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Mathematics, Semiotics, and the Growth of Social Knowledge

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Mathematics, Semiotics, and the Growth of Social Knowledge Mathematics, Semiotics, and the Growth of Social Knowledge MICHAEL OTTE Any epistemology, any methodology, and any learning the­ 1650): the geometry of mechanical-metric activity. The ory as well, is based on a metaphysics, on an idea of what straight line in Cartesian geometry corresponds to an axis makes up the world and what brings about the things which of rotation or to the stiffness of a measuring rod. The other exist in the world. In his 1898 Cambridge lecture on the geometrical style is represented in the work of Desargues .. logic of continuity Peirce said, for instance: "Every attempt (1591-1661). The straight line of Desarguesian geometry is to understand anything - every research - supposes, or the ray of light or the line of sight. It is a geometry on which, at least hopes, that the very objects of study themselves are among other things, perspective in painting is based. Each subject to a logic more or less identical with that which we system is based on different basic concepts and fundamental employ" [CP 6.189]. And vice-versa any logic or method­ objects. In Cartesian geometry, these include the concept of ology is to be grounded in some subject matter. There is no "distance", or the circle as a geometrical object. Other struc­ thought without an object of thought, thinking always tures or concepts are defined in terms of these. The infinite means thinking about something. straight line in a plane or the infinite plane within space Prevailing epistemology and methodology considers, in appear as geometrical loci of all points that have the same contrast to the above, method to be independent of subject distance from two fixed, given points. For Desargues' matter. It gained its plausibility from Newton's great geometry, however, the infinite straight line and the infinite achievement, making mechanics the basic science, and it plane are basic elements. For nearly two centuries these two conceives of the world in terms of particles in motion. It also sides of geometrical thinking developed relatively indepen­ introduced a distinction between the things themselves and dently from each other into two virtually exclusive the manner in which they come about or how they come to theoretical systems. There were no aporias, but mutual be known. exclusion. Each system completed itself as a closed, This fundamental idea of explaining the world in terms of consistent theory. particular definite bodies in motion and interaction looks By means of the new instrument of arithmetic and algebra back on a long history and it has its merits and its strengths. developed by the early modern society of the European As everyone knows, the problem of mechanical motion con­ cities, predominantly in Italy and in the Netherlands, which centrated the efforts of many brilliant minds from the days were oriented towards commerce and the market, Descartes, of Zeno and it has been the focus of mathematical sciences as expounded in his 1637 Geometrie, wished to achieve since Galileo and the Scientific Revolution. Now, great something the Greeks had not attained, that is to introduce achievements are also the source of great errors or biases. a common bound into the totality of mathematical knowl­ It is a common prejudice to try to explain everything in edge, and to create the basis for further generalization by mechanical terms. There are areas or questions where the this systematic. Descartes uses the structure of arithmetic, world might perhaps better be conceived of in terms of signs and in particular the fact that "the entire arithmetic contains and interpretations, rather than in terms of objects and inter­ but four types of calculations", to classify all the problems actions. Bateson has given the following description of these of geometry and to present them consistently. After having two options. "The difference between the Newtonian world explained the parallelity of geometric construction and of and the world of communication is simply this: that the arithmetic operation by means of some diagrams, and after Newtonian world ascribes reality to objects and achieves having stressed that "all problems of ordinary geometry can its imJ5 tcity oy excluding the content of the context­ be constructed by exclusive application of the few things excluding indeed all- meta-relationships - a fortiori contained in the figures explained", he proceeds to criticize excluding an infinite regress of such relations. In contrast, the "Ancients" for obviously not having realized this, "as the theorist of communication insists upon examining the they would otherwise have shied away from the effort of meta-relationships while achieving its simplicity by exclud­ writing so many voluminous books on it, books in which the ing all objects. This world, of communication, is a order of their theorems alone shows that they were not in Berkeleyan world" [Bateson, 1973, 221]. possession of the true method which supplies all these 1'ntlr ~ at follow I shall try to outline some con­ theorems, but merely had picked up those which they had sequences of these different views with respect to accidentally come across" (Descartes' Geometrie, our mathematics. translation). In this role, algebra functions like a logic, an idea which Descartes vs. Desargues on geometry Leibniz, rather than Descartes, because of his own emphasis During the seventeenth century geometrical perception on the priority of form over concrete content, was to became separated, so to say, into two relatively distinct develop. "Leibniz thought that Descartes had stopped short, forms of geometry, into two different geometrical styles. and did not see his way through to a completely general One of these is represented by the work of Descartes (1596- abstract Universal Characteristic" [Hacking, 1984, 213]. For the Learning of Mathematics 17,1 (February 1997) 47 FLM Publishing Association, Vancouver, British Columbia, Canada Now Leibniz wanted his characteristic because he thought Poncelet speak of the etres de raison of algebra they do nof that truth is constituted by proof. Descartes did not believe mean that algebra is just a syntactical game but intend to say this. Both, however, shared the view that science and math­ that geometry should imitate and even surpass algebra in ematics aim at a characterization of individual objects or its capacity to deal with objects that are not strictly individ­ substances, rather than being concerned with the general ual and are not completely and individually describable and the implicitly and incompletely defined. according to Leibniz' principle of indiscernibility. From Formal proof is related to form rather than to content, the point of view of philosophical realism (as opposed to and to subjugate content to form, or the particular to the nominalism) the relational generality of the .continuum is general, one had to geometrize algebra. This idea occurred prior to the particulars of the discrete. This was, with respect to Leibniz when he dealt with systems of linear equations, to mathematics (not logic), Desargues' idea. Granger for trying to design an algorithm for solving them. In a series instance, writes about Desargues: "A "conic", for the of letters to l'H6pital, written in 1693, Leibniz stresses that Ancients or for Descartes is but the generic name of a vir­ the variables occurring in the matrix derived from a system tual and incomplete object, which becomes specific - a of linear equations are nothing but place holders within a circle, a hyperbola, a parabola - when the parameters are certain spatial arrangement, so therefore algebra should be determined. For Desargues it is a real object, completely understood as a branch of combinatorics rather than as determined in its projective nature; specification occurs only generalized arithmetic. But Leibniz' · views remained for accidental reasons" [Granger, 1968, 69]. ambiguous. On the one hand, the ultimate goal of scientific Desargues' work grew out of the practical tradition of. enquiry, which is in general only to be accomplished by applied geometry, as exemplified by the Renaissance trea- God through an infinite analysis, lies in the determination of tises on linear perspective. These treatises in the beginning individual substances. On the other hand Leibniz introduced concentrated mainly on recipes for foreshortening such key · relational thinking and the idea of general proofs into math­ elements as cubes, column capitals, and the like. Graduall~ ematical thinking. they became concerned also with the manner in which reg­ Only during the 19th century were were these insights ularities in natural space become transformed inter pursued further. Peirce states: "Logical algebra ought to be regularities within the image space of paintings. Therefore entirely self-developed. Quantitative algebra, on the con­ artists occupied themselves with perspective-constructions trary, ought to be developed as a special case of logical of accurate patterns (like tiled floors, squared pavements) algebra" [CP 4.134]. Algebra gains such a position only if which could_be employed to give a sense of spatial depth linked to geometry, from the concentration on relational · as part of the design of the picture. This loosened the fixa- thinking made possible by this connection of sfructijre and tion on geometrical semantics, shifting emphasis to the rules space. One can say therefore that Descartes has arithmetized of transformation. Still, emphasis remained more with ~ ' geometry. But he should rather have geometrized algebra. remained invariant under the projection. "Desargues was to Peirce praised his own logical characteristic for "its being concentrate instead upon the elements that remained· veridically iconic, naturally analogous to the thing repre­ unchanged. His important original contribution seems to sented, and not a creation of conventions. It represents lqgic have been the concept of invariance" [Field & Gray, 1987, because it is governed by the same law. Still more 28]. In this manner he paved the way towards a matherp.ati- closely, it resembles the application of geometry to algebra.
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