Language Use in Mathematical Practice: An Ethnographic Perspective

Abstract

Mathematics is often seen as an epitome of cold objectivity and astounding infallibility. Particularly for the outsiders, it comes across as an extremely rigid and closed system which seems impenetrable owing to its very specific and technical language. This article problematises these assumptions and seeks to study mathematics as a social practice with insights drawn from an anthropology of language and concepts, Wittgenstein’s philosophy of mathematics and semiotics. Using the anthropological insight that a language is always embedded in a form of life, this article shows how mathematical practice generates its own conventions and forms of language use. In particular, two dimensions of language use in mathematics are delineated and their consequences for further research are drawn out. In the first part of the article, the role of concepts in the discourse of mathematics is explored and in the second it is shown how applying a rigid distinction between syntax and semantics to mathematical language obstructs our understanding of its fluid and dynamic character. The argument unfolds through an analyses of interviews, texts and classroom sessions and shows how mathematical practice is heavily context bound and mathematicians often display an ethnographic attentiveness towards their work. The general tenor of the description is such that it attempts to trace the ethical dimension latent in mathematical practice and suggests a possibility of exploring it as a form of life. Connected to this thought is the argument that like any other practice, mathematical practice generates its own forms of reflections which cannot simply be assimilated to philosophical/theoretical knowledge. The question whether this action knowledge regarding mathematics has some relation to the South Asian location where the ethnography unfolds is also tentatively explored.

Keywords

Incompleteness Gödel inadequacy context syntax semantics ethical

Introduction

Given the remarkable success of science and technology studies in demystifying scientific processes and laboratory practices, it is surprising that similar efforts as far as mathematics is concerned have been few and far between.¹

See Latour (1987) and Latour and Woolgar (1979) as representatives of the social studies of science paradigm. Also see Bloor (1973) and Livingston (1986) for early sociological investigations of mathematics. Where there are numerous remarkable studies of science that one could think of as following the lead of aforementioned studies, somehow mathematics hasn’t received a similar amount of attention from researchers in anthropology and sociology.

Although some interesting studies from a mathematics education perspective have come out but there has been a huge dearth of applying concepts and techniques drawn from the anthropological and ethnographic tradition to understand the nature of mathematical practice.²

For an out-and-out semiotic perspective, see Raymond Duval’s remarkable text, Under-standing the Mathematical Way of Thinking—The Registers of Semiotic Representations (2017). For an enlightening study of formal logic from a sociological perspective, in tune with the science studies paradigm, see Claude Rosental’s important work (2008). This article departs from these studies in bringing to bear upon the ethnography presented here insights drawn from the mainstream anthropological tradition. Also, it seeks to underscore a slightly different point from these studies in arguing not that there are social influences on the discourse of mathematics and that things in mathematics are often decided by factors which have nothing to do with mathematics itself but that mathematics is internally and pre-eminently social.

Neither have there been serious reflections on why mathematical practice must hold interest for sociologists and anthropologists and not just for mathematicians and mathematics educators. This article is offered as a contribution to address this gap and to demonstrate that an anthropological approach has much to add to our existing knowledge of mathematics as a form of social practice. Also it is shown by adding anthropological flesh to the idea that mathematics can be seen as a form of life, that it is not merely a closed sign system and though semiotic and literary approaches can go a long way in demythicising mathematics, they should be extended further to make sense of it as a distinctive form of social practice. With that end in sight let us first briefly review the approaches and directions taken in this article.

In this article, I have attempted to highlight two different dimensions of mathematical practice which might hold interest for anthropologists looking for ethnographic entry points into the same. The first is regarding how particular and specific mathematicians are in their usage of concepts. Given that some of the English words designating mathematical concepts can be used in general as well as philosophical contexts, working mathematicians continuously seek to delimit their precise mathematical sense to clearly specify the way in which they are used in the discourse of mathematics as against their other usages. In this article, the focus is on two words ‘incompleteness’ and ‘inadequacy’ because of the distinctive accent they acquired in the discussions with a well-known mathematician Professor S. M. Srivastava based at Indian Statistical Institute (ISI) Kolkata, conducted as part of my ongoing ethnographic study of mathematics.³

Professor Shashi Mohan Srivastava works on mathematical logic and is widely regarded as the foremost scholar working on the subject in India. He is part of the Stat-Math Unit at ISI Kolkata. This article draws heavily from the fieldwork conducted at ISI Kolkata in April–May 2016.

These discussions revolved around Kurt Gödel’s famous ‘incompleteness’ results, first published in 1931, and Srivastava’s attempt to present these results in an undiluted mathematical form in a book he has published on mathematical logic.⁴

The book by Srivastava is titled A Course on Mathematical Logic (2008). It gives a precise mathematical introduction to Gödel’s results and also serves as a concise introduction to mathematical logic for the university students.

Since Gödel’s ideas about ‘incompleteness’ easily lend themselves to philosophical and even cultural interpretations, in an interesting bit of conversation, Srivastava specified what this word precisely means in mathematics and even suggested a better word for the concept of ‘incompleteness’, lest one dilutes its mathematical sense by mixing it with general and philosophical meanings of the word. This better word pointed out by Srivastava was ‘inadequacy’. In the first part of the article, an attempt has been made to understand this crucial insight into this aspect of mathematical practice that mathematicians are so particular about the words they use, using two key approaches drawn respectively from anthropology and philosophy.

In a recent paper of his anthropologist Bhrigupati Singh has illuminatingly explored the theme of how concepts can influence the way in which we look at the world and life (Singh 2014: 159–87). He does it in the context of his efforts to look for concepts that could best respond to the pressures of his ethnography. In the process he eschews concepts that might impose a form of closure on ethnographic description for the concepts that are alive to the open endedness of life as it is lived and observed. In the first part of the article, I have attempted to understand the aforementioned preoccupation with English and concepts by drawing out the argument that mathematicians bring an ethnographer’s attentiveness to the fore when they speak about their practice. They specify their concepts in such a way which is true to the ongoing, dynamic and contextual character of their enterprise instead of straitjacketing it in a preconceived finality and context-free determinations. Further deepening this line of argument the ethnographic insight on the role of English in mathematical discourse has been put into conversation with Wittgenstein’s point that mathematical propositions are like rules of grammar or the sole function of mathematical propositions is to fix the sense of concepts they invoke (Mühlhӧlzer 2001; Schroeder 2014). Tying these two approaches together the first part of the article attempts to develop a comprehensive perspective on the usage of English/concepts in mathematics.

In the second part of the article, I shift my attention to another aspect of mathematical practice to underscore its dynamic nature. It is shown how there exists a creative interplay between syntactics and semantics in a mathematical text as also in the learning and transmission of mathematics and how this plays a very crucial role in mathematical semiosis.⁵

I refer to semantics and syntactics in the way in which they are generally referred to. Syntactics is about the formal relations between signs within a structure, while semantics is about the meaning/interpretation attributed to these signs and their relations. One could say that I am interested in a third aspect which we may refer to as pragmatics—the study of the interrelationships between signs and people who use them. This involves a study of ways in which context contributes both to syntax and to semantics of a system.

This theme is explored on two levels. One is more restrictedly a semiotic one. To unwind the semiotic knots of mathematical practice Derrida’s attempts at unravelling structural ambiguity in a system of writing are put to use. Of particular interest for this paper is his argument that such ambiguity is not accidental but results due to the essential crossing between syntax and semantics with in any such system (Derrida 1982; Livingston 2012). More specifically Derrida calls this procedure diagonalisation through which an element can be constructed within a system of writing through syntactical manipulations yet which remains undecidable or unprovable from the constitutive rules governing the system.⁶

The idea of diagonalisation used by Derrida has a mathematical origin. It refers to a style of argument and proof developed by the German mathematician Georg Cantor to show that real numbers are uncountably infinite. Students of anthropology will also be reminded that Derrida’s use of diagonalisation is implicit in his reading of Lévi Strauss’s treatment of the prohibition of incest. Derrida notes that although prohibition of incest inaugurates or governs the structure of kinship analysed by Lévi Strauss, it remains an element which itself ‘escapes structurality’ (Derrida 2006: 352).

To draw out this semiotic concern I have looked at some relevant technical details of the proof of the first incompleteness result to show how the proof creatively employs the tension between syntax and semantics to establish what it sets out to achieve. Further, in the article, this semiotic insight is extended to other fieldwork contexts, specifically to descriptions of certain scenes of learning mathematics. Broadly the argument demonstrates that the ever-dynamic and extending practitional matrix, that mathematics is, deems whether formal-syntactic or meaning-semantic considerations will have primacy in particular contexts and situations. One would readily recognise that this takes the semiotic argument that there is ambiguity inherent in the sign system of mathematics further, showing how mathematical practice generates and sustains this ambiguity.⁷

For an engaging discussion of how ambiguity is inherent in the tissues of mathematical practice, see Floyd (1991).

It is hoped that the arguments raised in the article also contribute towards recognising the ethical dimension latent in mathematical practice.⁸

Notion of the ethical invoked here is in sync with Veena Das’s notion of this concept in her various contributions. In particular, see her paper in the volume Four Lectures on Ethics: Anthropological Perspectives (Das et al. 2015). She sees the ethical not as a specific kind of object but as a dimension immanent in the clamour of everyday life.

It emerges that mathematical research is guided not by questions about some pre-semiotic mathematical reality but by efforts to develop new modes of descriptions and forging/inventing new connections between concepts and procedures within a relatively self-contained discourse.⁹

As Wittgenstein has also remarked, ‘The mathematician is an inventor, not a discover’ (Wittgenstein 1956: 47e).

In particular the ethical dimension emerges most clearly in the case of transmission and pedagogy of mathematics where the focus on right language use clearly overshadows the question of reality or non-reality of mathematical facts or objects. Thus it could be said that one could follow the trail of the ethical in mathematics by paying close attention to the way in which linguistic concerns problematise the very notion of mathematical reality. In fact mathematicians can be self-avowedly Platonists or otherwise but this doesn’t influence the way they go about doing mathematics.¹⁰

Well-known field medallist Timothy Gowers puts it in the following manner, …the point remains that if A is a mathematician who believes that mathematical objects exist in a Platonic sense, his outward behaviour will be no different from that of his colleague B who believes that they are fictitious entities, and hers in turn will be just like that of C who believes the very question of whether they exist is meaningless. (Gowers 2006: 198)

It is demonstrated in the article how the use of language in and language of mathematics problematises the very notion of reference alerting us to the possibility of exploring mathematics as a form of life in which contextualisation and the mutual absorption of syntax into semantics and vice versa play a very important role at every step. Moving on, let us first briefly review the idea of incompleteness in mathematics, for it is used in both the parts of this article as the technical background against which the ethnography presented takes shape.

Incompleteness

Before we try to understand the idea behind Gödel’s ‘incompleteness’, few general and historical comments are in order. The two incompleteness results created a lot of furore among mathematicians, especially among the ones interested in the foundational questions, when they were first published by Gödel in 1931.¹¹

The text by Ernest Nagel and James R. Newman Gödel’s Proof (1968) remains by far the best and most accessible introduction to the incompleteness results. I have used this text throughout this article to provide a context for Gödel’s work and also to summarise his arguments.

Although in this article I’ll just be dealing with the first incompleteness theorem, it is important to note that taken together these results form a significant chapter in the history of mathematics and logic known primarily for having struck a death blow to David Hilbert’s ambitious axiomatisation programme. Without going into the details of it, Hilbert’s programme was set out to address the crises in the foundations of mathematics. Hilbert rightly perceived that mathematics of his time was phrased in a language that could engender contradictions and inconsistencies and hence his proposal was to base all existing areas of mathematics on secure axiomatic foundations. This entailed a belief that all the branches of the discipline could be based on a finite set of axioms, consistency of which can be proved by standard proof-theoretic methods elaborated by him. The consistency of a system (say number theory) means that it is entirely free of contradictions. Putting it differently this means that given a system both a statement (say p) in it and its negation (-p) cannot be derived from the axioms on which the system is set. Gödel entered the picture at this point and established once for all that aims of Hilbert’s programme, at least as it was originally conceived, were unachievable. Gödel’s method was to construct a statement within a formal system that is both true and unprovable effectively establishing that formalising all of mathematics is an impossible goal because certain statements can always be found which remain undecidable from the given axioms (roughly the first incompleteness result). Moreover Gödel also established that given a set of axioms which is consistent, it is impossible for this set to establish its own consistency (roughly the second incompleteness result). It does not mean that its consistency cannot be proved but that in order to do so one will have to use certain criteria and elements of reasoning that do not, properly speaking, belong to the system whose consistency has to be proved.

Let us now try to understand the gist of the first incompleteness result by Gödel. Well-known mathematician John W. Dawson Jr. has pointed out that taken together the two incompleteness results establish ‘the syntactic incompleteness of formal number theory’ (Dawson Jr. 2008: 819). This I believe is an important characterisation of Gödel’s incompleteness results because it gives a precise cue regarding how one should understand the word ‘incompleteness’ in this context. Let me clarify this issue first. One of the ways in which Hilbert’s aforementioned programme could be understood is by saying that he attempted to make or at least articulated a vision for making all of mathematics syntactically transparent. This meant precisely specifying for each area of mathematics the set of symbols to be used for variables, constants, logical operators and so on and also fixing precise rules of combination and transformation so that syntactically correct statements could be made by manipulating these symbols. Through this vision Hilbert attempted to demonstrate the superfluity of the question of meaning in mathematics thinking of it rather as a game of manipulating meaningless marks on paper according to certain fixed criteria.¹²

In the philosophy of mathematics this position is known as formalism and Hilbert is its most well-known proponent.

What Gödel demonstrated through his incompleteness results was the inadequacy of this way of thinking about mathematics. He showed that given a definite logical system there are properly articulated expressions or statements whose truth or falsity cannot be arrived at purely through syntactical manipulations and that within the bare-bone and machinic framework that Hilbert envisioned for each area of mathematics, there can be located undecidable statements which can neither be proved nor disproved. Now let us see how my discussion with S. M. Srivastava evolved on this idea of ‘incompleteness’ into an interesting shred of insight into the weave of mathematical practice.

Inadequacy

Interestingly, we came to the word ‘inadequacy’ in our conversation when I a posed a rather bad question, quite certainly so from an ethnographic perspective, to SM Srivastava. I read out to him a small excerpt from a philosophical paper by Gödel which is commonly cited as a signature expression of his Platonism.¹³

The excerpt in question mentions the Cantor’s hypothesis which is that there is no set with cardinality or size between that of real numbers and integers. In 1940, Gödel showed that it cannot be disproved from the standard axioms of set theory and subsequently in 1963 Paul Cohen showed that it can’t be proved either from these axioms, thereby establishing its independence from the Zermelo Fraenkel axioms in set theory (ZFC). Now Gödel, being a Platonist, believed that truth and provability are two different things and the independence of continuum hypothesis from the ZFC axioms notwithstanding, its truth or falsity can be decided by a different set of axioms. See, Gödel (1995: 260).

After reading out this passage to him I asked Srivastava that on the one hand is the practice of mathematics which is all about syntactic correctness, semantic preciseness, rule following, proper usage and so on, that is, a complex web of activities/procedures, and on the other hand are such philosophical beliefs as in the actual existence of mathematical objects. So, what perspective does he have on this seeming gulf between the practice and philosophy of mathematics? How does one reconcile or even take a position on these two seemingly opposing tendencies? I can see now that it was a badly framed question—first, because I assumed that these really are two opposing tendencies and second because the question was too direct and hence went against the grain of general ethnographic practice. But quite surprisingly it yielded a very interesting response from Srivastava, not so because it directly addressed the question but because it alerted me of the assumptions hidden in my question and in a very subtle way bypassed the question itself. Interestingly Srivastava interpreted Gödel’s Platonism as a given premise which reminds the working mathematician of the context of her work and also inspires her to look for ways of expanding and developing her mode(s) of description.¹⁴

One can see how the meaning of Platonism is given a particular twist by Professor Srivastava by interpreting it as context sensitivity. This inflected sense boils down to the fact that one can only speak from within a context and for matters beyond the context in which one is working one can at best speak speculatively, without making specific claims on mathematical matters.

So, for example, when I commented that according to the quoted passage it seems that Gödel thinks that set theory describes a well-determined reality, he intervened by saying that,

but it all hinges on what do you mean by set theory. If you mean those, bunch of axioms, what you call Zermelo Fraenkel axioms then No! … but set theory … some kind of set theory is there in the mind of Cantor … which these axioms are still not sufficient to describe.¹⁵

ZFC axioms called so after the mathematicians Ernst Zermelo and Abraham Fraenkel are one of the many axiomatic systems devised to base set theory on secure and paradox-free foundations. For what these axioms are, see Srivastava (2008: 12) and for a rigorous and exhaustive history of mathematical logic and set theory see Grattan-Guinness (2000).

So immediately with this response I was reminded that, expression of or belief in a Platonistic viewpoint notwithstanding, what matters more is the domain of practice and a precise specification of the context in which one is working. What was interesting for me was that Srivastava seemed undeterred by the seeming gulf that I presented to him in an embarrassingly dramatic manner. It now occurs to me that perhaps it is we who take the notion of belief that seriously and attribute to it a representational character vis-à-vis action whereas what matters more for practitioners is the concreteness of their practices and the indubitable character of their discourse. I invite the readers to recognise this significantly watered down and a loose, working sense of Platonism which is very important for the tenor of my argument. Now, this is how Srivastava responded further:

for example look at number theory. Now even for number theory Gödel’s incompleteness theorem says that you cannot find a complete set of axioms which describes the entire world of numbers. Nonetheless we find it as concrete as anything else … so there is this understanding that there is a precise world of numbers where everything is either true or false … so Gödel’s incompleteness theorem is also that through first order logic you cannot capture everything … so actually in my class I do tell the students that, look this is better called the insufficiency or inadequacy of the first order logic in describing say the world of numbers … so I would say that more than incompleteness it is inadequacy….¹⁶

First-order logic also known as predicate logic is a formal system used in mathematics and other disciplines, distinguished by the use of quantifiers over non-logical objects.

Notice how with this response the seemingly demanding opposition between philosophy and practice is downplayed and concreteness of numbers and the ‘inadequacy’ of first-order logic are affirmed in the same breath.¹⁷

Note how the Platonist belief in the existence of mathematical objects is interpreted as ‘concreteness’ and not in terms of the notion of ‘transcendence’.

Srivastava was clear that ‘incompleteness’ might evoke several meanings whereas ‘inadequacy’ being a less poetic word immediately directs one to consider the inadequacy of something which in this case is first-order logic, distinguished by the fact that it can be done algorithmically and mechanically.¹⁸

Later in our conversation while we were discussing Gödel’s another ingenious technique of ‘constructability’ Professor Srivastava suggested that we are better off using the word ‘definability’ rather than ‘constructability’ because it might evoke the idea of mechanical construction and cooking up sets which is not what Gödel meant! It is not my purpose here to specify what ‘constructability’ and ‘definability’ mean but to point out that the concern with English words was not a detached concern confined to one particular instance but that it persisted throughout our conversation.

Consider his response further,

logic is actually science of reasoning in a certain environment. So as there are many environments there are many logics. For example, for computer scientists … they use just … OR, AND, etc. gates … for them propositional logic is good enough … okay … in other environments first order logic is sufficient … many times even that is not sufficient … I am being basically a mathematician, this is what it is.

Notice how interesting it is that a conversation which began with discussing Gödel’s Platonism was pruned down to a relativist standpoint on what is logic. It was in this context that the word ‘inadequacy’ was suggested by S. M. Srivastava to make it clear what Gödel’s well-known results actually achieved. It is important to recognise that what was important was not this word in itself but the fact that Srivastava came to this word in the middle of a conversation which opened with taking an account of a philosophical position on mathematics. This unaristocratic and plebeian view of mathematics is concurrent with way in which the first incompleteness result is presented and proved in Srivastava’s book in a simple but sharp manner after a barrage of mathematical results, theorems and examples drawn from model theory, recursion theory and so on (ibid.: 115). Also it is worthwhile to note that my conversations with S. M. Srivastava on Gödel’s results began with his straightforward and confident assertion that he takes considerable pride in the fact that he was able to present these results in an undiluted and neat mathematical form, completely free of established clichés and philosophical meanderings. To add one more example of how Srivastava’s responses were hitched to the mesh of mathematical practices it is worthwhile to note that his book opens not with defining mathematical logic and thus addressing its elusive aboutness but rather with enumerating what one studies in it (ibid.: 1). So it was in this context where Srivastava interpreted Gödel’s incompleteness as inadequacy and clarified repeatedly that it must be thought of as such. Incompleteness being a more negatively charged word might imply a sense of negative transcendence at the core of mathematics whereas inadequacy simply stands for the inadequacy, in this case, of first-order logic. And this was precisely what Srivastava was concerned with. Now having seen how the concern with words is not extra mathematical but integral to the discipline and its practitioners, let us strive for a preliminary understanding of this unique dimension of mathematical practice.

Write an Essay in English

As it has been suggested in the introduction, mathematicians bring an ethnographic attentiveness to the fore when they speak about their practice. Insofar as this is the case they choose their concepts carefully when they are asked to describe what they do. Thus they take every bit of care that they do not use words which might suggest an inherent limitation to their practice. Instead they look for words which attest to its ongoing, open-ended and advancing character. Bhrigupati Singh has argued, in the case of ethnographic writing, ‘a concept gives us coordinates along which to pay closer attention’ (ibid.: 168). Similarly in the case of mathematical enterprise, mathematicians suitably transform the concepts they use to outline the contours along which they work. Rather than imposing concepts mentioned earlier which might compromise the vitality of their engagements, by suggesting conceptual dead ends, they choose words which refer to the contexts in which they work and hence their rootedness in the mathematical discourse. Now doesn’t Srivastava’s use of inadequacy in place of incompleteness bring this very same attentiveness to the fore? I think it does but still questions remain like, ‘why inadequacy?’, ‘why not some other word’ and ‘why in this context?’

One way to understand why our conversation got strung around the word ‘inadequacy’ and why the discussions which followed drifted more and more towards the warp and woof of mathematical practice is to use Wittgenstein’s insight on mathematics, admirably highlighted by scholars such as Felix Mϋhlhӧlzer and Severin Schroeder, that mathematical propositions are like rules of grammar which fix the sense of concept words or the sole function of mathematical propositions is to determine the concepts they invoke. The crucial point to be noted here is that mathematical propositions do not report on facts, whether mathematical or otherwise, and in a certain sense they can be seen as independent of experience. Also, it must be noted that when Wittgenstein says that mathematical propositions are rules of grammar what is meant is not that they perform their grammatical function within mathematics but outside of it.¹⁹

As Wittgenstein points out in the Tractatus, ‘… in real life … we make use of mathematical propositions only in inferences from propositions that do not belong to mathematics to others that likewise do not belong in mathematics’ (quoted in Schroeder 2014).

So, for example, a simple mathematical proposition like 2 + 2 = 4 acts as a norm of representation and allows us to make an inference from the concept ‘2 + 2’ to the concept ‘4’. A simple arithmetical equation, like the one previously, provides a framework for using number words in normal English. Thus an equation provides us with a grammatical rule that one side of the equation can be used as a substitute for the other side and vice versa while using these concepts in natural language. This is how Schroeder clarifies this idea:

Thus, the equation ‘2 + 3 = 5’ is a grammatical rule for the use of number words in a natural language, licensing, for instance, the inference from ‘I have two coins in my right pocket and three coins in my left pocket’ to ‘I have five coins in my pockets’ (2014: 22).

But it is important to understand that there is a significant difference between mathematical propositions and ordinary grammatical propositions. Whereas a sentence such as ‘colourless green ideas sleep furiously’ does not make any sense a wrong mathematical proposition is not strictly speaking nonsensical but non-usable. Also mathematical propositions, although independent of experience because they act as norms, are rela-ted to it in a unique manner such that if a mathematical proposition is repeatedly contradicted by experience it thereby becomes not false but as already mentioned worthless.²⁰

Put two apples on a bare table now put another two apples on the table; now count the apples that are there. You have made an experiment; the result of the counting is probably four … this is how our children learn sums; for one makes them put down three beans and then another three beans and then count what is there. If the result at one time were five, at another 7 … then the first thing we said would be that, beans were no good for teaching sums. But if the same thing happened with sticks, fingers, lines and most other things that would the end of all sums. ‘But shouldn’t we than still have 2 + 2 = 4?’—This sentence would have become unusable (Wittgenstein 1956: 14e).

Moreover ordinary grammatical propositions add nothing to the existing meanings present, say in a sentence, and in that sense are analytical whereas mathematical propositions forge new connections between independently comprehensible concepts and are synthetic in nature. This leads to another very important insight from Wittgenstein, as formulated by Mühlhӧlzer, that the use of concept words carries a very important component of ‘decision’ such that empirical confirmations or falsifications do not change the way in which these words are used once their sense is ‘decisively’ fixed. So, for example, since we have the proof that a heptagon with equal sides and equal angles is non-constructible using a ruler and compass, the word ‘non-constructible’ has a marked conceptual content such that it carries a decisively precise meaning in the discourse of mathematics.²¹

See Mühlhӧlzer (2001) and Wittgenstein (1989: 45–91) on an extended discussion on the ‘non-constructability’ of heptagon.

Whereas if one remains only within the framework of Euclidean geometry, say, for example, in school mathematics, there is no sense to be made of the word ‘non-constructible’ because within the Euclidean language game one cannot make any sense of equal lines and equal angles simply because there is no other criteria apart from construction to establish the equality of figures or to even think of figures in the first place.²²

To see how particular mathematicians are with this notion of criteria in their everyday practices, see Gowers (2006). His description involves illuminating insights into the

various criteria mathematicians use while thinking about sets.

In the Euclidean framework one has a vague sense of ‘not yet constructed’ but not of ‘non-constructability’ which we know, from a very different area of mathematics, with its own techniques and rules, is a well-determined mathematical concept. We can of course think and prove the ‘non-constructability’ of a regular heptagon in the analytical framework and project this understanding back on to the Euclidean framework believing that we have simply discovered a possibility which was already latent there but as far as the practice of mathematics is concerned such mixing of language games is regarded as malpractice or bad mathematics by working mathematicians. In the same way with Gödel we get a very precise mathematical sense of ‘incompleteness’ as also of ‘completeness’, ‘consistency’ and so on. Drawing together these insights we can understand why Srivastava was so particular about the word ‘inadequacy’ in our discussions. It was as if the mathematician was acting under an injunction to specify the sense of the concept of ‘incompleteness’ in our conversations. His utterances appeared to be grammatically bound to an underlying calculus. But his desire to be specific about the context and to be precise was also accompanied by a sense of freedom in doing what he does best (I am being basically a mathematician this is what it is). Almost in a Kantian spirit this sense of freedom and surefootedness ensued from following the ‘ethical imperative’ of steadfastly following the demands of his practice. To end this section, I think the point I am trying to make will become clearer when we compare Srivastava’s usage of the word ‘incompleteness’ and his interpretation of it as ‘inadequacy’ with the usage of this word in a different language game. For example, philosophical concerns on Gödel’s two proofs often raise the question whether they show that human minds are not machines. Since Gödel’s results specify the limitations of a syntactically transparent system as, for example, a computer is, a mathematician like Roger Penrose argues in a popular and well-known take on the subject that unlike what the proponents of artificial intelligence believe, human minds are very different from computers for otherwise how one could broach the issues of truth, insight, creativity and so on in mathematics (Penrose 1989). So, for example, at one place he writes:

As I have stated earlier, a good part of the reason for believing that consciousness is able to influence truth judgements in a non-algorithmic way stems from considerations of Gödel’s theorem … surely we may be persuaded that such a non-algorithmic ingredient could be crucial also for the role of consciousness in more general (non-mathematical) circumstances (ibid.: 538).

Just note how such concerns with the workings of consciousness are so alien to the discourse of mathematics and here words are projected through very different trajectories and thus take on very different lives and meanings.²³

Note that in this context the word which becomes important for Penrose is neither ‘incompleteness’ nor ‘inadequacy’ but ‘non-algorithmic’.

It is not my purpose here to evaluate the philosophical merit of such arguments but to bring to the attention of readers how in different language games Gödel’s arguments are put to different uses whereas within the discourse of mathematics English words carry a very precise and almost inviolable conceptual sense and hence mathematicians take every bit of care to weigh their words carefully before they speak. It will be pertinent to point out that a well-known mathematician Geetha Venkatraman, an expert on group theory whom I interviewed during my fieldwork, told me that when she started writing her thesis at Oxford she was told by her supervisor, ‘first try to write an essay in English, and insert the symbols and equations later’. This again emphasises both the importance of words for mathematicians and the grammatical nature of mathematical propositions. Now before I end this section a brief comment must also be made on the loose working sense of Platonism that was inherent in Srivastava’s remarks in our discussions on incompleteness. I think the description I have offered so far makes it clear that my emphasis is not simply on the translation of the word ‘incompleteness’ but rather on the projection of this word into a new context. As Veena Das has argued it is only through projection into new contexts and ordinary exchanges that concepts come to have life.²⁴

As she has written, A further thought of Wittgenstein … about our life in language is that we learn to project words in new situations and in so doing we not only learn the nuances of our language but also the nuances of the world…thus for Wittgenstein, concepts acquire life in the give-and-take of ordinary life. (Das et al. 2015: 60–61)

It is this aliveness of concepts which imparts them an indubitable reality and in the case of mathematics enables the mathematicians to invoke a loose working sense of Platonism without at the same time being guided by the notion of mathematical reality in their research. Thus it can be said that what seems real or concrete to working mathematicians are not some elusive mathematical objects but concepts pulsating with life and vitality. What I have said so far also suggests an interesting possibility to be explored with further research. As a tentative hypothesis it can be suggested that mathematics in the South Asian or more broadly non-Western locations generate reflections which cannot simply be assimilated to the dominant Western philosophical thinking on mathematics. What constitutes these reflections? What are the possibilities and challenges and what relations do these have with specific institutional histories? These can be deeply rewarding questions to follow and can be taken up with further inquiries on the subject. Let’s now shift our attention to the other dimension of mathematical practice mentioned in the introduction. For that let us first try and understand the leading idea behind Gödel’s first incompleteness theorem. The following section attempts to provide the flavour of Gödel’s strategy involving two simple but remarkable ideas that of self-reference and the distinction between truth and provability.

Leading Principle²⁵
25
As Brian Rotman has argued, Proofs are arguments and, as Peirce has forcefully pointed out, every argument has an underlying idea—what he called a leading principle—which converts what would otherwise be merely an unexceptionable sequence of logical moves into an instrument of conviction … indeed, it is only by virtue of it that the sequence is an argument and not an inert, formally correct string of implications. (Rotman 1988: 14–15) Thus, leading principle is the underlying narrative of the proof or the ‘idea behind the proof’ without which the proof may be syntactically correct but would fail to be persuasive (ibid.).

Gödel developed a remarkable technique for mapping various statements or formulas of a formal system onto numbers such that each formula or any finite sign sequence is assigned a unique number through a systematic and mechanical procedure.²⁶

This technique by Gödel is known as the ‘arithmetisation of syntax’. For an extended discussion, see Nagel and Newman (1968: 76–85).

What this achieves is that various formal relations between formulas can be mirrored onto purely arithmetical relations between numbers. Now suppose there is a formula numbered y and another sign sequence which proves that this formula is numbered x. Gödel showed that a formula π (x, y) can be proved within the system which expresses precisely the aforementioned relationship, that is, the sign sequence numbered x proves the formula numbered y. Not only this but the formula -π(x, y), which says that the finite sequence numbered x fails to prove the formula numbered y, can also be proved in the system. Now quite ingeniously Gödel produced a number, say m, such that the formula is numbered m. Now as earlier, this formula states that there is no number x corresponding to the proof of the formula numbered m. But note that m is also the number assigned to this formula which asserts its own unprovability! Simply put the formula numbered m states that the formula numbered m is unprovable (the idea of self-reference). Now Gödel was able to demonstrate that the formula numbered m is indeed unprovable and thus the formula which asserts its own unprovability is true. Thus Gödel was able to construct or define a formula that was both true and unprovable in the system under consideration. In the original text of the proof he compares his line of reasoning with the statement of Richard’s antinomy and the liar’s paradox thus demonstrating how to make a ‘self-referential statement inside a theory’ (Srivastava 2008: 116).²⁷

‘The analogy of this conclusion with the Richard-antinomy leaps to the eye; there is also a close kinship with the liar antinomy… Hence, we have in front of us a theorem that states its own unprovability’ (Gödel 1988: 19).

What is noticeable in this informal presentation of the proof is that Gödel gives the language of the formal system an intended interpretation (by assigning numbers to formulas), which clearly is a semantic move, and based on this interpretation shows that there are formulae that are true but unprovable in the system.²⁸

Readers who are not interested in the twisted logic that Gödel uses just have to understand what he achieves through his pioneering work on incompleteness. In a nutshell he shows that there can be located statements within a formal system which are true but cannot be proven using the rules governing the system. And the technique that he uses, without going into its details, is that of assigning numbers to formulas of the system and then treating the formulas as statements about numbers.

We’ll see in the next section how this semantic move is not as arbitrary as it appears and is rather constrained by crucial syntactic considerations. Thus, having gained some idea about the proof of Gödel’s first incompleteness result, let us go into the other dimension of mathematical practice that I wish to delineate in this article.

Syntax and Semantics²⁹
29
The classic articulation of the complete separation between formal syntactic considerations from informal semantic ones is Rudolph Carnap’s The Logical Structure of Language (1937). In this section I seek to problematise this separation through an ethnographic perspective on mathematical texts and practices.

What I would like to bring to the attention of the readers is the play in Gödel’s text between purely syntactical moves on the one hand and on the other hand certain crucial semantic interpretations which give the proof its force. On reading the original text of the proof as well as the short summary I have tried to outline, one gets the sense that Gödel is playing with the syntactical framework. He certainly accepts casting all of mathe-matics in a formalised language as a possible framework for doing mathematics but at the same time subverts the ambitions of this programme by bringing in considerations of meaning and interpretation to bear on a purely formalised arithmetic. Consider what he says about a formal system, ‘a formal system consists only of symbols and mechanical rules relating to them, the meaning which we attach to the symbols is a leading principle in the setting up of the system’ (emphasis mine) (Gödel quoted in Wagner 2008: 202). However leaving it there would mean accepting Gödel as a meta-subject who simply assigns meaning to the formal apparatus, as it were, from the outside. As Roy Wagner has perceptively pointed out this clipping on (and off) of meaning to the formal apparatus or the effect of meaning in a mathematical text arises not from the agency of the mathematical subject as such but rather from the juxtaposition of different texts within a text (ibid.). Meaning is not added from the outside; rather, it arises from the clash of forces within a text. The mathematical technique of mapping enables Gödel to encode formal statements of the system into the language of number theory and this bringing together of two different registers creates an effect of meaning that undermines the pretensions of a purely formalist understanding of mathematics by destabilising the formal language from within.³⁰

Mapping is a well-known technique used in mathematics to correlate one to one elements from two different sets or registers. This is how Nagel and Newman define mapping in their text, ‘the basic feature of mapping is that an abstract structure of relations embodied in one domain of “objects” can be shown to hold between “objects” (usually of a different sort from the first set) in another domain’ (ibid.: 66).

However this bringing together of two different registers is guided or constrained by clearly laid out syntactic considerations. So, for example, assigning a unique number to a formula is not arbitrary but is guided or constrained by systematic and mechanical rules or as Wagner puts it incisively that this gluing together of two different systems is discursively constrained.³¹

Roy Wagner has analysed the text of Gödel’s proof from the perspective of post-structural semiotics. His enquiry is guided by the question ‘how does meaning operate in a mathematical text?’ (Wagner 2008: 196). Although I have made use of his perceptive comments but what I am interested in is not only textual analysis but also scenes of instruction and questions of pedagogy and how they shape mathematical semiosis.

Clearly both semantic and syntactic considerations come together to specify the inadequacy or limits of a mathematical system in Gödel’s text. Modulating Foucault’s comments on the classification from a Chinese encyclopaedia slightly one could say that what Gödel creates is not unlike a textual heterotopia which plays a great role in sterilising the lyricism of the formalist language (Foucault 1970: xviii). The foregoing comments point towards an interesting prospect for enquiry that the undecidable in Gödel’s text is produced as a consequence of the very technique of writing. It can be shown that writing and in this case writing of mathematics carries within itself the capacity to cause ripples in the syntactical pool thereby creating effects of meaning.

At this point I recall a conversation I had with my spouse over Cantor’s diagonalisation argument to establish that real numbers are uncountably infinite.³²

My spouse is a junior statistical officer at the Ministry of Statistics, Government of India. We often discuss and debate mathematics with each other and these conversations form an integral part of my ongoing ethnography of mathematical practice. There are several popular and simplified versions available of Cantor’s diagonal argument but for a good mathematical introduction, see Bartle and Sherbert (2000: 21–51). The technical aspect of Cantor’s proof is not necessary to appreciate the point I am making, it is sufficient to see that Cantor’s surprising result is produced as a consequence of a certain technique of writing.

The key element of Cantor’s proof was to number real numbers as 0, 1, 2 and so on and then to observe that this numbering notwithstanding real numbers can always be located which do not fall in the list. This procedure involves writing the real numbers in their decimal form one beneath the other and then choosing the digits which lie on the diagonal and forming a new number out of these which cannot be found in the order. I wrote this proof down and showed it to my wife and both of us felt that it is just an artefact of writing and not a mathematical proof. But later we realised or came to see that the proof worked precisely by creating this ambiguity between syntax and semantics or between the formal and informal aspects of the argument. Similarly in Gödel’s proof, undecidability or inadequacy is intricately tied with the syntactical praxis of mapping or mimesis in which syntax is made to reflect on itself through the technique of systematic coding. As Paul Livingston has perceptively pointed out through his reading of Derrida:

…the very act of inscription implies the failure of this reference to the totality or at least its constitutively paradoxical status. This amounts to a demonstration of the necessary existence of points or sentences (in particular, those that express the system’s own conditions of possibility) that cannot be decided (as true or false, or as present or absent) in terms of it. (2012: 118)

Thus the full import of Gödel’s procedure cannot be understood just by focusing on the formal inscriptional rules of a system or else just by focusing on the considerations of meaning alone. Rather, what gives us the undecidable is precisely this intertwining of the syntactic and the semantic registers. According to this reading Gödel’s procedure can be illuminatingly compared with Derrida’s strategy of deconstruction and the undecidable formula in Gödel’s system with such Derridean concepts as trace, différance and so on. Différance as Derrida has pointed out is ‘neither a word nor a concept’ but makes possible or structures linguistic presentation both through synchronic difference and through diachronic deferring but which itself cannot be represented in language.³³

As Derrida has written, Différer in this sense is to temporise … the temporal and temporising mediation of a detour that suspends the accomplishment or fulfilment of ‘desire’ or ‘will’ … the other sense of différer is the more common and identifiable one: to be not identical, to be other, discernible, etc. .… Différance as temporisation, différance as spacing. (ibid.: 8–9)

It is that point or location in the text where the structural logic of the system unravels and binaries like nature/culture, presence/absence and syntax/semantics are brought into question.

Let us now move towards some ethnographic examples to observe the interplay between syntax and semantics in a slightly displaced setting. The point being made here is that sometimes syntactical considerations and at others semantic ones play a very crucial role in mathematical semiosis and by privileging the one over the other we’ll be unable to capture the true movement of mathematical praxis taken as a whole.³⁴

Note that this is a different point than exploring the ambiguity between syntax and semantics within a mathematical text. Also, it is pertinent to note that this constant shifting between the syntactic and semantic registers rather than taking away from the reality of mathematical objects adds to their autonomy and obduracy. The idea of shift in registers is borrowed from Bruno Latour’s study of the processes through which various sciences produce an effect of reality. For example, in his study of Einstein’s text on relativity Latour locates twin processes of shifting out and shifting in at play in producing an internal referent for his physics (Latour 1988).

First, let me comment upon the significance of syntactic issues in mathematical practice. As far as the importance of syntactic considerations is concerned it is pertinent to note that Professor Srivastava’s book on mathematical logic also opens with laying out the syntactical framework for the rest of the text. The first chapter of S. M. Srivastava’s book is titled ‘Syntax of First-Order Logic’. It begins with precisely spelling out the nature and features of the language of first-order logic known as ‘first-order language’ (ibid.: 3). The first chapter also presents problems that relate to formal issues like, when is an expression rightly constructed, what is a formula and how to prove that a particular expression is a term of the language or not and also introduces, in the form of exercises and examples, standard procedures to address these problems. In fact, at one place, he revealingly points out that although such deliberations over the syntax might appear ‘useless’, quite often syntactical clarity over the ‘structure of a formula itself helps us to make inferences about the formula’ (ibid.: 10), thus underlining the importance of syntactic reasoning in the doing of mathematics.

As a further evidence of the importance of syntactic considerations, consider this very interesting example that I came across while attending a workshop on mathematics at Harish-Chandra Research Institute Allahabad as part of my fieldwork. The teacher was lecturing on one of the advanced areas of mathematics, field theory. But before commencing with the main topic the teacher called one of the students to write down the axioms of group theory on the whiteboard, a topic that they had done earlier.³⁵

A Group, say G, is an algebraic structure based on a binary operation, say *, such that if ‘a’ and ‘b’ are two elements belonging to G then a * b also belongs to G. Along with this there are certain axioms which have to be satisfied in order for G to be a group. Group identity is usually denoted by the letter ‘e’ and is such that if ‘a’ is any element belonging to G then a * e = a.

A student came up to the board and wrote the identity axiom for groups in the following way:

(\forall x \in G)(\exists 0 \in G)x .0 = 0 .x

(1)

(Taken literally this would read as for all x belonging to G, there exists 0 belonging to G such that x.0 = 0.x)

This no doubt meant that the student had understood the meaning of the identity axiom (which means that there is an element 0 belonging to the set G such that if ‘.’ denotes the binary operation on the elements of the set then,) correctly. However there is a slight problem with the aforementioned statement which reveals itself if we compare the aforementioned expression with the literal meaning of the statement:³⁶

This example was taken by the teacher herself. The student had a bewildered look on his face as the teacher suddenly shifted from the mathematics to a statement in English to highlight his mistake.

‘A person dies every five minutes in India’.

Taken literally this would mean:

There is a person who dies, then comes to life again, then dies again, then comes to life again and so on, ad infinitum, every five minutes.

The teacher said looking at the class, ‘now, in normal English, the above statement poses no problem, because the meaning of it is relatively clear but in mathematics statements like this are not valid because they are not syntactically correct as the quantifiers are not in the right order’.³⁷

Symbols like ∃ (existential quantifier, read as ‘there exists’) and ∀ (universal quantifier, read as ‘for all’) are known as quantifiers because they specify the range and quantity of variables in the given context.

The correct form, as she wrote down on the board, should be:

\exists e \in G s u c h t h a t x . e = e . x \forall x \in G

(2)

(Read as, there exists an element ‘e’ belonging to the set G such that for all elements belonging to G, represented by the variable x, x.e = e.x)

The teacher emphasised that it is the right form because it clearly specifies the uniqueness of identity element in the structure which is an essential property of groups.³⁸

As against this, students rendering of this property might have suggested as if each

element of a particular group has a separate identity.

One can see how the teacher concerned herself with a mistake, which seems so trivial at first sight, in an imaginative way to bring to light the importance of syntactic considerations in mathematics

On the other hand sometimes semantic framing or presentation of an idea enables the students to easily acquiesce or to be persuaded into the discourse. In one of the several classes I attended on group theory at St. Stephens College, the University of Delhi, I came across an interesting piece of insight on the nature of pedagogy and department of teaching and learning in abstract algebra. Usually the general tenor of the classes and the disposition of teaching were inflected towards the axiomatic but on the other hand the teacher frequently imported crucial semantic considerations into her discourse to make the syntactic procedures more compelling and persuasive for the students. In this class teacher wrote a property obeyed by any two elements (say ‘a’ and ‘b’) having inverses in a semi-group with identity ‘e’ that (a.b)–1 exists and is equal to b–1.a–1. Now this property went against the usual syntactic habit of the students to write (a.b)−1 = a−1.b−1 according to the general exponential law and hence many students in the class had a confused look on their faces while jotting down this property in their notebooks. The teacher it seems had expected this problem to arise and hence to elucidate the property better she called it ‘the shoe and socks property’. She said,

if you conceive the group product of two elements as putting on socks first and then shoes, then as you remove shoes first and then socks, the inverse of the group product is taken to mean multiplying the inverses of these elements in the reverse order.

With this presentation, the aforementioned property became immediately accessible to the students notwithstanding its initial appearance as a syntactic aberration. In fact I could draw various such examples from my fieldwork which attest to the importance of semantic considerations in the transmission of mathematical knowledge. If in the aforementioned case, the everyday act of wearing and removing shoes is invoked then in other examples simple and everyday objects like chairs, pillows or simple acts such as seating arrangements, sharing and so on are invoked no matter how complex or advanced the subject at hand is. Further research can shed light on the kind of examples used in semantic framing of mathematical arguments and how and why is it that often deceptively simple kinds of reasoning tend to reveal much about complex mathematical structures. Having commented upon the interplay between formal and informal aspects of mathematical arguments let us now move towards a tentative conclusion which reiterates as well as adds to the points raised in the article so far.

Conclusion

In this article, I have attempted to demonstrate the significance of linguistic considerations, involving both mathematical and non-mathematical language, in the learning or more broadly in the transmission of mathematical knowledge. I have attempted to show, through close description, how such considerations are responsible, broadly, for the inwardness of mathematical practice. Context determination through the projection of words and the constant play between formal and informal aspects of reasoning show how concerns with language use dominate over philosophical questions about mathematical reality, mathematical truth and so on in the doing of mathematics. So, for example, belief in or not thereof in the notion of mathematical reality plays no integral part in the organisation of mathematical research. Even though mathematicians might espouse Platonism as a working philosophy but it goes along with rigorous context boundedness and the highly relative character of their approach. It is to be remembered that when I talk about the self-referential character of mathematical discourse, I am not simply making a semiotic point but rather expressing the possibility of exploring mathematics as a form of life. What it means is that mathematical practice engenders forms and conventions of language use which are specific to it. Also, it has been shown in the article that these practices of language use create an effect of reality which is internal to mathematics. Thus it makes no sense to speak about an external domain of mathematical facts and objects which is independent of its practitioners. Rather if there is any reality to mathematics at all it lies within the complex web of practices that constitute it. However, as one can see from the picture presented in this article, these practices are multiform and diverse and we need several such strategies of description in further opening them up for anthropological investigation. It is within this context I have suggested that one could explore the ethical dimension of mathematical practice. This dimension concerns the way in which restraint, precision and deliberate avoidance of broad and sweeping generalisations are seen as virtues in mathematical research and taken by themselves both syntacticism and semanticism are regarded as malpractices. Perhaps the moral import comes out most vividly in the intertwining of words and practices in the seamless framework of mathematics and in the deportment of inwardness that characterises it. However, to what extent I was enabled in raising these issues by conducting fieldwork on mathematics in the South Asian context still remains an open-ended question and must be taken up later as an important epistemological issue. I would like to leave these issues here noting them as immensely productive possibilities for further exploration and clarification.

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