The growth of mathematical knowledge—Introduction of convex bodies

Studies in History and Philosophy of Science 43 (2012) 359–365

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The growth of mathematical knowledge—Introduction of convex bodies Tinne Hoff Kjeldsen a, Jessica Carter b a b

IMFUFA, Department of Science, Systems and Models, Roskilde University, Roskilde, Denmark Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

a r t i c l e

i n f o

Article history: Received 30 May 2011 Received in revised form 23 December 2011 Available online 17 February 2012 Keywords: Introducing mathematical objects Representing Contexts

a b s t r a c t The article addresses the topic of the growth of mathematical knowledge with a special focus on the question: How are mathematical objects introduced to mathematical practice? It takes as starting point a proposal made in a previous paper which is based on a case study on the introduction of Riemann surfaces. The claim is that (i) a new object first refers to previously accepted objects, and that (ii) reasoning is possible via a correspondence to the objects with reference to which it is introduced. In addition Riemann surfaces are geometrical objects, i.e., they are placed in a geometrical context, which makes new definitions possible. This proposal is tested on a case study on Minkowski’s introduction of convex bodies. The conclusion is that the proposal holds also for this example. In both cases we notice that in a first stage is a close connection between the new object and the objects it is introduced with reference to, and that in a later stage, the new object is given an independent definition. Even though the two cases display similarity in these respects, we also point to certain differences between the cases in the process of the first stage. Overall we notice the fruitfulness of representing problems in different contexts. Ó 2012 Elsevier Ltd. All rights reserved.

When citing this paper, please use the full journal title Studies in History and Philosophy of Science

1. Introduction A number of well-established positions address questions concerning the general development of the Natural Sciences. Indeed, most preliminary courses in philosophy of science cover the ‘‘Big Three’’: inductivism, Popper’s critical rationalism and Kuhn’s theory of paradigms and revolutions. There is hardly any such established and general account on the development of mathematics. Attempts have been made, e.g., in Lakatos’ (1976) Proofs and Refutations, and more recently Breger and Grosholz (2000) collected a number of articles addressing this topic. The topic of growth of mathematical knowledge is also the focus of Kitcher’s (1984) The Nature of Mathematical Knowledge. Kitcher argues that mathematical knowledge is empirically founded. He holds that our present knowledge derives—via a chain of ‘rational interpractice transitions’—from the simple practices of dealing with physical objects in ancient Egypt and Babylonia. Inspired

by Kuhn, Kitcher holds that any given mathematical practice can be described by the following 5 components: Language (L), a set of metamathematical views (M), a set of accepted reasonings (R), a set of accepted questions (Q), and finally, a set of accepted statements (S). A mathematical practice is characterised by the tuple (L, M, R, Q, S). An interpractice transition is a transformation from one practice (L, M, R, Q, S) to another ðL0 ; M0 ; R0 ; Q 0 ; S0 Þ. (Note that not all components need to change.) Defending an anti realist view, he omits objects from this matrix.1 In opposition, we address questions regarding the growth of mathematical knowledge with the focus on the introduction of mathematical objects. Since objects are accompanied by properties, we assume that production of knowledge is tightly connected with the introduction of objects. Note that when we talk about ‘objects’ or ‘introducing’ objects it is not meant to have any ontological import. In the language of mathematics there are clearly certain terms that can be construed as being about objects and sentences express that these objects have certain properties.2

E-mail addresses: [email protected] (T.H. Kjeldsen), [email protected] (J. Carter) According to Kitcher ‘‘mathematics consists in idealised theories of ways in which we can operate on the world’’ (Kitcher, 1984, p. 161). Thus the subject matter of mathematics concerns idealised dealings with physical objects by an idealised agent. 2 Although this exposition is intended to be ontologically neutral, we believe that studies like this could throw some light on the debate about realism/anti realism. Carter (in press), for example, uses the study of Riemann’s introduction of his surfaces to advance arguments for anti realism. The case study is employed to substantiate the notions of truth and reference of traditional fictionalist accounts. 1

0039-3681/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.shpsa.2011.12.031

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By addressing the topic of introduction of mathematical objects directly our case studies reveal more aspects of object introduction than is expressible in Kitcher’s account. We consider the introduction of objects by investigating and comparing two case studies on the introduction of mathematical objects. The case studies concern Riemann’s introduction of Riemann surfaces and the development of convex bodies in the work of Minkowski. It should be noted that these case studies have originally been done independently. The study of Riemann’s work was performed with the intent to investigate how it is possible to introduce mathematical objects. The resulting picture is published in Carter (in press). We shall summarise part of these results. In this article it is our intent to test this picture in the light of Kjeldsen’s (2008) case study on the introduction of convex bodies. We address the question on how objects are introduced by studying the work leading to the first formulations of the concept of a Riemann surface and a general convex body. Taking this approach we complement the picture described in Lakatos’ Proofs and Refutations. In this essay the formulation of a proof is described as an ever recurring process. In the main example a naive concept—a polyhedron—and a statement about such objects are gradually transformed to a proof generated theorem.3 Lakatos does not say much about how the concept of a polyhedron is first arrived at—and perhaps not much is needed to say, since one could argue that it is a natural extension in 3 dimensions of the concept of a polygon, which in its turn is a basic geometric object. However, as we shall see, much more can be said about the introduction of more advanced mathematical objects. The picture we investigate involves two claims. The first is that new objects (at first) are introduced by referring to previously accepted objects. Or the new objects are sometimes even representations of previously accepted objects. The second claim concerns how we are able to reason about the new objects. Theorems are proved establishing relations between the new object and the object with reference to which it is introduced. It is then possible to reason via this correspondence. Furthermore, the new object—or representation—is sometimes in a different context than the object it is introduced with reference to. Hence, new definitions may become possible. By a mathematical context (or domain, see (Grosholz, 2000)), we understand a certain collection of a type of objects together with ways of handling these objects, such as methods for constructing new objects and proving theorems about these kinds of objects. Paradigmatic examples are Euclidean geometry and (axiomatic) set theory. In general we find that the case study on the introduction of convex bodies fits the given description. In addition we shall note some of the important similarities as well as differences between our cases that allow us to address the question of growth in mathematics. Although the cases by no means are identical, we identify two overall stages in their development. This makes it possible to address differences between the cases. Among similarities, we note that in the first stage of the development, there is a close connection between (the objects of) the contexts at stake, whereas in the final stage, the new object is given an independent definition. As for observed differences, in the case of the introduction of convex bodies, the relations to previously existing objects and problems concerning these play a crucial role in their success. In contrast, the Riemann surfaces very quickly became an interest in themselves. In general we emphasise the importance of representations in mathematics, and the fruitfulness of placing

problems in different contexts. By stressing the role of representations in mathematics, we are in (partial) agreement with Emily Grosholz (2007) who—arguing against reductionism—demonstrates that ambiguous representation is a prerequisite for the growth in mathematics. 2. Introduction of mathematical objects Before presenting the case study on Minkowski’s introduction of convex bodies, we shall summarise the results of Carter (in press) based on Riemann’s introduction of his surfaces. For more details on Riemann’s work we refer to Riemann (1851), Riemann (1857) and Laugwitz (1999). As mentioned above the proposal involves two types of claims. One has to do generally with how objects are introduced. The other concerns how it is possible to reason with these objects, determining which properties they may have. It is proposed that new objects are introduced with reference to already accepted objects. In addition they sometimes ‘‘live’’ in a different context than the one the referred objects are considered to be in.4 It is possible to obtain knowledge about these new objects via the correspondences to the objects with reference to which they are introduced, and because of the properties they can be seen to have in the—possibly new— context. We consider the introduction of Riemann surfaces in a number of steps. A major motivation for introducing the surfaces was to make it possible to handle multivalued functions such as the Abelian functions. Abelian functions are defined via integrals of raR tional functions, i.e., functions of the form Rðu; v Þdz, where the variables are connected by a polynomial expression f ðu; v Þ ¼ 0. Note that when expressing such a polynomial as a function v of u, one would often get a multi valued function. A simple example pffiffiffi is v 2  u ¼ 0, leading to v ¼  u. As a first step, Riemann (due to Gauss) thought of functions as mappings and not as analytical expressions as had been the custom since Euler. First he proposed to consider them as mappings from the complex domain A taking values on a complex plane B. But soon he suggested that they take values on a surface spread over the complex plane. As noted, Abelian functions considered as functions over the complex plane are generally not single valued functions and this is required when employing the techniques of integration of complex valued functions developed by Cauchy. In order to obtain single valued functions, Riemann represented Abelian functions by such surfaces extended over the complex plane. Loosely, the surfaces are composed of as many copies of the complex plane as the function has values. These surfaces are connected at so called branch points where the function only takes a single value and at infinity. The surfaces are then cut up along some curve between these branch points and re-connected such that it is possible to move between the sheets. A simple example to illustrate this is the surface assopffiffiffi ciated with the function w2 ¼ z, or w ¼  z. Since the function has two values for every z, the surface has two sheets or branches. For z ¼ 0 there is just one value, so the two branches are connected at zero. They are cut up along the positive x-axis and re-connected with the other as shown in Fig. 1 below. When Riemann introduced his surfaces as corresponding to Abelian functions he also envisaged a theorem establishing a connection between them, namely that an Abelian function is algebraic if and only if the corresponding surface is compact (Riemann, 1851, p. 39 and Laugwitz, 1999, p. 90).

3 Even though we do not address the question on how the concepts of a convex body and a Riemann surface develop, much more could be said. One interesting example, would be to study the process from Riemann’s first introduction of his surface and his formulation of the Riemann-Roch theorem leading to the versions of this theorem as presented in Weyl’s books. 4 In a previous article, Carter (2004), it was noted that sometimes new objects are constructed from previously accepted objects. One example was the K-group introduced by Grothendieck. In this case the new object was introduced—by construction—to the same context as the objects it was introduced with reference to.

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pffiffiffi Fig. 1. A picture of w ¼  z.

In addition, a Riemann surface is a geometric object, and as such it is considered in a geometric context. In the way that it is constructed, i.e., cutting it up and reconnecting, a Riemann surface may contain a number of holes. This observation led Riemann to introduce the concept of a connectivity number (which later led to the notion of the genus of a surface).5 In Riemann (1851) the connectivity number of a surface is defined as the number of cross-cuts, i.e., non intersecting cuts between boundary points, minus number of resulting components of the surface. Riemann proved this number to be a constant, and argued that the number of cross-cuts that it takes to make a surface simply connected is an even number, 2p. The p here was later defined, by Clebsch as the genus of the surface, and can be visualised as the number of holes of the surface. The concept of a connectivity number of a surface turned out to be a characterizing property of Riemann surfaces, and thus played a central role in Riemann’s fruitful question: ‘‘How many functions can be defined on a given surface?’’ which led to the famous Theorem of Riemann-Roch. More precisely Riemann’s version of this theorem is: Given a surface with a certain connectivity number 2p + 1 and specified number of points, m, where the function can have poles of first order, then there are at least m  p + 1 such functions. Commenting on the propositions made above, we claim that at first Riemann surfaces were defined with reference to Abelian functions—or more generally multivalued functions. Riemann himself wrote that a surface is a geometric representation of a multivalued function (Riemann, 1857, p. 90). Connections were then found, so that it is possible to translate properties about Riemann surfaces into properties of Abelian functions. Furthermore the surfaces are in a geometric context, which makes it possible, for example, to define the concept of a connectivity number. This concept again proves to be an invariant of such surfaces (which means that it is possible to deduce properties of all surfaces with a certain connectivity number). Finally, we note that the concept of a Riemann surface later was defined without reference to Abelian functions, see for example the definition of a Riemann surface as a certain manifold in Weyl (1913). (An n-dimensional manifold is a surface that locally looks like Rn .) We now turn to a comparison of these points in relation to Minkowski’s introduction of convex bodies. 3. Introduction of convex bodies To test the two general philosophical proposals on the introduction of mathematical objects we consider a detailed case study by Kjeldsen (2008, 2009) on the history of Hermann Minkowski’s introduction of general convex bodies into mathematics at the turn of the 20th century. First we present an outline of the results of the historical study, i.e., an analysis of Minkowski’s work from the point of view of how and why he introduced a concept of convex bodies. The analysis shows that it is possible to identify three phases in the develop5

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ment of the idea of a convex body into a research object in itself within parts of Minkowski’s work from 1887 to 1903. The transition from one phase to the next is characterized by shifts in what functioned as question generators in Minkowski’s mathematical practice. It is also seen that the production of knowledge in this episode can be described as a dynamical process, as an interplay between treating known or old problems in new ways by a change in context and posing and answering new questions. We then discuss the findings in terms of the two general claims about how it is possible to introduce mathematical objects and how it is possible to reason with them. 3.1. Three phases The trajectory of Minkowski’s research that led him to introduce convex bodies as a new mathematical object leads back to his work on the minimum problem for positive definite quadratic forms in n-variables, i.e. for the following forms:

f ðxÞ ¼

X

ahk xh xk ;

x ¼ ðx1 ; . . . xn Þ; ahk 2 R; ahk ¼ akh ;

f ðxÞ > 0 for x–0: The minimum problem was an important problem for the theory of reduction for positive definite quadratic forms. It is the problem of finding the minimum value of such quadratic forms for integer values of the variables not all zero. Regarding the minimum problem, Minkowski was inspired primarily by two sources: The letter of Charles Hermite to Jacobi, where Hermite proved that there exists an integer point x such that

p ffiffiffiffi 1 n f ðxÞ < ð4=3Þ2ðn1Þ D; where D is the determinant of the form (Hermite, 1850, p. 263); and a paper by Dirichlet, where he developed a geometrical foundation for the theory of positive definite quadratic forms in three variables (Dirichlet, 1850). Hermite’s letter and Dirichlet’s paper were published in the same volume of Crelle’s Journal in 1850. Inspired by Dirichlet’s work Minkowski changed the context for the minimum problem for forms in n variables: from conceiving and studying positive definite quadratic forms as algebraic objects to interpreting and studying them in a geometrical context. Actually, the geometrical interpretation of quadratic forms goes back to Gauss in 1831 (Gauss, 1863). He associated a positive definite quadratic form in two variables:

f ðx; yÞ ¼ axx þ 2bxy þ cyy with a lattice built of congruent parallelograms (see Fig. 2). In a rectangular coordinate system ðx; yÞ the level curves f ðx; yÞ ¼ k form ellipses. By a linear coordinate transformation the form f can be reduced to a sum of squares. For the form in Fig. 2 we have:

f ðx; yÞ ¼ 2x2 þ 6xy þ 5y2 ¼ ðx þ yÞ2 þ ðx þ 2yÞ2 The determinant of the coefficient matrix for the linear transformation is 1. Hence, X = x + y, Y = x + 2y defines a 1-1 correspondence between integer pairs (x, y) and (X, Y). The angle / between the two coordinate axes in the (X, Y) coordinate system is determined as

b cos / ¼ pffiffiffiffiffi ac with b = 3, a = 2 and c = 5 for the example in Fig. 2. For integer pffiffiffi pffiffiffi values of x and y the points ðx a; y cÞ are the lattice points in the skew coordinate system (X, Y), and they form the lattice. These lattice points are the vertices of the congruent (standard)

In Riemann (1857), the connectivity number is the least number of cuts along closed curves on the surface that it takes for the surface to be simply connected.

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of the lattice and associated bodies as well as their extensions to arbitrary dimensions. According to the summary,

Fig. 2. Geometrical interpretation of the quadratic form f ðx; yÞ ¼ 2x2 þ 6xy þ 5y2 . The smallest distance in the lattice is M1=2 ¼ 1. The bold lines represent the two coordinate axes.

parallelograms. The determinant of the quadratic form is equal to the square of the area of such a standard parallelogram. In the skew coordinate system (X, Y) (Fig. 2) f ðx; yÞ ¼ k forms circles since f(x, y) is equal to the square of the distance between the origin and the pffiffiffi pffiffiffi point ðx a; y cÞ. Hence, the quadratic form measures the distance in the lattice. In this interpretation the minimum problem becomes the problem of determining the smallest distance between the origin and a lattice point, or, which is the same, the smallest distance between two points in the lattice. Minkowski explained the geometrical interpretation of such forms in three variables in his probationary lecture for his habilitation in Bonn in March 18876. Here he reached an upper bound for the minimum through a geometrical argument. Around each lattice point, he placed spheres with the smallest distance in the lattice as diameter. Then he argued that since these spheres do not overlap and since they do not fill out a whole parallelotope the volume of such a sphere is less than the volume of a standard parallelotope (a standard parallelotope is the three dimensional version of the standard parallelogram for the two dimensional case which is illustrated in Fig. 2). From this inequality, Minkowski was able to derive an upper bound for the smallest distance in the lattice, thereby reaching an upper bound for the minimum problem. He claimed that the result could be generalized to n-dimension. He published a geometrical proof four years later, in 1891, along the same lines for the upper bound for the minimum for forms in n-variables. In this first phase of Minkowski’s work that eventually led to his introduction of the notion of a general convex body, Minkowski translated the minimum problem into a geometrical problem. In doing so, Minkowski found that he could translate the minimum problem for a positive definite quadratic form into a geometrical problem concerning lattices and thus obtain a shorter (and more natural—he claimed7) geometrical proof of the upper bound. Around the same time, in 1891, Minkowski began to investigate his geometrical method that had been so fruitful for his work on the minimum problem. This marks the transition into the second phase. Here he studied the lattice and bodies associated with the lattice, as he explained in the talk ‘‘Über Geometrie der Zahlen’’, which he presented in Halle in 1891. A summary of the talk is published in Minkowski’s Collected Works (Minkowski, 1891, vol. I, (1891)). In the talk Minkowski introduced the three-dimensional lattice as the collection of points with integer coordinates in space with orthogonal coordinates, and he defined the term ‘‘Geometrie der Zahlen’’ (geometry of numbers) as geometrical investigations

6 7 8

Minkowski’s manuscript for the lecture is published in Schwermer (1991). Minkowski (1887) in Schwermer (1991), p. 87–88. Minkowski used the German term ’’Eichkörper’’ in stead of the term ’’unit ball’’.

‘‘The speaker [Minkowski] asked mainly two questions concerning the lattice; they supplement each other in certain ways and they have the following in common: It is always, especially when the talk concerns space, about a very general category of bodies that are constructed in such a way that they circumscribe a particular lattice point—for instance the origin—in a certain way, and for such bodies, through the size of the volume of the body alone, a certain property with respect to the lattice shall always be achieved. The first category of bodies consists of all those bodies that have the origin as middle point, and whose boundary towards the outside is nowhere concave; and for that category the property in question reads: When the volume of a body of that category is P 23 then this body necessarily contains additional lattice points in addition to the origin’’ (Minkowski, 1891, vol. I, pp. 264–265, (1891)). Minkowski investigated the lattice and the associated bodies, not because they were interesting in themselves, but because they constituted a method that was useful in number theory as he explicitly pointed out: ‘‘Every statement about the number grid [lattice] has of course a purely arithmetic core’’ (Minkowski, 1891, vol. I, p. 264, (1891)). We can nonetheless see that Minkowski is beginning to single out ‘‘a very general category of bodies’’. At this point he did not give a precise definition of those bodies. They have not yet been crystallized into clearly definable mathematical objects. In the second part of the above quote we witness the fruitfulness of the change of context from algebraic to geometric treatment of positive definite quadratic forms. Here Minkowski referred to what later became known as his lattice point theorem with which he connected the geometrical property of volume of a certain body with the arithmetical question of whether such a body contains points with integer coordinates. In a manuscript that was read two years later at the International Mathematical Congress at Chicago in 1893, Minkowski developed his geometry of numbers much further. About the lattice he wrote that: ‘‘The deeper properties of the lattice are connected with a generalization of the concept of the length of a straight line by which only the theorem, that the sum of two of the sides in a triangle is never less than the third side, is maintained’’ (Minkowski, 1893, p. 272–73). He introduced what he called a radial distance (strahledistanz) SðabÞ between two points, a and b, and, what we today would call its associated unit ball Sð0uÞ 6 1, consisting of all points u for which the radial distance to the origin is less than or equal to 1. And he continued: ‘‘If moreover SðacÞ 6 SðabÞ þ SðbcÞ for arbitrary points a, b, c [. . .] Its unit ball8 then has the property that whenever two points u and v belong to the unit ball then the whole line segment uv will also belong to the unit ball. On the other hand every nowhere concave body, which has the origin as an inner point, is the unit ball of a certain [. . .] radial distance function [for which the triangular inequality SðacÞ 6 SðabÞ þ SðbcÞ holds for arbitrary points a, b, c]. .. . SðabÞ is called reciprocal if SðbaÞ ¼ SðabÞ without exceptions. This is the case when and only when the unit ball has the origin as middle point’’ (Minkowski, 1893, vol. I, pp. 272–273, (1893)).

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Today we would recognize a reciprocal radial distance for which the triangular inequality holds as a metric that also induces a norm. In this second phase where Minkowski investigated his geometrical method, we have seen how Minkowski conceived of nowhere concave bodies with middle point as geometrical objects that were interesting as useful tools to link geometry and arithmetic and to generalize the concept of distance. The transition from the second to the third phase is marked by yet another shift of focus in Minkowski’s work which appears clearly in a paper published in 1897. This was the first paper where Minkowski explicitly investigated nowhere concave bodies, or convex bodies as he then called them, for their own sake. This can, among other things, be seen from his introduction to the paper where he actually defined the concept of a convex body: ‘‘A convex body is completely characterized by the properties that it is a closed set of points, has inner points, and that every straight line that takes up some of its inner points always has two points in common with its boundary’’ (Minkowski, 1897, p. 103). Minkowski published four papers where convex bodies were investigated for their ‘‘own’’ sake, and he began to work on a foundation of the theory for convex bodies of which a longer unfinished manuscript was found among his papers when he died suddenly in 1909 of a ruptured appendix. In this third phase Minkowski defined convex bodies as a certain (geometric) object, a set, that has the property that was found important in the study of lattices, for each two points in the set, the line segment connecting these points will also be in the set. At this phase convex bodies were completely detached both from the context of number theory and from the objects introduced in the process of creating the notion of nowhere concave bodies. 4. Growth of mathematical knowledge Three phases were distinguished in the analysis of Minkowski’s work with respect to how and why he introduced the concept of a general convex body into mathematics. In the first phase he treated the minimum problem for positive definite quadratic forms geometrically, i.e., he represented quadratic forms by lattices. The quadratic forms were the objects under investigation. They functioned as question generators whereas the lattices through the geometrical interpretation, were used to get answers. In the second phase he turned his attention towards the geometrical method. Instead of asking questions about positive definite quadratic forms he asked questions about the lattices and their associated bodies with their connections to radial distances, that is he investigated the geometrical method. The lattice and the associated bodies were the objects under investigation. They functioned as question generators, studied in order to obtain answers in number theory. Finally, in the third phase he asked questions concerning the concept of a convex body at this point detached from quadratic forms and from lattices. In this phase Minkowski began a systematic investigation of convex bodies. Here the convex bodies functioned as question generators. They had become a research object in themselves and in this phase Minkowski initiated the beginning of a modern theory of convexity. During these three phases with their shifts in focus, Minkowski’s idea of a convex body developed in a dynamical process of treating problems (the minimum problem) in new ways, and asking new questions (about the lattice and its associated bodies). Returning to the claims made in the introduction, we note that in the first phase lattices are introduced with reference to previously accepted objects (which was our first claim), namely

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the positive definite quadratic forms (so that Minkowski could translate the problem of finding a minimum of the quadratic form into a question about the least distance between lattice points). However, in the third phase a convex body is defined as a set with certain properties found by studying lattices and associated bodies. Addressing the claim concerning the possibility of reasoning, it is also clear that certain properties of the lattices correspond, at first, to properties of quadratic forms and later, more generally, to results within number theory. Finally, also in this case the objects (the positive definite quadratic forms) are represented in a new context which makes new definitions possible. Thus, overall, the two cases display similarities. In the next section we consider what we see as important differences that will allow us to address further issues on the development of mathematics. 5. Discussion Overall both cases can be construed as having three phases. In the first (pre-) phase there is the study of Abelian functions or the study of problems concerning positive definite quadratic forms. In the latter case these objects or problems concerning them are represented in a new context. For Abelian functions this happens in the second phase. The motivation for a shift in context can be either that there are problems that cannot be solved in the original setting, as with the problem of the multiple values of Abelian function; or because the new context is more natural as was claimed by (Minkowski (1887), in (Schwermer, 1991, p. 87–88)). In the second phase objects in the new context are studied—or even the new context itself. In the last phase, certain characterising properties are identified and used to define the new objects. We noted above that this was the case in Weyl’s book. Also in the last phase Minkowski introduced and studied convex bodies. Thus we see that in the beginning of the process—the first stage— reasoning can take place by shifting between the different contexts. In the case of Riemann there is a shift between considering the algebraic descriptions of Abelian functions and their corresponding surfaces. In the case of Minkowski, there are positive definite quadratic forms that are represented by lattices from which certain properties are extracted which later led Minkowski to define the concept of a convex body. In the last stage the newly defined concepts are removed from the original setting of the introduction, as illustrated by the stipled line in Fig. 3 below. However, it is clear that the cases also display a number of differences. We discuss three such differences. The first concerns our claim that a new object is introduced with reference to a previously accepted object. The kind of reference relation between these objects may vary. In some cases the relation is a representation, in others the relation is an identity relation, based on the identification of certain properties that the new object is taken to have. Second, we note that the motivation for representing objects or problems in a different context varies. Overall the motivation for representing is to solve problems—or produce knowledge. The difference concerns which context these problems belong to, the ‘‘old’’ or the new context. Finally there is a profound difference between the originality of the contexts. We address these differences in more detail below. First we turn to the different type of relations that may hold between objects. Addressing the difference between the relations between a Riemann surface and its corresponding Abelian function on the one hand, and the convex body and lattices on the other, we need to be a bit more precise. It was claimed that the Riemann surface refers to Abelian functions. Recall that an Abelian function R is defined as an integral, Rðu; v Þdz, where R(u, v) is a rational function and the variables are connected by a polynomial expression f(u,v) = 0, which can be construed as a certain multi-valued

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Riemann surfaces

Abelian functions

Quadratic forms

Lattices

Covex bodies

Fig. 3. An illustration of the two stages involved in the introduction of objects.

function in the variable u. The Riemann surface is a representation of this function. It is thus a representation of part of the expression of an Abelian function. A convex body was singled out as the set that has certain properties that were deemed essential when studying the representations of number theoretical expressions in Minkowski’s geometrical context. So the relation between a convex body and a radial distance in the lattice, the so called Eichkörper (the unit ball), is that they have similar properties. In addition it makes sense to say that in the first stages of Minkowski’s work he represented the problem of finding the minimum value of a quadratic form in a geometric context. Thus problems are represented as well as objects. Our claim that ‘new objects are introduced with reference to previous accepted objects’ is similar in spirit to Kitcher’s project of arguing that interpractice transitions are rational. Recall that according to him, a mathematical practice can be described as a tuple (L, M, P, Q, S). Growth of knowledge is described as a transition from one practice to another, where change can take place in some components, but not necessarily all. That such a transition is rational means that ‘‘one must recognize the relation of the modified component to other components of the previous practice’’ (Kitcher, 1984, 193). We have just pointed to such relations between objects in different practices. Since Kitcher does not acknowledge objects among his components, such changes must take place in the language component, L. Thus at a first glance it seems that Kitcher’s model is able to account for the introductions that we describe, so that our account is a supplement to Kitcher’s project. Indeed, Kitcher does not write much about changes in the language component. When addressing changes involved in ‘questioninganswering’ (which is also involved in the examples we treat), he gives two examples. In the first, there is no addition to language, and even though he writes that there are additions to the language component in the second example, these additions are discussed in a footnote.9 However by addressing the question of object-introduction directly we point to factors that he miss. First we see that sometimes objects—the convex bodies—are introduced by stipulation. This is rational since they have properties that turned out to be useful when solving problems in number theory. Second, objects are introduced as representations of others. In particular, representing is missing from Kitcher’s account as well as the focus on contexts.

Furthermore all implications of ‘methods’ or techniques is not discussed by Kitcher. ‘Techniques’ is by Kitcher taken to be included in the ‘Reasoning’ component. In our case study on Minkowski’s work we note that methods play a decisive role. Recall that in Minkowski’s second phase the object of study was the lattices, the method that had been fruitful in finding minimum values. Thus a method—or a tool—can become the object of study. Returning to the differences displayed by the two cases, we noted that in both cases it is possible to talk about a stage of shifting between contexts; however the motivation for change of context is different. In the case of Riemann surfaces, these very quickly became the object of study for their own sake.10 For example, the investigation of how many functions can be defined on a given surface can be regarded independently of questions concerning Abelian functions. Of course, much of Riemann (1857) also concerns a generalisation of Abel’s work on these. However a striking feature of Riemann’s new geometric or ‘‘visual’’ approach is that notions from Abel’s theorem take a new meaning, as for example, the number p of integrals that can be interpreted in terms of the connectivity of the associated Riemann surface. See Laugwitz (1999), pp. 133– 139. In Minkowski’s work the lattices and nowhere concave bodies and the geometric study of these were mainly seen as a tool to obtain information about number theory. Minkowski is explicit in the first and second phase that the purpose of the shift in context is that it solves problems in number theory. It is only later that it becomes clear that properties from the studied context can be used to define a new object, a convex body. This has implications for the initial proposal we set out to test. One of the claims was that theorems are proved establishing relations between the new objects and the objects they are representations of, and that this enables reasoning about the new objects. In Minkowski’s case the ‘‘direction’’ of this reasoning is opposite: The correspondence is used to derive properties about number theory. Thus we have demonstrated the usefulness of representing— either by creating a whole new discipline as did Riemann—or by creating a new tool for solving problems in number theory, (a tool that actually also developed into a new discipline, namely Geometry of Numbers). We hasten to add that not all types of representing turn out to be fruitful. One interpretation of the conclusions of Kjeldsen (2009) is that it is precisely the relation to number theoretical questions that made Minkowski’s convex body a success. Conversely one can explain the failure of a similar concept introduced by Brunn because of the limitations of the context and the objects his objects were introduced with reference to. Brunn introduced ‘‘ovals and egg-forms’’ inspired by quasi-empirical two- and three-dimensional geometrical objects. The further development of Brunn’s theory was inhibited, partly because of his methods, and partly because this context did not provide interesting new problems. As we have seen, the correspondence between number theoretic problems and a geometric way of formulating these problems opened the way for considering convex sets in a n-dimensional space, and generated a number of problems that could be solved. Kjeldsen (2009), writes ‘‘in Minkowski’s case it [the geometric context] broadened the investigations and opened new possibilities, because methods from one field—geometry—were applied to problems from another field— number theory. This mixing of different mathematical disciplines, . . . is another significant difference between Brunn and Minkowski, which is a second aspect regarding the possibility of generating new knowledge’’ (p. 110).

9 The second example concerns the introduction of analytic geometry by Fermat and Descartes. The introduced language terms are, for example, a2 ; a3 referring to line segments. It is not mentioned how this introduction of the correspondence between algebraic equations and geometric objects made the way for an extension of the concept of a curve. 10 In the beginning only a few mathematicians can be said to have studied Riemann surfaces. Riemann’s student Roch completed the theorem that Riemann had begun, but he also died young. Among Riemann’s other ‘‘followers’’ were Dedekind who pursued other interests. Later mathematicians commented that it was indeed very difficult to understand his ideas. See (Gray, 1998).

T.H. Kjeldsen, J. Carter / Studies in History and Philosophy of Science 43 (2012) 359–365

By pointing to the fact that representing may not always be fruitful, we differ from Grosholz (2007). The main focus of Grosholz’s book is to argue for the fruitfulness—or even the requirement—of ambiguous representation in mathematics. She claims that both iconic and symbolic representation—accompanied by text—is essential to producing results in mathematics. With the above reservation, we agree on the usefulness of representing. However we would not stress the use of symbolic representation. If anything we would emphasise (with Peirce) the iconic character of the representations used here. A special kind of icons are diagrams, i.e., signs that represent relations. As an example—when representing the problem of finding the (minimum) value of a quadratic form in a lattice, certain relations between points in the lattice correspond to properties of the quadratic form (i.e., f ðx; yÞ ¼ k represents circles in the lattice). The lattice is therefore an iconic diagram, in the sense that it represents certain relations. In addition we would note that all representations presented here are not fully articulated. This is most notable in the case with Riemann’s first introduction of his surface. It is more like an ‘‘imaginary object’’ in need of further articulation. Last—but not least—a difference that would strike most people concerns the originality of the context in which the Abelian functions were studied.11 As noted, Minkowski used a context already developed by Gauss in order to obtain geometrically the theorem on the minimum of positive definite quadratic forms. Later this context was developed and extended to several dimensions. In the work of Riemann both the idea of the surfaces and their context is completely new. This meant that, in addition to developing the concept of these surfaces, their geometrical context needed to be developed in tandem. Scholz (1982) argues that Riemann’s notion of a Manifold presented in his 1854 paper ‘‘On the Hypotheses which lie at the Foundations of Geometry’’ can be seen as an attempt to found geometrically his concept of a Riemann surface (see also Ferreirós, 2007, p. 60). In addition it is widely acknowledged that many of Riemann’s ideas—even if brilliant—were vague and his general foundation lacking. Examples include his idea of a ‘analysis situs’ and his use of the so-called Dirichlet principle. In the light of the above, we emphasise that when we talk about objects as being in a certain context, it is often merely a way to talk about mathematical practice. Within the practice of dealing with mathematical objects, it may not be entirely clear which context is at stake. This is especially clear in the work of Riemann where the context is developed alongside the study of objects within it. After noting several differences, we wish to point out important aspects of the picture that remain. In both case studies it is the case that new objects, Riemann surfaces and convex sets, can be construed as being introduced at first with reference to previously accepted objects. In addition in the two processes there is reasoning based on correspondence between two different contexts. Overall we wish to emphasize the fruitfulness of studying problems in dif-

11

Laugwitz (1999) calls it a ‘‘flash of genius’’.

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