Sociological Approaches to Mathematics

SOCIOLOGICAL APPROACHES TO MATHEMATICS zyxwvutsrqponmlkjihgfe Richard Startup, University College, Swansea Summary The underdeveloped state of this field does no credit to sociology. Some sociologists are more interested in using quantification as a source of legitimation than in confronting mathematics as a central element of culture. Yet mathematics may be viewed as an institution with a normative core developing alongside experience of the physical world, susceptible of both "internalist" and "externalist" programmes of investigation. There is a need to examine how the symbolic system of mathematics structures the development of thought and how the acceptance of mathematical ideas is related to their utility. The use of the evolutionary approach has generated some distinctive concepts which can assist cross-cultural work. Smaller-scale studies continue to provide insight into the relation between mathematical theory and social structure. A Neglected Field The sociology of mathematics is an important but underdeveloped field. That it is important can hardly be doubted, for mathematics is a vast and strategically vital cultural phenomenon in Western industrial societies. Mathematics forms an integral part of the technology which underlies industrial organisation. The discipline of mathematics is a prominent ingredient of school and university curricula, which touches us all directly or indirectly. In some respects the cultural position of mathematical ideas bears comparison with that of religious ideas, although their functions differ. Sociological studies of religion are frequent and familiar enough: why then the relative neglect of mathematics? The answer is that mathematics and the related field of logic tend to be reified, i.e. they come to possess the "thing-like" quality of a material object. They appear fixed and inviolable: part of our situation of action which we rest upon and take for granted in the deepest sense. Paradoxically, religion-whatever its transcendent and universal claims-is unlikely to be reified in the same way. at least by sociologists. This is because religion varies from one culture (or subculture) to another. If Christianity differs from Buddhism we are at least inclined to ask how and why does it differ and inquire about the differential effects on the social and economic structure. By contrast mathematical and logical ideas appear to differ from society to society only in the extent of their development, not in their essential content, so those ideas do not seem to invite parallel sociological scrutiny. Despite this impediment to investigation, perusal of the socioligical literature on mathematics reveals that the cupboard is by no means bare, but research work is patchy and its development in some respects distorted. What one tends to find are iimited "externalist" accounts which typically focus on events at some distance in the past. "Externalist" as opposed to "internalist" accounts are those which seek to analyse the influences on mathematics of ideas and activities which belong outside it (Barnes, 1974:99-124). An example would be the way in which the idea of perspective 64 among Renaissance artists led on to projective geometry (Wilder. 1979:59-60). In the same context one can note Merton's well known work in the sociology of science (1938; 1968:585-681; 1973) which continues to be a potent influence. He and other writers (Hessen, 1931; Zilsel, 1941; Kargon. 1966; Needham. 1969) have analysed socio-economic changes which are implicated in the phenomenon of the "Scientific RevoIution"--that period from Copernicus to Newton which was marked by the mathematisation of science (and in mathematics especially by the introduction of coordinate geometry and calculus). Sociological work relating to more recent periods is substantially lacking and few even of these externalist sociological accounts are forthcoming. It is as though mathematics since the time of Newton is seen as a subject developing simply on the basis of its own "internal logic"--perhaps with some additional promptings from physics. The subject is not simply reified but its vast scope and abstract technical nature contribute to its overwhelming character. As religion becomes demystified--at any rate to sociologists-mathematics becomes mystifying. How to respond? Well, if you can't beat them join them, as they say. Since sociology is in many respects an insecure discipline there is a discernible tendency for sociologists to turn to the most secure discipline of mathematics for support rather than making that cultural phenomenon an object of sociological scrutiny. Quantification has become both a methodologyzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA and a source of legitimation, a phenomenon to which the pages of the American Sociological Review bear voluble though not eloquent witness. zyxwvutsrqponml Realism and Relativism How can one make progress from this unsatisfactory position and open up mathematics to deeper sociological inquiry? Apparently the most direct route is to attempt to de-reify the subject and one or two braver souls have chosen this course. Bloor (1973), for instance, makes use of Wittgenstein's ideas (1956) to develop a distinctive sociological perspective. Bloor finds fault with the "realist" view of mathematics (which influenced Mannheim, 1936:239-40) in which mathematical and logical forms are seen as being, "like material objects in the sense that they are set over and against the knowing subject who moves (in thought) amongst and through them" (1973:176). Instead Bloor follows Wittgenstein in stressing the centrality of the essentially social process of rule following involved in basic mathematical operations such as counting and using formulae. Wittgenstein asserts that, "mathematics forms a network of norms" (1956: V-46). Bloor judges that Wittgenstein is "proposing a non-realist theory of the objectivity of mathematics" (Bloor, 1973:189) and concludes that mathematics may be appropriately viewed as an institution with no part excluded from sociological scrutiny. (For Wittgenstein's position, see also Philips, 1977:11941.) Elsewhere Bloor (1976) elaborates further his conception of the sociology of knowledge and seeks to demonstrate its applicability to mathematics. He characterises his approach to that subject as "naturalistic" for he develops a sociologically extended version of J.S. Mill's empiricist account (1976:74-94). Mill (1856) viewed mathematics essentially as a collection of beliefs--or mental events-concerned with the physical world and arising from experience of it. Bloor sees this as a useful starting point, but he is critical of Mill for ignoring the objectivity of mathematical ideas (a feature stressed particularly by Frege. 1959). However. Bloor builds on Mill by introducing objectivity in the form of institutionalised belief (Bloor, 1976:87). In 65 addition, he employs the notions of model and metaphor to account for the wide range of application of mathematical ideas (1976:87-92). Bloor proceeds to identify a striking contrast amongst those who have approached mathematics sociologically. If mathematical activity and its normative structure simply form an institution then one may expect that varying forms of mathematics will be observable in different cultures. Alternatively, if what Bloor characterises as the "realist" view is adopted, then a single version of mathematics will be found everywhere, although no doubt there will be variations in the extent of its development. So on this view it becomes something of a test case to determine whether or not there are "alternative" mathematics. Bloor (1976:95-116) firmly believes there are and he points to instances such as the "arithmetic" (or what we characterise as algebra) of the Alexandrian Diophantus. in which there are some (to us) strange concepts and processes. Despite the suggestiveness of his ideas it is necessary to take issue with Bloor's relativist position. In order to decide whether or not there are alternative mathematics one needs to have a clear conception of what "a mathematics" is--in a way which goes beyond the experience of Western mathematics itself. Since such a definition is lacking, the quest for alternatives is misguided. In connection with this issue one may reasonably inquire what might be meant if it is suggested (for instance) that 2 + 2 = 5. Short of saying that one or more of the symbols involved in this statement is being used in an unfamiliar way, one would simply conclude that it was false and has no place in mathematics. This is in no way to deny that the history of mathematics has been marked by conceptual variation, a phenomenon which is a fit subject for sociological analysis. In fairness to Wittgenstein it should also be pointed out that both the relativist and from his notes on mathematics he may be taken to be critical ofzyxwvutsrqponmlkjihgfedcbaZYX anti-relativist positions. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE Symbolic Systems As well as considering the essential character of mathematics the sociologist must also focus on mehtods of numeration, for in this formal subject there is a symbiotic relation between particular designations and that which is designated. The classification of number words and symbols is particularly well advanced (Menninger, 1969) and it has proved possible to analyse many practical, social and scientific implications ever, a basic unsolved problem concerns the specification of the social circumstances in which particular kinds of numeration become institutionalised. Within number systems the more important types of variation include the use of a particular base, the presence or absence of a symbol for zero, the employment of positional notation ("place value") and the means of fractional representation (Resnikoff and Wells. 1973: Chapter 1). Different systems have their relative strengths and weaknesses, for instance the Babylonian possessed the advantages of positional notation but allowed ambiguities owing to the lack of a symbol used consistently to indicate the absence of a number (Kline, 1972:5-7; Resnikoff and Wells, 1973:23), while the Ancient Egyptian system used the base ten but was not positional (Struik, 1967:20-25; Kline, 1972:16-18; Gillings, 1972:20-23). The contemporaneous Chinese (Shang) numeral system had the capacity to express any desired number, however large with no more than nine numerals and Needham (1959:5-15) judges that it provided a more positive influence on scientific development than those of Babylonia 66 and Egypt. In each of the ancient civilisations the number system had structural implications for numerology (Needham, 1954-1976: Volume 2, 271). mysticism (Kline 1972:13) and astrology. Classical Greek mathematics was marked by the existence of a close connection between the stratification of people and of types of knowledge. The distinction emphasised was that between calculation used for practical purposes in the marketplace (logistika) and the intellectual contemplation of number (arithmetica). a dichotomy corresponding to the social division between those in commerce and the "lovers of wisdom" (philosophers). Numerations which were employed for practical purposes made extensive use of letters of the alphabet (Menninger, 1969: 262-63; Kline. 1972: 131-34) and in the Alexandrian period whole numbers appear to have been written to a decimal base but without positional notation and with limited use of a symbol for zero (Kline 1972: 132-33). Since the Ancient Greek period mathematicians have successively modified (or extended) the number concept. These extensions have been accompanied by disputes about whether particular types of numbers can properly be said to exist, e.g. irrational numbers, the number zero, negative numbers, complex numbers, and infinity (Nagel, 1935). These controversies are important to the sociologist for the light which they shed on prevailing ontologies and cosmologies, but it must also be asserted that it is the way in which numbers are represented and hence used which is most closely related to their acceptance. Sometimes the use is to be considered as internal to mathematics itself, sometimes external in the sense indicated above. For instance, the Greek stress on abstract truth and the playing down of the commercial utility of mathematics helped to produce the pure geometrical emphasis for which Euclid is renowned, together with the failure to conceptualise the irrational number. Later on the Hindus, who were more oriented to practical and commercial concerns, began to operate with the latter type of quantity and also introduced negative numbers, probably through their usefulness in representing debts as opposed to assets. In this short account of symbolic usages, it must be stressed that one is fundamentally concerned with a system of manipulating numbers, operating with them and so on, mathematical development being structured by the elaboration of symbolism which assists this process. For example, involved in the transition from arithmetic to algebra was the need to distinguish particular expressions and formulae from entire classes of expressions and formulae. Of considerable importance in this connection was the use of literal coefficients in forms and equations, an innovation dating from the Sixteenth Century (Kline, 1972; 259-63). In the further development of generalising power-perhaps the essence of mathematics--the two basic but also polar ideas were variable and function; notions which were clarified only in the Nineteenth Century. There have also emerged such generalising procedures as mathematical induction and the duality principle, which either confirm or generate results while also providing insight. It has been much emphasised that in the early development of calculus symbolic usages were employed which would now be forbidden and advances made despite an absence of vigour (Boyer, 1959:169) and sometimes it has been claimed that this reflects a different value system from that evident in modern mathematics. However, the latter assertion is doubtful since so many Eighteenth Century mathematicians 67 expressed disssatisfaction with a lack ofrigourand/or strove (albeit unsuccessfully) to supply the missing element. However, if values have been relatively stable, what is undeniable is that the history of mathematics has been marked by periods of creative (if somewhat insecure) innovation and subsequent consolidation. Evolutionism In the sphere of symbolism--as elsewhere within mathematics-there is a strong tendency to think in terms of a progression starting with the Classical Greek period. Certainly from that time on mathematics acquired a persistent cumulative character. This has led several authors — notably Wilder (1974) and Fang and Takayama (1975)--to adopt the evolutionary perspective, and it is important to evaluate the success of an approach which is familiar from its use elsewhere in sociology. Wilder holds the refreshing view that not only the researches of anthropologists on early mathematics, but also the conceptual categories and theories of anthropology itself, can be of great assistance to historians and sociologists of mathematics. He maintains the position that mathematics is a human artifact evident in many cultures in which pragmatic rather than idealist criteria of acceptability prevail. In a central section of his book the author sets out eleven "forces of mathematical evolution" (Wilder, 1974: 163) and ten "laws governing the evolution of mathematical concepts" (1974: 199-201). Wilder's eleven forces are environmental and hereditary stress, symbolisation, diffusion, abstraction, generalisation, consolidation, diversification, cultural lag and resistance, and selection. Many of these concepts were introduced by anthropologists but an exception seems to be his own notion of "hereditary stress"-or tension between ideas within mathematics itself--which he puts to effective use in analysing such phenomena as the reaction of the Pythagoreans to the possibility of incommensurables. One problem in connection with Wilder's (overambitiously labelled) "laws" is whether to evaluate their significance in descriptive or explanatory terms (Crowe, 1978: 101-102). There are difficulties with the latter view since in many of his illustrations that which is doing the "explaining" is not identifiable independently of that which is explained. However, if the emphasis is on description then the account is enlightening as when Wilder's law 4 links the intriguing phenomenon of simultaneous multiple discovery in mathematics to systematic changes in the conceptual structure of a theory. Law 6 is similarly suggestive for it seeks to point up the impetus given to mathematical innovations by developing social institutions (e.g. art). However, Wilder does take his emphasis on pragmatic criteria of acceptability in mathematics too far, for he refers to but does not systematically incorporate aesthetic considerations among mathematicians and he neglects to analyse changing perceptions of mathematical truth (Richards. 1980). In their turn, Fang and Takayama (1975) formulate a theory applying to the sciences which is one of sociocultural evolution, based upon innovation (via invention, discovery, alteration and diffusion), continuation (via socialism) and extinction. It is claimed that evolution raises "the upper level of societal potentiality" (Fang and Takayama, 1975: 127). They stress that elements of both solidarity and conflict are involved in changes and the basic idea of evolution is refined with the useful distinction between "local" and "global" developments which operates at both a conceptual and an institutional level (1975: 136-46). In a way which is suggested by a considera68 tion of Wilder's work the theory in its explicitly stated form seems in need of supplementation from structuralist ideas, but these are referred to (1975: 137) without being systematically incorporated. Fang and Takayama illustrate their theory using strategically important examples: Classical Greece, seen as the birthplace of the exact science; a critical phase in the development of Japanese mathematics; and striking innovations in Western mathematics (particularly the introduction of non-Euclidean geometries). In considering Greek mathematics the authors make considerable use of Weber's rationalisation theme. An original phase of rationalisation is viewed as concurrent with "a latent state of equalisation" and "the initial stage of mathematisation" (1975: 152). Part of the argument consists of saying that certain politico-legal developments and economic innovations were accompanied by equalisation (and presumably individuation) of citizens. The process of equalisation must be considered in conjunction with the existence of certain values, such as a stress on abstract intellectual ability and a public conscience. These were manifested in debate in the market-place, the social role of groups such as the sophists and the emergence, implicit and later explicit, of rules of inference expressed in logic and also in geometry (cf. Szabo, 1967). Geometry was favoured because it lent itself to axiomatic deductive formulation, thus supplying the truth or "certainty" sought by Greek philosophers. This attempt to develop concepts which can assist the analysis of an emergent rational viewpoint is certainly worth pursuing and suggestive of further comparative work (see also, Richter: 62-85). One such possibility is implied by Fang and Takayama themselves, for they confront Weber's claim that in China there was "no rational science, no rational practice of art natural science or technology" (Weber, 1951: 151) with Needham's work (1954-1976) and conclude that Weber's view has at least partially been refuted. The need to investigate the conceptual and institutional basis of Chinese mathematics--partly to assist further comparison with the Greek--seems pressing. However, more attention must be paid to the different senses in which Weber uses his key notion than is given by Fang and Takayama. In particular rational (or axiomatic) systems of concepts (as in Euclid) must be sharply distinguished om the rational employment of mathematics in the pursuit of practical ends and in empirical studies (in which the Chinese engaged). The approach of Wilder (1974) and Fang and Takayama (1975) may be characterised as historical analysis on the grand scale. For the sociology of mathematics to prosper there is a need for this type of study (with its tendency to disregard the strictest canons of verification) to be complemented by micro-studies of groups of mathematicians. Particularly effective in the latter connection is the work of C.S. Fisher (1966; 1967; 1973). Mathematics and Social Structure Focusing on some late Nineteenth Century mathematical work Fisher has provided an analysis of "the death of a mathematical theory" (1966: 137-59). He is concerned to account for the virtual disappearance of the "Theory of Invariants" a theory which he views--surely correctly--as "a social category within the world of mathematics" (1967: 217). Among algebraists this theory was believed to have been effectively killed off by work of the German mathematician Hilbert in 1893. but Fisher considers that Hilbert's paper took on symbolic meaning after the event and assumed the status 69 of an explanation for the theory's demise (Fisher. 1966: 158). Fisher provides evidence that, in fact, the beginnings of its decline originated earlier and are mainly to be accounted for by various structural factors which are necessary conditions for the maintenance and transmission of a theory. He directs attention towards: "the general zyxwvutsrqp environment in which the mathematics is done, the specialists' commitment to the theory, their relationship to their students, and the places in which they worked" (1967: 218: italics in original). He affirms that a speciality is practised in different institutional contexts and its fate becomes linked to the social conditions of the scholars working in those settings. Fisher analyses, for instance, the position in Great Britain where the theory started its life and points out that there were no schools of mathematical thought, for conditions in the universities were exceptionally unhelpful. On the other hand in the United States a productive school of invariant theory developed (at Chicago) but when graduate students left, they found little encouragement in other institutions towards research. Fisher reckons that Invariant Theory could arise again, but in general he ascribes its decline and subsequent dormant condition to structural factors. Some later writers (e.g. Fang and Takayama, 1975: 226-33) have contested the detail of Fisher's account, but its central importance is that it seeks to highlight such factors as the consequences of the way in which mathematicians are grouped, patterns of communication and--at any rate implicitly--the way power is exercised (e.g. by eminent mathematicians over their research students). In short one can say Fisher attempts to relate mathematical theory and social structure. The work of Bloor is also germane to the same theme. For as part of his general thesis that logic and mathematical inference can be grasped sociologically, he argues that theories and proofs--and therefore presumably also the demise of theories-are frequently arrived at by "processes of negotiation" among mathematicians (Bloor, 1976: 117-40). He illustrates by reference to Lakatos's analysis of the protracted debate over Euler's formula for polyhedra (Lakatos, 1963-64) which can be taken to show that the interaction between mathematicians is directly related to the successive modifications in the formulation of a result and its changing theoretical status. My own studies of the academic role in the U.K. and U.S. over the period 1971-81 also seek to analyse micro-structures. With particular reference to the disciplines of mathematics, civil engineering, psychology and classics, I have examined patterns in the role performance of university teachers, how they evaluate their job and what students expect of them (Startup, 1972; 1976, 1979; 1979b). With respect to research activity in departments in both countries one can show that while civil engineering tends to produce organic solidarity in the form of an elaborate division of labour and psychology facilitates a fluid pattern of research relationships, mathematics generates a modified tree structure with low researcher mobility and poor communication between subfields (Startup, 1979a: 66-75; c.f. Hargens, 1975). This kind of evidence is suggestive of the conception that the structure of the discipline--and for a scientific subject especially its theoretical structure-is interdependent with the social processes involved in the production and communication of disciplinary ideas. Most recently I have sought to examine this particular relation within mathematics, taking in both the research and teaching spheres. To illustrate with respect to teaching: in the mathematics department of a U.S. university there was controversy concerning the most effective way of training research mathematicians. Limited use had been made of the teaching method of the mathematician R.L. Moore (Moise, zyxwvutsrq 70 1965; Whyburn, 1970), a method which is axiomatic (Wilder, 1967) while placing little or no reliance on books. Instead the instructor challenges students on an individual basis by presenting them with problems and statements of possible theorems (perhaps numbered in order of increasing difficulty) for evaluation and, where appropriate, solution. The method involves the inculcation of self-reliance, competitiveness and pleasure in creative endeavour. Nevertheless, the experience with the method had been mixed. Significantly, most of the favourable experience with the method had been in topology and it seems that the relative narrowness of that field and the fact that it has a short "past" is highly relevant. In specialised areas of historically more developed fields such as algebra the use of the method tended to frustrate the instructor, for student progress was too slow. In summary, the utility of a particular kind of pedagogical relationship varied between fields of mathematics. Discussion and Conclusion The relatively underdeveloped state of work on mathematics does no credit to sociology. It seems that some sociologists are rather more interested in quantification as a source of legitimation for their own discipline than they are in seriously confronting mathematics as a central element of modern culture. Perhaps they are also overwhelmed by the vast edifice of mathematics-certified as objective truth--and made me feel that only limited "externalist" programmes of investigation are possible. The directness of Bloor's (1973; 1976) approach to mathematics with a perspective derived from the sociology of knowledge presents a refreshing contrast. That author develops a view of mathematics as an institution developing alongside experience of the physical world, no part of which is withdrawn from sociological analysis. However, it may be judged that Bloor does go astray in respect of his articulation of a relativist position. The quest for "alternative mathematics" is misguided in the absence of a conception of mathematics which goes beyond that evident in the experience of Western mathematics itself. This is in no way to deny the usefulness of exploring the ontological and cosmological perspectives within a culture from which mathematics is viewed (Richards, 1980; Restivo, 1981) and the extensive conceptual variation within the discipline. The essential feature of mathematics is that it constitutes a system at both a formal symbolic and conceptual level with immense generalising power. Certain elements which have received comparative analysis are purely conventional e.g. the base of a number system. In proceeding further there is a need to examine the connection between the general acceptance of mathematical ideas--for example "imaginary" numbers--and their utility. Putman (1975) has gone so far as to state that the criterion of truth in mathematics is the success of its ideas in practice. However, that a priori grounds and therefore can form a focal issue cannot be decided exclusively onzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ point for sociological analysis. Judging from the work of Wilder (1974) and Fang and Takayama (1975) the evolutionary approach has considerable life left in it, although these accounts are probably better construed as descriptive rather than exploratory in function. For the sceptic it may be worth emphasising the usefulness of some of the concepts employed: by Wilder, for instance, in examining tensions within and between systems of ideas, and by Fang and Takayama, in the analysis of an emergent rational viewpoint. Additional impetus may be provided to comparative work encompassing preliterate 71 societies but also--particularIy promising-China (Needham, 1956) and Japan (suggested by Shimodaira, 1972). The smaller scale studies of Fisher (1966; 1967; 1973) which analyse environment, group life and patterns of communication, are fundamentally concerned with the relation between mathematical theory and social structure. A still more general focus of empirical work is provided by the idea that the theoretical structure of mathematics—and one might add the inherent difficulty of its organising concepts (e.g. differential ratio and real number)--is inter dependent with the social processes involved in the production and communication of disciplinary ideas. An illustration is provided by the way in which particular fields of mathematics lend themselves to a special type of pedagogical relationship. A topical way of developing this theme even further is provided by the investigation of the impact of new technologies, e.g. the effect of electronic calculators on the teaching of arithmetic and the advent of computer assisted proofs (De MillozyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF et al., 1980). 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