Mathematical equations as Durkheimian social facts?

Mathematical equations as Durkheimian social facts

Mathematical relativism: logic, grammar, and arithmetic in cultural comparison

Cultural relativism is supposed to be a bold and provocative thesis. In this paper we challenge the idea that it is an empirical thesis, i.e., one that is supported through anthropological and historical examples. We focus on mathematical relativism, the view that a mathematics from another culture or time might be so radically divergent from our mathematics that ‘theirs’ would stand in direct conflict with ‘ours’ (and in that sense constitute an alternative mathematics). We question in what sense the examples given to support the general thesis are relativistic about mathematics and argue that on close inspection they are not, and certainly not in any radical sense. We do not contest the fact that there can be great mathematical diversity between cultures, but wonder whether it makes sense to talk of ‘the same’ mathematical forms in heterogeneous mathematical environments. Finally, while relativists see the later Wittgenstein as providing support for their own thesis, we claim that Wittgenstein argues against both realism and relativism

Logical relativism: logic, grammar, and arithmetic in cultural comparison

Logical relativism is the claim that different cultures may think according to different logical laws. For example, it is often argued that whereas ‘we’ (Westerners) operate according to the law of the excluded middle, ‘they’ (e.g., the Chinese or the Azande) may not. In this article, we question whether logical relativism is an empirical thesis, i.e., a thesis that is substantiated through anthropological examples. We distinguish two forms of logical relativism, both of which try to account for alleged contradictions in the beliefs of other cultures. The ‘alternative logic’ approach suggests that contradictions only appear if we judge beliefs according to classical logic, but do not exist if we judge them according to an alternative (non-standard) logic. The ‘symmetric treatment’ suggests that whether there is a contradiction or not is itself a culture-specific matter, such that what may be a contradiction ‘for them’ may not be a contradiction ‘for us’ and vice versa. We question whether either of these arguments really involves relativism and show that problems arise in the treatment of the examples, firstly, in terms of questionable preconceptions made about the status of logic as a standard of comparison and, secondly, in the ways in which relevant beliefs are formulated

Where do the limits of experience lie? Abandoning the dualism of objectivity and subjectivity

The relationship between 'subjective' and 'objective' features of social reality (and between 'subjectivist' and 'objectivist' sociological approaches) remains problematic within social thought. Phenomenology is often taken as a paradigmatic example of subjectivist sociology, since it supposedly places exclusive emphasis on actors' 'subjective' interpretations, thereby neglecting 'objective' social structures. In this article, we question whether phenomenology is usefully understood as falling on either side of the standard divides, arguing that phenomenology's conception of 'subjective' experience of social reality includes many features taken to be 'objective' elements of it. We illustrate our argument by a critical examination of Jean Lave's attempt to differentiate social practice theory from phenomenology. We show that many theoretical positions that want to overcome the subjective-objective dualism retain an objectivist conception of the 'subjective' features of social reality. © 2008 SAGE Publications

Linguistic relativism: logic, grammar, and arithmetic in cultural comparison

Linguistic relativism is the thesis that the grammatical structures of different languages imply different conceptions of reality. In this paper we critically discuss one form of linguistic relativism, which argues that grammatical differences between the English and Yoruba language exhibit differences in how English and Yoruba speakers ‘see’ reality (namely in terms of ‘spatiotemporal particulars’ and ‘sortal particulars’ respectively). We challenge the idea that linguistic relativism is an empirical thesis, i.e., a thesis that is substantiated through anthropological examples. We show that linguistic relativism is based on two assumptions: firstly, that the purpose of language is to describe the world; secondly, that being able to speak presupposes an ontological theory of the ultimate constituents of the world. We argue that the attempt to extract the outline of that theory from the language inevitably distorts the portrayal of language-using practice itself

Two concepts of attachment to rules

In this paper, we discuss the implications of John Rawls’ (1955) paper “Two concepts of rules” for social science. We argue that Rawls’ notion of ‘practice’ is not a straightforward contribution to sociological theory, but rather re-orients the idea of what understanding social actions might consist of. We explicate how Rawls’ distinction between ‘summary’ and ‘practice’ views of rules might play out in approaching mathematical practice and mathematical expressions. We argue that social constructivists like Bloor hold on to a ‘summary’ conception of rules while Wittgenstein adopts the more radical ‘practice’ conception

Linguistic relativism: logic, grammar, and arithmetic in cultural comparison

Linguistic relativism is the thesis that the grammatical structures of different languages imply different conceptions of reality. In this paper we critically discuss one form of linguistic relativism, which argues that grammatical differences between the English and Yoruba language exhibit differences in how English and Yoruba speakers ‘see’ reality (namely in terms of ‘spatiotemporal particulars’ and ‘sortal particulars’ respectively). We challenge the idea that linguistic relativism is an empirical thesis, i.e., a thesis that is substantiated through anthropological examples. We show that linguistic relativism is based on two assumptions: firstly, that the purpose of language is to describe the world; secondly, that being able to speak presupposes an ontological theory of the ultimate constituents of the world. We argue that the attempt to extract the outline of that theory from the language inevitably distorts the portrayal of language-using practice itself

Sources for myths about mathematics: on the significance of the difference between finished mathematics and mathematics-in-the-making

Sources for myths about mathematics: on the significance of the difference between finished mathematics and mathematics-in-the-makin

Gestures in the blackboard work of mathematics instruction

Lectures in mathematics consists almost entirely of the lecturer writing definitions, theorems, and proofs on the blackboard (often reproducing word-by-word what is distributed in advance in lecture notes) while simultaneously commenting on what is being written. The writing, talking, and gesturing conjointly formulate the cohesive logic of the mathematical argument that the formulae instantiate. In the first part we examine the blackboard organization of the exposition: what is written is not just written ‘anywhere’, but the physical structure of the blackboard is organized into segregated fields so as to re-order the formulae on the board in a way that displays their mathematical role amongst the interrelated constituents of the mathematical argument put forward. The second part focuses on how gestures are used in conjunction with and coordination of what is being written on the blackboard and what is being said

Does mathematics look certain in the front, but fallible in the back?

In this paper we re-examine the implications of the differences between 'doing' and 'writing' science and mathematics, questioning whether the way that science and mathematics are presented in textbooks or research articles creates a misleading picture of these differences. We focus our discussion on mathematics, in particular on Reuben Hersh's formulation of the contrast in terms of Goffman's dramaturgical frontstage-backstage analogy and his claim that various myths about mathematics only fit with how mathematics is presented in the 'front', but not with how it is practised in the 'back'. By investigating examples of both the 'front' (graduate lectures in mathematical logic) and the 'back' (meetings between supervisor and doctoral students) we examine, first, whether the 'front' of mathematics presents a misleading picture of mathematics, and, second, whether the 'front' and 'back' of mathematics are so discrepant that mathematics really does look certain in the 'front', but fallible in the 'back'