Mathematical Practice and Conceptual Metaphors: On cognitive studies of historical developments in mathematics

Transcription

1 Mathematical Practice and Conceptual Metaphors: On cognitive studies of historical developments in mathematics Dirk Schlimm December 31, 2011 (Revised) Word count: 6450 Keywords: History of mathematics, mathematical cognition, mathematical practice, metaphors, set theory.

2 Abstract The aim of this paper is to look at recent work in cognitive science on mathematical cognition from the perspective of philosophy of mathematical practice. The discussion is focused on the work of Lakoff and Núñez, because this is the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. Building on a distinction between mathematics as it is presented in textbooks and as it presents itself to the researcher, I argue that the focus of cognitive analyses of historical developments of mathematics has been primarily on the former.

3 1 Introduction The Symposium on Mathematical Practice and Cognition, held March 2010 at De Montfort University in Leicester, UK had the explicit aim of bringing together scientists from various disciplines who study aspects of mathematical cognition. In this spirit, the goal of the present paper is to look at work in cognitive science from the perspective of a philosophy of mathematics that takes seriously both mathematical practice and the history of mathematics. There has been considerable interest lately in cognitive studies of mathematics, which resulted in well-known monographs such as (Dehaene 1997), (Butterworth 1999), and (Lakoff and Núñez 2000). More recent publications, like (Campbell 2005) and (Carey 2009), show that this interest still persists. However, despite subtitles such as How the Mind Creates Mathematics (Dehaene 1997), the subject matter that is discussed by most authors is not mathematics in general, but it is almost always restricted to representations of numbers, counting, and basic arithmetic. A notable exception is the work of Lakoff and Núñez, which is explicitly aimed at high-level mathematical ideas like infinity, continuity, and set theory (Lakoff and Núñez 2000, Núñez 2005; 2008). This allows me to use Lakoff and Núñez s reconstruction of some developments in the history of mathematics in terms of metaphors as the focus of my discussion, but I believe that my observations are pertinent to any cognitive study of advanced mathematics and its historical development. In this paper I draw upon Hersh s distinction between the two aspects of mathematics that he calls the front and the back of mathematics, which I shall also refer to as textbook mathematics and research mathematics. Taking some clues from Lakatos I argue that mathematical practice often oscillates between rigorous deductive presentations and more tentative, exploratory reasoning, i. e., between the front and the back. This back and forth takes place at several levels: In developments of an individual mathematician, of a mathematical discipline, and of particular mathematical ideas and results. Using the example of mathematical sets, I discuss Lakoff and Núñez s technique of mathematical idea analysis and I show the neglect of research mathematics in these cognitive studies of historical developments. 2 Textbook mathematics and research mathematics Any account of mathematics, be it philosophical, logical, or cognitive, must first address the question What is mathematics?. In other words, it must be specified what one wants 1

4 to give an account of in the first place. Most take something like the current state of mathematics as their target, by which they understand the stable body of mathematical knowledge that one can find in contemporary textbooks. Hersh (1991) calls this the front of mathematics, i. e., how mathematics is presented, and he contrasts this with the back of mathematics, i. e., how it is done. He also describes these aspects as the public and private aspects of mathematics. Similarly, Polya speaks of the two faces of mathematics, contrasting rigorous, systematic, and deductive presentations with the more experimental and inductive mathematics in the making (Polya 1973, vii). I shall refer to these two aspects of mathematics respectively as textbook mathematics and research mathematics. Textbook mathematics is a stable body of knowledge that is organized logically. Single disciplines are based on a few primitive notions 1 that are related by basic propositions (axioms). The mathematical disciplines themselves are related in a hierarchy, in which a foundational discipline like arithmetic, logic, or set theory is supposed to be the basis upon which the other disciplines are built. Textbook mathematics is static and its concepts are definite and fixed. In contrast, for a researcher who is developing a new theory or introducing new concepts, mathematics is dynamic and fluid. A well-known presentation of this latter aspect of mathematics is Lakatos Proofs and Refutations (1976). Here, conjectures are entertained, proved, or refuted, concepts are stretched and narrowed down, novel constructions and proof techniques are explored. Both aspects capture some important features of the human enterprise of mathematics, and any account of mathematics that focuses on a single one of them at the expense of neglecting the other runs the risk of overemphasizing certain features and is likely to miss other characteristic ingredients of mathematical practice. My argument in (Schlimm 2012) that the two most prominent, but seemingly incompatible, philosophical notions of mathematical concepts (Fregean and Lakatosian concepts) are motivated by the different demands of textbook and research mathematics illustrates this point. While the distinction between the front and back of mathematics is useful for highlighting certain features of mathematics, it should be noted that mathematical activities (like learning, creating, and presenting mathematics) often involve both aspects, i. e., moving back and forth between front and back. The novice often begins with listening to standard material presented in lectures, reading textbooks, and solving exercises, i. e., she is explicitly confronted with textbook mathematics and uses this to develop her own conception of mathematical objects and relations between them. In 1 I deliberately use the somewhat ambiguous term notion as a general term that comprises ideas, concepts, and formal definitions. 2

5 terms of Hersh s characterization, learning mathematics thus consists in appropriating public knowledge and turning it into something private. When this knowledge is extended and new mathematics is developed, this is always done with the goal in mind of formulating proofs for the new results (see Lakatos 1976 for an illustrative case study). In other words, the work in the back is always carried out with an eye towards the front. Finally, once the new material is revised and consolidated it finds its way into research articles and possibly also into neat textbook presentations, the former aimed at communicating it to other mathematicians and the latter for teaching it to novices. It seems to me that a considerable part of the appeal and excitement of mathematics consists in crossing the boundary between textbook and research mathematics: Learning new ideas is a move from the front to the back, while creating new mathematics and teaching it involves moving from the back to the front. Based on the different aims of textbook and research mathematics, two different uses of metaphors in mathematics can also be distinguished. On the one hand, heuristic or research metaphors are employed to express some pre-theoretical and preliminary understanding of mathematical ideas. These metaphors can explicitly guide the development of new mathematics and they reflect how the mathematician conceptualizes the subject matter under investigation. On the other hand, pedagogical or textbook metaphors serve to convey a basic understanding of fully developed mathematical notions to the novice in the field and to encapsulate some part of mathematics in a vivid and informative image. Such metaphors are typically adopted by the mathematical community only some time after the related mathematical innovations were introduced. Axiom systems often stand at the crossroads between research mathematics and textbook mathematics. The formulation of a set of axioms and the study of its consequences and models is part of mathematical research, while the results of these investigations reformulated, cleaned-up, and rearranged often find their way into textbook presentations. However, not everything that plays a role in research mathematics finds its expression in the front of mathematics. In modern mathematics theories are frequently presented on the basis of axiomatic systems that contain otherwise undefined primitives and that determine the inferential structure of the mathematical objects in question. From a metaphysical standpoint this leaves the mathematical objects (or structures) underdetermined. What is of primary concern for the mathematician is what one can say about these structures, i. e., what properties can be inferred from the axioms. This view is expressed, for example, by the mathematician Abraham Robinson: To understand a theory means to be able to follow its logical development and not, necessarily, to interpret, or give a denotation for, its individual 3

6 terms (Robinson 1965, 235; italics in original). A discussion of the exact nature of mathematical objects can thus be left to philosophy. However, this does not mean that philosophical beliefs play no role in the individual s decisions about what to research in mathematics or how to pursue that research, nor that mathematicians must not employ any analogies or metaphors in arriving at an understanding of axioms or in their daily work; rather, what it means is that only the inferential relationships are mathematically relevant (for the front of mathematics, that is). This allows, at least to a certain extent, for the development and communication of mathematical results among people with incompatible philosophical views on the nature of mathematics. Historical accounts of mathematics often emphasize its cumulative character by reinterpreting older work in light of more recent insights (Kitcher 1983, 161). Similarly, Kuhn emphasized the role of textbooks for consolidating a body of knowledge after scientific revolutions (Kuhn 1996, ). For him, textbooks expound the body of accepted theory, illustrate many or all of its successful applications, and compare these applications with exemplary observations and experiments (Kuhn 1996, 10; italics by DS). Because of this it is easily overlooked that the historical development of mathematics is not a straightforward one-way affair that leads from research to textbooks. As mentioned above, a mathematician s career often begins with the study of textbooks and then moves on to the realm of research mathematics. So, while textbook mathematics presents in a certain sense the end product of a line of research, it is frequently also the starting point for new developments, which may force the re-evaluation of previously held propositions. Moreover, the oscillation between back and front is not only characteristic for the individual development of mathematicians, but also for that of mathematical ideas and entire disciplines. 3 Metaphorical sets and mathematical sets 3.1 Mathematical cognition and conceptual metaphors Various approaches can be applied in the study of mathematical cognition. Introspective accounts (e. g., Hadamard 1945), in vitro observations or experiments (e. g., Marghetis and Núñez 2012, this volume), and in vivo 2 observations of actual mathematicians can all yield insights into mathematical thinking (e. g., Carter 2010). Finally, the artifacts that result from mathematical activities, like articles and books, can also be used as 2 See (Dunbar and Blanchette 2001) for a discussion of in vivo/in vitro approaches to the study of cognition. 4

7 data for investigating the nature of the underlying and often implicit thought processes. When it comes to cognitive accounts of historical developments in mathematics, the task is more complicated, because any direct access to the protagonists is impossible. Thus, one has to rely mainly on published materials, which belong for the most part to the front of mathematics. Only in rare occasions can one also find reflections on the thought processes underlying research mathematics in the form of letters, accounts of conversations, or in parenthetical published remarks. In other words, the study of the historical back of mathematics is difficult and, since it often relies almost exclusively on published sources, it can easily be confused with the front of mathematics. Abstract concepts in science and in everyday life are often thought of in terms of metaphors (Lakoff and Johnson 2003). Moreover, I take it as fairly uncontroversial that metaphors are also used to conceptualize and understand mathematics. Indeed, the language of mathematics is full of terms that were borrowed from human activities in the real world: Three divides six, the natural numbers go on and on, and points lie on a line. Over a decade ago this observation was worked out in detail by Lakoff and Núñez on the basis of their previous work in linguistics and cognitive science. In Where Mathematics Comes From (2000) they argue that conceptual metaphors and conceptual blending are the main cognitive mechanisms used to conceive of mathematical objects. Conceptual metaphors are grounded, inference-preserving cross-domain mappings (Lakoff and Núñez 2000, 6; italics in original), which are considered to be a phenomenon of thought and not of purely linguistic character. Lakoff and Núñez s work offers many insights for understanding certain mathematical inferences and reasoning processes involved in the development of mathematics. For example, the concept of an actual infinity can be understood as a completed iterative process, i. e., though the Basic Metaphor of Infinity (Núñez 2005). According to Lakoff and Núñez s theory of embodied mathematics, both research and textbook mathematics must ultimately be explained by embodied cognitive mechanisms, of which conceptual metaphor is a central example. As I argue next, however, it seems that in regard to historical developments their mathematical idea analysis has so far been directed mainly at textbook mathematics, which has been mistaken for research mathematics. As mentioned in the introduction, I take this to be merely an example for a more general phenomenon. Another instance of this are the treatments of Roman numerals in (Dehaene 1997, 98) and (Butterworth 1999, 96 97), which are based on popular books on the history of mathematics that do not do justice to the historical and conceptual details afforded by the notation (Schlimm and Neth 2008). 5

8 3.2 Conceptual metaphors for sets We now turn to Lakoff and Núñez s cognitive analysis of the development of set theory. As an answer to the question What is a set? Núñez writes: Intuitively, many people (including mathematicians) would say that a set is some kind of collection or aggregate [DS: Footnote with a quotation of Cantor s definition (Cantor 1895, 282); see next block quote, below.]. Many authors speak of sets as containing their members and most students think of sets this way. Even the choice of the word member suggests such a reading, as do the Venn diagrams used to introduce the subject [... ]. Implicit in this form of understanding of sets is the conceptual metaphor sets are container schemas [... ]. (Núñez 2008, 340) According to this metaphor sets are conceptualized as container schemas that have an interior, a boundary, and an exterior, with the entities inside the container being the elements of the set (Lakoff and Núñez 2000, 30 33). This grounding metaphor, which is frequently only implicit, is used to conceptualize the mathematical notion of set as determined by the well-known Zermelo-Fraenkel axioms with Choice. A difference between metaphorical sets (i. e., those brought forth by the conceptual metaphor) and mathematical sets (i. e., those that are the intended subject matter of mathematics) is that the metaphor of sets as container schemas rules out sets that can contain themselves, whereas such sets are not ruled out by the standard ZFC axioms. By restricting the structure of axiomatically defined sets through the introduction of a further condition, the axiom of foundation, the structure of mathematical sets can be brought closer to that of the metaphorical sets. Mathematical sets of this kind are called well-founded sets. However, one might also want to explicitly allow sets to contain themselves. Such sets (also called hypersets ) must then be conceptualized differently, e. g., by accessible pointed graphs (Barwise and Moss 1991). According to Núñez, the origins of the corresponding two axiomatic systems for set theory are two internally consistent but mutually inconsistent metaphorical conceptions of sets: one in terms of container schemas and another in terms of graphs (Núñez 2008, 344; italics in original). 3.3 Mathematicians conceptions of sets Giving a detailed historical analysis of the early notions of sets and the metaphors that were used to conceptualize them would certainly be an interesting project, but would 6

9 go far beyond the scope of this paper. Instead, I just want to give a few examples to illustrate the wealth of conceptions that have arisen between Cantor s first papers on set theory and contemporary textbook presentations to illustrate the gap between the back and the front of set theory. As mentioned above, Núñez quotes the following definition of sets given by Cantor in support of the popularity of the view that a set is some kind of collection or aggregate. By a set we understand every collection to a whole M of definite, welldifferentiated objects m of our intuition or our thought. (We call these objects the elements of M.) (Cantor 1895, 282) 3 Notice that Cantor does not choose the term member, but rather uses element to denote the entities that belong to a set. More importantly, Cantor also gave three other definitions of sets. The earliest one, put forward well over a decade before the one just quoted, does not refer to collections, but states more generally that the elements belong to a conceptual sphere by which is meant the extension of a concept (Hallett 1984, 45): I call a manifold (a totality, a set) of elements which belong to some conceptual sphere well-defined, if on the basis of its definition and as a consequence of the logical principle of excluded middle it must be seen as internally determined both whether some object belonging to the same conceptual sphere belongs to the imagined manifold as an object or not, as well as whether two objects belonging to the set are equal to one another or not, despite formal differences in the way they are given. (Cantor 1882, 150; italics in original) The importance of imagination and thought as linking the elements of a set is emphasized again one year later, but Cantor now also mentions a law that determines the elements that belong to a set. By a manifold or set I understand in general any many which can be thought of as one, that is, every totality of definite elements which can be united to a whole through a law. By this I believe I have defined something related to the Platonic εἰδος or ἰδέα. (Cantor 1883, 204, n. 1) Finally, after having realized that some definitions of sets can lead to contradictions, Cantor adds a requirement of non-contradiction for elements of a set in 1899: 3 This and the following quotations from Cantor are taken from (Hallett 1984, 33 34, 45). 7

10 When... the totality of elements of a multiplicity can be thought without contradiction as being together, so that their collection into one thing is possible, I call it a consistent multiplicity or a set. (Letter to Dedekind, 28 July 1899; in Cantor 1932, 443, italics in original) The reason for quoting Cantor s definitions in full is to show that it is not entirely obvious how the father of modern set theory conceptualized sets, but it seems that his understanding of sets was more sophisticated than simple container schemas, and that over time his views changed in important respects. To make matters worse, in regard to the imagery associated with sets, Cantor is reported to have said A set I imagine as an abyss (Dedekind 1932, 449). 4 In the years following Cantor s pioneering work the understanding of sets diverged even further, so that in 1905 the mathematician König observed that the term set was being used indiscriminately for completely different notions (König 1905, 145). By this time Cantor s definitions were not considered to be precise enough to dispel ambiguity. The intuitions that had developed about sets varied substantially and did not allow for unanimous agreement with regard to their properties. In particular with regard to the validity of the axiom of choice a heated debate ensued among mathematicians (see Ewald 1996, for some of the main arguments) and a few years later Poincaré referred to Cantor s set theory itself as an interesting pathological case (Poincaré 1909, 182). 5 The distinction between well-founded and non-well-founded sets, which was mentioned in the previous section, was made explicit by Mirimanoff in 1917, who had no difficulty accepting sets that contain themselves as genuine sets (Aczel 1988, ). Some of his contemporaries, however, objected to this understanding of sets. The axiomatic formulation of this distinction was put forward by von Neumann in 1925, and the explicit connection to accessible pointed graphs seems to have been developed only much later. To sum up the early development of set theory the historian of mathematics Gregory H. Moore speaks of the appearance of a potpourri of set-theoretic principles which some accepted and others denied during the first years of the 20th century (Moore 1982, 142). Whether all of these notions, the tentative ones as well as those that became well-established, can be accounted for by the two conceptual metaphors for 4 An anonymous reviewer noted that mental imagery and conceptual metaphors are distinct processes. 5 One also finds the claim that Poincaré considered set theory to be a disease, but this does not seem to be accurate (Gray 1991). 8

11 sets discussed by Lakoff and Núñez is an open question, but it appears to be somewhat unlikely. 3.4 Conceptual metaphors and mathematical sets After this glimpse into the variety of historical conceptions of sets we return to Lakoff and Núñez s account of sets. Recall, the aim of this paper is not to criticize their analysis of mathematical ideas as a cognitive science theory, but to discuss it from a perspective that takes mathematical practice and the history of mathematics seriously. In the previous section we saw various nuances in Cantor s conception of sets from just considering four different ways in which he introduced this notion. Any explanation of why mathematics is the way it is should be able to address these differences in one way or another. For this, the ideas employed in research mathematics need to be considered and the actual inferences of mathematicians, not only those in polished textbook presentations, should be explained. To illustrate the subtle differences between these reasoning processes, let us consider Cantor s proof of the denumerability of the rational numbers. It is discussed in (Núñez 2005, 1725) as being based on a clever infinite array [... ] which displays all possible fractions (italics in original). A particular path through this array then leads to the well-known enumeration of the rational numbers. Cantor s original paper of 1874, which Núñez refers to in his discussion, however, neither contains such an array nor an explicit enumeration of the rational numbers. Instead, it presents an enumeration of the real algebraic numbers, which are those real numbers that satisfy an equation of the form a 0 x n + a 1 x n a n 1 x + a n = 0, for natural numbers a 0 to a n. It is on the basis of this representation that Cantor argues that the real algebraic numbers can be arranged in a linear sequence ω 1, ω 2, ω 3 and so on. Because the rational numbers are a subset of the real algebraic numbers it immediately follows that they can also be put in a one-toone correspondence with the natural numbers, but Cantor does not state this explicitly in the paper (although he is aware of this, as becomes clear in his correspondence with Dedekind). The resulting enumeration is not the same as that obtained from following the path in the array shown by Núñez. Of course, this does not matter for the purpose of proving that a one-to-one mapping between the natural and rational numbers is possible, since here one enumeration is as good as any other. But the difference might matter for the analysis of the underlying cognitive processes. My point is that there is no evidence that Cantor s own argument relied on an arrangement of the rational numbers in the well-known array, which is the basis of Núñez s discussion 9

12 of the application of the Basic Metaphor of Infinity (Núñez 2005, 1737). On the one hand, this leaves open the possibility of providing an alternative account of Cantor s reasoning in terms of conceptual metaphors. On the other hand, it is worth pointing out that the argument Cantor presents is based on a particular symbolic representation of the real algebraic numbers. This suggests that such representations might also play a crucial role in mathematical reasoning that takes place in the back of mathematics, not only for presentations in the front of mathematics. 6 Núñez continues his discussion of historical material by noting that Cantor introduced the notion of having the same cardinal number for sets that can be put into one-to-one correspondence and that he called the smallest infinite cardinal number ℵ 0, as if these developments were immediately connected with his results of 1874, namely the denumerability of the rationals and the non-denumerability of the reals. While they certainly are connected at the level of contemporary abstract ideas, it took Cantor himself several years to come up with them: He introduced cardinal numbers in 1878 and the aleph notation in 1895! Again, the point of these remarks is not to argue that Núñez s discussion is flawed. Rather, it is to show that his analysis is based on contemporary textbook mathematics, where the organization of mathematical ideas differs considerably from the way in which they were originally conceived. Indeed, the proof he presents is exactly the one that is found in most expositions of this material (e. g., Courant et al. 1996, 80). As a consequence, a mathematical idea analysis that is based on such a textbook presentation cannot take into consideration Cantor s groundbreaking thought processes or his conceptual struggles. Cantor himself has reported that he arrived at the proof only with difficulty after several failed attempts (Cantor 1932, 118). Rare insights into the development of the theory of sets can be found in Cantor s correspondence at the time with Dedekind (Ewald 1996, ): For instance, that Cantor s main question was the denumerability of the continuum and that the cardinality of the set of rational numbers was merely a by-product of these considerations and that he was well aware of two different conceptions for sameness of cardinality (discussed below). This correspondence also reveals Cantor s initial hesitation to develop the theory since he saw no practical interest in it, the great influence of Dedekind s ideas on Cantor s thinking, and that Cantor did not follow Dedekind s advice of presenting a stronger result in his 1874 argument, namely an enumeration of the complex algebraic numbers. 6 On the role of symbolic representations for arithmetical cognition, see (De Cruz et al. 2010); for more general approaches for the use of external resources in cognition, see (Hutchins 1995) and (Clark 1997). 10

13 The brief overview regarding the various notions of sets that were held in the development of set theory leads to another concern regarding Lakoff and Núñez s analysis of mathematical ideas, which often takes as its starting point a privileged everyday conception. For example, they claim that there is an intuitive premathematical notion of classes, which is usually and normally conceptualized as container schemas, which is our natural, everyday unconscious conceptual metaphor for what a class is (Lakoff and Núñez 2000, ; in this context class and set are used synonymously). The metaphor of sets as containers can be found on the first pages of some introductory textbooks, but only to be quickly discarded. Indeed, more careful analyses have revealed crucial shortcomings of such metaphors, as Potter points out: A collection, by contrast, does not merely lump several objects together into one: it keeps the things distinct and is a further entity over and above them. Various metaphors have been used to explain this a collection is a sack containing its members, a lasso around them, an encoding of them but none is altogether happy. [Footnote: See (Lewis 1991) for an excellent discussion of the difficulty of making good metaphysical sense of such metaphors.] (Potter 2004, 22) Even if we grant that there is such a shared premathematical understanding of sets, which is conceptualized by the metaphor of sets as containers, it is not clear how these meanings carry over from metaphorical sets to properties of mathematical sets. Given the historical disagreements about the characteristic properties of genuine sets, there must be ample wiggle room. Nevertheless, Lakoff and Núñez allow for the application of common sense notions to mathematical objects, for example, when discussing the comparison of the cardinalities of the natural, even, and rational numbers. They contrast Cantor s definition of sameness of cardinality in terms of a one-to-one mapping or pairability with our ordinary everyday understanding of more than, where [g]roup B has more objects than group A if, for every member of A, you can take away a member of B and still have members left in B (Lakoff and Núñez 2000, ), and conclude: According to our usual concept of more than, there are more natural numbers than even numbers, so the concepts Same Number As and pairability are different concepts. (Lakoff and Núñez 2000, 143;italics in original) According to our ordinary concept of More Than, there are more rational numbers than natural numbers, since if you take the natural numbers away from the rational numbers, there will be lots left over. (Lakoff and Núñez 2000, 144) 11

14 Two issues are worth pointing out in this analysis. First, both processes of comparing cardinalities in terms of stepwise subtracting elements and of pairability are equivalent for finite collections. Indeed, the process of subtracting elements can be seen as establishing a pairing between those two members that are taken away in each step (one from A and the other from B). Thus, it remains unclear on what basis one process is singled out as reflecting our ordinary understanding of more than. Further evidence for accepting both processes as contributing to our understanding of this relation (which is, after all, based on experiences with finite collections) is given by Galileo Galilei. 7 In 1638 he famously discussed both conceptions of sameness of cardinality to determine whether there are more natural numbers or square numbers (1, 4, 9, 16, 25,... ). According to one conception there are more natural numbers, because 2, 3, 5, 6, etc. are natural numbers that are not squares, but according to the other conception there are as many natural numbers as there are squares, because each square number can be mapped one-to-one to a natural number (1 2, 2 2, 3 2, 4 2, 5 2,... ). Galileo concluded that the attributes equal, greater, less are not applicable to infinite, but only to finite, quantities (Galilei 1954, 33). Second, Lakoff and Núñez s conclusion crucially depends on how the taking away procedure is carried out. For example, if one alternately takes away one rational number and one natural number, at each step there will always be infinitely many left over in either set. To reach their conclusion, Lakoff and Núñez must assume that the process of taking away an infinite set of natural numbers from an infinite set of rational numbers is carried out in one single step. But, this does not necessarily strike me as a process that is more basic and intuitive. (It would also require an application of the Basic Metaphor of Infinity, I suspect.) On the contrary, Lakoff and Núñez s own description of the ordinary understanding of more than (quoted above, just before the last block quote) is more in line with a stepwise process, since they describe a step of this process as take away a member and not as taking away all members. These considerations cast doubt on there being a privileged understanding of more than and show that the relation between metaphorical notions and mathematical ones is more problematic than it might at first seem. 8 The last remark leads to the question of how to distinguish between good and bad conceptual metaphors. Here, mathematical practice can yield an answer: The good metaphors are simply the ones that work best. Prima facie, there are no reasons for not accepting that there are more natural numbers than even numbers in mathematics. In 7 As mentioned above, Cantor was also aware of both conceptions. 8 The tension between human mathematical ideas and formal definitions is also expressed in (Núñez 2006, ). 12

15 this sense, I agree with Núñez that there is nothing wrong with our intuition per se (Núñez 2005, 1736). However, mathematics stands in the way. As it turned out, the inferential structure of one conception of more than does not support a fruitful and consistent theory of infinite sets, while the other one does. Cantor s achievement was to work out the mathematical details of a theory that is based on an understanding of more than in terms of pairings. His insight, however, was not just the outcome of a fortuitously chosen metaphor, but was the result of deep mathematical investigations. A similar example is the set-theorists choice of using the set {{a}, {a, b}} as standing for the ordered pair a, b. This Ordered Pair Metaphor (Lakoff and Núñez 2000, 141) was not the starting point, but rather the outcome of Kuratowski s efforts to improve on earlier suggestions (Scott and McCarty 2008). The close connection between mathematical investigations and conceptual metaphors alluded to in the previous paragraph raises the question of what comes first, conceptual metaphors or mathematical concepts? I contend that, similarly to the relation between textbook and research mathematics discussed in Section 2, this is not a clear-cut affair. Rather, a mathematical idea might be suggested by a vague metaphor, a hunch, or an accidental feature of the chosen notation. It can then be elaborated further in a process that involves trials and errors (recall Cantor s comment on the genesis of his proof) and possibly also interactions with other mathematicians (recall Cantor s correspondence with Dedekind), leading to more sophisticated metaphors, notations, and so on. This back and forth between various intellectual activities is characteristic of mathematical practice. Indeed, despite the fact that Lakoff and Núñez have a strong tendency to emphasize the priority of metaphors, one can also find passages that suggest the priority of mathematical concepts. We have seen that they discuss two conceptual metaphors for sets, container schemas for well-founded sets and accessible pointed graphs for non-well-founded sets. From these metaphors, as the following quote from Núñez suggests, the axioms for the corresponding mathematical notions were concocted : And in both cases, corresponding axioms have been especially concocted to organize the inferential structure (theorems) of both kinds of set theory, namely, the Axiom of Foundation and the Anti-Foundation Axiom, respectively. (Núñez 2008, 344) We might call this the metaphors first view. However, historically, set theorists conceived of accessible pointed graphs only long after non-well-founded sets had been investigated and axiomatized. The following account by Lakoff and Núñez, thus squares better with the actual historical developments: 13

16 Set theorists have realized that a new noncontainer metaphor is needed for thinking about sets, and they have explicitly constructed one: hyperset theory [... ]. (Lakoff and Núñez 2000, 147) According to this ( concepts first ) view, the need was felt for a specific metaphor to conceptualize hypersets, of which many properties were already known, and set theorists concocted a new metaphor for this specific purpose. It has been argued that the development of the iterative conception of sets also followed this pattern (Ferreirós 2007, ). I do not think that one has to settle for only one of these views to be correct, i. e., that either metaphors always come first or that concepts always do. Rather, both views seem to reflect certain historical developments. The distinction between textbook and research mathematics might be employed fruitfully also in this case: If metaphors are used mainly for communication and pedagogical purposes, this suggest that they are developed after the mathematical results have been obtained and accepted; the use of metaphors in the exploration of new mathematical ideas suggest that in these cases the metaphors come first. To conclude, I d like to raise one last concern for Lakoff and Núñez s mathematical idea analysis that arises from mathematical practice. Their analysis aims at explaining the meanings of empirically observed expressions in terms of embodied cognitive mechanisms (Lakoff and Núñez 2001). Mathematicians are often painfully aware of the tension between intuitive terminology that invokes associations or metaphors and the mathematical entities they intend to speak about; in particular, they find that associations stemming from the use of the terminology in non-mathematical contexts can be outright misleading. A striking example of an attempt to avoid such situations is Pasch s deliberate choice of terminology: In (Pasch 1930) he introduced the German term Rotte, which is extremely unusual for mathematics and which means pack or gang, instead of Reihe, the usual term for sequence that he had used earlier. The point of this change of terminology was to use a term that did not already have a meaning in similar contexts that could be misleading in the present investigation. Indeed Pasch warned repeatedly against using terminology with other pre-established mathematical meanings, because the reader can be easily lead astray. Again, I agree with Núñez that this is not a problem of everyday language per se, or with its alleged vagueness and imprecision (Núñez 2005, 1736); rather, the problem arises with the application of one conceptual structure to a domain to which it does not fully or appropriately apply. 14

17 4 Outlook Lakoff and Núñez s aim is to analyze how mathematical ideas are conceptualized and grounded in everyday experience. Their work was groundbreaking, since it provides us with the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. With regard to the latter, the main objects of investigation have been mathematical ideas that can be found in textbook mathematics, either contemporary textbooks or general accounts of the history of mathematics. However, the ideas that occur in these contexts might well be too coarse to accurately account for actual historical developments of mathematics and for giving a realistic cognitive account of these developments, e. g., the transitions between Cantor s various conceptions of sets. What is desired from a perspective of philosophy of mathematical practice is a more fine-grained and nuanced analysis of the cognitive processes that are involved in mathematical practice. In philosophy of science such careful analyses of creative scientific work have been undertaken for some time now, for example in Nersessian s cognitive-historical analyses (1992, 2002, 2008). In light of the great diversity of activities that occur in mathematical practice and of the specific differences between the front and the back of mathematics, I think it is wise to resist the temptation of trying to analyze all mathematical ideas into unique conceptualizations, but to allow for different, incompatible ones to be used side by side (like the conceptualizations of light as waves and a particles). As an example, a thorough engagement with scientific and mathematical practice has recently led to a careful reconsideration of too simple characterizations of analogies that were put forward in philosophy and cognitive psychology (Bartha 2010). Lakoff and Núñez s work on metaphors in mathematics has opened the door for similar investigations in mathematics and (Núñez 2005) is another step in this direction. I hope to have shown that a cognitive account of mathematical practice would profit from clear distinction between textbook mathematics and research mathematics, in particular when historical developments are investigated. This would be the natural continuation of previous research towards a historically informed cognitive science of mathematics and a cognitive history of mathematics as it is actually practiced. Acknowledgements. The author would like to thank the organizers of the Symposium on Mathematical Practice and Cognition, Markus Guhe, Alison Pease, and Alan Smaill, as well as the participants of the symposium for many interesting discussions that informed the present paper. I am also grateful to Rachel Rudolph and three anonymous reviewers of this journal for helpful comments on an earlier version. Work on this 15

18 paper was funded in part by Fonds Québécois de Recherche sur la Société et la Culture (FQRSC). References Aczel, P. (1988). Non-well-founded Sets. CSLI Lecture Notes. University of Chicago Press, Chicago. Bartha, P. F. A. (2010). By Parallel Reasoning: The Construction and Evaluation of Analogical Arguments. Oxford University Press, New York. Barwise, J. and Moss, L. (1991). Hypersets. The Mathematical Intelligencer, 13(4): Butterworth, B. (1999). The Mathematical Brain. Papermac, London. Campbell, J. I. D. (2005). Handbook of Mathematical Cognition. Psychology Press, New York. Cantor, G. (1874). Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Journal für die reine und angewandte Mathematik, 77: Reprinted in (Cantor 1932, ). Cantor, G. (1878). Ein Beitrag zur Mannigfaltigkeitslehre. Journal für die reine und angewandte Mathematik, 84: Reprinted in (Cantor 1932, ). Cantor, G. (1882). Über unendliche, lineare Punktmannigfaltigkeiten, 3. Mathematische Annalen, 20: Reprinted in (Cantor 1932, ). Cantor, G. (1883). Über unendliche, lineare Punktmannigfaltigkeiten, 5. Mathematische Annalen, 21: Reprinted in (Cantor 1932, ). Cantor, G. (1895). Beiträge zur Begründung der transfiniten Mengenlehre, 1. Mathematische Annalen, 46: Reprinted in (Cantor 1932, ). Cantor, G. (1932). Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Springer, Berlin. Edited by Ernst Zermelo. Carey, S. (2009). The Origin of Concepts. Oxford series in cognitive development. Oxford University Press, Oxford. Carter, J. (2010). Diagrams and proofs in analyis. International Studies in the Philosophy of Science, 24(1):

19 Clark, A. (1997). Being There: Putting brain, body, and world together again. The MIT Press, Cambridge, MA. Courant, R., Robbins, H., and Stewart, I. (1996). What is Mathematics? An Elementary Approach to Ideas and Methods. Oxford University Press, 2 edition. De Cruz, H., Neth, H., and Schlimm, D. (2010). The cognitive basis of arithmetic. In Löwe, B. and Müller, T., editors, Philosophy of Mathematics: Sociological Aspects and Mathematical Practice, pages College Publications, London. Dedekind, R. (1932). Gesammelte mathematische Werke, volume 3. F. Vieweg & Sohn, Braunschweig. Edited by Robert Fricke, Emmy Noether, and Öystein Ore. Dehaene, S. (1997). The Number Sense: How the Mind Creates Mathematics. Oxford University Press, New York. Dunbar, K. and Blanchette, I. (2001). The invivo/invitro approach to cognition: The case of analogy. Trends in Cognitive Sciences, 5: Ewald, W. (1996). From Kant to Hilbert: A Source Book in Mathematics. Clarendon Press, Oxford. Two volumes. Ferreirós, J. (2007). Labyrinth of Thought: A history of set theory and its role in modern mathematics. Birkhäuser, Basel, Switzerland, 2nd rev. ed edition. Galilei, G. (1954). Dialogues Concerning Two New Sciences. Dover Publications, New York. Translated by Henry Crew and Alfonso De Salvio. Original publication Gray, J. (1991). Did Poincaré say Set theory is a disease? The Mathematical Intelligencer, 13(10): Hadamard, J. (1945). The Psychology of Invention in the Mathematical Field. Princeton University Press. Reprinted by Dover Publications. Hallett, M. (1984). Cantorian Set Theory and Limitations of Size. Claredon Press, Oxford. Hersh, R. (1991). Mathematics has a front and a back. Synthese, 88(2): Hutchins, E. (1995). Cognition in the Wild. The MIT Press, Cambridge, MA. Kitcher, P. (1983). The Nature of Mathematical Knowledge. Oxford University Press, Oxford. 17

20 König, J. (1905). Über die Grundlagen der Mengenlehre und das Kontinuumproblem. Mathematische Annalen, 61: English translation by Stefan Bauer-Mengelberg in (van Heijenoort 1967), Kuhn, T. S. (1996). The Structure of Scientific Revolutions. University of Chicago Press, Chicago, 3 edition. Lakatos, I. (1976). Proofs and Refutations. Cambridge University Press, Cambridge. Edited by John Worrall and Elie Zahar. Lakoff, G. and Johnson, M. (2003). Metaphors We Live By. University of Chicago Press, Chicago,Ill., 2 edition. Lakoff, G. and Núñez, R. E. (2000). Where Mathematics Comes From. How the Embodied Mind Brings Mathematics into Being. Basic Books, New York. Lakoff, G. and Núñez, R. E. (2001). A reply to Bonnie Gold s review of Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Mathematical Association of America Online. Originally online at http: // Retrieved 29 April 2011 from http: // Lewis, D. K. (1991). Parts of Classes. B. Blackwell, Oxford, UK. Marghetis, T. and Núñez, R. (2012). The motion behind the symbols: A vital role for dynamism in the conceptualization of limits and continuity in expert mathematics. Topics in Cognitive Science. Moore, G. H. (1982). Zermelo s Axiom of Choice. Its Origins, Development, and Influence. Springer Verlag, Berlin, Heidelberg, New York. Nersessian, N. J. (1992). How do scientists think? Capturing the dynamics of conceptual change in science. In Giere, R. N., editor, Cognitive Models of Science, volume XV of Minnesota Studies in the Philosophy of Science, pages 3 44, Minneapolis. University of Minnesota Press. Nersessian, N. J. (2002). Maxwell and the Method of Physical Analogy : Model-based reasoning, generic abstraction, and conceptual change. In Malament, D., editor, Essays in the History and Philosophy of Science and Mathematics to Honor Howard Stein on his 70th Birthday, pages Open Court, La Salle, Il. Nersessian, N. J. (2008). Creating Scientific Concepts. MIT Press, Cambridge, Mass. 18

21 Núñez, R. (2005). Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals. Journal of Pragmatics, 37(10): Núñez, R. (2006). Do real numbers really move? Language, thought, and gesture: The embodied cognitive foundations of mathematics. In Hersh, R., editor, 18 Unconventional Essays on the Nature of Mathematics, pages Springer, New York. Núñez, R. (2008). Mathematics, the ultimate challenge to embodiment: Truth and the grounding of axiomatic systems. In Calvo, P. and Gomila, A., editors, Handbook of Cognitive Science: An Embodied Approach, chapter 17, pages Elsevier, Amsterdam. Pasch, M. (1930). Der Ursprung des Zahlbegriffs. Springer, Berlin. Poincaré, H. (1909). L avenir des mathematiques. In Castelnuovo, G., editor, Atti del IV congresso internazionale dei matematici, Roma, 6-11 aprile 1908, volume I, pages , Roma. Tipografia della R. Accademia dei Lincei. Polya, G. (1973). How to Solve It: A New Aspect of Mathematical Method. Princeton University Press, Princeton, NJ. First edition Potter, M. D. (2004). Set Theory and its Philosophy: A Critical Introduction. Oxford University Press, Oxford. Robinson, A. (1965). Formalism 64. In Bar-Hillel, T., editor, Logic, Methodology, and Philosophy of Science, pages Schlimm, D. (2012). Mathematical concepts and investigative practice. In Steinle, F. and Feest, U., editors, Scientific Concepts and Investigative Practice, volume 3 of Berlin Studies in Knowledge Research. De Gruyter, Berlin. (Forthcoming). Schlimm, D. and Neth, H. (2008). Modeling ancient and modern arithmetic practices: Addition and multiplication with Arabic and Roman numerals. In Sloutsky, V., Love, B., and McRae, K., editors, Proceedings of the 30th Annual Meeting of the Cognitive Science Society, pages , Austin, TX. Cognitive Science Society. Scott, D. and McCarty, D. (2008). Reconsidering ordered pairs. Bulletin of Symbolic Logic, 14(3): van Heijenoort, J. (1967). From Frege to Gödel: A Sourcebook of Mathematical Logic. Harvard University Press, Cambridge, Massachusetts. 19