A Role for Representation Theorems

A Role for Representation Theorems Abstract I argue that the construction of representation theorems is a powerful tool for creating novel objects and theories in mathematics, as the construction of a new representation introduces new pieces of information in a very specific way that enables a solution for a problem and a proof of a new theorem. In more detail I show how the work behind the proof of a representation theorem transforms a mathematical problem in a way that makes it tractable and introduces information into it that it did not contain at the beginning of the process. 1. REPRESENTATION THEOREMS Representation theorems play a pivotal role in the advancement of mathematical knowledge: the construction of a new representation, or a change of representation, is a typical move that mathematicians employ to approach and solve problems (see, e.g., [Alexander, 1923; Erdős et al., 1966]). A representation theorem, in a broad sense, is a theorem that states that a given structure with certain properties can be reduced, or is somehow isomorphic, to another structure. Thus, it aims at establishing a mapping between elements and relations of a certain kind and other elements and relations of similar or different kind. In this way, it enables the transfer of certain properties from a source to a target, making it possible to treat problems in the latter with knowledge about the former. This construction, and the consequent transfer, requires a manipulation of the target whereby pieces of information are introduced into the target that it did not contain at the beginning of the process. Such a construction might end up with a novel concept or even a new theory — algebraic topology is a paramount example in this sense. This construction unfolds in a way that Imre Lakatos [1976, pp. 89–91] sketches in his account of theoretical novelty in mathematics and fits Emily Grosholz’s [2007] account of productive ambiguity in mathematics — that is, the use of multiple representations in mathematical problem-solving. Not only is the construction of a new representation of mathematical entities a powerful engine for advancing mathematical knowledge (see [Byers, 2007; Grosholz, 2007]), but a new representation is the endpoint of a process of assimilation of mathematical entities [Thomas, 2011]. The assimilation transforms the entities of the problems and eventually leads to a new representation theorem, which often establishes a formal reduction or an isomorphism, or a partial isomorphism. An entity is transformed and remodeled in ways that change its original content and meaning, as it no longer ‘denotes what it set out to denote: that its naïve meaning has disappeared and that now it is used [...] for a totally different, novel concept’ [Lakatos, 1976, p. 90]. So a new representation produces a multivalent entity, a ‘hybrid’ in Grosholz’s terms, which can be employed as a hypothesis to solve problems in the several fields from which it emerges and partially merges: the multiplication and juxtaposition of representations of a mathematical object enable us to integrate features of multiple fields that ‘provoke discovery in unexpected ways’ [Grosholz, 2000, p. 82]. Thus in this paper I set out to investigate the role of representation theorems in a way that fills some gaps in Lakatos’s account of theoretical novelty in mathematics and that specifies some of the heuristic processes involved in the growth of mathematical knowledge at a fine-grained level so enriching the heuristic processes detected by Grosholz in her account of productive ambiguity in mathematics. 2. INTRA-DOMAIN REPRESENTATION THEOREMS In a more technical sense a representation theorem is one stating that an abstract structure with certain features is mapped, or reduced, to another more concrete structure and vice versa. Representation theorems are widespread in mathematics: in algebra, functional analysis, or geometry, just to mention few cases. A remarkable example in algebra is Cayley’s theorem [Cayley, 1854; Burnside, 1897]. This theorem, whose origin and relevance have been investigated in detail (see in particular [Chowdhury, 1995; Chakraborty and Chowdhury, 2005; Pengelley, 2005]), establishes an intra-domain result: the mapping is between entities within the same domain and of the same kind — groups, that is, algebraic ones. More precisely, Cayley’s theorem originally states that: if the entire group is multiplied by any one of the symbols, either as further or nearer factor, the effect is simply to reproduce the group; or what is the same thing, that if the symbols of the group are multiplied together so as to form a table [as in Figure 1], that as well each line as each column of the square will contain all the symbols l, $${\alpha,\beta}$$ ... . [Cayley, 1854, p. 124] Fig. 1. View largeDownload slide Multiplication table [Cayley, 1854, p. 124]. Fig. 1. View largeDownload slide Multiplication table [Cayley, 1854, p. 124]. In modern terminology, Cayley’s theorem states that every group of order $$n$$ is isomorphic to a group of permutations on $$n$$ letters, or that every finite group is isomorphic to a subgroup of the symmetric group. In order to establish this theorem, he introduces the multiplication table of a finite group, Figure 1, and states that an abstract group is provided by its multiplication table. It is worth noting that the modern formulations of Cayley’s theorem employ the notion of isomorphism, which is not contained in its original formulation. Nonetheless Cayley seems to be aware of it when he notes that the cyclic group of order $$n$$ “is in every respect analogous to the system of the roots of the ordinary equation $$x^n-1=0$$” (ibid., p. 125). As a matter of fact, the very notion of group was not there at that time and it is just with this theorem that we see the very first attempt of defining the modern (abstract1) concept of ‘group’. In Cayley’s words: ‘A set of symbols $$1, \alpha, \beta, \dots$$, all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a group’ (ibid., p. 124). In this sense, Cayley provides a remarkable conceptual novelty in mathematics, ‘the birth of one of the most essential concepts of modern mathematics’ [Pengelley, 2005, p. 7]. Groups, and later an abstract view on them, arise from the study of two phenomena. The first one is the study of permutations: ‘clearly permutation groups served Cayley as a prime motivating example’ [Chakraborty and Chowdhury, 2005, p. 277]. More specifically the problem at stake was explicitly the action of some functions on the roots of some polynomials, as emerging from Galois’s work. As Cayley himself notes, ‘the idea of a group as applied to permutations or substitutions is due to Galois, and the introduction of it may be considered as marking an epoch in the progress of the theory of algebraic equations’ [1854, p. 124]. More generally, the study of collections of permutations of a set of specific entities is the very first step for the construction of the notion of group and, later, of an abstract view of groups. The second phenomenon comes from outside algebra and mathematics, namely from physics — geometrical optics: ‘Cayley’s unexpected discovery of a non-abelian group of order 6 in the practical context of geometrical optics, served as the trigger for generalizing the group concept’ [Chakraborty and Chowdhury, 2005, p. 278]. In more detail this discovery was suggested by ‘Cayley’s discovery of the six transformations that leave the equation of the secondary caustic unchanged, and his realization that these transformations form a group under the composition of mappings’ (ibid., p. 277).2 In turn this new finding has been reasonably made possible by a previous, and more general, issue concerning transformations of variables and their effect on the equation: ‘for one who had discovered the non-associative algebra of octaves (Cayley numbers) very soon after the discovery (1843) of the division ring of quaternions by Sir William Rowan Hamilton and had worked on invariants, it would have been natural for Cayley to ask himself the question: what, if any, transformation of variables leaves the equation unchanged?’ (ibid.) In order to get his theorem, Cayley put forward first of all a process of assimilation3 of mathematical entities that have been treated as distinct so far — invertible matrices (under multiplication), permutations, Gauss’s quadratic forms, quaternions (under addition), and various kinds of elliptic functions. In modern terms, these are more concrete structures, while a group is an abstract one, which is mapped onto them. In order to produce this notion of group, Cayley’s theorem builds upon the seminal idea of studying the action of an entity on itself: in effect the symmetries of an entity become a subject of inquiry in themselves. This notion is the one that triggers a new representation: it is the property by means of which similarities among these entities are highlighted, generalized, and then unified into a new concept. A representation requires that we highlight certain features of the entities of the problem, search for similarities among them — and deliberately neglect others. Which features to highlight and which to neglect is governed by the problem to be solved and the corpus of existing knowledge. In this case, as we have seen, the highlighted features are permutations and symmetries. This crucial move generates not only a new representation and viewpoint on known entities, but also a new technical concept (i.e., ‘group’) by means of which other mathematical entities can be investigated. This new approach and representation enable Cayley to advance mathematical knowledge and problem solving in algebra: just to mention a few cases, he determined all the groups of orders four and six, showing that there are two of each (again, by exhibiting their multiplication tables). Moreover, he showed that there is exactly one group of a given prime order. In this sense the notion of group enables a classification of mathematical entities. Of course, today we know that Cayley’s theorem is not so effective for studying a group, nevertheless Cayley’s steps towards an abstract view of groups can be legitimately seen as ‘a remarkable accomplishment at this time in the evolution of group theory’ [Kleiner, 2007, p. 31]. This theorem exhibits well the several roles a representation theorem can play in mathematics. First of all, it provides a generalisation and unification of properties and entities. It makes a domain more compact by giving a uniform treatment of previously unconnected or heterogeneous mathematical entities. Accordingly, it boosts mathematical problem solving, for not only can a more general concept be used to approach distinct entities, but local results and techniques can be transferred from one kind of entity to another kind, for instance from elliptic functions to invertible matrices. Thus it also provides a way of classifying entities and, accordingly, of seeking and establishing relations among them. Moreover, it enables an abstraction that may end up with an axiomatization, as happened in the case of the abstract theory of groups. Last but not least, it suggests lines of research: for instance it seems that Cayley was tacitly assuming that his groups were finite and it was not clear if his results could hold also for infinite groups — in effect, Cayley’s results were later proved to hold also for infinite groups. So the final outcome, the theorem, is important, but the very construction that leads to the assimilation of entities is the key and this representation theorem shows how conceptual novelty and change can take place in mathematics.4 It brings to light both a new object — the notion of group — and new features of already known objects — i.e., invertible matrices under multiplication, permutations, Gauss’s quadratic forms, quaternions under addition and various kinds of elliptic functions — by means of gradual steps that remodel the entities by adding pieces of information (properties, relations) to them. In his account of mathematical novelty, Lakatos [1976, p. 89] notes that during such a construction ‘the original “naïve” realm’ of a problem and its entities ‘dissolves, and remodelled conjectures reappear’ in several fields. At that point also the original concept, in Lakatos’s case a ‘polyhedron’, in our case a notion like the one of ‘permutations’, is something new: it denotes something that has changed content, meaning and modus operandi. In effect, what a permutation, a matrix, or an elliptic function is and how it works after the construction and introduction of the notion of group, is different from what it was before: their properties, their relations and inter-relations have changed and been remodelled in a way that makes them work in a new way and serve new purposes in mathematical problem solving. Lakatos’s analysis focuses on ‘proof-generated concepts’, that is, the ones that are stretched, contracted, or dissolved by means of proofs and refutations. In this way his analysis neglects all the inferential and rational work before the proof (or the refutation), which is essential in order to understand how a mathematical novelty can be introduced and, eventually, a theorem proved. For instance his account could not explain how the concept of group could be introduced since it requires building and adjusting the idea of studying the action of a structure on itself and that is not part of a proof, nor does it explains how and why the search for new representations of a mathematical entity is so important. In this paper, I try to account for how the informal heuristic work that takes place before a proof is put forward, how new information is introduced in a field or a problem, and possibly adjusted, as the case of Cayley’s theorem shows. Cayley’s theorem, of course, accomplishes an intra-domain result, but a similar approach by using representations can be put forward also for mathematical entities that are in different domains. In this case, the formal results will establish a broader, conceptual, bridge. 3. INTER-DOMAIN REPRESENTATION THEOREMS In a broader conceptual sense, a representation theorem is one stating that a given structure with certain properties can be partially reduced, or is isomorphic, to another structure. Such an isomorphism or reduction can be put forward between entities that are in different domains, for instance algebra and topology. In this sense these theorems can serve as a kind of bridge law between distinct areas of mathematics. In order to show what role these theorems can play between domains, I will examine the fundamental group and the construction that led to it. 3.1. The Example of the Fundamental Group The power of the construction of a new representation is displayed by the examination of a long-standing problem in topology, namely the classification of 3-manifolds. This problem is hard, as traditional tools employed to approach it, such as genus, orientability, or boundary components, do not offer a satisfactory description of 3-manifolds. Thus, a key idea put forward is to classify them by means of invariants, but traditional invariants, like the Euler characteristic, turn out to be useless for 3-manifolds, since their values all became zero.5 Therefore, the need for a new approach to the problem emerged. Several approaches have been put forward to find appropriate invariants for topological entities (see also [Ippoliti, 2016]) and one of them was developed employing algebra, or better an algebraic representation. The move to using algebra draws on a simple seminal idea: an invariant is something that does not change under given operations — and in this sense it is a structure. Moreover, algebra can be conceived as just the study of structures, in the sense of something preserved under specific functions or operations. It is worth noting that this approach is not new (see also [Grosholz, 2000]): ‘elementary analytic geometry provides a good example of the application of formal algebraic techniques to the study of geometrical concepts’ [Crowell and Fox, 1963, p. 13]. The hard part of this attempt is to find a way of associating algebraic structures to topological entities, that is, of assimilating them. This process transforms the entities of the problem in a way that makes them treatable by certain tools. In effect, if an algebraic structure can be uniquely associated with a topological space and is ‘preserved’, then it can tell manifolds apart, and we could be in a position to attack and solve the problem of classifying 3-manifolds. Then, if we produce a representation theorem that maps, or bridges, algebra and topology, we can use algebraic techniques to approach and in some cases solve a topological problem. The first problem to solve here is to produce such a construction, that is, to show how and which algebraic structure, if any, can be associated with topological entities. Poincaré [1895] puts forward just this approach, ending up with the construction of the fundamental group (the first homotopy group). There are a number of ways of associating algebraic structures with topological spaces, many possible algebraic interpretations of a topological entity, and the data not only do not determine, but also do not uniquely suggest the most appropriate algebraic structure to obtain it. So the assimilation of algebraic and topological entities is quite complex; in order to accomplish it, we have to find a way of introducing a specific topological counterpart for each algebraic entity and each relation that is not there at the beginning of this process. It is by means of this representation that new elements are introduced in a topological space: the use of a specific representational source guides us in the construction of new mathematical entities and relations into the target. The fundamental group is the endpoint of such a step-by-step construction. As the name suggests, the algebraic structure associated with a topological space at the end of this construction is a group: a set of elements with a composition map and three properties — associativity, an identity element, and inverse elements. Essentially, it provides an algebraic ‘metric’ that can tell apart shapes of topological spaces. To build a bridge and an assimilation between algebra and topology that can eventually end up with representation theorems, we have to solve two preliminary and intertwined problems: first, to construct a topological counterpart of the set of elements of an algebraic structure; and second, to construct a topological counterpart of operations over algebraic structures. A first step is to note that since the Euler characteristic is of no use for differentiating 3-manifolds, we have to employ a new form of equivalence. A first form of such an equivalence is homotopy, which is a kind of continuous deformation: in addition to pulling and stretching, it enables us also to compress the geometrical entities. The employment of homotopy introduces a new topological feature, since it implies that is possible to change dimension.6 At this point it is possible to construct several algebraic representations of topological entities [Crowell and Fox, 1963, pp. 10–15]. This construction is governed by the purpose we are trying to achieve, that is, the features of the entities involved in the problem we are trying to solve. For instance a first attempt is the one that looks at the set of all paths $$a_i$$ of a topological space $$X$$ between two given points $$p_0$$ and $$p_1$$, and then defines a kind of composition for them, e.g., their product. The paths would keep track of information about the shape of the space, and their product would model the behaviour of these paths under homotopy. Unfortunately, the algebraic structure provided by the set of all paths of a topological space with respect to the product does not provide a solution for our problem — it cannot tell even simple distinct surfaces apart. Thus, we need a different way of constructing topological entities in algebraic terms. One way to improve this approach is to use loops as basic elements (see Figure 2) instead of simple paths — that is, we select ‘an arbitrary point $$p$$ in $$X$$ and restrict our attention to paths which begin and end at $$p$$’ (ibid., p. 13). Moreover by introducing an orientation for paths, we can define the inverse of a loop and then extend the assimilation between an algebraic and a topological structure. An orientation for paths is simple to define: it is given by allowing a loop to be followed in the opposite direction (see the arrows in Figure 2). The employment of loops rather than simple paths provides at least three advantages. First, since a loop can be continuously deformed into another one under homotopy, we can consider them as a single ‘element’ in our space — in our structure. A loop that cannot be continuously deformed into another will define a different basic entity. In this way loops provide a way of telling specific regions of the topological structure apart. Fig. 2. View largeDownload slide A loop $$a$$ over a topological space $$T$$ that has no holes. Fig. 2. View largeDownload slide A loop $$a$$ over a topological space $$T$$ that has no holes. Secondly, loops keep track of crucial topological properties of a manifold, such as holes in a surface — a property that cannot be tracked by means of simple paths. Third, it is easy to define a function of composition and a product for loops: since there are loops starting and ending at the same place (base point), their composition is certainly defined at their base point. Moreover the base point acts as the topological counterpart of the identity element $$e$$ in algebra — in effect the identity element $$e$$ can be thought of, in topological terms, as a loop simply standing at the base point. Hence, the identity path $$e$$ is also a multiplicative identity. In this way we have constructed a semi-group with identity for a topological space $$T$$: the set of all $$p$$-based loops in $$T$$. This algebraic representation is a better one than the previous — the set of all paths of a topological space with respect to the product. But it can be improved, in the sense that it accounts for more properties of a topological space. A way to improve it is to consider the set of all paths and ‘equivalent paths’: to ‘consider a new set, whose elements are the equivalence classes of paths. The fundamental group is obtained as a combination of this construction with the idea of a loop’ [Crowell and Fox, 1963, p. 17] Therefore we are now in a position to associate a group to a topological space — known as the fundamental group (see Figure 3). In order to define the fundamental group of a topological space $$T$$ with respect to a point $$p$$ in $$T$$, we consider the set of equivalence classes of loops at $$p$$, i.e., paths starting and ending at $$p$$, under the equivalence relation of homotopy. Fig. 3. View largeDownload slide Two elements of the fundamental group associated with a topological space $$T$$ with one hole. Fig. 3. View largeDownload slide Two elements of the fundamental group associated with a topological space $$T$$ with one hole. The class of a loop $$a$$ is denoted by [$$a$$]. Then we define a product on this set by setting $$[a]\bullet [b]=[a*b]$$, where $$a*b$$ is the loop obtained by composing $$a$$ and $$b$$. It turns out that the product is well defined and associative (the two loops ($$a*b)*c$$ and $$a*(b*c)$$ are homotopic). Therefore we obtain a group in which the identity is the class $$[e_p]$$ of the constant loop $$e_p$$ and the inverse of a class $$[a]$$ is the class $$[\overline{a}]$$ of the loop obtained by traversing $$a$$ in the opposite direction. If the topological space $$T$$ is pathwise connected, then the fundamental group is independent of the choice of the base point $$p$$ because any loop through $$p$$ is homotopic to a loop through any other point $$q$$ in $$T$$. The product is associative but it is not always commutative, that is, [$$a*b$$] is not homotopic to [$$b*a$$]. In order to illustrate this, let us consider two circles $$A$$ and $$B$$ intersecting at just one point (Figure 4). In this case the fundamental group is the free group generated by $$a$$ and $$b$$, which is a non-commutative group: the loop [$$a*b$$] (traversing them in left-to-right order) is not homotopic to the loop [$$b*a$$] (traversing them in right-to-left order), i.e., the former cannot be deformed into the latter by maintaining the given orientation. This construction is shaped by the assimilation of topology and algebra and gradually introduces information about topology that was not there at the beginning of the process. In this sense the fundamental group is a bridge between two domains and lays the foundation for algebraic topology: it is a basic tool for forming algebraic representations of topological spaces and telling them apart. The heuristic power of the fundamental group is that it integrates (not simply juxtaposes or multiplies) properties of several objects and fields and so it leads to a partial unification of those fields, showing how ‘some of the most significant advances in mathematical knowledge take place in the context of such partial unification’ [Grosholz, 2000, p. 82]. Fig. 4. View largeDownload slide Example of non-commutativity of the fundamental group. Fig. 4. View largeDownload slide Example of non-commutativity of the fundamental group. It is worth noting that while a large part of algebraic representations of topological entities are groups, different structures like rings or modules can be associated with topological entities. The choice of one representation over another is governed by the purpose we are trying to achieve, that is, the features of the entities involved in the problem we are trying to solve. So we have that algebraic topology ‘associates algebraic structures with purely topological, or geometrical, configurations. The two basic geometric entities of topology are topological spaces and continuous functions mapping one space into the other. To the spaces and continuous maps between them are made to correspond groups and group homomorphisms’ [Crowell and Fox, 1963, p. 13]. With this new tool we are in a position to calculate the fundamental group, denoted by $$\pi$$, of several surfaces such as the sphere, torus, annulus,7 and to tell them apart.8 Unfortunately, the fundamental group does not provide us with a complete invariant for 3-manifolds, and more generally ‘the algebra of topology is only a partial characterization of the topology’ (ibid.). More specifically, algebraic topology will produce one-way results or representation theorems, since it states that ‘if topological spaces $$X$$ and $$Y$$ are homeomorphic, then such-and-such algebraic conditions are satisfied. The converse proposition, however, will generally be false. Thus, if the algebraic conditions are not satisfied, we know that $$X$$ and $$Y$$ are topologically distinct. If, on the other hand, they are fulfilled, we usually can conclude nothing’ (ibid.). But even if the representation theorems9 that we get are limited, and so are the relations and kind of isomorphism bridging algebra and topology, we can build a lot upon them. After an algebraic representation of a topological entity has been constructed, we can use algebraic tools and results as a means for investigating topological entities. The fundamental group is paradigmatic in this sense: as it improves mathematical problem solving in topology it allows us both to solve previously unsolved problems and to find new ones. On one hand, the fundamental group has been successfully employed to solve existing problems. I recall just two examples. The first one is provided by Wirtinger [1905], who demonstrated that a trefoil10 is really knotted by proving that the fundamental group of the trefoil is the symmetric group on three elements. Tellingly, Wirtinger extended his method [Brauner, 1928] to constructing the fundamental group of an arbitrary link — now known as the Wirtinger presentation. The second one is provided by Dehn [1910], who develops an algorithm for constructing the fundamental group of the complement of a link. He shows that a knot is nontrivial when its fundamental group is non-abelian, and that a trefoil knot and its mirror image are topologically distinct. On the other hand, the Poincaré conjecture is a stock example of a new problem that has been generated from the fundamental group. Its original formulation, in effect, is built on the notion of fundamental group: ‘Est-il possible que le groupe fondamental de $$V$$ se réduise à la substitution identique, et que pourtant $$V$$ ne soit pas simplement connexe?’ [Poincaré, 1904, p. 110].11 This problem, as is well known, took about one hundred years to be solved [Perelman, 2002; 2003a;,b]. 3.2. Integration of Representations These two examples from algebra and algebraic topology exemplify the central role played by representations and representation results in the advancement of mathematical knowledge. The advancement they provide is the outcome of the construction of a new representation that transforms the problem and introduces new information into it. This information (which takes the form of new stipulations, functions, entities) is not contained in it at the beginning of the process of assimilation that shapes the construction of a new representation. The assimilation of distinct mathematical entities, in turn, involves specific manipulations of the entities at stake, in the sense of adapting or changing them to serve a given purpose or advantage. It enables us to approach a problem, or a sub-problem of it, more easily or even in a way that would not be possible otherwise. The first step of assimilation it is to find similar features between entities. Similarity is defined with respect to a given viewpoint, i.e., a specific property or a set of properties. In Cayley’s case the property is symmetry, in Poincaré’s case invariance under a kind of homeomorphism. The features of the problem to be solved guide us in the choice of these properties. Thus the building of a new representation is a step-by-step process, which highlights certain features of the entity and deliberately neglects other ones. This construction, when successful, creates an information surplus that can open the door to a solution to the problem. Moreover, this assimilation might produce also a conceptual novelty, just like the notion of group or fundamental group, and might occur between entities of the same kind (intra-domain, e.g., algebraic in Cayley’s case) or of a different kind (inter-domain). The construction of a new representation might end up with a representation theorem that formalizes a kind of isomorphism or reduction between the mathematical entities at stake. As a matter of fact, this is the very aim of this construction. Even when such an isomorphism or reduction is partial or one-way, as in the case of the fundamental group, it generates several benefits, that is, formal results and techniques that contribute to building at least a part of the solution for the problem at hand, just like the examples of classification of 3-manifolds or the classification of knots. This explains why the search for a proof of a representation theorem is so important and so intensely pursued by mathematicians. A new representation theorem establishes and formalizes a bridge between two mathematical entities and fields that boost problem solving by enabling us both to solve problems and to find new problems. As noted by Emily Grosholz [2000, p. 4] not only the use of ‘modes of representation is typical of reasoning in mathematics’, but ‘reductive methods are successful at problem-solving not because they eliminate modes of representation, but because they multiply and juxtapose them; and this often creates what I call productive ambiguity’ (ibid., p. xii). Grosholz examines many examples of the kind of productive multiplication of representation in mathematics and the sciences. My analysis of representation theorems and their construction enriches her account by showing some fine-grained dynamics that are employed in such an informal, before-the-proof construction and how, not only the multiplication of juxtaposition, but also of representation is essential. I show how a new representation of an object is built, how pieces of information are introduced and adjusted during the search for a solution of a problem. The construction of a new representation, as Grosholz also notes, is not simply a passive, static transfer of properties that leaves the target unaltered, but a dynamic, active construction that gradually transforms the target. I have shown how we have to model the entities in the target in specific ways in order to get a new useful representation of them. The example of the search for an appropriate algebraic representation of a topological entity is a stock example — for instance, simple paths are not sufficient, while loops can serve the purpose. Since the assimilation and the representation that it produces are partial, a problem of sensitivity to representation can arise. In effect, since the construction of the representation highlights certain features of the entities and deliberately neglects other ones, one could argue that the results derived from a new representation might be considered as holding only for it, and accordingly to depend on it. That is, the formal results holding for that specific representation might not be extended to the original entities. A remarkable example is provided by the use of projections onto a plane (i.e., a 2-D representation) in a branch of topology, that is knot theory. Since several 3-D features of knots are lost and others added by employing a 2-D projection (a typical example of the latter is the overcrossings in 2 dimensions), we cannot be sure that results obtained for the 2-D representation will hold also for the 3-D entity. The use of a specific representation (a 2-D projection) might produce results that are valid only for this representation and could not be extended to the original 3-D mathematical knot. In effect in this case we need Reidemeister’s [1927] theorem to establish that, and when, a set of operations over 2-D projections are valid also for the original 3-D knot. Even more tellingly, the employment of a specific representation, in this case a 2-D projection, opens the door to the use of other representations and tools that otherwise would be impossible to put to use. A recent example is the employment of coloring in knot theory,12 which requires a 2-D projection. In effect, an interesting feature of the application of coloring to knot theory is the fact that in principle there are no components for knots. In this sense, a 2-D representation is introducing new information into the problem. A knot is a single strand in 3-D space, and as such it has no crossings and strands: accordingly, it cannot be discretized as required by coloring. No labels for items can be identified for coloring, for there is only one item — the single string in 3D. Thus, in principle, coloring could not even be defined for a knot. On the other hand, a 2-D entity, like a knot diagram, can be discretized and hence colored. So even if colorability cannot be defined for knots in 3D, it ends up revealing interesting properties of knots, and not simply of their diagrams. The important issue here is to understand that all these properties and results are sensitive to the representation employed to obtain them. Topology, in particular knot theory, is exemplary in this sense. Not only coloring, but other approaches put forward to solve the problem of classification of knots so far, like the one based on representations with braids,13 [Alexander, 1923; Artin, 1947] numbers [Gauss, 1798; Schubert, 1949; Mazur, 1973], or graphs [Listing, 1847; Tait, 1877; Yajima and Kinoshita, 1957; Kinoshita and Terasaka, 1957], do not solve the problem of classification of knots: they offer an answer to fractions of it by revealing partial classes of equivalence. In this respect, representation theorems are a way not only of connecting different areas of a domain or different domains, but different pieces of the same puzzle. Each of these representation theorems connects pieces of the puzzle in a way that provides local answers to the problem and in this sense enables the advancement of mathematical knowledge. They create virtuous circles, so to speak, whereby each representation contributes to the construction of part of the solution. Alexander’s theorem [Alexander, 1923] is a remarkable example of this. By establishing that every knot has a closed-braid presentation — namely that every knot is isotopic to a closed braid — not only does this fact formalize a reduction or a partial isomorphism between two entities, i.e., knot diagram and the closure of a braid, but since braids can be treated in algebraic terms (i.e., group theory), it turns out that algebra can be used to study knots too. Braids provide a bridge by which algebra can be employed in knot theory, so that a variety of algebraic tools can be employed to approach problems or sub-problems about knots. As a new representation is constructed and a representation result is proved in a given domain, it can be employed to deal with problems in other parts of mathematical knowledge: it is open to the application to other problems in that domain (intra-domain) and also in another domain (inter-domain) through the construction of other suitable representations. When this application succeeds, the target domain will benefit from it. In return, the knowledge produced in the target domain can be used to approach the source domain. That is just what, for example, Foisy [2002; 2003] did in graph theory: after the successful application of graphs to knot theory using a representation of knots in terms of planar graphs [Tait, 1877; Yajima and Kinoshita, 1957; Kinoshita and Terasaka, 1957], we can go the other way by studying knotted graphs, in other words we can use knots to understand graphs better. Of course these moves do not always work, and in general heuristics aiming at integrating several representations could fail: one set of techniques might not work, while another will. For instance, Euler characteristics fail while the fundamental group and the concept of homotopy work. There are at least two explanations for this difference in their expressive and heuristic power.14 The first is that a representation works for a given problem and the way it is formulated. A change in representation reshapes the problem, its objects and features, making it possible to approach that problem in a way not possible before. The second is that new techniques are put forward on the basis of the failures of previous ones: they try to overcome the weaknesses that led to failure. These new techniques emerge from the failures or weaknesses of the previous ones, benefiting from them. For instance, the fundamental group built its notion of an invariant taking advantage of the failures of previous approaches. The role of representations highlights an important historical dimension of mathematical research and growth of knowledge. A new or changed representation draws on the corpus of available knowledge, which is a historical entity. As the corpus of our knowledge grows and changes, new representations can be built or old ones modified, so that new integrations of objects and fields become possible. A new representation of a known object that was previously blocked becomes possible, and so the toolbox for approaching that problem expands, sometimes opening up the road to a solution. The dynamics of this construction reveal not only the importance of the integration of representations — what Cavaillès calls ‘transversal relations’ and Grosholz ‘constructive ambiguities’ — but also of their interplay with vertical relations for the construction of new knowledge in mathematics. The ‘vertical relations’, which expand our knowledge inside a domain, enable us to bring out properties and specific configurations of objects that make them ‘approachable’ via a suitable integration of representations from other fields, advancing mathematical knowledge. Footnotes 1 Cayley, and the other nineteenth-century researchers, did not connect the term abstract with the group concept. The term abstract group is a later development (see [Miller, 1916; Chakraborty and Chowdhury, 2005; Pengelley, 2005]). 2 Studying the caustic equation for reflection and refraction Cayley notes [1857, § XXVIII] how the equation of a secondary caustic may be expressed indifferently in any one of the four forms (ibid., p. 299):   \begin{align*} \sqrt{\left(x-\frac{a}{\mu^2}\right)^2+y^2} &=\frac{1}{\mu}\sqrt{(x-a)^2+y^2} + \frac{c}{\mu}\left(\mu-\frac{1}{\mu}\right)\!;\\ \sqrt{\left(x-\frac{c^2}{a}\right)^2}+y^2&=\frac{c}{a}\sqrt{(x-a)^2+y^2}+\frac{1}{\mu}\left(a-\frac{c^2}{a}\right)\!;\\ \sqrt{\left(x-\frac{c^2}{a}\right)^2}+y^2&=\frac{c\mu}{a}\sqrt{(x-\frac{a}{\mu^2})^2+y^2}+\frac{a}{\mu}\frac{c^2\mu}{a};\\ \begin{split} c\left(\mu-\frac{1}{\mu}\right) \sqrt{\left(x-\frac{c^2}{a}\right)^2+y^2} &+\left(-a+\frac{c^2}{a}\right)\sqrt{\left(x-\frac{a}{\mu^2}\right)^2+y^2}\\ & + \left(\frac{a}{\mu}-\frac{c^2\mu}{a}\right)\sqrt{(x-a)^2+y^2} = 0. \end{split} \end{align*} and from them he shows that ‘the same caustic is produced by six different systems of a radiant point and refracting circle’ (ibid.). 3 See [Thomas, 2011] for an analysis of the role of assimilation in mathematics. 4 See also [Gillies, 1992] on this topic. 5 This follows from a well-known theorem (see, e.g., [Hatcher, 2002]) stating that the Euler characteristic $$\chi$$ of an odd-dimensional compact manifold is zero. Intuitively, this is a consequence of the fact that the Euler characteristic $$\chi$$ for a 2-manifold is obtained by triangulation of its surface — it can be proved that it is independent of the specific triangulation. Such a computable characterization of a 2-manifold is not available for 3-manifolds via the Euler characteristic, so we need some different computable way of characterizing 3-manifolds and telling them apart. 6 In effect under homotopy, a disk, a ball, and a point are all equivalent. 7 Intuitively speaking an annulus is the topological space between concentric circles, while a torus is a topological space that is a product of two circles. 8 For example, the fundamental group of a sphere $$S^2$$ is: $$\pi_1(S^2)= 0.$$ The fundamental group of the annulus $$X^2$$ is: $$\pi_1(X^2)= Z.$$ The fundamental group of a torus $$T^2$$ is: $$\pi_1(T^2) = \pi_1(S^1) \times \pi_1(S^1)\cong Z\times Z = Z^2.$$ 9 An example is Van Kampen’s theorem [Seifert, 1931; Van Kampen, 1933]. 10 A mathematical knot is like an ordinary knot whose ends are joined together so that it cannot be undone. A trefoil is obtained by joining together the two loose ends of the usual overhand knot. 11 In more modern terms, the conjecture asks: consider a compact 3-dimensional manifold $$V$$ without boundary. Is it possible that the fundamental group of $$V$$ could be trivial, even though $$V$$ is not homeomorphic to the 3-dimensional sphere? 12 Coloring is a way of assigning distinct labels (a color, a number, a letter, etc.) to each component of a discrete entity, such as a plane or a graph. In knot theory, coloring has been introduced by R. Fox [Crowell and Fox, 1963] in order to tell knots apart, and since then it has been extensively used (see [Kauffman, 1991; Montesinos, 1985]). 13 Intuitively a braid is an intertwining of some number of strings attached to top and bottom parallel lines such that each string never ‘turns back up’. 14 My thanks go to an anonymous referee for pointing out this issue. It deserves its own paper; so I will only attempt to outline an answer here. 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Deutschen Mathematiker Vereinigung  14, 517. Yajima T., and Kinoshita S. [ 1957]: ‘On the graphs of knots’, Osaka Math. J.  9, 155– 163. © The Author [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Philosophia Mathematica Oxford University Press

A Role for Representation Theorems

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Abstract

Abstract I argue that the construction of representation theorems is a powerful tool for creating novel objects and theories in mathematics, as the construction of a new representation introduces new pieces of information in a very specific way that enables a solution for a problem and a proof of a new theorem. In more detail I show how the work behind the proof of a representation theorem transforms a mathematical problem in a way that makes it tractable and introduces information into it that it did not contain at the beginning of the process. 1. REPRESENTATION THEOREMS Representation theorems play a pivotal role in the advancement of mathematical knowledge: the construction of a new representation, or a change of representation, is a typical move that mathematicians employ to approach and solve problems (see, e.g., [Alexander, 1923; Erdős et al., 1966]). A representation theorem, in a broad sense, is a theorem that states that a given structure with certain properties can be reduced, or is somehow isomorphic, to another structure. Thus, it aims at establishing a mapping between elements and relations of a certain kind and other elements and relations of similar or different kind. In this way, it enables the transfer of certain properties from a source to a target, making it possible to treat problems in the latter with knowledge about the former. This construction, and the consequent transfer, requires a manipulation of the target whereby pieces of information are introduced into the target that it did not contain at the beginning of the process. Such a construction might end up with a novel concept or even a new theory — algebraic topology is a paramount example in this sense. This construction unfolds in a way that Imre Lakatos [1976, pp. 89–91] sketches in his account of theoretical novelty in mathematics and fits Emily Grosholz’s [2007] account of productive ambiguity in mathematics — that is, the use of multiple representations in mathematical problem-solving. Not only is the construction of a new representation of mathematical entities a powerful engine for advancing mathematical knowledge (see [Byers, 2007; Grosholz, 2007]), but a new representation is the endpoint of a process of assimilation of mathematical entities [Thomas, 2011]. The assimilation transforms the entities of the problems and eventually leads to a new representation theorem, which often establishes a formal reduction or an isomorphism, or a partial isomorphism. An entity is transformed and remodeled in ways that change its original content and meaning, as it no longer ‘denotes what it set out to denote: that its naïve meaning has disappeared and that now it is used [...] for a totally different, novel concept’ [Lakatos, 1976, p. 90]. So a new representation produces a multivalent entity, a ‘hybrid’ in Grosholz’s terms, which can be employed as a hypothesis to solve problems in the several fields from which it emerges and partially merges: the multiplication and juxtaposition of representations of a mathematical object enable us to integrate features of multiple fields that ‘provoke discovery in unexpected ways’ [Grosholz, 2000, p. 82]. Thus in this paper I set out to investigate the role of representation theorems in a way that fills some gaps in Lakatos’s account of theoretical novelty in mathematics and that specifies some of the heuristic processes involved in the growth of mathematical knowledge at a fine-grained level so enriching the heuristic processes detected by Grosholz in her account of productive ambiguity in mathematics. 2. INTRA-DOMAIN REPRESENTATION THEOREMS In a more technical sense a representation theorem is one stating that an abstract structure with certain features is mapped, or reduced, to another more concrete structure and vice versa. Representation theorems are widespread in mathematics: in algebra, functional analysis, or geometry, just to mention few cases. A remarkable example in algebra is Cayley’s theorem [Cayley, 1854; Burnside, 1897]. This theorem, whose origin and relevance have been investigated in detail (see in particular [Chowdhury, 1995; Chakraborty and Chowdhury, 2005; Pengelley, 2005]), establishes an intra-domain result: the mapping is between entities within the same domain and of the same kind — groups, that is, algebraic ones. More precisely, Cayley’s theorem originally states that: if the entire group is multiplied by any one of the symbols, either as further or nearer factor, the effect is simply to reproduce the group; or what is the same thing, that if the symbols of the group are multiplied together so as to form a table [as in Figure 1], that as well each line as each column of the square will contain all the symbols l, $${\alpha,\beta}$$ ... . [Cayley, 1854, p. 124] Fig. 1. View largeDownload slide Multiplication table [Cayley, 1854, p. 124]. Fig. 1. View largeDownload slide Multiplication table [Cayley, 1854, p. 124]. In modern terminology, Cayley’s theorem states that every group of order $$n$$ is isomorphic to a group of permutations on $$n$$ letters, or that every finite group is isomorphic to a subgroup of the symmetric group. In order to establish this theorem, he introduces the multiplication table of a finite group, Figure 1, and states that an abstract group is provided by its multiplication table. It is worth noting that the modern formulations of Cayley’s theorem employ the notion of isomorphism, which is not contained in its original formulation. Nonetheless Cayley seems to be aware of it when he notes that the cyclic group of order $$n$$ “is in every respect analogous to the system of the roots of the ordinary equation $$x^n-1=0$$” (ibid., p. 125). As a matter of fact, the very notion of group was not there at that time and it is just with this theorem that we see the very first attempt of defining the modern (abstract1) concept of ‘group’. In Cayley’s words: ‘A set of symbols $$1, \alpha, \beta, \dots$$, all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a group’ (ibid., p. 124). In this sense, Cayley provides a remarkable conceptual novelty in mathematics, ‘the birth of one of the most essential concepts of modern mathematics’ [Pengelley, 2005, p. 7]. Groups, and later an abstract view on them, arise from the study of two phenomena. The first one is the study of permutations: ‘clearly permutation groups served Cayley as a prime motivating example’ [Chakraborty and Chowdhury, 2005, p. 277]. More specifically the problem at stake was explicitly the action of some functions on the roots of some polynomials, as emerging from Galois’s work. As Cayley himself notes, ‘the idea of a group as applied to permutations or substitutions is due to Galois, and the introduction of it may be considered as marking an epoch in the progress of the theory of algebraic equations’ [1854, p. 124]. More generally, the study of collections of permutations of a set of specific entities is the very first step for the construction of the notion of group and, later, of an abstract view of groups. The second phenomenon comes from outside algebra and mathematics, namely from physics — geometrical optics: ‘Cayley’s unexpected discovery of a non-abelian group of order 6 in the practical context of geometrical optics, served as the trigger for generalizing the group concept’ [Chakraborty and Chowdhury, 2005, p. 278]. In more detail this discovery was suggested by ‘Cayley’s discovery of the six transformations that leave the equation of the secondary caustic unchanged, and his realization that these transformations form a group under the composition of mappings’ (ibid., p. 277).2 In turn this new finding has been reasonably made possible by a previous, and more general, issue concerning transformations of variables and their effect on the equation: ‘for one who had discovered the non-associative algebra of octaves (Cayley numbers) very soon after the discovery (1843) of the division ring of quaternions by Sir William Rowan Hamilton and had worked on invariants, it would have been natural for Cayley to ask himself the question: what, if any, transformation of variables leaves the equation unchanged?’ (ibid.) In order to get his theorem, Cayley put forward first of all a process of assimilation3 of mathematical entities that have been treated as distinct so far — invertible matrices (under multiplication), permutations, Gauss’s quadratic forms, quaternions (under addition), and various kinds of elliptic functions. In modern terms, these are more concrete structures, while a group is an abstract one, which is mapped onto them. In order to produce this notion of group, Cayley’s theorem builds upon the seminal idea of studying the action of an entity on itself: in effect the symmetries of an entity become a subject of inquiry in themselves. This notion is the one that triggers a new representation: it is the property by means of which similarities among these entities are highlighted, generalized, and then unified into a new concept. A representation requires that we highlight certain features of the entities of the problem, search for similarities among them — and deliberately neglect others. Which features to highlight and which to neglect is governed by the problem to be solved and the corpus of existing knowledge. In this case, as we have seen, the highlighted features are permutations and symmetries. This crucial move generates not only a new representation and viewpoint on known entities, but also a new technical concept (i.e., ‘group’) by means of which other mathematical entities can be investigated. This new approach and representation enable Cayley to advance mathematical knowledge and problem solving in algebra: just to mention a few cases, he determined all the groups of orders four and six, showing that there are two of each (again, by exhibiting their multiplication tables). Moreover, he showed that there is exactly one group of a given prime order. In this sense the notion of group enables a classification of mathematical entities. Of course, today we know that Cayley’s theorem is not so effective for studying a group, nevertheless Cayley’s steps towards an abstract view of groups can be legitimately seen as ‘a remarkable accomplishment at this time in the evolution of group theory’ [Kleiner, 2007, p. 31]. This theorem exhibits well the several roles a representation theorem can play in mathematics. First of all, it provides a generalisation and unification of properties and entities. It makes a domain more compact by giving a uniform treatment of previously unconnected or heterogeneous mathematical entities. Accordingly, it boosts mathematical problem solving, for not only can a more general concept be used to approach distinct entities, but local results and techniques can be transferred from one kind of entity to another kind, for instance from elliptic functions to invertible matrices. Thus it also provides a way of classifying entities and, accordingly, of seeking and establishing relations among them. Moreover, it enables an abstraction that may end up with an axiomatization, as happened in the case of the abstract theory of groups. Last but not least, it suggests lines of research: for instance it seems that Cayley was tacitly assuming that his groups were finite and it was not clear if his results could hold also for infinite groups — in effect, Cayley’s results were later proved to hold also for infinite groups. So the final outcome, the theorem, is important, but the very construction that leads to the assimilation of entities is the key and this representation theorem shows how conceptual novelty and change can take place in mathematics.4 It brings to light both a new object — the notion of group — and new features of already known objects — i.e., invertible matrices under multiplication, permutations, Gauss’s quadratic forms, quaternions under addition and various kinds of elliptic functions — by means of gradual steps that remodel the entities by adding pieces of information (properties, relations) to them. In his account of mathematical novelty, Lakatos [1976, p. 89] notes that during such a construction ‘the original “naïve” realm’ of a problem and its entities ‘dissolves, and remodelled conjectures reappear’ in several fields. At that point also the original concept, in Lakatos’s case a ‘polyhedron’, in our case a notion like the one of ‘permutations’, is something new: it denotes something that has changed content, meaning and modus operandi. In effect, what a permutation, a matrix, or an elliptic function is and how it works after the construction and introduction of the notion of group, is different from what it was before: their properties, their relations and inter-relations have changed and been remodelled in a way that makes them work in a new way and serve new purposes in mathematical problem solving. Lakatos’s analysis focuses on ‘proof-generated concepts’, that is, the ones that are stretched, contracted, or dissolved by means of proofs and refutations. In this way his analysis neglects all the inferential and rational work before the proof (or the refutation), which is essential in order to understand how a mathematical novelty can be introduced and, eventually, a theorem proved. For instance his account could not explain how the concept of group could be introduced since it requires building and adjusting the idea of studying the action of a structure on itself and that is not part of a proof, nor does it explains how and why the search for new representations of a mathematical entity is so important. In this paper, I try to account for how the informal heuristic work that takes place before a proof is put forward, how new information is introduced in a field or a problem, and possibly adjusted, as the case of Cayley’s theorem shows. Cayley’s theorem, of course, accomplishes an intra-domain result, but a similar approach by using representations can be put forward also for mathematical entities that are in different domains. In this case, the formal results will establish a broader, conceptual, bridge. 3. INTER-DOMAIN REPRESENTATION THEOREMS In a broader conceptual sense, a representation theorem is one stating that a given structure with certain properties can be partially reduced, or is isomorphic, to another structure. Such an isomorphism or reduction can be put forward between entities that are in different domains, for instance algebra and topology. In this sense these theorems can serve as a kind of bridge law between distinct areas of mathematics. In order to show what role these theorems can play between domains, I will examine the fundamental group and the construction that led to it. 3.1. The Example of the Fundamental Group The power of the construction of a new representation is displayed by the examination of a long-standing problem in topology, namely the classification of 3-manifolds. This problem is hard, as traditional tools employed to approach it, such as genus, orientability, or boundary components, do not offer a satisfactory description of 3-manifolds. Thus, a key idea put forward is to classify them by means of invariants, but traditional invariants, like the Euler characteristic, turn out to be useless for 3-manifolds, since their values all became zero.5 Therefore, the need for a new approach to the problem emerged. Several approaches have been put forward to find appropriate invariants for topological entities (see also [Ippoliti, 2016]) and one of them was developed employing algebra, or better an algebraic representation. The move to using algebra draws on a simple seminal idea: an invariant is something that does not change under given operations — and in this sense it is a structure. Moreover, algebra can be conceived as just the study of structures, in the sense of something preserved under specific functions or operations. It is worth noting that this approach is not new (see also [Grosholz, 2000]): ‘elementary analytic geometry provides a good example of the application of formal algebraic techniques to the study of geometrical concepts’ [Crowell and Fox, 1963, p. 13]. The hard part of this attempt is to find a way of associating algebraic structures to topological entities, that is, of assimilating them. This process transforms the entities of the problem in a way that makes them treatable by certain tools. In effect, if an algebraic structure can be uniquely associated with a topological space and is ‘preserved’, then it can tell manifolds apart, and we could be in a position to attack and solve the problem of classifying 3-manifolds. Then, if we produce a representation theorem that maps, or bridges, algebra and topology, we can use algebraic techniques to approach and in some cases solve a topological problem. The first problem to solve here is to produce such a construction, that is, to show how and which algebraic structure, if any, can be associated with topological entities. Poincaré [1895] puts forward just this approach, ending up with the construction of the fundamental group (the first homotopy group). There are a number of ways of associating algebraic structures with topological spaces, many possible algebraic interpretations of a topological entity, and the data not only do not determine, but also do not uniquely suggest the most appropriate algebraic structure to obtain it. So the assimilation of algebraic and topological entities is quite complex; in order to accomplish it, we have to find a way of introducing a specific topological counterpart for each algebraic entity and each relation that is not there at the beginning of this process. It is by means of this representation that new elements are introduced in a topological space: the use of a specific representational source guides us in the construction of new mathematical entities and relations into the target. The fundamental group is the endpoint of such a step-by-step construction. As the name suggests, the algebraic structure associated with a topological space at the end of this construction is a group: a set of elements with a composition map and three properties — associativity, an identity element, and inverse elements. Essentially, it provides an algebraic ‘metric’ that can tell apart shapes of topological spaces. To build a bridge and an assimilation between algebra and topology that can eventually end up with representation theorems, we have to solve two preliminary and intertwined problems: first, to construct a topological counterpart of the set of elements of an algebraic structure; and second, to construct a topological counterpart of operations over algebraic structures. A first step is to note that since the Euler characteristic is of no use for differentiating 3-manifolds, we have to employ a new form of equivalence. A first form of such an equivalence is homotopy, which is a kind of continuous deformation: in addition to pulling and stretching, it enables us also to compress the geometrical entities. The employment of homotopy introduces a new topological feature, since it implies that is possible to change dimension.6 At this point it is possible to construct several algebraic representations of topological entities [Crowell and Fox, 1963, pp. 10–15]. This construction is governed by the purpose we are trying to achieve, that is, the features of the entities involved in the problem we are trying to solve. For instance a first attempt is the one that looks at the set of all paths $$a_i$$ of a topological space $$X$$ between two given points $$p_0$$ and $$p_1$$, and then defines a kind of composition for them, e.g., their product. The paths would keep track of information about the shape of the space, and their product would model the behaviour of these paths under homotopy. Unfortunately, the algebraic structure provided by the set of all paths of a topological space with respect to the product does not provide a solution for our problem — it cannot tell even simple distinct surfaces apart. Thus, we need a different way of constructing topological entities in algebraic terms. One way to improve this approach is to use loops as basic elements (see Figure 2) instead of simple paths — that is, we select ‘an arbitrary point $$p$$ in $$X$$ and restrict our attention to paths which begin and end at $$p$$’ (ibid., p. 13). Moreover by introducing an orientation for paths, we can define the inverse of a loop and then extend the assimilation between an algebraic and a topological structure. An orientation for paths is simple to define: it is given by allowing a loop to be followed in the opposite direction (see the arrows in Figure 2). The employment of loops rather than simple paths provides at least three advantages. First, since a loop can be continuously deformed into another one under homotopy, we can consider them as a single ‘element’ in our space — in our structure. A loop that cannot be continuously deformed into another will define a different basic entity. In this way loops provide a way of telling specific regions of the topological structure apart. Fig. 2. View largeDownload slide A loop $$a$$ over a topological space $$T$$ that has no holes. Fig. 2. View largeDownload slide A loop $$a$$ over a topological space $$T$$ that has no holes. Secondly, loops keep track of crucial topological properties of a manifold, such as holes in a surface — a property that cannot be tracked by means of simple paths. Third, it is easy to define a function of composition and a product for loops: since there are loops starting and ending at the same place (base point), their composition is certainly defined at their base point. Moreover the base point acts as the topological counterpart of the identity element $$e$$ in algebra — in effect the identity element $$e$$ can be thought of, in topological terms, as a loop simply standing at the base point. Hence, the identity path $$e$$ is also a multiplicative identity. In this way we have constructed a semi-group with identity for a topological space $$T$$: the set of all $$p$$-based loops in $$T$$. This algebraic representation is a better one than the previous — the set of all paths of a topological space with respect to the product. But it can be improved, in the sense that it accounts for more properties of a topological space. A way to improve it is to consider the set of all paths and ‘equivalent paths’: to ‘consider a new set, whose elements are the equivalence classes of paths. The fundamental group is obtained as a combination of this construction with the idea of a loop’ [Crowell and Fox, 1963, p. 17] Therefore we are now in a position to associate a group to a topological space — known as the fundamental group (see Figure 3). In order to define the fundamental group of a topological space $$T$$ with respect to a point $$p$$ in $$T$$, we consider the set of equivalence classes of loops at $$p$$, i.e., paths starting and ending at $$p$$, under the equivalence relation of homotopy. Fig. 3. View largeDownload slide Two elements of the fundamental group associated with a topological space $$T$$ with one hole. Fig. 3. View largeDownload slide Two elements of the fundamental group associated with a topological space $$T$$ with one hole. The class of a loop $$a$$ is denoted by [$$a$$]. Then we define a product on this set by setting $$[a]\bullet [b]=[a*b]$$, where $$a*b$$ is the loop obtained by composing $$a$$ and $$b$$. It turns out that the product is well defined and associative (the two loops ($$a*b)*c$$ and $$a*(b*c)$$ are homotopic). Therefore we obtain a group in which the identity is the class $$[e_p]$$ of the constant loop $$e_p$$ and the inverse of a class $$[a]$$ is the class $$[\overline{a}]$$ of the loop obtained by traversing $$a$$ in the opposite direction. If the topological space $$T$$ is pathwise connected, then the fundamental group is independent of the choice of the base point $$p$$ because any loop through $$p$$ is homotopic to a loop through any other point $$q$$ in $$T$$. The product is associative but it is not always commutative, that is, [$$a*b$$] is not homotopic to [$$b*a$$]. In order to illustrate this, let us consider two circles $$A$$ and $$B$$ intersecting at just one point (Figure 4). In this case the fundamental group is the free group generated by $$a$$ and $$b$$, which is a non-commutative group: the loop [$$a*b$$] (traversing them in left-to-right order) is not homotopic to the loop [$$b*a$$] (traversing them in right-to-left order), i.e., the former cannot be deformed into the latter by maintaining the given orientation. This construction is shaped by the assimilation of topology and algebra and gradually introduces information about topology that was not there at the beginning of the process. In this sense the fundamental group is a bridge between two domains and lays the foundation for algebraic topology: it is a basic tool for forming algebraic representations of topological spaces and telling them apart. The heuristic power of the fundamental group is that it integrates (not simply juxtaposes or multiplies) properties of several objects and fields and so it leads to a partial unification of those fields, showing how ‘some of the most significant advances in mathematical knowledge take place in the context of such partial unification’ [Grosholz, 2000, p. 82]. Fig. 4. View largeDownload slide Example of non-commutativity of the fundamental group. Fig. 4. View largeDownload slide Example of non-commutativity of the fundamental group. It is worth noting that while a large part of algebraic representations of topological entities are groups, different structures like rings or modules can be associated with topological entities. The choice of one representation over another is governed by the purpose we are trying to achieve, that is, the features of the entities involved in the problem we are trying to solve. So we have that algebraic topology ‘associates algebraic structures with purely topological, or geometrical, configurations. The two basic geometric entities of topology are topological spaces and continuous functions mapping one space into the other. To the spaces and continuous maps between them are made to correspond groups and group homomorphisms’ [Crowell and Fox, 1963, p. 13]. With this new tool we are in a position to calculate the fundamental group, denoted by $$\pi$$, of several surfaces such as the sphere, torus, annulus,7 and to tell them apart.8 Unfortunately, the fundamental group does not provide us with a complete invariant for 3-manifolds, and more generally ‘the algebra of topology is only a partial characterization of the topology’ (ibid.). More specifically, algebraic topology will produce one-way results or representation theorems, since it states that ‘if topological spaces $$X$$ and $$Y$$ are homeomorphic, then such-and-such algebraic conditions are satisfied. The converse proposition, however, will generally be false. Thus, if the algebraic conditions are not satisfied, we know that $$X$$ and $$Y$$ are topologically distinct. If, on the other hand, they are fulfilled, we usually can conclude nothing’ (ibid.). But even if the representation theorems9 that we get are limited, and so are the relations and kind of isomorphism bridging algebra and topology, we can build a lot upon them. After an algebraic representation of a topological entity has been constructed, we can use algebraic tools and results as a means for investigating topological entities. The fundamental group is paradigmatic in this sense: as it improves mathematical problem solving in topology it allows us both to solve previously unsolved problems and to find new ones. On one hand, the fundamental group has been successfully employed to solve existing problems. I recall just two examples. The first one is provided by Wirtinger [1905], who demonstrated that a trefoil10 is really knotted by proving that the fundamental group of the trefoil is the symmetric group on three elements. Tellingly, Wirtinger extended his method [Brauner, 1928] to constructing the fundamental group of an arbitrary link — now known as the Wirtinger presentation. The second one is provided by Dehn [1910], who develops an algorithm for constructing the fundamental group of the complement of a link. He shows that a knot is nontrivial when its fundamental group is non-abelian, and that a trefoil knot and its mirror image are topologically distinct. On the other hand, the Poincaré conjecture is a stock example of a new problem that has been generated from the fundamental group. Its original formulation, in effect, is built on the notion of fundamental group: ‘Est-il possible que le groupe fondamental de $$V$$ se réduise à la substitution identique, et que pourtant $$V$$ ne soit pas simplement connexe?’ [Poincaré, 1904, p. 110].11 This problem, as is well known, took about one hundred years to be solved [Perelman, 2002; 2003a;,b]. 3.2. Integration of Representations These two examples from algebra and algebraic topology exemplify the central role played by representations and representation results in the advancement of mathematical knowledge. The advancement they provide is the outcome of the construction of a new representation that transforms the problem and introduces new information into it. This information (which takes the form of new stipulations, functions, entities) is not contained in it at the beginning of the process of assimilation that shapes the construction of a new representation. The assimilation of distinct mathematical entities, in turn, involves specific manipulations of the entities at stake, in the sense of adapting or changing them to serve a given purpose or advantage. It enables us to approach a problem, or a sub-problem of it, more easily or even in a way that would not be possible otherwise. The first step of assimilation it is to find similar features between entities. Similarity is defined with respect to a given viewpoint, i.e., a specific property or a set of properties. In Cayley’s case the property is symmetry, in Poincaré’s case invariance under a kind of homeomorphism. The features of the problem to be solved guide us in the choice of these properties. Thus the building of a new representation is a step-by-step process, which highlights certain features of the entity and deliberately neglects other ones. This construction, when successful, creates an information surplus that can open the door to a solution to the problem. Moreover, this assimilation might produce also a conceptual novelty, just like the notion of group or fundamental group, and might occur between entities of the same kind (intra-domain, e.g., algebraic in Cayley’s case) or of a different kind (inter-domain). The construction of a new representation might end up with a representation theorem that formalizes a kind of isomorphism or reduction between the mathematical entities at stake. As a matter of fact, this is the very aim of this construction. Even when such an isomorphism or reduction is partial or one-way, as in the case of the fundamental group, it generates several benefits, that is, formal results and techniques that contribute to building at least a part of the solution for the problem at hand, just like the examples of classification of 3-manifolds or the classification of knots. This explains why the search for a proof of a representation theorem is so important and so intensely pursued by mathematicians. A new representation theorem establishes and formalizes a bridge between two mathematical entities and fields that boost problem solving by enabling us both to solve problems and to find new problems. As noted by Emily Grosholz [2000, p. 4] not only the use of ‘modes of representation is typical of reasoning in mathematics’, but ‘reductive methods are successful at problem-solving not because they eliminate modes of representation, but because they multiply and juxtapose them; and this often creates what I call productive ambiguity’ (ibid., p. xii). Grosholz examines many examples of the kind of productive multiplication of representation in mathematics and the sciences. My analysis of representation theorems and their construction enriches her account by showing some fine-grained dynamics that are employed in such an informal, before-the-proof construction and how, not only the multiplication of juxtaposition, but also of representation is essential. I show how a new representation of an object is built, how pieces of information are introduced and adjusted during the search for a solution of a problem. The construction of a new representation, as Grosholz also notes, is not simply a passive, static transfer of properties that leaves the target unaltered, but a dynamic, active construction that gradually transforms the target. I have shown how we have to model the entities in the target in specific ways in order to get a new useful representation of them. The example of the search for an appropriate algebraic representation of a topological entity is a stock example — for instance, simple paths are not sufficient, while loops can serve the purpose. Since the assimilation and the representation that it produces are partial, a problem of sensitivity to representation can arise. In effect, since the construction of the representation highlights certain features of the entities and deliberately neglects other ones, one could argue that the results derived from a new representation might be considered as holding only for it, and accordingly to depend on it. That is, the formal results holding for that specific representation might not be extended to the original entities. A remarkable example is provided by the use of projections onto a plane (i.e., a 2-D representation) in a branch of topology, that is knot theory. Since several 3-D features of knots are lost and others added by employing a 2-D projection (a typical example of the latter is the overcrossings in 2 dimensions), we cannot be sure that results obtained for the 2-D representation will hold also for the 3-D entity. The use of a specific representation (a 2-D projection) might produce results that are valid only for this representation and could not be extended to the original 3-D mathematical knot. In effect in this case we need Reidemeister’s [1927] theorem to establish that, and when, a set of operations over 2-D projections are valid also for the original 3-D knot. Even more tellingly, the employment of a specific representation, in this case a 2-D projection, opens the door to the use of other representations and tools that otherwise would be impossible to put to use. A recent example is the employment of coloring in knot theory,12 which requires a 2-D projection. In effect, an interesting feature of the application of coloring to knot theory is the fact that in principle there are no components for knots. In this sense, a 2-D representation is introducing new information into the problem. A knot is a single strand in 3-D space, and as such it has no crossings and strands: accordingly, it cannot be discretized as required by coloring. No labels for items can be identified for coloring, for there is only one item — the single string in 3D. Thus, in principle, coloring could not even be defined for a knot. On the other hand, a 2-D entity, like a knot diagram, can be discretized and hence colored. So even if colorability cannot be defined for knots in 3D, it ends up revealing interesting properties of knots, and not simply of their diagrams. The important issue here is to understand that all these properties and results are sensitive to the representation employed to obtain them. Topology, in particular knot theory, is exemplary in this sense. Not only coloring, but other approaches put forward to solve the problem of classification of knots so far, like the one based on representations with braids,13 [Alexander, 1923; Artin, 1947] numbers [Gauss, 1798; Schubert, 1949; Mazur, 1973], or graphs [Listing, 1847; Tait, 1877; Yajima and Kinoshita, 1957; Kinoshita and Terasaka, 1957], do not solve the problem of classification of knots: they offer an answer to fractions of it by revealing partial classes of equivalence. In this respect, representation theorems are a way not only of connecting different areas of a domain or different domains, but different pieces of the same puzzle. Each of these representation theorems connects pieces of the puzzle in a way that provides local answers to the problem and in this sense enables the advancement of mathematical knowledge. They create virtuous circles, so to speak, whereby each representation contributes to the construction of part of the solution. Alexander’s theorem [Alexander, 1923] is a remarkable example of this. By establishing that every knot has a closed-braid presentation — namely that every knot is isotopic to a closed braid — not only does this fact formalize a reduction or a partial isomorphism between two entities, i.e., knot diagram and the closure of a braid, but since braids can be treated in algebraic terms (i.e., group theory), it turns out that algebra can be used to study knots too. Braids provide a bridge by which algebra can be employed in knot theory, so that a variety of algebraic tools can be employed to approach problems or sub-problems about knots. As a new representation is constructed and a representation result is proved in a given domain, it can be employed to deal with problems in other parts of mathematical knowledge: it is open to the application to other problems in that domain (intra-domain) and also in another domain (inter-domain) through the construction of other suitable representations. When this application succeeds, the target domain will benefit from it. In return, the knowledge produced in the target domain can be used to approach the source domain. That is just what, for example, Foisy [2002; 2003] did in graph theory: after the successful application of graphs to knot theory using a representation of knots in terms of planar graphs [Tait, 1877; Yajima and Kinoshita, 1957; Kinoshita and Terasaka, 1957], we can go the other way by studying knotted graphs, in other words we can use knots to understand graphs better. Of course these moves do not always work, and in general heuristics aiming at integrating several representations could fail: one set of techniques might not work, while another will. For instance, Euler characteristics fail while the fundamental group and the concept of homotopy work. There are at least two explanations for this difference in their expressive and heuristic power.14 The first is that a representation works for a given problem and the way it is formulated. A change in representation reshapes the problem, its objects and features, making it possible to approach that problem in a way not possible before. The second is that new techniques are put forward on the basis of the failures of previous ones: they try to overcome the weaknesses that led to failure. These new techniques emerge from the failures or weaknesses of the previous ones, benefiting from them. For instance, the fundamental group built its notion of an invariant taking advantage of the failures of previous approaches. The role of representations highlights an important historical dimension of mathematical research and growth of knowledge. A new or changed representation draws on the corpus of available knowledge, which is a historical entity. As the corpus of our knowledge grows and changes, new representations can be built or old ones modified, so that new integrations of objects and fields become possible. A new representation of a known object that was previously blocked becomes possible, and so the toolbox for approaching that problem expands, sometimes opening up the road to a solution. The dynamics of this construction reveal not only the importance of the integration of representations — what Cavaillès calls ‘transversal relations’ and Grosholz ‘constructive ambiguities’ — but also of their interplay with vertical relations for the construction of new knowledge in mathematics. The ‘vertical relations’, which expand our knowledge inside a domain, enable us to bring out properties and specific configurations of objects that make them ‘approachable’ via a suitable integration of representations from other fields, advancing mathematical knowledge. Footnotes 1 Cayley, and the other nineteenth-century researchers, did not connect the term abstract with the group concept. The term abstract group is a later development (see [Miller, 1916; Chakraborty and Chowdhury, 2005; Pengelley, 2005]). 2 Studying the caustic equation for reflection and refraction Cayley notes [1857, § XXVIII] how the equation of a secondary caustic may be expressed indifferently in any one of the four forms (ibid., p. 299):   \begin{align*} \sqrt{\left(x-\frac{a}{\mu^2}\right)^2+y^2} &=\frac{1}{\mu}\sqrt{(x-a)^2+y^2} + \frac{c}{\mu}\left(\mu-\frac{1}{\mu}\right)\!;\\ \sqrt{\left(x-\frac{c^2}{a}\right)^2}+y^2&=\frac{c}{a}\sqrt{(x-a)^2+y^2}+\frac{1}{\mu}\left(a-\frac{c^2}{a}\right)\!;\\ \sqrt{\left(x-\frac{c^2}{a}\right)^2}+y^2&=\frac{c\mu}{a}\sqrt{(x-\frac{a}{\mu^2})^2+y^2}+\frac{a}{\mu}\frac{c^2\mu}{a};\\ \begin{split} c\left(\mu-\frac{1}{\mu}\right) \sqrt{\left(x-\frac{c^2}{a}\right)^2+y^2} &+\left(-a+\frac{c^2}{a}\right)\sqrt{\left(x-\frac{a}{\mu^2}\right)^2+y^2}\\ & + \left(\frac{a}{\mu}-\frac{c^2\mu}{a}\right)\sqrt{(x-a)^2+y^2} = 0. \end{split} \end{align*} and from them he shows that ‘the same caustic is produced by six different systems of a radiant point and refracting circle’ (ibid.). 3 See [Thomas, 2011] for an analysis of the role of assimilation in mathematics. 4 See also [Gillies, 1992] on this topic. 5 This follows from a well-known theorem (see, e.g., [Hatcher, 2002]) stating that the Euler characteristic $$\chi$$ of an odd-dimensional compact manifold is zero. Intuitively, this is a consequence of the fact that the Euler characteristic $$\chi$$ for a 2-manifold is obtained by triangulation of its surface — it can be proved that it is independent of the specific triangulation. Such a computable characterization of a 2-manifold is not available for 3-manifolds via the Euler characteristic, so we need some different computable way of characterizing 3-manifolds and telling them apart. 6 In effect under homotopy, a disk, a ball, and a point are all equivalent. 7 Intuitively speaking an annulus is the topological space between concentric circles, while a torus is a topological space that is a product of two circles. 8 For example, the fundamental group of a sphere $$S^2$$ is: $$\pi_1(S^2)= 0.$$ The fundamental group of the annulus $$X^2$$ is: $$\pi_1(X^2)= Z.$$ The fundamental group of a torus $$T^2$$ is: $$\pi_1(T^2) = \pi_1(S^1) \times \pi_1(S^1)\cong Z\times Z = Z^2.$$ 9 An example is Van Kampen’s theorem [Seifert, 1931; Van Kampen, 1933]. 10 A mathematical knot is like an ordinary knot whose ends are joined together so that it cannot be undone. A trefoil is obtained by joining together the two loose ends of the usual overhand knot. 11 In more modern terms, the conjecture asks: consider a compact 3-dimensional manifold $$V$$ without boundary. Is it possible that the fundamental group of $$V$$ could be trivial, even though $$V$$ is not homeomorphic to the 3-dimensional sphere? 12 Coloring is a way of assigning distinct labels (a color, a number, a letter, etc.) to each component of a discrete entity, such as a plane or a graph. In knot theory, coloring has been introduced by R. Fox [Crowell and Fox, 1963] in order to tell knots apart, and since then it has been extensively used (see [Kauffman, 1991; Montesinos, 1985]). 13 Intuitively a braid is an intertwining of some number of strings attached to top and bottom parallel lines such that each string never ‘turns back up’. 14 My thanks go to an anonymous referee for pointing out this issue. It deserves its own paper; so I will only attempt to outline an answer here. 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Deutschen Mathematiker Vereinigung  14, 517. Yajima T., and Kinoshita S. [ 1957]: ‘On the graphs of knots’, Osaka Math. J.  9, 155– 163. © The Author [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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Published: Apr 27, 2018

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