate predictions like ‘x + y = y +x’ or ‘the square root of 2 is irrational’ unless he carries out infinitely many operations. Instead, as we have observed, he must act indirectly and set up an imagined experience – a thought experiment – in which not he but his Agent, the skeleton diagram of himself, is required to perform the appropriate infinity of actions. By observing his Agent performing in his stead, by ‘reflective observation’, the Mathematician becomes convinced – persuaded somehow by the thought experiment – that were he to perform these actions the result would be as predicted.
Now the Mathematician is involved in scribbling as well as thinking. The process of persuasion that a proof is supposed to achieve is an amalgam of fictive and logical aspects in dialectical relation to each other: Each layer of the thought experiment (that is, each stage of the journey undertaken by the Agent) corresponds to some written activity, some manipulation of written signs performed by the Mathematician: so that, for example, in reading/writing an inclusive imperative the Mathematician modifies or brings into being a suitable facet of the Agent, and in reading/writing an exclusive imperative he requires this Agent to carry out the action in question, observes the result, and then uses the outcome as the basis for a further bout of manipulating written signs. These manipulations form the steps of the proof in its guise as a logical argument: any given step is taken either as a premise, an outright assumption about which it is agreed no persuasion is necessary, or is taken because it is a conclusion logically implied by a previous step. The picture offered so far, then, is that a proof is a logically correct series of implications that the Mathematician is persuaded to accept by virtue of the interpretation given to these implications in the fictive world of a thought experiment.
Such a characterization of proof is correct but inadequate. Proofs are arguments and, as Peirce forcefully pointed out, every argument has an underlying idea – what he called a leading principle – which converts what would otherwise be an unexceptionable sequence of logical moves into an instrument of conviction. The leading principle, Peirce argued, is distinct from the premise and the conclusion of an argument, and if added to these would have the effect of requiring a new leading principle and so on, producing an infinite regress in place of a finitely presentable argument. Thus, though it operates through the logical sequence that embodies it, it is neither identical nor reducible to this sequence; indeed, it is only by virtue of it that the sequence is an argument and not an inert, formally correct string of implications.
The leading principle corresponds to a familiar phenomenon within mathematics. Presented with a new proof or argument, the first question the mathematician (but not, see below, the Mathematician) is likely to raise concerns ‘motivation’: he will in his attempt to understand the argument, that is, follow and be convinced by it – seek the idea behind the proof. He will ask for the story that is belng told, the narrative through which the thought experiment or argument is organized. It is perfectly possible to follow a proof, in the restricted, purely formal sense of giving assent to each logical step, without such an idea. If in addition an argument is based on accepted familiar patterns of inference, its leading principle will have been internalized to the extent of being no longer retrievable: it is read automatically as part of’ the proof. Nonetheless a leading principle is always present, acknowledged or not – and attempts to read proofs in the absence of their underlying narratives are unlikely to result in the experience of felt necessity, persuasion, and conviction that proofs are intended to produce, and without which they fail to be proofs.
Now mathematicians – whether formalist, intuitionist, or platonist – when moved to comment on this aspect of their discourse might recognize the importance of such narratives to the process of persuasion and understanding, but they are inclined to dismiss them, along with any other ‘motivational’ or ‘purely psychological’ or merely ‘esthetic’ considerations, as ultimately irrelevant and epiphenomena1 to the real business of doing mathematics. What are we to make of this?
It is certainly true, as observed, that the leading principle cannot be part of the proof itself: it is not, in other words, addressed to the subject who reads proofs that we have designated as the Mathematician. Indeed, the underlying narrative could not be so addressed, since it lies outside the linguistic resources mathematics makes available to the Mathematician. We might call the total of all these resources the mathematical Code, and mean by this the discursive sum of all legitimately defined signs and rigorously formulated sign practices that are permitted to figure in mathematical texts. At the same time let us designate by the metaCode the penumbra of informal, unrigorous locutions within natural language involved in talking about, referring to, and discussing the Code that mathematicians sanction. The Mathematician is the subject pertaining to the Code – the one who reads/writes its signs and interprets them by imagining experiments in which the actions inherent in them are performed by his Agent. We saw earlier that mathematicians prohibit the use of any deictic terms from their discourse: from which it follows that no description of himself is available to the Mathematician within the Code. Though he is able to imagine and observe the Agent as a skeleton diagram of himself, he cannot – within the vocabulary of the Code – articulate his relation to that Agent. He knows the Agent is a simulacrum of himself, but he cannot talk about his knowledge. And it is precisely in the articulation of this relation that the semiotic source of a proof’s persuasion lies: the mathematician can be persuaded by a thought experiment designed to validate a prediction about his own symbolic actions only if he appreciates the resemblance – for the particular mathematical purpose at hand – between the Agent and himself. It is the business of the underlying narrative of a proof to articulate the nature of this resemblance. In short, the idea behind a proof is situated in the meta-Code; it is not the Mathematician himself who can be persuaded by the idea behind a proof, but the Mathematician in the presence of the Person, the natural language subject of the metaCode for whom the Agent as a simulacrum of the Mathematician is an object of discourse.
What then, to return to the point above, is meant by the ‘real business’ of doing mathematics? In relation to the discussion so far, one can say this: if it is insisted that mathematical activity be described solely in terms of manipulations of signs within the Code, thereby restricting mathematical subjectivity to the Mathematician and dismissing the metaCode as an epiphenomenon, a domain of motivational and psychological affect, then one gives up any hope of a semiotic view of mathematical proof able to give a coherent account – in terms of sign use – of how proofs achieve conviction.
There are two further important reasons for refusing to assign to the meta-Code the status of mere epiphenomenon. The first concerns the completion of the discussion of the indicative. As observed earlier, there are, besides the assertions within the Code that we have characterized as predictions, other assertions of undeniable importance that must be justified in the course of normal mathematical practice. When mathematicians make assertions like ‘definition D is well-founded’, or ‘notation system N is coherent’, they are plainly making statements that require some sort of justification. Equally plainly, such assertions cannot be interpreted as predictions about the Mathematician’s future mathematical experience susceptible of proof via a thought experiment. Indeed, indicatives such as ‘assertion A is provable’ or ‘x is a counter-example to A’ where A is a predictive assertion within the Code, cannot themselves be proved mathematically without engendering an infinite regress of proofs.
It would seem that such metalingual indicatives – which of course belong to the metaCode – admit ‘proof’ in the same way that the proof of the pudding is in the eating: one justifies the statement ‘assertion A is provable’ by exhibiting a proof of A. The second reason for treating the metaCode as important to a semiotic account of mathematics relates to the manner in which mathematical codes and sign usages come into being, since it can be argued (though I will not do so here) that Code and metaCode are mutually constitutive, and that a principal way in which new mathematics arises is through a process of catachresis – that is, through the sanctioning and appropriation of sign practices that occur in the first place as informal and unrigorous elements, in a merely descriptive, motivational, or intuitive guise, within the metaCodc.
The model I have sketched has required us to introduce three separate levels of mathematical activity corresponding to the sub-lingual imagined actions of an Agent, the lingual Coded manipulations of the Mathematician, and the metalingual activities of the Person, and then to describe how these agencies fit together. Normal mathematical discourse does not present itself in this way; it speaks only of a single unfractured agency, a ‘mathematician’, who simply ‘does’ mathematics. To justify the increase in complexity and artificiality of its characterization of mathematics, the model has to be useful; its picture of mathematics ought to illuminate and explain the attraction of the three principal ways of regarding mathematics we alluded to earlier.
Formalism, intuitionism, platonism
We extracted the idea of proof as a kind of thought experiment from Peirce’s general remarks on reflective observation; we might have gotten a later and specifically mathematical version straight from Hilbert, from his formalist conception of metalogic as amounting to those ‘considerations in the form of thought-experiments on objects, which can be regarded as concretely given’ (Hilbert and Bernays 1934: 20). But this would have sidetracked us at that point into a description of Hilbert’s metamathematical program, which it was designed to serve.
The object of this progam was to show by means of mathematical reasoning that mathematical reasoning was consistent, that it was incapable of arriving at a contradiction. In order to reason about reasoning without incurring the obvious circularity inherent in such an enterprise, Hilbert made a separation between the reasoning that mathematicians use – which he characterized as formal manipulations, finite sequences of logically correct deductions performed on mathematical symbols – and the reasoning that the metamathematician would use, the metalogic, to show that this first kind of reasoning was consistent. The circularity would be avoided, he argued, if the metalogic was inherently safe and free from the sort of contradiction that threatened the object logic about which it reasoned. Since the potential source of contradiction in mathematics was held to he the occurrence of objects and processes that were interpreted by mathematicians to be infinitary, the principal requirement of metalogic was that it be finitary and that it avoid interpretations, that is. that it attribute no meanings to the subject matter about which it reasoned. Hilbert’s approach to mathematics, then, was to ignore what mathematicians thought they meant or intended to mean, and instead to treat it as a formalism, as a system of meaningless written marks finitely manipulated by the mathematician according to explicitly stated formal rules. It was to this formalism that his rnetalogic, characterized as thought experiment on things, was intended to apply.
The first question to ask has to concern the ‘things’ that are supposed to figure in thought experiments: what are these concretely given entities about which thought takes place? The formalist answer – objects concretely given as visible inscriptions, as definite but meaningless written marks – would have been open to the immediate objection that meaningless marks, while they can undoubtedly be manipulated and subjected to empirical (that is, visual) scrutiny, are difficult to equate to the sort of entities that figure in finitary arithmetical assertions that form the basis for distinguishing experiments from thought experiments. Thus, in order to validate a finitary assertion ‘2 + 3= 3+2’. the formalist mathematician supposes that a direct experiment is all that is needed: the experimenter is convinced that concatenating ‘11’ and ‘111’ is the same as concatenating ‘111’ and ’11’through the purely empirical observation that in both cases the result is the assemblage of marks ‘11111’. But such an observation is a completely empirical validation — a pure ad oculo demonstration free of any considerations of meaning – only if the mathematical mark ‘1’ is purely and simply an empirical mark, a mark all of whose significance lies in its visible appearance: and this, as philosophical critics of formalism from Frege onward have pointed out, is manifestly not the case. If it were, then arithmetical assertions would lack the generality universally ascribed to them; they would be about the physical perceptual characteristics of particular inscriptions – their exact shape, color, and size; their durability; the depth of their indentation on the page; the exact identity of one with another; and so on – and would need reformulating and revalidating every time one of these variables altered – a conclusion that no formalist of whatever persuasion would want to accept.
In fact, regardless of any considerations of intended meaning, the mathematical symbol ‘I’ cannot be identified with a mark at all. In Peirce’s terminology ‘I’ is not a token (a concretely given visible object), but a type, an abstract pattern of writing, a general form of which any given material inscription is merely a perceptual instance. For us the mathematical symbol ‘I’ is an item of thinking/scribbling, a sign, and we can now flesh out one aspect of our semiotic model by elaborating what is to be meant by this. Specifically, we propose: the sign ‘I’ has for its signifier the type of the mark ‘1’ used to notate it, and for its signified that relation in the Code – which we have yet to explicate – between thought and writing accorded to the symbol ‘1’ by the Mathematician. This characterization of ‘l’ has the immediate consequence that mathematical signifiers are themselves dependent on some prior signifying activity, since types as entities with attributes – abstract, unchanging, permanent, exact; and so on – can only come into being and operate through the semiotic separation between real and ideal marks. Now neither the creation nor recognition of this ideality is the business of what we have called the Mathematician: it takes place before his mathematical encounter with signs. Put differently, the Mathematician assumes this separation but is not, and cannot be, called upon to mention it in the course of interpreting signs within the Code; it forms no part of the meaning of these signs insofar as this meaning is accessible to him as the addressee of the Code. On the contrary, it is the Person, operating from a point exterior to the Mathematician, who is responsible for this ideality; for it is only in the metaCode, where mathematical symbols are discussed as signs, that any significance can be attributed to the difference between writing tokens and using types.
We can apply this view of signs to the question of mathematical ‘experience’ as it occurs within formalism’s notion of a thought experiment. The pressure for inserting the presence of an Agent into our model of mathematical activity came from the fact that whatever unitary actions upon signs the Mathematician might in fact be able to carry out, such as concatenating ‘11’ and ‘111’, these were the exception; for the most part the actions which he was called upon to perform (such as evaluating x + y for arbitrary integers x and y) could only be carried out in principle and for these infinitary actions he had to invoke the activity of an Agent. I suggested earlier that for the former sort of actions there appeared to be no need for him to invoke an Agent and a thought experiment: that the Mathematician might experience for himself by direct experiment the validity of assertions such as ‘2 + 3 = 3 +2′. This idea that (at least some) mathematical assertions are capable of being directly ‘experienced’ is precisely what formalists – interpreting experience as meaning visual inspection of objects in space –claim. One needs therefore to be more specific about the meaning of direct mathematical ‘experience’. From what has been said so far, no purely physical manipulation of purely physical marks can, of itself, constitute mathematical persuasion. The Mathematician manipulates types: he can be persuaded that direct experiments with tokens constitute validations of assertions like ‘2 + 3 = 3+2’ only by appealing to the relationship between tokens and types, to the way tokens stand in place of types. And this he, as opposed to the Person, cannot do. The upshot, from a semiotic point of view, is that in no case can thought experiments be supplanted by direct experirnentation; an Agent is always required. Validating finitary assertions is no different from validating assertions in general, where it is the Person who, by being able to articulate the relation between Mathematician and Agent within a thought experiment, is persuaded that a prediction about the Mathematician’s future encounter with signs is to be accepted.
This way of seeing matters does not dissolve the difference between finitary and non-finitary assertions. Rather it insists that the distinction – undoubtedly interesting, but for us limitedly so in comparison to its centrality within the formalist program – between mathematical actions in fact and in principle (in reified form: how big can a ‘small’ concretely surveyable collection of marks be before it becomes ‘large’ and unsurvcyable?) males sense only in terms of what constitutes mathematical persuasion, which in turn can only be explicated by examining the possible relationships between Agent, Mathematician, and Person that mathematicians are prepared to countenance as legitimate.
Hilbert’s program for proving the consistency of mathematics through a finitary metamathematics was, as is well known, brought to an effective halt by Gödel’s Theorem. But this refutation of what was always an overambitious project does not demolish formalism as a viewpoint; nor,without much technical discussion outside the scope of this essay, can it he made to shed much light on how formalism’s inadequacies arise. Thus, from a semiotic point of view the problems experienced by formalism can be seen to rest on a chain of misidentifications. By failing to distinguish between tokens and types, and thereby mistaking items possessing significance for pre-semiotic ‘things’, formalists simultaneously misdescribe mathematical reasoning as syntactical manipulation of meaningless marks and metamathematical reasoning as thought experiments that theoreticalize these manipulations – in the sense that the formalists’ ‘experiment’, of which the thought experiment is an imagined theoretical version, is an entirely empirical process of checking the perceptual properties of visible marks.. As a consequence, the formalist account of mathematical agency – which distinguishes merely between a mathematician who manipulates mathematical symbols as if they were marks and a ‘metamathematician’ who reasons about the results of this manipulation – is doubly reductive of the picture offered by the present model, since at one point it shrinks the role of Mathematician to that of Agent and at another manages to absorb it into that of Person.
If formalism projects the mathematical amalgam of thinking/scribbling onto a plane of formal scribble robbed of meaning, intuitionism projects it onto a plane of thought devoid of any written trace. Each bases its truncation of sign on the possibility of an irreducible mathematical ‘experience’ which is supposed to convey by its very directness what it takes to be essential to mathematical practice: formalism, positivist and suspicious in a behaviorist way about mental events, has to locate this experience in the tangible written product, surveyable and ‘real’; intuitionism, entirely immersed in Kantian apriorism, identifies the experience as the process, the invisible unobservable construction in thought, whereby mathematics is created.
Brouwcr’s intuitionism, like Hilbert’s formalism, arose as a response to the paradoxes of the infinite that emerged at the turn of the century within the mathematics of infinite sets. Unlike Hilbert, who had no quarrel with the platonist conception of such sets and whose aim was to leave mathematics as it was by providing a post facto justification of its consistency in which all existing infinitary thought would be legitimated, Brouwer attacked the platonist notion of infinity itself and argued for a root and branch reconstruction from within in which large areas of classically secure mathematics – infinitary in character but having no explicit connection to any paradox – would have to he jettisoned as lacking any coherent basis in thought and therefore meaningless. The problem, as Brouwer saw it, was the failure on the part of orthodox – platonist-inspired – mathematics to separate what for him are proper objects of mathematics (namely constructions in the mind) from the secondary aspect of these objects, the linguistic apparatus that mathematicians might use to describe and communicate about in words and symbols — the features and results of any particular such construction. Confusing the two allowed mathematicians to believe, Brouwer argued, that linguistic manipulation was an unacceptable route to the production of new mathematical entities. Since the classical logic governing such manipulation has its origins in the finite states of affairs described by natural language, the confusion results in a false mathematics of the infinite: verbiage that fails to correspond to any identifiable mental activity, since it allows forms of inference that make sense only for finite situations, such as the law of excluded middle or the principle of double negation, to appear to validate what are in fact illegitimate assertions about infinite ones.
Thus the intuitionist approach to mathematics, like Hilbert’s for metamathematics, insists that a special and primary characteristic of logic lies in its appropriateness to finitary mathematical situations. And though they accord different functions to this logic – for Hilbert it has to validate unitary metamathematics in order to secure infinitary mathematics, while for Brouwer it is an after-the-fact formalization of the principles of correct mental constructions finitary and infinitary – they each require it to be convincing: formalism grounding the persuasive force of its logic in the empirical certainty of ad oculo demonstrations, intuitionism being obliged to ground it, as we shall see, in the felt necessity and self-evidence of the mathematician’s mental activity.
If mathematical assertions are construed platonistically, as unambiguous, exact, and precise statements of fact, propositions true or false about some objective state of affairs,
Brouwer’s rejection of the law of excluded middle (the principle of logic that declares that an assertion either is the case or is not the case) and his rejection of double negation (the rule that not being not the case is the same as being the case) seem puzzling and counterintuitive: about the particulars of mathematics conceived in this exact and determinate way there would appear to be no middle ground between truth and falsity, and no way of distinguishing between an assertion and the negation of the negation of that assertion. Clearly, truth and falsity of assertions will not mean for intuitionists – if indeed they are to mean anything at all for them – what they do for orthodox mathematicians: and, since platonistic logic is founded on truth, intuitionists cannot be referring to the orthodox process of deduction when they talk about the validation of assertions.
For the intuitionist, an assertion is a claim that a certain mental construction has been carried out. To validate such claims the intuitionist must either exhibit the construction in question or, less directly, show that it can be carried out by providing an effective procedure, a finite recipe, for executing it. This effectivity is not an external requirement imposed on assertions after they have been presented, but is built into the intuitionist account of the logical connectives through which assertions are formulated. And it is from this internalized logic that principles such as the laws of excluded middle, double negation, and so on are excluded. Thus, in contrast to the platonist validation of an existential assertion (x exists if the assumption that it doesn’t leads to a contradiction), for the intuitionist x can only be shown to exist by exhibiting it, or by showing effectively how to exhibit it. Again, to validate the negation of an assertion A, it is not enough – as it is for the platonist – to prove the existence of a contradiction issuing from the assumption A: the intuitionist must exhibit or show how one would exhibit the contradiction when presented with the construction that is claimed in A. Likewise for implication: to validate ‘A implies B’ the intuitionist must provide an effective procedure for converting the construction claimed to have been carried out in A into the construction being claimed in B.
Clearly, the intuitionist picture of mathematical assertions and proofs depends on the coherence and acceptability of what it means by an effective procedure and (inseparable from this) on the status of claims that mental constructions have been or can he carried out. Proofs, validations, and arguments, in order to ‘show’ or ‘demonstrate’ a claim, have before all else to be convincing; they need to persuade their addressees to accept ‘what is claimed. Where then in the face of Brouwer’s characterization that “intuitionist mathematics is an essentially languageless activity of the mind having its origin in the perception of a move in time” (1952), with its relegation of language – that is, all mathematical writing and speech – to an epiphenomenon of mathematical activity, a secondary and (because it is after the fact) theoretically unnecessary business of mere description, are we to locate the intuitionist version of persuasion? The problem is fundamental. Persuading, convincing, showing, and demonstrating are rhetorical activities whose business it is to achieve intersubjective agreement. But for Brouwer, the intersubjective collapses into the subjective: there is only a single cognizing subject privately carrying out constructions – sequences of temporally distinct moves – in the intuition of time. This means that, for the intuitionist, conviction and persuasion appear as the possibility of a replay, a purely mental reenactment within this one subjectivity: perform this construction in the inner intuition of time you share with me and you will – you must – experience what I claim to experience.
Validating assertions by appealing in this way to felt necessity, to what is supposed to be self-evident to the experiencing subject, goes back to Descartes’ cogito, to which philosophers have raised a basic and (this side of solipsism) unanswerable objection: what is self-evidently the case for one may be not self-evident – or worse, may be self-evidently not the case – for another; so that, since there can be no basis other than subjective force for choosing between conflicting self-evidence, what is put forward as a process of rational validation intended to convince and persuade is ultimately no more than a refined reiteration of the assertion it claims to he validating. That intuitionism should be unable to give a coherent account of persuasion is what a semiotic approach which insists that mathematics is a business with and about signs, conceived as public, manifest amalgams of scribbling/thinking. would lead one to believe The inability is the price intuitionism pays for believing it possible to first separate thought from writing and then demote writing to a description of this prior and languageless – pre-semiotic – thought. Of course, this is not to assert that intuitionism’s fixation on thinking to the exclusion of writing is not useful or productive; within mathematical practice, by providing an alternative to classical reasoning, It has been both. And indeed, insofar as a picture of mathematics-as-pure-thought is possible at all, intuitionism in some form or other could be said to provide it.
From a semiotic viewpoint, however, any such picture cannot avoid being a metonymic reduction, a pars pro toto which mistakes a part – the purely mental activities that seem undeniably to accompany all mathematical assertions and proofs – for the whole writing/thinking business of manipulating signs, and in so doing makes it impossible to recognize the distinctive role played by signifiers in the creation of mathematical meaning. Far from being the written traces of a language that merely describes prior mental construction, as pre-semiotic events accessible only to private introspection, signifiers mark signs that are interpreted in terms of imagined actions which themselves have no being independent of their invocation in the presence of these very signifiers. And it is in this dialectic relation between scribbling and thinking, whereby each creates what is necessary for the other to come into being, that persuasion – as a tripartite activity involving Agent, Mathematician, and Person within a thought experiment – has to be located.
These remarks about formalism and intuitionism are intended to serve not as philosophical critiques of their claims about the nature of mathematics, but as means of throwing into relief the contrasting claims of our semiotic model. From the point of view of mainstream mathematical practice, moreover, the formalist and intuitionist descriptions of mathematics are of relatively minor importance. True, formalism’s attempt to characterize finitary reasoning is central to metamathematical investigations such as proof theory, and intuitionistic logic is at the front of any constructivist examination of mathematics; but neither exerts more than a marginal influence on how the overwhelming majority of mathematicians regard their subject matter. When they pursue their business mathematicians do so neither as formalist manipulators nor solitary mental constructors, but as scientific investigators engaged in publicly discovering objective truths. And they see these truths through platonistic eyes: eternal verities, objective irrefutably-the-case descriptions of some timeless, spaceless, subjectless reality of abstract ‘objects’.
Though the question of’ the nature of these platonic objects – what are numbers? – can be made as old as Western philosophy, the version of platonism that interests us (namely, the prevailing orthodoxy in mathematics) is a creation of nineteenth-century realism. And since our focus is semiotic and not philosophical, our primary interest is in the part played by a realist conception of language in forming and legitimizing present-day mathematical platonism.
For the realist, language is an activity whose function is that of naming: its character derives from the fact that its terms, locutions, constructions, and narratives are oriented outward, that they point to, refer to, denote some reality outside and prior to themselves. They do this not as a byproduct, consequentially on some more complex signifying activity, but essentially and genetically so in their formation: language, for the realist, arises and operates as a name for the pre-existing world. Such a view issues in a bifurcation of linguistic activity into a primary act of reference concerning what is ‘real’, given ‘out there’ within the prior world waiting to be labeled and denoted – and a subsidiary act of describing, commenting on, and communicating about the objects named. Frege, who never tired of arguing for the opposition between these two linguistic activities – what Mill’s earlier realism distinguished as connotation/denotation and he called sense/reference – insisted that it was the latter that provided the ground on which mathematics was to be based. And if for technical reasons Frege’s ground – the pre-existing world of pure logical objects – is no longer tenable and is now replaced by an abstract universe of sets, his insistence on the priority of reference over sense remains as the linguistic cornerstone of twentieth-century platonism.
What is wrong with it? Why should one not believe that mathematics is about some ideal timeless world populated by abstract unchanging objects; that these objects exist, in all their attributes, independent of any language used to describe them or human consciousness to apprehend them; and that what a theorem expresses is objectively the case, an eternally true description of a specific and determinate states of affairs about these objects?
One response might be to question immediately the semiotic coherence of a pre-linguistic referent:
Every attempt to establish what the referent of a sign is forces us to define the referent in terms of an abstract entity which moreover is only a cultural convention. (Eco 1976: 66)
If such is the case, then language – in the form cultural mediation – is inextricable from the process of sense-making. This will mean that the supposedly distinct and opposing categories of reference and sense interpenetrate each other, and that the object referred to can neither be separated from nor ante-date the descriptions given of it. Such a referent will be a social historical construct; and, notwithstanding the fact that it might present itself as abstract, cognitively universal, pre-semiotic (as is the case for mathematical objects), it will be no more timeless, spaceless, or subjectless than any other social artifact. On this view mathematical platonism never gets off the ground (i.e. never touches any ground), and Frege’s claim that mathematical assertions are objectively true about eternal ‘objects’ dissolves into a psychologistic opposite that he would have abhorred, namely that mathematics makes subjective assertions – dubitable and subject to revision – about entities that are time-hound and culturally loaded.
Another response to platonism’s reliance on such abstract referents – one which is epistemological rather than purely semiotic, but which in the end leads to the same difficulty – lies in the questions ‘How can one come to know anything about objects that exist outside space and time?’ and ‘What possible causal chain could there be linking such entities to temporally and spatially situated human knowers?’ If knowledge is thought of as some form of justified belief, then the question repeats itself on the level of validation: what manner of conviction and persuasion is there which will connect the plantonist mathematician to this ideal and inaccessible realm of objects? Plato’s answer – that the world of human knowledge is a shadow of the eternal ideal world of pure form, so that by examining how what is perceivable partakes of and mimics the ideal, one arrives at knowledge of the eternal – succeeds only in recycling the question through the metaphysical obscurities of how concrete and palpable particulars are supposed to partake of and be shadows of abstract universals. How does Frege manage to deal with the problem?
The short answer is that he doesn’t. Consider the distinctions behind Frege’s insistence that ‘the thought we express by the Pythagorean theorem is surely timeless, eternal, unchangeable’ (1967: 37). Sentences express thoughts. A thought is always the sense of some indicative sentence; it is ‘something for which the question of truth arises’ and so cannot be material, cannot belong to the ‘outer world’ of perceptible things which exists independently of truth. But neither do thoughts belong to the ‘inner world’, the world of sense impressions, creations of the imagination, sensations, feelings, and wishes. All these Frcgc calls ‘ideas’. Ideas are experienced, they need an experient, a particular person to have them whom Frege calls their ‘bearer’; as individual experiences, every idea has one and only one bearer. It follows that if thoughts are neither inner ideas nor outer things, then
’A third realm must be recognized. What belongs to this corresponds to ideas, in that it cannot be perceived by the senses, but with things, in that it needs no bearer to the contents of whose consciousness to belong. Thus the thought, for example, which we expressed in the Pythagorean theorem is timelessly true, true indcpendently of whether anyone takes it to be true. It needs no bearer. It is not true for the first time when it is discovered, but is like a planet which, already before anyone has seen it, has been in interaction with other planets.’ (Frege 1967: 29)
The crucial question, however, remains: what is our relation to this non-inner, non-outer realm of planetary thoughts, and how is it realized? Frege suggests that we talk in terms of seeing things in the outer world, having ideas in the inner world, and thinking or apprehending thoughts in this third world; and that in apprehending a thought we do not create it but come to stand ‘in a certain relation … to what already existed before’. Now Frege admits that while ‘apprehend’ is a metaphor, unavoidable in the circumstances, it is not to be given any subjectivist reading, any interpretation which would reduce mathematical thought to a psychologism of ideas:
’The apprehension of a thought presupposes someone apprehends it, who thinks it. He is the bearer of the thinking but not of the thought. Although the thought does not belong to the consciousness yet something in his consciousness must be aimed at that thought. But this should not be confused with the thought itself. ‘(Frege 1967: 35)
Frcge gives no idea, explanation, or even hint as to what this ‘something’ might be which allows the subjective, temporally located bearer to ‘aim’ at an objective, changeless thought. Certainly he sees that there is a difficulty in connecting the eternity of the third realm to the time-bound presence of bearers: he exclaims. ‘And yet: What value could there be for us in the eternally unchangeable which could neither undergo effects nor have effect on us?’ His concern however, is not an epistemological one about human knowing (how we can know thoughts), but a reverse worry about ‘value’ conceived in utilitarian terms (how can thoughts be useful to us who apprehend them). The means by which we manage to apprehend them are left in total mystery.
It does look as if Platonism, if it is going to insist on timeless truth, is incapable of giving a coherent account of knowing, and a fortiori of how mathematical practice comes to create mathematical knowledge. But we could set aside platonism’s purely philosophical difficulties about knowledge and its aspirations to eternal truths (though such is the principal attraction to its adherents), and think semiotically: we could ask whether what Frege wants to understand by thoughts might not be interpreted in terms of the amalgamations of thinking/scribbling we have called signs. Thus, can one not replace Frcge’s double exclusion (thoughts are neither inner subjectivities nor outer materialities) with a double inclusion (signs are both materially based and mentally structured signifieds), and in this way salvage a certain kind of semiotic sense from his picture of mathematics as a science of objective truths? Of course, such ‘truths’ would have to relate to the activities of a sign-interpreting agency; they would not be descriptions of some non-temporal extra-human realm of objects but laws – freedoms and limitations – of the mathematical subject. A version of such an anthropological science seems to have occurred to Frege as a way of recognizing the ‘subject’ without at the same time compromising his obsessive rejection of any form of psychologism:
‘Nothing would be a greater misunderstanding of mathematics than its subordination to psychology. Neither logic nor mathematics has the task of investigating minds and the contents of consciousness whose bearer is a single person. Perhaps their task could be represented rather as the investigation of the mind, of the mind not minds.’(Frege 1967: 35)
But in the absence of any willingness to understand that both minds and mind are but different aspects of a single process of semiosis, that both are inseparable from the social and cultural creation of meaning by sign-using subjects, Frege’s opposition of mind/minds degenerates into an unexamined Kantianism that explains little (less than intuitionism, for example) about how thoughts – that is, in this suggested reading of him, the signifieds of assertion signs – come to inhabit and be ‘apprehended’ as objective by individual subjective minds.
In fact, any attempt to rescue platonism from its incoherent attachment to ‘eternal’ objects can only succeed by destroying what is being rescued: the incoherence, as we said earlier, lies not in the supposed eternality of its referents but in the less explicit assumption, imposed by its realist conception of language, that they are pre-linguistic, pre-semiotic, pre-cultural. Only by being so could objects – existing, already out there, in advance of language that comes after them – possess ‘objective’ attributes untainted by ‘subjective’ human interference. Frege’s anti-psychologism and his obsession with eternal truth correspond to his complete acceptance of the two poles of the subjective/objective opposition – an opposition which is the sine qua of nineteenth-century linguistic realism. If this opposition and the idea of a ‘subject’ it promotes is an illusion, then so too is any recognizable form of platonism.
That the opposition is an illusion becomes apparent once one recognizes that mathematical signs play a creative rather than merely descriptive function in mathematical practice. Those things which are ‘described’ – thoughts, signifieds, notions – and the means by which they are described – scribbles – are mutually constitutive: each causes the presence of the other; so that mathematicians at the same time think their scribbles and scribble their thoughts. Within such a scheme the attribution of ‘truth’ to mathematical assertions becomes questionable and problematic, and with it the platonist idea that mathematical reasoning and conviction consists of giving assent to deductive strings of truth-preserving inferences.
On the contrary, as the model that we have constructed demonstrates, the structure of mathematical reasoning is more complicated and interesting than any realist interpretation of mathematical language and mathematical ‘subjectivity’ can articulate. Persuasion and the dialectic of thinking/scribbling which embodies it is a tripartite activity: the Person constructs a narrative, the leading principle of an argument, in the metaCode; this argument or proof takes the form of a thought experiment in the Code; in following the proof the Mathematician imagines his Agent to perform certain actions and observes the results; on the basis of these results, and in the light of the narrative, the Person is persuaded that the assertion being proved – which is a prediction about the Mathematician’s sign activities – is to be believed.
By reducing the function of mathematical signs to the naming of presemiotic objects, platonism leaves a hole at precisely the place where the thinking/scribbling dialectic occurs. Put simply, platonism occludes the Mathematician by flattening the trichotomy into a crude opposition: Frege’s bearer – subjective, changeablc immersed in language, mortal – is the Person, and the Agent – idealized, infinitary – is the source (though he could not say so) of objective eternal ‘thoughts’. And, as has become clear, it is precisely about the middle term, which provides the epistemological link between the two, that platonism is silent.
If platonic realism is an illusion, a myth clothed in the language of some supposed scientific ‘objectivity’, that issues from the metaphysical desire for eternal truth rather than from any non-theistic wish to characterize mathematical activity, it does nonetheless – as widespread belief in it indicates – answer to some practical need. To the ordinary mathematician, unconcerned about the nature of mathematical signs, the ultimate status of mathematical objects, or the semiotic basis of mathematical persuasion, it provides a simple working philosophy: it lets him get on with the scientific business of research by legitimizing the feeling that mathematical language describes entities and their properties that are ‘out there’, waiting independently of mathematicians, to be neither invented nor constructed nor somehow brought into being by human cognition, but rather discovered as planets and their orbits are discovered.
It is perfectly possible, however, to accommodate the force of this feeling without being drawn into any elaborate metaphysical apparatus of eternal referents and the like. All that is needed is the very general recognition – familiar since Hegel – that human products frequently appear to their producers as strange , unfamiliar, and surprising: that what is created need bear no obvious or transparent markers of its human (social, cultural, historical, psychological) agency, but on the contrary can, and for the most part does, present itself as independent of if not alien and prior to its creator.
Marx, who was interested in the case where the creative activity was economic and the product was a commodity, saw in this masking of agency a fundamental source of social alienation, whereby the commodity appeared as a magical object, a fetish, separated from and mysterious to its creator; and he understood that in order to be bought and sold commodities had to be fetishized, that it was a condition of their existence and exchangeability within capitalism. Capitalism and mathematics are intimately related: mathematics functions as the grammar of techno-scientific discourse which every form of capitalism has relied upon and initiated. So it would be feasible to read the widespread acceptance of mathematical platonism in terms of the effects of this intimacy, to relate the exchange of meaning within mathematical languages to the exchange of commodities, to see in the notion of a ‘time-less, eternal, unchangeable’ object the presence of a pure fetishized meaning, and so on: feasible, in other words, to see in the realist account of mathematics an ideological formation serving certain (techno-scientific) ends within twentieth-century capitalism.
But it is unnecessary to pursue this reading. Whether one sees realism as a mathematical adjunct of capitalism or as a theistic wish for eternity, the semiotic point is the same: what present-day mathematicians think they are doing – using mathematical language as a transparent medium for describing a world of pre-semiotic reality – is semiotically alienated from they are, according to the present account, doing: namely, creating that reality through the very apparatus which claims to ‘describe’ it.
What Is mathematics ‘about’?
To urge as I have done that mathematical thought and scribbling enter into each other, that mathematical language creates as well as talks about its worlds of objects, is to urge a thesis antagonistic not only to the present-day version of mathematical platonism, but to any interpretation of mathematical signs, however sanctioned and natural, that insists on the separateness of objects from their descriptions.
Nothing is nearer to mathematical nature than the integers, the progression of those things mathematicians allow to be called the ‘natural’ numbers. And no opposition is more sanctioned and acknowledged as obvious than that between these numbers and their names, the numerals which denote and label them. The accepted interpretation of this opposition runs as follows: first (and the priority is vital) there are numbers, abstract entities of some sort whose ultimate nature, mysterious though it might be, is irrelevant for the distinction in hand; then there are numerals – notations, names such as 1111111111, X, 10, and so on – which are attached to them. According to this interpretation, the idea that numerals might precede numbers, that the order of creation might be reversed or neutralized, would be dismissed as absurd: for did not the integers named by Roman X or Hindu 10 exist before the Romans took up arithmetic or Hindu mathematicians invented the place notation with zero? And does not the normal recognition that X, 10, ‘ten’, etc., name the same number require one to accept the priority of that number as the common referent of these names?
Insisting in this way on the prior status of the integers, and with it the posterior status of numerals, is by no means a peculiarity of Fregean realism. Hilbert’s formalism, for all its programmatic abolition of meaningful entities, had in practice to accept that the whole numbers are in some sense given at the outset – as indeed does constructivism, either in the sense of Brouwcr’s intuitionism, where they are a priori constructions in the intuition of time, or in the version urged by Kronecker according to which ‘God made the integers, all the rest being the work of man’. In the face of such a universal and overwhelming conviction that the integers – whether conceived as eternal platonic entities, pre-formal givens, prior intuitions, or divine creations – are before us, that they have always been there, that they are not social, cultural, historical artifacts but natural objects, it is necessary to he more specific about the semiotic answer to the fundamental question of what (in terms of sign activity) the whole numbers are or might be.
However possible it is for them to be individually instantiated, exemplified, ostensively indicated in particular, physically present. pluralities such as piles of stones, collections of marks, fingers, and so on, numbers do not arise, nor can they be characterized, as single entities in isolation from each other: they form an ordered sequence, a progression. And it seems impossible to imagine what it means for ‘things’ to be the elements of this progression except in terms of their production through the process of counting, And since counting rests on the repetition of an identical act, any semiotic explanation of the numbers has to start by invoking the familiar pattern of figures
1. 11, 111, 1111, 11111, 111111, etc.
created by iterating the operation of writing down some fixed but arbitrarily agreed upon symbol type. Such a pattern achieves mathematical meaning as soon as the type ‘1’ is interpreted as the signifier of a mathematical sign and the ‘etc.’ symbol as a command, an imperative addressed to the mathematician, which instructs him to enact the rule: copy previous inscription then add to it another. Numbers, then, appear as soon as there is a subject who counts. As Lorenzen – from an operationalist viewpoint having much in common with of the present project – puts it: ‘Anybody who has the capacity of producing such figures can at any time speak of numbers’ (1955). With the semiotic model that we have proposed, the subject to whom the imperative is addressed is the Mathematician, while the one who enacts the instruction, the one who is capable of this unlimited written repetition, is his Agent. Between them they create the possibility of a progression of numbers, which is exactly the ordered sequence of signs whose signifiers are ‘1’, ‘11’, and so on.
Seen in this way, numbers are things in potentia, theoretical availabilities of sign production, the elementary and irreducible signifying acts that the Mathematician, the one-who-counts, can imagine his Agent to perform via a sequence of iterated ideal marks whose paradigm is the pattern 1, 11, 111, etc. The meaning that numbers have – what in relation to this pattern they are capable of signifying within assertions – lies in the imperatives and thought experiments that mathematics can devise to prove assertions; that is, can devise to persuade the mathematician that the predictions being asserted about his future encounters with number signs are to be believed.
Thus, the numbers are objects that result – that is, are capable of resulting – from an amalgam of two activities, thinking (imagining actions) and scribbling (making ideal marks), which are inseparable: mathematicians think about marks they themselves have imagined into potential existence. In no sense can numbers be understood to precede the signifiers which bear them; nor can the signifiers occur in advance of the signs (the numbers) whose signifiers they are. Neither has meaning without the other: they are coterminous, co-creative, and co-significant.
What then, in such a scheme, is the status of numerals? Just this: since it seems possible to imagine pluralities or collections or sets or concatenates of marks only in the presence of notations which ‘describe’ these supposedly prior pluralities, it follows that every system of numerals gives rise to its own progression of numbers. But this seems absurd and counter-intuitive. For is it not so that the ‘numbers’ studied by Babylonian, Greek, Roman, and present-day mathematicians, though each of these mathematical cultures presented them through a radically different numeral system, are the same numbers? If they are not, then (so the objection would go) how can we even understand, let alone include within current mathematics, theorems about numbers produced by, say, Greek mathematics? The answer is that we do so through a backward appropriation: mathematics is historically cumulative not because both we and Greek mathematicians are thinking about the same timeless ‘number’ – which is essentially the numerals-name-numbers view – but because we refuse to mean anything by ‘number’ which does not square with what we take them to have meant by it. Thus, Euclid’s theorem ‘given any prime number one can exhibit a larger one’ is not the same as the modern theorem ‘there exist infinitely many prime numbers’ since, apart other considerations, the nature of Greek numerals makes it highly unlikely that Greek mathematicians thought in terms of an infinite progression of numbers. That the modern form subsumes the Greek version is the result not of the timelessness of mathematical objects, but of a historically imposed continuity – an imposition that is by no means explicitly acknowledged, on the contrary presenting itself as the obvious ‘fact’ that mathematics is timeless.
In relation to this issue one can make a more specific claim, one which I have elaborated elsewhere (Rotman 1987/93) that the entire modern conception of integers in mathematics, was made possible by the system of signifiers provided by the Hindu numerals based on zero; so that it was the introduction of the sign zero – unknown to either classical Greek or Roman mathematicians – into Renaissance mathematics that created the present-day infinity of numbers.
Insofar, then, as the subject matter of mathematics is based on the whole numbers, we can say that its objects – the things which it countenances as existing and which it is said to be ‘about’ – are possibles, the potential sign productions of a counting subject who operates in the presence of a notational system of signifiers. Such a thesis, though. is by no means restricted to the integers. Once it is accepted that the integers can be characterized in this way, essentially the same sort of analysis is available for numbers in general. The real numbers, for example, exist and are created as signs in the presence of the familiar extension of Hindu numerals – the infinite decimals – which act as their signifiers, Of course, there are complications involved in the idea of signifiers being infinitely long, but from a semiotic point of view the problem they present is no different from that presented by arhitrarily long finite signifiers. And moreover, what is true of numbers is in fact true of the entire totality of mathematical objects: they are all signs – thought/scribble amalgams – which arise as the potential activity of a mathematical subject.
Thus mathematics, characterized here as a discourse whose assertions are predictions about the future activities of its participants, is ‘about’ – insofar as this locution makes sense – itself. The entire discourse refers to, is ‘true’ about, makes sense in relation to, nothing other than its own signs. And since mathematics is entirely a human artifact, the truths it establishes – if such is what they are – are attributes of the mathematical subject: the tripartite agency of Person/Mathematician/Agent which reads and writes mathematical signs and suffers its persuasions.
But in the end, ‘truth’ seems to be no more than a psychologically explicable but semiotically unhelpful relic of the platonist obsession with a changeless eternal heaven. The question of whether a mathematical assertion, a prediction, can be said to be ‘true’ (or accurate or correct) collapses into a problem about the tense of the verb. A prediction – about some determinate world for which true and false make sense – might in the future be seen to be true, but only after what it foretold has come to pass; for only then, and not before, can what was be predicted be dicted. Short of’ fulfillment, as is the condition of all but trivial mathematical cases, predictions can only be believed to be true. Mathematicians believe because they are persuaded to believe; so that what is salient about mathematical assertions is not their supposed eternal truth about some world that precedes them, but the inconceivability of persuasively creating a world in which they are denied. Thus, instead of a picture of logic and reasoning as a form of truth-preserving inference, a semiotics of mathematics would see it as a species of rhetoric, an inconceivability-preserving mode of persuasion – with no mention of ‘truth’ anywhere.
Berry, Margaret 1975. Introduction to Systemic Linguistics I. London: T.J. Batsford.
Brouwer, L.E.J. 1952. “Historical background, principles and methods of intuitionism”, South African Journal of Science 49, 139-46.
Eco, Umberto 1976. A Theory of Semiotics. London: Macmillan.
Frege, Gottlob 1967. “The Thought: A logical enquiry” in Philosophical Logic, P.F. Strawson (editor), Oxford: Oxford U. Press.
Hilbert, D and Bernays, P. 1934. Grundlagen der Mathematik, vol 1. Berlin: Springer Verlag.
Lorenzen, P. 1955. Einfuhrung in die operative Logik und Mathematik. Berlin: Springer Verlag.
Rotman, Brian 1987/93. Signifying Nothing: The Semiotics of Zero. London: Macmillan/ Stanford: Stanford U. Press.
Weyl. H 1949. Philosophy of Mathematics and Natural Science. Princeton: Princeton U. Press.