Coming from a non-mathematician, this may seem audacious, even arrogant, so let me first briefly address the position from which I am proposing this.
It is now close to five years since I submitted my PhD thesis on sociology of mathematical practices and have since continued to think about mathematical knowledge as a sociological-anthropological object of inquiry. Even though I have drifted onto other topics of research, the production and producers of mathematics have never ceased to intrigue me.
If we overlook the school years, my ‘official’ tryst with mathematics began in the late 2000s, when I finished my undergraduate degree (BSc Honours) in mathematics from the University of Delhi. It then continued in a different form when, after advanced training in sociology and some years of teaching, I decided to return to mathematics, but this time to explore it as an area of research from a sociological perspective.
I believe this gives me some credibility to be asking the above question.
However, I must point out that I am not raising this question as a sociologist, but as a former student of mathematics who has been exposed to different traditions of philosophising and historicising mathematical practice. Furthermore, my response to the above question is primarily addressed to educators and learners of mathematics, rather than solely to those in the humanities and social sciences.
As somebody who has experienced ‘doing’ mathematics, it was exhilarating to encounter, after moving to Sociology, thinkers as diverse as Gottlob Frege, Ludwig Wittgenstein, Ernst Cassirer, Claude Imbert, Andrea Nye, Herbert Mehrtens and others who were either trying to ‘think’ or ‘think with’ logic and mathematics.
While I was not learning mathematics anymore, I could better understand the sense of mathematical writing and the force behind its conventions. That even such basic symbols as ∃, ∀, ¬, and expressions such as ∀x ∃y: y > x, and the order of symbols thereof, involve commitments to universality, generality, or alternatively to relativity, uniqueness, etc., allowed me to view mathematical formulae as laced with subtleties and not just as cut-and-dried representational schemas.
Even though a lot of what I read and taught was only elliptically concerned with the actual practice of mathematics, it set me on the path to wonder how a practice carries an imprint of its history and how we can learn to recognise it. The philosophical and historical discourses on mathematics helped me understand the nature of mathematical formalism better – the signs we make on paper, and the distinctions we create through them, obey an inscriptional logic, similar to practices such as calligraphy and woodcarving, and hence alerting one to the craft-like character of mathematics.
As a sociologist, once I decided to explore mathematical practices as a research area, writings explicitly concerned with mathematics and debating the nature of mathematical truth (Kurt Gödel), intuition (Luitzen Egbertus Brouwer), formalism (David Hilbert), abstraction (Emmy Noether) and experiment (Imre Lakatos) opened up new burrows and alleys in the landscape of mathematics.
It must be pointed out that it is mistaken to view the philosophy of mathematics as a meta-commentary on mathematics. Its relationship to the latter should not be construed as that of theory to practice, but as a discourse which is adjacent to it. In my view, philosophy of mathematics helps less in clarifying mathematical issues and more in creating a subjective disposition of ‘thinking’ and ‘doing’ mathematics critically and reflexively. Readers should also note that what I call the philosophy of mathematics must be broadly and creatively understood and interpreted. It is not a narrow, specialised endeavour largely divorced from contemporary and everyday mathematics, but one which not only dares to think through and with mathematics, but also attempts to capture the thought(s) and form(s) of thinking unique to mathematics. The most illustrious examples of this style of philosophising are the philosophies practised by French thinkers, Gilles Deleuze and Alain Badiou.
Thus, drawing on my personal experience, I wish to suggest that engaging with philosophical and historical discourses related to mathematics may enable young students and even mathematicians to, firstly, develop a critical and reflexive relationship with their practice. This is not achieved by providing a ready-made framework of thought but by inducing reflections suggested by the practice itself, which may be obscured by mechanical and repetitive forms of learning and dominant ways of sense-making in mathematics.
Secondly, practitioners may profitably organise their thoughts on the relationship between mathematics and intuition, to see how the ‘intuitive’ and the ‘social’ are braided together, rather than the former being self-evident and in some privileged way connected with mathematics.
Thirdly, familiarity with philosophical discourse on mathematics is bound to induce some sensitivity in learners on how a problem in mathematics functions; which means to understand how truly interesting problems survive their solutions and are intimately tied to the evolution of forms. For example, understanding the Zermelo-Fraenkel axioms in the light of paradoxes in set theory completely changes how these axioms are received and understood. Taken thus, they attest to the materialism of practice and one begins to understand why they are not to be seen merely as escapist conventions designed to bypass paradoxes.
Fourthly, by understanding metaphysical choices which affect the making of mathematics, such as in Abraham Robinson’s non-standard analysis, students’ reception of mathematical discourse will undoubtedly become more informed as they’ll be aware that they are not reading neutral and ethically indifferent material, but a discourse with stakes. This will also have an impact on classroom exchanges.
Fifthly, reading histories of mathematics such as the stunning Mathematics and Society by Senthil Babu D can make young students and practitioners confront what is designated by the common name mathematics? With practices of mathematics embedded in regional cultures and forms of caste-labour, as Babu’s work painstakingly and admirably demonstrates, students may be led to ask questions about how ‘modern-university’ mathematics too is shaped by different institutional cultures and thus designates a radical heterogeneity that bringing it under one umbrella can only be a form of work; whether of the minutiae of hegemonic power or global politics of knowledge. Further, histories of mathematics can also show us how context can be read from mathematical forms, not merely through representation and narration, but in the form of gestures carrying an indelible imprint of reference without being subordinated to it. I can go on and on with this list, but let me stop and address a legitimate question I anticipate at this point.
It might reasonably be said that all this may really amount to a form of indulgence on the part of educators as well as learners of mathematics. The latter should rather be studying proper mathematics instead of wasting time with philosophy and history. Studying the latter in connection with mathematics could be a legitimate concern for people from the domain of humanities and social sciences, but for budding mathematicians it can only be a distraction in which they are best advised not to drift in.
To such objections I can only respond by saying that they stem from a purely instrumentalist understanding of mathematics where the subject is seen as a highly complex and multi-dimensional set of skills to be picked up with great labour, overcoming the many gate-keeping hurdles. Under such an achievement-centric vision of the subject, what gets left behind is the possibility of taking mathematics as a branch of critical thought.
So, for example, working through complex and tedious truth tables, if one is not amazed and alerted by how logic reveals the limits of and thus problematises intuition, results in incomplete and superficial learning. Entering a knowledge-domain like mathematics must be accompanied with an understanding of how it is constituted and differentiates itself from other modes of inquiry. The analogy which comes to mind is that of how film-appreciation and exposure to world-cinema changes how filmmakers view the making of films, encouraging them to constantly push the bounds of the sensible. Just like art goes hand-in-hand with art-criticism, math-criticism too will only enrich the ways of doing mathematics, generating critical and creative reflections on how it shapes its objects and handles them. Let me now conclude with a caveat and note of caution.
It might be construed by many that what I am proposing falls squarely within the changes wrought by National Education Policy’s (NEP’s) push towards interdisciplinary approach and is, in a way, already underway. In fact, what I am suggesting aims at restoring the integrity of the discipline by integrating it with what it should not have been separated from in the first place.
As several commentators have noted, NEP has led to a further instrumentalisation of mathematical knowledge and a severe pedagogisation which makes it akin to a survival skill that children must possess to become citizen-subjects, fit for a neoliberal world-order. The interdisciplinarity which is often touted as the way ahead leads to a watering down of content and amounts to little more than a mixing or juxtaposition of different disciplines, along with the reduction of mathematics to a set of quirky-smart techniques (instead of seeing it a form of thought), as mentioned above. The perspective I offer suggests that the appreciation and love of mathematics should also foster the understanding that it is no more or less necessary than, say, painting, and why we must be in a position to think and evaluate this necessity in order to make the learning of mathematics more inclusive and representational.
That new frameworks such as the NEP, establish the need and necessity of mathematics, also in a bid to Indianise it, by making it akin to music, doesn’t help much, except for further exoticising these two creative endeavours. The perspective I am emphasising upon should help us to explore the resonances between mathematics and other creative practices without in the process naturalising these connections.
How this can happen in an education system where semesters whizz past as daydream frenzies, is something people who love and live through mathematics, including first and foremost the math-educators of this country, have to think about and work their way around. I am sure what I am suggesting is not radically new, but the pragmatics of it surely has to be worked out by critically-situated and capable stakeholders with more deliberation and open communication on mathematics education and pedagogy.
Saumya Malviya is Assistant Professor (Sociology), School of Humanities and Social Sciences, Indian Institute of Technology Mandi, Himachal Pradesh, India.
