Decolonizing Mathematical Knowledge
What happens after typing “sociology of mathematics” into the google search bar? The very first “suggested” result is mathematical sociology. A very curious outcome, considering the fact that mathematical sociology is in a way the exact opposite of a sociology of mathematics. Sociology of mathematics is a discipline that investigates the conditions that enable mathematicians to practice mathematics. Mathematical sociology is, on the other hand, one of many possible applications of mathematics. In the first case, the human lifeworld is ontologically primary; people engage in symbolic interaction in order to “do mathematics”, whatever that means. In the latter, mathematical entities are ontologically primary providing epistemic grounding for the investigation of human relationships. These frameworks are incompatible. This is precisely the sense in which they confront each other as opposites. The goal of this paper is not a comparative analysis between these paradigms, in fact, it will not address mathematical sociology in any rigorous manner at all. Instead, the aim will be to analyze mathematics as a social practice. Neither is it the role of mathematics in society that is at question here, nor its particular use or effectiveness, but in a way quite the contrary. The text aims to investigate the social conditions of possibility for mathematics and how mathematics is “played out” or perhaps even “performed” in and throughout the social body. Can we go as far as to say that mathematics is a form of social domination? Perhaps. In order to embark on this project, we will draw on four different social theorists and their work to see how each can be used as a tool to shed a new light on mathematics as an institution; and a coercive one at that! While at the same time, comparing each of them with the ultimate sociologist of mathematics and a notorious figure in the philosophy of mathematics: Ludwig Wittgenstein.
I will begin by consulting Anthony Giddens’s Magnum Opus: The Constitution of Society: Outline of the Theory of Structuration to show how the multiplicity of behaviors that render mathematics practicable can be seen as a kind of moving structure. Or a structure with movable parts. Unlike functionalism, which is based on biological accounts of human behavior or structuralism which analyzes society in terms of dehumanized and fixed abstract structures, the theory of structuration is neither one that ascribes priority to the subject, as for instance, the hermeneutical tradition, nor to the concrete or abstract whole, as is the case with the already mentioned functionalist and structuralist accounts respectively. “The basic domain of study of the social sciences, according to the theory of structuration, is neither the experience of the individual actor, nor the existence of any form of societal totality, but social practices ordered across space and time” (Giddens 1984, 2). This is precisely one of the frameworks through which we could interrogate mathematics. It is as if by some strange coincidence that at the very beginning of the book Giddens mentions recursivity, when he states that social action is recursive. Recursion is a very important term in mathematics. Perhaps we will be comparing mathematical sociology with the sociology of mathematics to a certain degree after all. Imagine how psychedelic this may turn out to be. Recursive functions produced by recursive actions of mathematicians in the physical world. A fractal doubling! But of course things are not that simple. Recursive entities are only one of many types of objects encountered in mathematics. More to the point: Giddens’s diagram[1] can be summarized in the following way: Human actions are incited by motives, motives can be reflected upon, monitored and justified. There is therefore a field of human behavior which is conscious, as opposed to the unconscious or unintended effects of the very same, which then reproduce the unconscious milieu of social action, which in turn serves as the enabling as well as the limiting condition for future action.
What are the intended effects of mathematics? This is not a question that could be covered in a single paper, or even a book. Even the reduced question: “What is the purpose of pure mathematics?” remains fairly complex. Let us say this much, mathematics has a self-directed teleology. As opposed to applied mathematics (i.e. mathematical sociology), pure mathematics does not seek to solve practical empirical problems. There is a powerful aesthetical motivation behind finding a theorem that is broad in its application, non-trivial and continuous with the rest of the mathematical corpus. This can be posed as a problem or even an elaborate puzzle, however it remains completely internal to mathematics itself. In this sense, mathematicians would justify their work by referring to the very immanence of symmetry as well as the satisfaction brought about by its realization as a recursive formation of self-referential patterns. “Order as such” one could say.
A fascinating parallel may be observed between the work of Giddens and Wittgenstein: Between the notions of monitoring and surveillance. Admittedly, their use of the terms is quite different, nonetheless we can still claim with certain confidence that the role that the surveyability of a mathematical proof plays in Wittgenstein’s account of what it means to do mathematics, is the same as the role played by the notion of monitoring one’s behavior in the theory of structuration. It is the process by which an action, a social practice, be that of the mathematician or any other social actor, is made to conform to a normative rule. This would be the essential feature of mathematics as a social practice. But what are the unintended consequences of mathematical practices?
This question introduces a new dimension of complexity. One would think, it is difficult enough to identify the structures produced through unconscious agency under “normal” circumstances, but to identify the very same objects within the institution of mathematics and to argue further that these very same hidden mechanisms may turn out to be oppressive, would be taking things to a whole new level. It is ambitious at best, but this is precisely why we should draw on Wittgenstein’s work. The Remarks on the Foundations of Mathematics (RFM from now on) is our primary mediator between the sociologists we will discuss here and the highly complex work of mathematicians.
Continuing our relay race across the sociology of mathematics with Wittgenstein as referee, Giddens now passes the baton to Luc Boltanski who dashes forward in order to interrogate mathematics as a potential field of social domination. Once again, it is almost surprising how well mathematics fits into Boltanski’s definition of domination. First, Boltanski emphasizes that domination operates as a totality. This is entirely true for mathematics. The entire discipline revolves around the notions of unity and completeness (despite Gödel’s work), but it also operates as totality in a subtler sense. Mathematics is so deeply intertwined within our lifeworld that we barely notice its normative injunctions and hidden commands. We are constituted to varying degrees as mathematicians through social conditioning. Even those who panic and convulse at the sight of an equation can only retaliate with a “I cannot do it!”, “it is beyond me” or an occasional “my brain is simply not wired this way”. But none of them are bold enough to say that mathematics is just one of many different ways of going about things, that there is something arbitrary about it or that it is a form of social domination. Even Wittgenstein dares not going so far, despite the fact that he has performed the most difficult work for us. In RFM Wittgenstein demonstrates that mathematics is not a descriptive science aiming to uncover the abstract structures tucked away in some Platonic realm of pure forms, but that it is nothing more than a very useful social practice; a convention that has proven incredibly helpful to us from time immemorial.
In this sense, despite Wittgenstein’s radical conventionalism about mathematics, he remains quite conservative about our common sense and our everyday human practices. It is precisely the latter that justify mathematics, not the other way around. Both Boltanski and Giddens help us take that additional, perhaps somewhat dangerous step towards questioning our common-sense practices and indirectly extending them into a social critique of mathematics. For lack of space and particular institutional constraints imposed on us at the moment, we cannot develop this particular line of reasoning to its required level of complexity. We must therefore pass the baton to Sartre and Franz Fanon.
And who is to say, that what is being written here is not an introduction; a preparatory experiment for what is soon to be a post-colonial reading of mathematics? Colonial administrators may not have had Hegel on their reading list[2], but they sure knew how to crunch numbers. Let us not forget where the word Statistics originated from. “Stat” meaning — State. Another word for statistics is Political Arithmetic. Today, statistics has indeed become the norm. But what have we been discussing so far, if not the fact that the norm (or the “new norm” as is the fashion to speak these days) is nothing but a sedimented, structural and the common-sense, taken-for-granted mode of violence?
One would assume that mathematics is the holy grail of system-building and (therefore) Imperialism. But this is only one of several readings one could provide. It is in fact the colonial reading; the one that needs to be exorcised. Without delving into the vast literature concerning various oriental exoticizations of i.e. Indian mathematicians like Ramanujan, who in the end, only served to re-enforce the numerous claims to mathematical universality that the western culture tends to take for granted, we can simply stick to Wittgenstein’s account to show that mathematics is nothing but a particularly useful set of discontinuous techniques of calculation. We are simply taught, drilled, trained and otherwise conditioned to calculate in a certain way. Would this be a post-modern “foundation” for mathematics? It could show some promise. It would seem then, that mathematics is not founded, but it is in fact de-centered and mobile. Mathematics is performed.
But the performance is oftentimes an effect, or an unintended consequence of how much power has been delegated to its authority as an institution. The dictatorship of numbers has become a regime of truth and a type of governmentality[3] by way of standardized testing and the contemporary undisputed faith in science that has indeed reached religious dimensions today. Let alone a corrective for common sense, mathematics seems to be a colonizer of common sense. Let us see then, whether Franz Fanon can offer us a way to decolonize mathematics. The first thing that needs to be recognized is that mathematics in its pure form (as opposed to the way it is applied for commercial or even scientific purposes) does not aim to secure wealth. We could at least say this much, that the privileged object of mathematical research is not profit, but Truth with a capital “T”. At the end of the day, most scientists, when hard-pressed with epistemic questions, will eventually rely on mathematics and common sense to account for a grounding or a firm starting point for any scientific method. Mathematicians therefore seem to be the de facto supporters of a very particular type of colonialism. One that is much more difficult to identify, expose, criticize and reform. A monopoly on truth.
Following Fanon’s reasoning: Decolonization will have disruptive effects. It must. In our case, this would imply precisely the kind of scandal that Wittgenstein’s work had brought about among mathematicians, logicians and the entire Vienna Circle. The difference being, as we already mentioned above, that we must go further. Not only do we bear the burden of proving Wittgenstein right (which is already an anathema among mathematicians), but in addition we must show that the convention of doing mathematics may not be nearly as useful as we have thought. It may in fact be the case that mathematics is illegitimately imposed on us as part of the standard curriculum. This is bound to be met by serious opposition. Fanon’s lesson would then be to have daring courage for this endeavor. To anticipate the steep, uphill battle. Passing on the baton…
Last but not least, let us take a look at Hannah Arendt’s famous essay On Violence and see if it can shed a light on what kind of violence we are dealing with precisely, as we unmask the colonial power and the unconscious episteme of mathematics as a Will to Knowledge[4]. Arendt performs an interesting dissection when she separates power from violence as two different supersets of social action which may nonetheless overlap in significant ways. But they can also stand outside of each other. The mathematical discipline is a clear example of power without violence[5], it is power made anonymous, mobile, productive and constitutive of the subject’s identity (whether they are “good” or “bad” at calculating is irrelevant). Last but not least it is the kind of modern occurrence of power as a rational force. A structure which excludes all those who “fail to understand”, those who “remain irrational”, as though it was merely a frequency that one needed to attune oneself to.
The goal of this paper was to articulate a decolonial possibility within the sociological deconstruction of mathematics as a powerful, yet entirely conventional form of a state-like institution, which plays an invaluable role in the operations of power within the contemporary milieu. We are governed as mathematical subjects, or at least subjects that cannot disavow the “Truth” professed by mathematicians. Pure mathematics is the central node that links up the contemporary episteme into a (pseudo)comprehensive whole. Whereas it is itself nothing but a diverse “motley of calculating techniques” (Wittgenstein 1956, 10), which has perhaps proven more useful in governing bodies, rather than “deciphering the secrets of the universe”.
Notes:
[1] I use the word in the Deleuzian sense.
[2] A point made by Sartre in the preface to The Wretched of the Earth
[3] Referring to the Foucaultian notion of governmentality as an elaborate, hidden and productive (as opposed to a limiting, prohibitive and repressive) mode of governance. Hinting at the fact that mathematics is in fact a state-like institution.
[4] Once again, employing Foucaultian terminology. “Epistemè” refers to the condition of possibility for a particular system of thought, a discipline or science. “Will to Knowledge” on the other hand refers to the hidden intertwinement between power and knowledge that often goes unexamined (the term is inspired by Nietzsche).
[5] We must be clear to define violence as direct domination or unmediated violence in this case. We are after all, arguing that mathematics is a subtler form of violence, but Arendt defines such pervasive forms of violence precisely as power.
Bibliography:
1. Anthony Giddens. The constitution of society: Outline of the theory of structuration. University of California Press, 1984.
2. Arendt, Hannah. On violence. Houghton Mifflin Harcourt, 1970.
3. Boltanski, Luc. On critique: A sociology of emancipation. Polity, 2011.
4. Fanon, Frantz. The wretched of the earth. Grove/Atlantic, Inc., 2007.
5. Wittgenstein, Ludwig. Remarks on the Foundations of Mathematics. Blackwell Oxford, 1956.
taken from here
Foto: Sylvia John