Cultures of Mathematics IV
22-25 March 2015
New Delhi, India

Accepted Contributed Papers.

Isabel Cafezeiro (Universidade Federal Fluminense, Brazil) and Ricardo Kubrusly (Universidade Federal do Rio de Janeiro, Brazil).
Paulo Freire and mathematics.

Paulo Freire was a Brazilian noticeably interested in understanding human issues and the overcoming of oppression. Initially, his educational practice consisted of talking and presenting concepts to popular groups. But even in the early 1960s, he realized that the abstract speech, full of stabilized concepts, did not cause in the oppressed/oppressor a different attitude with respect to situations of oppression. This failed to stimulate in each individual the perception with respect to his social role. Thus, he inverted his educational approach: instead of conducting speech and concepts to popular groups as he used to do at first, he began to look for ways to encourage individuals and groups in the expression of their own speech. Emerged a situated approach: issues of that particular time and of that particular place came to the foreground. The later work of Paulo Freire shows an abundance of narrated cases in which one realizes the effort to approximate closely the reader's perception and the experience lived by him. The formulation of concepts follows directly from the reported situations.

The national and worldwide recognition of Paulo Freire as a great thinker in the field of pedagogy multiplied references and citations to his work. In the way of modern scientific practices, knowledge acquires major importance to the extent which their links with worldly things gradually disappear. Thus, it is common today to perceive the work of Paulo Freire reported in a purified, neutral, and universal speech, which leaves no space for inspiring stories. Strengthened by the mechanisms of scientific legitimacy, but weakened in their nesting grounds in the world, the work of Paulo Freire becomes a jumble of jargon, what triggers the various criticisms and misunderstandings. However, this abstract body of knowledge may be subject of further transformations when rethought from other life situations (new problematizations). This is a process of traduction/trahison: the abstract knowledge is changed and renews as a result of its reshaping on the things of life.

This same process of detachment from the world can also be seen in mathematics. The distanced narrative, free from links to the things of the world, is frequent in mathematical texts produced in the modern era and legitimized by the scientific community. This creates a body of knowledge that, becoming stable, is detached from its connections with the original problems. It acquires an enunciation as autonomous formulations, ahistorical entities. The autonomous body of knowledge caters to a certain configuration of power, making math a subject for a few. At the same time, it sets the hegemonic mathematics as the referential culture for mathematics, making difficult to recognize as mathematics other constructions of life. When addressing other cultures, one expect find in them similar processes of counting, measuring, among other naturalized elements of hegemonic mathematics, forgetting that they are part of another historical process. Thus, the identification of mathematical cultures requires situated approaches, as the focus in local problems tends to keep out the referential mathematics. Besides this, it requires a broader understanding of mathematics, giving rise to a conceptualization of mathematics that can, as long as possible, be stated without resorting to the elements of homogeneous mathematical.

We consider mathematics as an ability to construct concepts that help to propose solutions to the requirements of life, together with the ability to conceive mechanisms to operate these concepts in a way that they can scale to the fluctuations of things of life. In the light of this, we focus the possibilities of comprehension of Paulo Freire's work in the field of mathematics, as well as the possibilities of comprehension of mathematics in the light of Paulo Freire's approach.

Slides (57 MB!)

Santanu Chacraverti (Society for Direct Initiative for Social and Health Action, India) and Mihir Chakraborty (Calcutta University, India).
Śubhańkarī: Exploring some features of a popular mathematical culture in pre-colonial Bengal

The term Śubhańkarī denotes a corpus of computational instruction enshrined in verses, most commonly used for mathematical instruction in the pāthśālās—e elementary schools in rural Bengal. The system hails back to the pre-colonial period (mainly the 17th-18th centuries) and the verses were composed by different and occasionally anonymous authors. The most famous of such authors was one Śubhańkar, from whom the tradition apparently got its name. The authors and their methods were so successful that the term Śubhańkarī, in common parlance, came to stand for mathematical skill in general.

The verses were often doggerel, rarely remarkable for beauty of composition. That did not stand in the way of their popularity. For, the language of the verses had a casual and homespun quality that carried a certain appeal. More importantly, the content was valuable.

The first category of content was patently practical. Bengal, like other places in the subcontinent, had a denary system of numerical notation using distinct symbols for numbers 0 to 9, which combined to create the remaining elements in the class of whole numbers. However, the weights, measures, and monetary units were proportioned on other numbers, mostly 4 and 16. Thus, computation could be a headache, similar to what one faces today with sexagesimal measure of time and angle, but much more severe as the proportions often varied as one went up or down the scale. Having quick computational formulas or algorithms was thus not merely a desirability but necessity. Many Śubhańkarī verses were just that, quick computational procedures set in doggerel rhyme. Their metrical form rendered them easy to remember and utilize, often in oral calculations. In fact, the computational culture encouraged oral computation, though one was expected to resort to "kali" (ink) in the case of lengthy computations.

The second category consisted of rhymed versions of arithmetic and simple algebraic problems. Occasionally, the Śubhańkarī manuscripts also contained problems involving indeterminate equations. A famous Śubhańkarī problem, oft-repeated in the manuscripts and given here in translation, is as follows:

A goat for a tākā, a cow for a siki
A buffalo for tākā five
Hundred tākā fetches hundred animals
Says the great Sadaślib

[Explanatory Note:
tākā = The pre-colonial predecessor of the present Indian rupee
siki = A quarter of a tākā]

Here, the question that one needs to answer is how many of each animal (goat, cow, and buffalo) must be there so that the total number of animals is 100 and the sum of animal prices is 100 tākā. [Note:The verses often had a riddle-like incompleteness or opacity and understanding the question was an integral aspect of Śubhańkarī expertise.]

The third category consisted of computational fun. Often, one played around with extraordinarily large numbers, indicating a typical Indian fascination with immense magnitudes. On other occasions, the numbers dealt with had interesting properties explored in the verses. For example, there is an understandable fascination with primes and using them to generate numbers with interesting features and symmetries. We shall discuss the specific examples in the main paper.

These verses constituted the basics of mathematical instruction in the pāthśālās—the elementary schools in rural Bengal. More interesting is that these verses (along with non-versed computational problems and riddles) had a role in the social and cultural life of the people. The words and terms used in the verses had strong cultural roots and associations with mythical tales. One hears of Śubhańkarī problems posed in āddās and wedding parties as part of the usual fare of exchanging riddles. [Note: The term āddā in Bengali usually denotes a gathering of people with similar backgrounds and interests, sharing gossip, ideas, jokes, and so on. One must remember that āddā denotes both the gathering and the act of sharing.]

In the paper that has been conceived but still not quite composed, we seek to explore some features of the computational and numerical culture in rural Bengal in the pre-colonial period. Further, we seek to show that computational tradition had features that were not only culturally distinctive but also mathematically interesting. Finally, we also seek to explore the possible reasons of how this tradition gradually died out.

Karen François (Vrije Universiteit Brussel, Belgium), Eric Vandendriessche ( Université Paris Diderot, France).
Reassembling mathematical practices. A philosophico-anthropological approach.

A major issue in the field of the philosophy of mathematical practices is to better understand how far mathematical practices are related to the cultural contexts within which they are developed. In this paper, we bring together two complementary directions of research to tackle this issue. First, we show how Wittgenstein's philosophy affords conceptual tools to discuss the possibility of simultaneous existence of culturally different mathematical practices. Wittgenstein abandons the essentialist concept of language and therefore denies the existence of a universal language. Languages—or 'language games' as Wittgenstein calls them—immerse in a form of life, in a cultural or social formation and are embedded in a totality of communal activities. This idea gave rise to the notion of understanding rationality as an invention or as a construct that emerges in specific local contexts. Relaying on the later work of Wittgenstein and his concept of 'family resemblance' (Familienähnlichkeit), we will give meaning to the existence of different kinds of mathematical knowledge and the coexistence of ethnomathematical practices.

Throughout the last decades, many activities practiced in non-western cultures (and in societies with an oral tradition in particular) have been analyzed by ethnomathematicians as related to mathematics. These activities still need to be further compared to one another, in an attempt to bring to light invariant and distinguishing features from one cultural context to another, and in order to better characterize mathematical practices (including Western ones) in sociological and epistemological terms. In that perspective, we draw on recent studies in ethnomathematics—integrating anthropological approaches and using ethnographical methods—which analyze data about 'geometrical' activities involved in the creation of artefacts (such as string figures, sand drawings, textile production...). These studies contribute to reaching a better understanding of the cognitive acts that underlie these activities, as well as the ways they are embedded into a cultural environment. We suggest that these works in ethnomathematics afford both new materials and fundamental outcomes worth analyzing in a comparative way to reflect on the forms of interrelations between mathematics and cultures.

Norma B. Goethe (National University of Cordoba & CONICET, Argentina), Gustavo Morales (National University of Cordoba & CONICET, Argentina).
Guiding ideas, cognition and working tools against the background of a variety of mathematical cultures.

In our paper we will rely on recent research relevant to the study of Leibniz's mathematical practice. In this context, our aim will be to discuss the epistemic value of iconicity in diagrammatic representations in geometry, as well as the importance of methodological guiding ideas in the design of working tools for problem-solving activities. In particular, we aim to show that iconic aspects of diagrams reveal structural relations underlying the method to solve quadrature problems developed by Leibniz (1675/1676), the fruitful outcome of the years of his mathematical studies in Paris where he became familiar with innovative mathematical research stemming from a variety of mathematical cultures. In the context of our case-study, we shall conclude with some remarks about the requirements the reader is facing in order to be able to establish a meaningful relationship between the information supplied by the diagram and the relevant background knowledge which often remains implicit.

Kenji Ito (Graduate University for Advanced Studies, Japan).
Mathematical Physics and Cultural Practices in Japan: The Question of Cultural Explanations.

Japan was relatively late to endorse the idea that mathematics is essential to understand the physical world. Modern physics began to be known in Japan around the beginning of the 19th century, but formal education and institutionalized research in physics started in the mid-19th century. Nevertheless, by the beginning of the 20th century, the idea to use mathematics in understanding nature was fully incorporated into physics in Japan. Moreover, mathematics was not just emphasized, but overemphasized in physics in Japan in the early decades of the 20th century. Most of early Japanese theoretical physicists indulged themselves in lengthy calculations and exotic mathematics. Instead of discussing fundamental principles of theoretical physics, they applied established physical principles on increasingly more complex phenomena. In short, Japanese theoretical physicists at that time often considered their discipline as mathematical physics. There was a culture among Japanese physicists in which calculation and use of advanced mathematics were highly regarded.

In this paper, I discuss methodological issues to explain how this happened. Japan has a longer tradition of mathematics than that of physics. One might be tempted to explain the dominance of mathematics in early theoretical physics in Japan in terms of this mathematical tradition. The problem of this kind of explanation is that it does not say much about how the presumed tradition was translated into actual scientific practices. Instead of considering the calculational culture of theoretical physics in Japan in terms of continuation or influence of traditional practices of Japanese mathematics, I claim it was a part of emerging cultural practices of physics, in which physics was closely linked with engineering. Since physics was practices and educated in conjunction with engineering, it was natural to apply, rather than discuss, established principles, to more complicated problems. At the same time, mathematical physicists had to distinguish themselves from engineering, hence they were motivated to move away from physical reality to artificial mathematical models.

Thus, it is not that local cultures influenced mathematical practices in theoretical physics. Rather, mathematical culture in theoretical physics evolved in Japan in the given social and cultural environment. The existence of traditional mathematics was only one element that constituted such environment.

Danielle Macbeth (Haverford College, U.S.A.).
Mathematical Meaning in Mathematics Pedagogy.

A good mathematical symbolism can relieve one of the burden of thinking insofar as one can manipulate the symbols according to rules without attending to the mathematical meaning of the symbols. And this is, or at least can be, enormously powerful in mathematical practice. But as any mathematics educator knows, it can also be a curse insofar as the manipulation of signs according to rules without understanding or insight into mathematical meaning can lead one inexorably into absurdities that will go unrecognized as long as the manipulations remain merely mechanical: why not say that (loga + log b)/log c = (a + b)/c? But what happens to mathematics pedagogy when there is no symbolism within which to work? How in that case are meanings conveyed and chains of reasoning communicated to the uninitiated? The pedagogic culture of college classes in, for example, analysis and abstract algebra hold the key to an answer, and promise to shed light on why college-level mathematics is so difficult, even for very bright students. This in turn can be made to yield crucial insights into the nature and roles of symbolism in mathematical practice—in the classroom, in the lecture hall, and in the study.

Nikhil Maddirala (Deloitte, Hyderabad).
Cultures of logic: an empirical investigation into the aims, goals and values of a scientific discipline.

Inspired by the philosophy of mathematical practice (PMP) movement, this paper seeks to advance an analogous domain of inquiry known as "philosophy of logical practice" (PLP) and to provide a concrete example of original research in this field by way of a case study in applied logic. Research in philosophy of mathematical practice is to a large extent about using techniques from the social sciences to shed light on the culture of mathematical research. One particular topic that has been sidelined by traditional philosophy of mathematics is the discussion of the aims, goals and values of a discipline and of the people working in that discipline; in particular, what are the criteria (if any) for the "success" or "failure" of a discipline? This paper seeks to discuss these questions in the analogous domain of PLP and also hopes to lay the foundations for similar research in PMP. The paper centers on a case study in formal semantics, which is a particular form of logical practice. In 2011, Martin Stokhof and Michiel van Lambalgen (two prominent formal semanticists) provocatively raised the question: "is formal semantics a failed discipline?" The question sparked an intense debate among leading researchers in the field in a special issue of the journal "Theoretical Linguistics." My case study discusses this question by drawing primarily on the methodological framework of qualitative research in the social sciences—in particular, the case study is structured as an interview study featuring interviews with critics, insiders and outsiders of formal semantics. Major themes that emerge from the case study and interviews are: (1) the underlying aims, goals and values of formal semantics, (2) the role of logical / mathematical modeling in formal semantics and (3) the criteria for success or failure of formal semantics as a discipline. Hopefully such research will encourage more logicians, philosophers and mathematicians to reflect critically upon aims, goals, values and standards for success or failure in their respective disciplines—especially concerning the way in which they make use of formal models to describe real world phenomenon.

Krishnamurthi Ramasubramanian (Indian Institute of Technology Bombay, India).
Mathematics in Metrical form: Its pros and cons

The art of blending mathematics with poetry seems to have its origin in India at least from the time of Aryabhata as evidenced from his seminal work Aryabhatiya (499 CE). This trend had been successfully taken forward by the later mathematicians like Mahaviracarya, Sridhara, Lalla and a host of others, and it reached its pinnacle by 12th century with the compositions of Bhaskaracarya (b.1114), whose 900th anniversary celebrations are on for paying tributes to his immortal compositions.

Be it principles of arithmetic, algebra, geometry, mensuration or combinatorics Indian mathematicians, over several centuries, seem to have developed the skills of couching them in the form of beautiful verses, with high poetic value. So much so, when the mathematicians of the Kerala School appeared in the arena in a big way around 14th century, they easily managed to succinctly present even the infinite series for Pi and other trigonometric functions in the form of nice metrical compositions. Presenting mathematics in the form of metrical compositions had its own advantages and limitations. During the talk we will try to discuss some of these aspects and also possibly touch upon the notion of proof as conceived by Indian mathematicians.

Slides

Smita Sirker (Jadavpur University, India).
Why Look Beyond The Given Information?: The Effect of Evidentiality.

Studies amongst illiterate subjects, in more ways than one show that their world knowledge or world-view can be different from the ways that we predict them to be. Shape recognition, colour perception and categorisations can mean completely different things other than abstract generalisation and categorisation of similar shapes and similar colour shades. The response of unschooled subjects to reasoning tasks shows that were reluctant to draw conclusions about unsure, unknown situations and were not ready to go by the information provided in the premises of the reasoning tasks. They often neglected the premises and also had difficulty in recalling the content of the premises. They did not understand that there was a need to retain the premise information or that the answer to the question could be drawn from the premises. They relied more on their own knowledge base. One would find it difficult to explain why the subjects show a strong resistance to solve syllogistic tasks, unless one looks into the response that the subjects give in other cognitive tasks like categorisation and perception. According to Greenfield (1966) exposure to formal educational experiences produces abstracting and generalizing ability. What is it that favours the development of generalizing and abstracting ability? Bruner (1966) considers the use of written form of language in school as an important factor that facilitates linguistic competence and symbolic functions in general. Greenfield holds that the structure of written language used in school facilitates context-independent thinking and thus the ability for generalisation and abstraction.

In this paper, I would like to analyse one of the key factors why reasoning patterns of illiterate people seems to be different from people who are exposed to formal education. The crucial question that I would address in this paper is - whether evidentiality (either grammatical or lexical) affect the interpretation of the information that is provided by the experimenter to the subjects?

Slides

Fenner Tanswell (University of St Andrews, Scotland).
Proof in Mathematics and the Open Texture of Mathematical Concepts.

One important lesson that can be drawn from Lakatos's "Proofs and Refutations" is that mathematical concepts often display the feature of having "open texture", a term from Waismann meaning that the concepts are not fully determined or settled for all potential applications. In the case of totally formalised proofs, however, open texture is not present. I will argue, in the Lakatosian spirit, that while open texture is only found in informal mathematics, it is vital for modern mathematical development, such as through posing new problems and forcing the extension of existing domains, meaning that any formalising, mechanising or reductionist approach to informal proofs is misguided. Crucially, invoking the notion of open texture will allow us to better understand how informal proofs can nonetheless be rigorous, despite the essential practical and cultural components they introduce, because the invoked concepts can still be exact and definite within certain boundaries. Finally, I will consider to what extent this means that the practice of mathematics involves a process of conceptual engineering.

Sunita Vatuk (City University of New York, U.S.A.).
Mathematical Thinking among Experts in Kolam.

The range of mathematical connections found in kolams—striking designs made every morning on threshholds by women in Tamil Nadu—make it a particularly rich arena in which to explore the nature of mathematical thinking outside of academia. Mathematicians and computer scientists familiar with kolam view it as a mathematical art form, both because the learning, appreciation, and creation of kolams can be aided by mathematical knowledge or techniques, and because kolams have inspired some mathematical research. This suggests that proficiency in kolams may be accompanied by an affinity for mathematics, even though most of the best practitioners have little or no formal mathematical training.

The voices of the makers have not been prominent in the literature about kolams and math, prompting this study. It is based on open-ended, structured interviews with approximately 80 women considered by their community to be experts in kolam about how they learn, organize, create and recall these designs. The researcher, a mathematician, learned hundreds of kolams and created many others, taking note of mathematical strategies. Analyses proceeded in parallel—the bulk focusing on the thinking of the kolam experts, with additional analysis focusing on that of the university-trained mathematician.

While kolam cannot be considered a branch of academic mathematics, the expert kolam-maker and the mathematician do share some patterns of thought. For example, classification is an important mathematical activity, and both experts and mathematicians viewed structure as more important than surface features. In addition, both viewed the process of variation—such as expansion—as separate from the particular kolam the variation was based on. Ideas about which kolams were "interesting" provided another point of comparison: both kolam-maker and mathematician took pleasure in solving difficult kolam "problems" that had no utility beyond the satisfaction gained from making something beautiful. Other mathematical issues—such as the unarticulated but strict rules governing what makes a kolam "correct" are also discussed.