Chistophe Heintz

http://christophe.heintz.free.fr

In J. Goggin & R. Burkes (eds.)(2002) Travelling Concepts II: Frame, Meaning and Metaphor, Amsterdam: ASCA Press .

Although our present understanding of concepts leads us to a cultural and historical analysis of their meaning, such an analysis has rarely been explored in the domain of mathematics. It is because the discipline is believed to derive from reason only, that it is assumed that cultural analysis can have no bearing on mathematical concepts. Against this commonly held notion, I shall argue that the cultural analysis of mathematical concepts and their meaning is both possible and fruitful. Concepts are socially constructed and their meaning is the result of social interactions combined with cultural and historical phenomena. It is because scientific concepts, and in particular mathematical concepts, are no exceptions that cultural analysis can indeed take place in this purportedly most rational of domains.

I will argue for a cultural analysis of mathematical concepts by showing that the concept of infinitesimals, as it appeared in late 17th-century France, is susceptible to such an approach. I will explain the social, cultural and historical context from which the concept of infinitesimals arose and analyse the social processes through which it acquired its meaning. This will also involve a comparison of cultural analysis and the traditional history of mathematics, showing that the former has much more explanatory power than the latter. Finally, I hope to demonstrate how an appropriate concept of mathematical meaning renders the cultural study of mathematics possible. In doing so I hope to show that the concept of infinitesimals, like other mathematicals, has properties that allow for a cultural analysis of their meaning.

The concept of infinitesimals, as any mathematical concept, did not derive with a precise, definitive meaning from of an individual mind at a precise juncture in history. The early history of the concept of infinitesimals determined its future use, and contributed to the French 17^th-century calculus. The concept of infinitesimals travelled through time and its beginnings can be traced to Zeno’s paradoxes of the 5^th century B.C. The concept has also travelled though the disciplines of theology, natural philosophy (mechanics), geometry and arithmetic. Infinitesimals have also had currency in schools of thought from Leibnizian formalism to Malebranche’s Cartesianism.

But of course, mathematical concepts do not travel alone; they travel with people. The analysis of the history of mathematical concepts must therefore provide an account of mathematicians’ actions and thoughts. The migration of infinitesimals from Lower Saxony to France, for instance, was only possible as a result of Leibniz’s correspondence with Malebranche. However, the real introduction of the calculus in France is due to J. Bernoulli’s visit to Paris. When he arrived in 1691, he went directly to Malebranche. This move was decisive, for in Malebranche’s room, he met the Marquis de L’Hopital, whom he taught the calculus during the winter of 1691-1692. The result of this tuition was the Analyse des infiniments petits, which became the French reference book in the calculus for a century. Malebranche played an essential role in all of the above. He was a catalyst in the process of the ‘conversion’ of French mathematicians to the calculus, although he did not contribute to it in any way. One can distinguish two moments in the process of ‘conversion’. The first of these is the moment at which a given mathematician becomes acquainted with the calculus, and the second is the point at which the mathematician becomes convinced of its worth and begins to work with it. Malebranche proved to be indispensable to both stages.

Although the calculus was available to French mathematicians as early as 1684, with Leibniz’ Nova Methodus, it was only after Leibniz personally convinced Malebranche of the importance of the calculus that contemporary mathematicians began to consider this new theory. Because of his status as a follower of Mersenne, Malebranche was an important European figure and a promoter of sciences. As well, his interest in mathematics made him the link between the source of the calculus and French mathematicians, so that he became the leader of the movement to promote the calculus in France. All of these events can thus be the object of cultural and social analysis.

The introduction of the calculus in France was accomplished through what the sociologist Boudon (1979) has called the ‘imitative process’. Boudon concludes from Hagerstrand’s study of the diffusion of agricultural innovation in Sweden that the adoption of a new technique is a process that requires the ‘confidence’ of social actors. This confidence can only be attained by the exposition of a ‘personal influence’, following which the new technique spreads because of the ‘imitative dimension’ of social actions. In 1690, the calculus was a new technique and one may recognised in Malebranche, and later in the Infinitesimalists, the personal influence which was essential to its dissemination. As was the case with the Swedish agricultural innovation, the existence of the new technique alone was not sufficient to overcome the intrinsically convincing traditions that were the Cartesian and synthetic practices of mathematics. The imitative process must have been decisive in academicians’ progressively changing opinion on the subject and later, that of a wider community of mathematicians.

The social nature of the debate that leads to the recognition or rejection of mathematical concepts can be seen in the infinitesimal’s ‘conquest’ of the French Royal Académie des Sciences. The concept of the infinitesimal was not in line with Cartesian principles. The concept’s introduction in France therefore, met with strong opposition, which culminated in the dispute at the French Académie between the Malebranchists, or Infinitesimalists, and the Finitists. The former group of mathematicians was so named for their association with Malebranche, who received them regularly in his room at the Oratoire, which became their headquarters. By 1699, they had developed so much interest in the calculus that the Malebranchists and the Infinitesimalists became one group which struggled for the recognition of the calculus. The recognition, largely due to Leibniz, that the calculus constitutes an independent field, gave the Infinitesimalists a definite object for which to fight. The ultimate basis of their unity was that they constituted an oppositional body, so that the debate had a politically strategic side. As Merthens has argued (1994) mathematics, like any other science, is bound to be political. Despite the Academy’s policy to maintain ‘the purity of science’ by forbidding theological and philosophical discussion, the debate on the calculus contained obvious political strategies meant to win the approval of the scientific community (see Mancosu, 1989:239-40). In 1706, the commission that judged the disputes had to take into account the composition of forces within the Academy (predominantly Infinitesimalists) and, therefore, the Finitists were asked to stop the dispute.

Hence, the introduction of the calculus in France was the outcome of a dispute led by groups of mathematicians. Consequently, the success of the calculus is the result of the processes and strategies of the Infinitesimalists. As Paolo Mancosu has written, "mathematics and its development are due to human efforts and not only to the soundness of the ideas involved" (1989:245). An analysis of human actions involved in the introduction of the calculus in France reveals them to be the actual causes behind of the success of the calculus. Thus we are led from the notion that mathematical concepts do travelle to the social and cultural analysis of mathematicians’ behaviour. Mathematicians as social actors succeeded in socially imposing the concept of infinitesimals as a genuine mathematical concept. What is more, all of these social events and the strategies employed are cultural in nature and cannot be understood in any other way.

The strategies mentioned above are cultural and are executed in culturally appropriate settings. They acquire applicability through the use of elements of culture. The victory of the calculus against what Varignon called ‘old-style mathematicians’ is attributable to existent values at the time. The calculus was a new theory, and its recognition by the community of mathematicians was controversial. Mathematicians in the 17^th century were resistant to the notion of infinite quantity. The lack of clarity of the concept of infinity also had some consequences for the foundation of the calculus. Comment: this note should end the following sentence In particular, infinitesimal quantities have no rigorous meaning Comment: the note 2 should be here. Sometimes they are used as finite quantities, in equations of the type (y.dx)/dx=y, and sometime as zeros, such as in x+dx=x. Such difficulties were overlooked because of the correspondence between the calculus' usefulness in solving a wide number of problems, both mathematical and physical, and the values which were common in the late 17^th and 18^th century France.[Please, do not go to the line] over the course of the scientific revolution it became apparent that mathematics could tell us something about the world. And indeed the calculus proved to be readily applicable in mechanics. This gave to the calculus the advantage of being useful, while, at the same time, the utility of science was thought to be the best reason for doing scientific research. Hence, the calculus developed in spite of its lack of rigour. The advent of the Enlightenment also saw a new philosophy of mathematics, more confident and formalistic, germane to the growth of analysis. The cultural climate of confidence in the progress of mathematics and its applications account for the success of the calculus and the outcome of the dispute which took place at the French Academy. The concept of the infinitesimal in the 17^th century was, therefore, born out of the cultural context in which it was conceptualised..

Socio-cultural components do more than just allow mathematical concepts to come into being. These components also act on the content of the concept. Ramifications, traditions and histories are conflated in the current usage of concepts, and mathematical concepts are no exception. The causal role of socio-cultural components on the meaning of mathematical concepts can be clearly seen in the case of the 17^th-century concept of infinitesimals. At that time, the status of the infinitely small was problematic, yet while Leibniz sometimes gives it a purely formal status, the French Infinitesimalists adopt a very realistic stance. This stance has socio-cultural causes. In this respect, I would argue that the Infinitesimalists emerged as a special interest group within a social context, which determined their constitution and their behaviour toward the calculus.

one might characterise the Infinitesimalists of the late 17^th century as a small, threatened group. This explains their ‘pollution-conscious’ behaviour toward infinitesimals: they adopted a categorical stance which asserted the real existence of infinitely small quantities, and strongly lamented Leibniz’ hesitations with regard to the nature of those quantities. Their eagerness to go forward and to show ever more of the potential of the calculus may be understood as a function of their social situation. This may also account for the fact that they barely took the time, under the pressure of the Finitists, to stop and think about the foundational problem. Their strategy of justification may be likened to a strategy frequently adapted by members of what is known as the nouveaux riches who aspire to aristocratic status through ostentatious displays of wealth. Infinitesimalists, having shown the usefulness and applicability of the calculus , called also on previously well-known mathematicians to support their claims for recognition. Thus, Varignon asserts that "Mr. de Fermat luy-même" used approximation.

The use of concepts and methods determines their meaning, hence the Infinitesimalists determined the meaning of concepts and methods of the calculus. In the present case this is all the more apparent as these concepts and methods were not yet clearly defined and their use was not strictly regulated. More concretely, the above strategies clearly influenced the practice and notion of the calculus. The realist philosophy on infinitesimals allowed for the bold development of equations with evanescent quantities. The legitimisation of their approach through references to canonical works forced them to establish their continuity with tradition. Moreover, the emphasis on results granted the continuation of the development of the theory, while controversy over the calculus constituted a social event that determined its content. In order to answer Rolle’s arguments against the foundations of the calculus, Varignon gave a dynamic explanation, by drawing on Newton’s fluxion. Newton’s work, therefore, influenced the French calculus well before his physics had arrived in France. Varignon justified operations with infinitesimals with the intuitive idea of continuously decreasing and vanishing quantities, and this intuition sustains the contemporary concept of limit. Rolle presented three counter examples to the methods of the calculus, and although he made mistakes in his proof, one can see in Rolle and Varignon’s arguments what Lakatos calls the dialectic of proofs and refutations. The meaning of the concept of infinitesimals is framed by precisely this dialectic. And Rolle’s counter examples are indeed special cases for the calculus. For instance, Varignon’s answer to the second counter example could be qualified as ‘concept-stretching’, for the method of the calculus is applied in a case not previously accounted for. The social context, that is the mathematical values of efficiency, and the fight for recognition, induced the Infinitesimalists to make of the 18^th century calculus an aggressively assertive conqueror who, at the same time, was slowly framing notions and rules.

The idea that mathematical concepts are the product of socio-cultural elements is very unorthodox. Mathematics is commonly seen as being pure, necessary and thus out of the reach of contingent historical events. And mathematics indeed seems totally unrelated to historical, social and cultural events. An account of mathematical concepts that portrays them as socio-cultural products must also account for the strong intuition that they are not. I have shown that different cultures, disciplines and historical periods have framed the meaning of the concept of infinitesimals, but what makes it specifically a mathematical concept? The concept of infinity migrated from theology, philosophy and physics to mathematics. The mathematical revolution of the calculus corresponds to the creation of a meaning for the concept of infinity that is proper to mathematics. The mathematical concept of infinity is created thanks to the emancipation of the use of the word ‘infinity’ in mathematics from the use of the same word in other disciplines. This emancipation occurred as a result of the creation of a specifically mathematical discourse on infinity. The debate at the Académie Royale des Sciences is a major event in this emancipation. Vocabulary and arguments specific to the debate situated the concept of infinity within the corpus of mathematics. The introduction of concepts to the mathematical corpus is also realised with the use of symbols, called mathematical symbols, which allow for a specific mathematical manipulation. In the case of infinitesimals, mathematical symbols were introduced by Leibniz, hence his historical importance in the development of the calculus.

A concept acquires mathematical meaning by emancipating itself from the discourses of other disciplines and from socio-historical objects and interests. Note however, that the process of the mathematisation of concepts is still a socio-cultural one and that a concept’s emancipation can only be relative since mathematical concepts are cultural objects. Concepts are accredited with having entered the cultural corpus of mathematics through social interactions and negotiations as well as through those socio-cultural processes which determine the meaning of the mathematised concept. And this is precisely what I have been arguing for in the case of the concept of the infinitesimal.

The necessity of mathematics motivated many philosophers and historians in the field to adopt a view that denies the possibility of culturally and socially analysing its history. This view, which Bloor referred to as the ‘realist-teleological philosophy of mathematics’, describes the history of mathematics as being nothing other than the necessary development of initial notions. It is realist because it assumes that there is a mathematical reality that is described by mathematics, and it is teleological because the clear description of mathematical reality is the target of mathematics and the goal of its practitioners. In this view, the calculus was bound to exist in its contemporary shape as soon as numbers were applied to geometrical objects. Further developments of the calculus over time would be seen as intuitive approximations of mathematical reality, which would inevitably have been discovered by one great mathematician or another. From the realist-teleological point of view, the cultural and social analysis of mathematics would be restricted to the inquiries of mathematicians or to the historical ‘destiny’ of mathematics defeating cultural prejudices. This is to say that the idea intrinsic to the calculus succeeded to convince mathematicians even though they were culturally prejudiced by theological and Cartesian ideas. Against this view, I have argued that socio-cultural elements effect the very content of mathematics, and indeed, the meaning of mathematical concepts.

Two facts render socio-cultural analysis relevant for the understanding of the history of the content of mathematics. First, mathematical developments are under-determined by any state of mathematics. Second, mathematical concepts are not descriptive. mathematics could have evolved differently than it has. With the calculus, we have an example of an alternative development of the concepts of the 17^th century. In the 19^th and 20^th centuries, the concept of infinitesimals was replaced by the concept of limit. According to realist-teleological versions of the history of mathematics, this was the discovery of the real foundations of the calculus. In the 1960s however, Robinson provided an alternative rigorous foundation to the Calculus. The theory, called non-standard analysis, provides a definition of a set *R that contains both real numbers and infinitesimal numbers. It is an alternative development of the calculus that does not reject Infinitesimalists’ assertions relative to the existence of infinitesimal quantities. Indeed, a number of contingent historical events prompted the calculus of the 19^th century to discard the concept of infinitesimals for the concept of limit, and to evolve toward the classical calculus rather than toward non-standard analysis. One of these events was Varignon’s argument during the debate at the French Academy, where he defined infinitesimal quantities as ‘evanescent quantities’, thereby privileging Newton’s interpretation of the calculus rather than Leibniz’. There is under-determinism in the development of mathematics, and this has allowed socio-cultural factors to play a crucial role in its growth.

Furthermore, the fact that mathematical concepts are not simply descriptive and are never used in precisely in the same way makes it possible for socio-cultural factors to play a crucial role. This is also true for non-mathematical concepts because when a concept it is used, it is done so in a new context or situation that has never happened before. Situations in which a given concept has been used can only be identical to a certain degree. Judgements concerning the degree to which a new situation corresponds to previous ones in which the concept was used, are precisely judgements concerning the correctness of the concept’s use. such judgements are made by people who are culturally positioned and who interact socially, they are socio-culturally biased judgements.

These considerations also apply to mathematical concepts. Sometimes, identity judgements are not entirely obvious to the community of mathematicians. It is in such cases that socio-cultural factors are most apparent. Such is the use of the concept of infinitesimal dx in the equations (y.dx)/dx=y and x+dx=x. For Rolle, dx was not given the same meaning in the both equations, while for Infinitesimalists, dx had precisely the same meaning in both cases and was used in accordance with its definition of requisite identity. A judgement of identity is thus being questioned among professional mathematicians. As I have tried to show, the outcome of this questioning has been informed by socio-cultural factors. This is the case because a definition cannot be more than a special use of a concept—in a specific situation, in a single sentence—even if this sentence is later treated as a reference for the future use of that concept. It cannot be descriptive, even in mathematics.

A mathematical definition, however, is highly programmatic, because it must be referred to at each step of theproofs where the concept defined is involved. These considerations have two consequences. The first consequence is that the meaning of mathematical concepts will be determined to a certain degree by all uses of these concepts as well as by their definitions. Any use of a concept is, to a certain degree, programmatic because it gives indications concerning its proper use. Cases where a proof based on the use of a specific mathematical concept acquires the status of canonical procedure for future demonstrations, are instances of highly programmatic usages of a concepts, although the proof is not a definition. It is therefore, in the everyday practices of Infinitesimalists that we see the meaning of the concept of infinitesimals being framed, rather than being definitively set by a canonical definition taken out of a referential text. This becomes all the more clear when one considers that these first uses of the concept were intended to fix norms for its use in proof procedures. The second consequence is that past uses of a mathematical concept cannot completely determine its future uses, so social and cultural factors must intervene to complete the determination. In sum therefore, I hope to have shown that cultural factors did indeed impact on the case of the concept of infinitesimals in the 17^th century.

The socio-cultural analysis I have sketched above is justified for two reasons. First, such forms of analysis are grounded in a concept of meaning that is successfully used and argued for in cultural analysis. Second, the socio-cultural analysis has more explanatory power than a traditional history of mathematics is able to provide. The concept of meaning I have used and developed is based on the following notions:

1) the meaning of a given concept is constituted by its history. This history may involve the migration of the concept through cultures (cultures germane to historical periods, geographical areas, disciplines, schools of thought, specific institutions and etc.). These cultures therefore frame the meaning of the concept.2) Socio-cultural factors act directly on the meaning of concepts, because their ‘proper’ use is socially established. Ultimately, their meaning is the outcome of culturally situated social interactions. 3) Concepts’ meaning is to be found in the use we make of them. each use of a concept is programmatic: it determines, though not fully, its future uses.

These characteristics of the concept of meaning are not new. What may appear more controversial is their application to mathematics. If they are grounded and well argued, however, and I think they are, then there is no reason why they should not be applicable to mathematical meaning.

I have also sketched three specificities of mathematical meaning: 1) because mathematics constitutes an autonomous discourse, mathematical meaning is rendered almost entirely independent from the discourses of other disciplines. 2) Definitions in mathematics are given programmatic strength because mathematical discourse makes constant reference to them. This has no direct equivalent in day to day discourse or in any other field of knowledge. 3) Symbols are central to modern and contemporary mathematics and determine a particular way to use concepts.

because of its ‘teleological-realist’ bias, the traditional history of mathematics tends to reduce history to the genealogy of mathematical notions. Social history and cultural analysis go further, in that they assert that mathematical concepts themselves are to be explained by the use mathematicians make of them. The uses of concepts are in turn explained by the socio-historical situation of those who use them. It is not Reason that speaks through mathematicians, but rather it is the accountable speech of mathematicians that makes reason.

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