Posted on 22 July, 2007 in category: Interviews

*The following interview between Arkady Plotnitsky and Amir Alexander was conducted on July 22, 2007. The interview as it appears here was transcribed and edited only slightly for clarity.*

**Alexander**: The first question that I have is about how you arrived at the main subject of your essay, the question of non-Euclidean mathematics, and specifically the use of narrative in it. How does this fit in with what is your main field of interests and your general research program?

**Plotnitsky**: The key epistemological problematic of the paper, that of nonclassical epistemology, is a subject which I have extensively explored in my earlier work. There are, however, significant new aspects of this problematic in this essay, especially those related to the question of narrative. Most of this earlier work was related to the 20th-century philosophy and the epistemology of quantum theory. I’ve also done some work on mathematics along these lines, and I developed a concept of non-Euclidean mathematics earlier as well, although in a more preliminary way. It is, however, in working on non-Euclidean mathematics that I came realized the significance of the relationships between nonclassical epistemology, defined by the irreducibly inaccessible nature of the ultimate objects considered by a given nonclassical theory (be it mathematical, physical, or philosophical), and the multiplicity, which appears equally irreducible, of different fields in working with these types of objects. These relationships appear more pronounced in mathematics than in quantum physics, and they have emerged earlier in the history of mathematics, sometime in the early-nineteenth century, although the epistemology in question was not perceived until the twentieth century. A really new dimension of this paper for me is its narrative dimension, which was not addressed in my previous work on physics or mathematics. I have written on narrative in the context of postmodern philosophy and literature, where the relationships between epistemological nonclassicality and discursive multiplicity are found as well.

**
Alexander**: Before we get to the narrative issue, tell us a little more about this multiplicity of fields, how this fits into this notion of non-Euclidean mathematics.

**Plotnitsky**: This is not an easy question to answer. While it is not surprising to me intuitively, I’m not altogether sure in analytical terms why nonclassicality (the irreducibly inaccessible nature of certain objects) and theoretical or discursive multiplicity combine in mathematics, or elsewhere. Both, however, jointly characterize non-Euclidean mathematics, as defined in my paper. Historically, this type of relationships appeared in number theory sometime around 1800, when, more generally, what I call non-Euclidean mathematics appears to have emerged, although as I said, the radical (nonclassical) epistemological implications of this type of mathematics were not initially perceived by mathematicians. Indeed they still appear to be less perceived by mathematicians or concern them less, as against the way physicists react to this type of epistemological situations, because quantum theory and, to some degree, relativity brought them into sharper focus and made them a subject of an intense debate in physics. The epistemological questions, including those with nonclassical implications, have been significant in mathematical logic and set-theoretical foundations of mathematics, especially in the wake of Gödel’s theorem and related findings. In any event, around the time of Gauss, in his own work and that of others, mathematical fields of arithmetic (number theory), algebra (specifically theory of polynomial equations), geometry, and analysis, or their particular subfields, were deployed jointly and interactively in approaching certain problems that would seem to belong to a particular single field or subfield. The mathematics of complex numbers, my main subject in the paper, had a special significance in this context. But there were other developments contributing to this, as it might be called, new mathematical situation, the rise non-Euclidean geometry, for example. Mathematicians such as Gauss or, especially, Riemann (a student of Gauss) were able to use and connect seemingly heterogeneous and distant mathematical areas and concepts. Indeed, part of Riemann’s way of doing mathematics was to work with and to relate certain properties of mathematical objects without necessarily fully specifying these objects themselves. One could, on the other hand, believe in the existence of such objects for various reasons, specifically because of their capacity to have effects upon other objects, including those that we can expressly describe. In the case of complex numbers, for example, however controversial the claim of their existence might have been at a certain point, it was difficult and even impossible to account for certain mathematical “effects” without introducing them.

**Alexander**: The image that I gather from you is that there are certain junctures in the development of mathematics at which new concepts do not fit into previously developed moulds, resulting in the introduction of entities that are, as you said, “unthinkable.” Nevertheless, historically these “unthinkables” become the focus of a whole range of new practices that cluster around them.

**Plotnitsky**: Yes, that is correct. Now, one could argue that at some later point such objects could be properly defined or constructed, thus resolving the situation on more classical lines. Sometimes this indeed happens, as, for example, in the case of irrational numbers, which were arithmetically ungraspable to the ancient Greeks, but eventually became (more) tangible mathematical objects. On the other hand, it is also possible that some objects of this type would remain inconceivable indefinitely, as is the case, thus far, with quantum objects in physics (at least in certain interpretations). Consider infinite-dimensional spaces. Can we possibly have a rigorous mathematical concept of such objects, especially as a space? I am not sure we can. Nevertheless, we can still work with such spaces in mathematics or in physics, which is one of my main points: nonclassical epistemology does not disable mathematical practice, and in certain circumstances, it might enable it.

**Alexander**: The issue here is how much you want to require of mathematical objects to be perceived in that sense. Physics is naturally about the world, and if our theoretical constructs don’t fit the world then we face a problem. This is different in the case of mathematics in which mathematicians are not as conscious of such problem. They think: Well, I’ve defined an object, at least formally and I didn’t find any contradictions, then that’s really all I need.

**Plotnitsky**: Sometimes it’s indeed like that. It might have something to do with the Platonist view of mathematics. (I am not saying Plato’s view, because I believe Plato himself held a different view.) I think that there could be a belief on the part of some mathematicians that, once you define enough properties consistently, such an object probably exists and, for some, even definitively exists as thus defined in a certain Platonic realm. But it is not necessary to have such a point of view of mathematical objects, and there are alternative views (intuitionist, constructivist, etc.). Indeed one of my main points is that our inability to define or even conceive of such objects doesn’t stop mathematics. Of course, encountering the impossibility of even conceiving the objects with which one is concerned, and building one’s theory in spite of it or, especially, as based on it is not easy and requires a kind of “negative capability,” as Keats called it.

**Alexander**: I keep thinking about examples from areas that I’ve worked on, such as the development of the early forms of the calculus the 17th century, which involved deliberate immersion in known paradoxes. For example Evangelista Torricelli, who was Galileo’s student, was constantly trying out paradoxes of the infinite, demonstrating how subversive they were to his work on infinitesimals. And yet, he continued developing his infinitesimal method without ever defining what the objects he was dealing with actually were. Is this equivalent to the moments in the history of physics and mathematics that you talk about?

**Plotnitsky**: I am not sure about the epistemological equivalence, but I would agree that the situations are epistemologically similar; they might be just about equivalent in terms of the “negative capabilities” required to confront them. To make a radical claim, we really know next to nothing about most mathematical objects. In a certain sense, mathematics may have been nonclassical all along, even if without realizing it. Frege once said that it is a scandal that we don’t really know what numbers are. I certainly do not think that we have an access to complex number qua complex numbers via their geometrical representation as points in the Gauss-Argand plan. This is the most controversial claim of my paper, since the commonly accepted view now is that the Gauss-Argand plane properly represents complex numbers geometrically. This was not always the case. As I note in my paper, Gauss himself was ambivalent about this issue, and others, Cauchy, for example, were even more skeptical. During the subsequent history of mathematics, the Gauss-Argand plane became so ingrained in the mathematicians’ thinking that they came to believe, nearly universally, that it is a geometrical representation of complex numbers, while it may be (I argue it is) just a diagram that helps us to work with them rather than geometrically represents them.

**Alexander**: So in your view this isomorphism between the complex number field and the Gauss-Argand plane is fundamentally different from, say, mapping the real numbers on the real line. Or is that also a diagram?

**Plotnitsky**: This is not easy to answer by yes or no. For, as I say in my paper, and as appears from Alain Berthoz’s work discussed in Bernard Teissier’s paper, that there are complexities in the case of real numbers as well. So, we may ultimately not have a rigorous concept of real numbers either, or any numbers, although a geometrical representation of, say, natural numbers (by discrete sets of points on the line) is more accessible to our intuition. But we certainly have mathematics of real numbers, or of complex numbers, a very rich and beautiful mathematics. In this sense, the situation becomes similar to quantum mechanics, where, however, we, again, deal with certain undefinable, inconceivable entities materially existing in nature and detectable with experiments. On the other hand, mathematics, too, may be more experimental than we think. As Federica La Nave’s paper shows, Bombelli was thinking along these lines, and his mathematics was kind of experimental. He accepted the reality of negative roots, the roots of negative numbers, because he could do things with them, without knowing what they are.

**Alexander**: There is a tradition in the 17th century of what can be called experimental mathematics. John Wallis for example is very influenced by Baconian empiricism, and his idea of a proof is: here’s a statement; I try it once, I try it twice, I try it three times, it works. OK, proven. He literally experiments. Along the same lines Bernard de Fontenelle wrote that our knowledge of the calculus is experimental.

**Plotnitsky**: In this case, however, unlike that of physics, we also create things that we are testing, or more so than in physics. For, I take a more Aristotelian (rather than Platonist) view, according to which mathematics and its objects are not pre-existing somewhere but are the products of who we are as human animals.

**Alexander**: That leads me to an issue that you’ve mentioned several times in your paper, which is that “Euclidean thinking may reflect the essential workings of our biological and specifically neurological machinery born with our evolutionary emergence as human animals and enabling our survival”. This suggests that Euclidean thinking is somehow “natural” to us. But Euclidian proof seems very unnatural to me. The very notion that you could prove something was completely remarkable to the ancients, and it only arose in a single historical context, that of ancient Greece. In fact, it’s very difficult for people in general to learn to think in terms of proof.

**Plotnitsky**: This is to some degree true. I would, however, be inclined to qualify the case. Euclidean geometry may seem unnatural because of its (especially by current mathematical standards) relatively abstract nature. On the other hand, it is reasonably natural, as the word geometry suggests: measuring the earth. Euclidean geometry or, in part correlatively, classical physics, as Bohr and Heisenberg note, represents the refinement of the way we generally think in everyday life, in which the proximity between both is rather transparent. Elsewhere in mathematics the distance between both becomes much greater and could eventually lead to a near unbridgeable separation, a divorce, between mathematical and everyday thinking. That is one of your own research subjects. Essentially, I think that whatever we can form a conception of and especially visualize is classical. On the other hand, quantum objects and behavior, or the ultimate nature of certain mathematical objects, is not something we can really conceive of in our thought. Thus, nature and our mathematical, or philosophical, thought (which is still the product of nature) bring us to the point where our thinking constructs the possibility and indeed necessity of “objects” that we cannot think, cannot conceive of. This is another crucial point of my paper, beginning with the question of the diagonal of the square, with the fact that the Greeks arrived at this type of rigorous mathematical construction of the irrational from and through the rational. That was, at the time, a kind of “scandal,” at least, as I say in the paper, in some modern “reconstitutions” of the case (and there, I admit, alternative views of the situation). Mathematics was also a model, the model of rational thinking. Geometry helped to avoid this particular scandal, albeit in a kind of proto-nonclassical way, while arithmetic was restricted to its proper domain of dealing with proportions. That this subtle proto-nonclassicality and hence the scandalous nature of geometry were not perceived at the time helped both ancient Greek mathematics and culture, or a certain Socratic Greek culture. The preceding, tragic, Greek culture, as Nietzsche calls it, was more open to and even defined by such conflicts, since the conflicting entities—being and becoming, the Apollonian and the Dionyssian, and so forth—were accepted and even celebrated in their coexistence, and hence were not seen as scandalous.

**Alexander**: Let me recapitulate so we can get back to the question of classical mathematics as against non-classical, non-Euclidean mathematics. It seems that the crisis of the irrational was the first scandal, and that it was settled, at least provisionally, by splitting geometry and algebra apart. But in the 16th century the trouble emerges again, with Bombelli and others: there is the scandal of imaginary numbers as well as an ongoing scandal of infinity, which becomes the central object of mathematics in the 17th century. This pattern seems to question whether there is just one big break in the development of mathematics, that between the Euclidean and the non-Euclidean, or whether perhaps such breaks are endemic to the development of mathematics.

**Plotnitsky**: It’s a good question. As our earlier discussion suggests, the history in question is even more multiple and multifaceted than this picture suggest. There are both “scandalous” breaks and more continuous developments, involving various degrees of both key features of non-Euclideanism: nonclassical epistemology and the interaction between different fields involved. The combination of both is crucial to my conception of non-Euclideanism, and, the second aspect of the phenomena emerges with an eruptive force around 1800, with the rise of many new subfields and branches of mathematics, and of new forms of interaction between them. On the other hand, it took a while to realize the radical, nonclassical epistemological features of this mathematics and some of the mathematical objects where these features are more pronounced have indeed appeared later (so there is a history in this respect as well). Of course, the multiplicity in question has its history as well, a long and rich history. As Barry Mazur astutely observed in commenting on my paper, the scandal of the diagonal reflects, including in epistemologically proto-nonclassical terms, the relationships between arithmetic (and by implication algebra) and geometry, and friction or obstructions in trying to bringing them together. The harmonious bringing them together is a kind of hope, an eternal and eternally unfulfilled hope, of mathematics. While it’s not quite clear to me (not being a historian of mathematics) to what degree the ancient Greeks were thinking in these terms, they must have been thinking about the connections between both. This thinking is manifest in the Pythagoreans’ discovery of the irrationals, or, to begin with, in the idea of measurement, which defines geometry. With Descartes and his project of analytic geometry that dream reemerged with a new force and appeared possible to realize, now with the help of algebra. But then a much richer architecture of multiplicity (and with more radical epistemological implications) has emerged in the nineteenth century.

**Alexander**: Right, you can think of the case of analytic geometry in these terms — geometry and algebra are two fields, and they are combined into a single field. What I think you are referring to in the 19th century is more radical — truly different fields that somehow converge to inform each other on a particular question. This seems to be a later phenomenon, and Barry Mazur’s paper demonstrates it very clearly.

**Plotnitsky**: Yes. Thus, while, as Federica La Nave’s paper, again, beautifully demonstrates, Bombelli and others already showed earlier how you could do things with imaginary roots by combining algebra and geometry, the explosive power of the mathematics of complex numbers and its impact elsewhere in mathematics and physics only emerged in the nineteenth century. So much was gained as a result in number theory, algebra, analysis, algebraic geometry, and so forth.

**Alexander**: Do you see, then, a fundamental difference between the use that Bombelli makes of imaginary numbers, and the field of complex analysis as it was developed 200 years later?

**Plotnitsky**: Yes, very much so. Bombelli’s approach is important, conceptually and historically, but the difference is essential, all the same. I don’t think that one could really speak of Bombelli’s mathematics as non-Euclidean in the present sense, even though one can locate certain elements (the link between algebra and geometry, for example) of non-Euclideanism in it.

**Alexander**: It seems to me that there is a qualitative difference here. Bombelli was simply saying that we can solve this equation by acknowledging ”imaginary” roots. But the field of complex analysis emerges at the period that you talk about at the nexus of your story.

**Plotnitsky**: That is right. It is not only a matter of gains for mathematics (and one can also speaks of losses, insofar as certain things previously useful were no longer possible), but also of the emergence of a different type of mathematics. Other developments were also crucial, for example and in particular, non-Euclidean geometry.

**Alexander**: Right, Non-Euclidean geometry is one of the cases in which, as you say in your paper, the problem becomes the solution. The problem that you cannot prove Euclid’s 5th postulate becomes the starting point for a new mathematics and a new physics. I find this historically fascinating because it brings out the aspect of multiplicity that emerges at this moment. To me this is very strong evidence for the break that you write about between non-Euclidean and Euclidean thought in this period. As Barry Mazur notes, the issues of friction between geometry and algebra have a long history, but nevertheless it seems that something critical happens at this point.

**Plotnitsky**: Well, first of all, one also had analysis, a field that is algebraic in the sense of formal manipulation of its symbols but that deals with things, such as continuity, that are not part of algebra. So, by that time, there are three main areas of mathematics rather than only two: algebra, geometry, and analysis. And, again, one also had complex numbers. These “resources” allowed mathematicians to give very rich structures to the mathematical objects involved, to enrich old ones and to create new ones.

**Alexander**: So it seems that this process was very gradual. Geometry is of course very ancient as is arithmetic, modern algebra dates to the late 16th and early 17th century, the calculus to around 1700, and then general analysis to the 18th century. One can conclude that it is a slow process in which each crisis generates a new object, a new field of discovery.

**Plotnitsky**: I would say, along the lines I suggested earlier, that the process is long, because, while at certain junctures things may accelerate fast and punctuations may occur near to each other, it takes time to cohesively integrate new ideas even within the original field of their emergence and, all the more so, on broader scales. I would also argue, however, that this integration is often productive and sometimes necessary, it is in general partial, especially on broader scales. The history of mathematics continues to bring its diverse concepts and fields together, but, it appears, without the possibility of an ultimate unification (partial or provisional unifications are possible). Such an ultimate unification may not be possible even within a more limited configurations or, in the language of Barry Mazur’s paper, templates, such as those created by programs like that of Kronecker or, later, that of Langlands. For, as these programs historically develop, things not only come together but also break apart, creating new separate trajectories, sometimes new programs. It is a great interplay of continuities and discontinuities, interconnections and disconnections, necessities and chances, and so forth, which becomes richer and more complex as we move to greater configurations of knowledge within and beyond mathematics. It’s like an archipelago, where you can build bridges between islands. In fact, I don’t even like the metaphor of bridges, I prefer that of ferries that move between different islands and sometimes run aground or arrive at unexpected destinations. Hegel already thought in terms of similarly multi-component and multi-interconnected architecture of knowledge and its history. But he still appears to have believed that it is possible, in principle, to bring it into a form of unity through the dialectical synthesis of difference and contradictions. I think, however, that even for mathematics, let alone for knowledge in general, that does not appear possible, and non-Euclidean mathematics is a manifestation of both this impossibility and of the productive use of interconnections without an ultimate synthesis.

**Alexander**: Could you say something about the work that narrative does in your theory, and what it adds to your methodology. Is it essential to it?

**Plotnitsky**: It may be more essential methodologically than I thought before working on this paper, although I have always recognized the essential role of narrative in the epistemological problematic in question. Narrative is obviously essential for this paper, both conceptually (it is about the narrative) and methodologically. For example, although it is not strictly a historical paper, I was interested in certain historical situations that have played themselves out in cases where the construction of certain narratives was crucial, because this is how culture and politics work. In your own paper, the figure of the mathematical martyr became part of a certain narrative operative in culture. The scandal of the irrational is also part of a certain cultural narrative. I stress “part,” because such situations are not reducible to narrative, even though their narrative component is irreducible. One can also flip the coin. The story a person goes and drowns himself, or herself, after discovering the irrationals may be an allegory of the shipwreck of the Pythagorean arithmetic; but it may have also been true. Gauss had famously concealed his discovery of non-Euclidean geometry for year for fear (justified) of being laughed at by philistines. It should be noted, however, that it is disputable and even doubtful that Gauss had ever fully established the existence of non-Euclidean geometry. Although I’m ambivalent of many of Socrates’ ideas, he was executed for them after all. Nietzsche even says that the Greeks had good reasons to prosecute, even if not execute, Socrates because he threatened to replace the tragic culture of the Greeks with a new culture, the despotic logical culture, with an androidal culture, if I may say so.

**Alexander**: True, his political propositions do not seem appealing at all… Thinking along your lines about narrative, and the role it plays in those schemes, it seems that once you start to think about this not just abstractly but temporally, about crises that occur historically at particular times with particular agents and certain stories, it becomes a very powerful (as well as convenient) way of talking about these transitions.

**Plotnitsky**: It plays a major role. It is not only a matter of it being convenient or powerful, although this is also true. Even though other ways of thought remain important and irreducible, there may not be another way to think about temporality and history without narrative.

**Alexander**: You bring in the narrative in the crisis moments that we talked about. But I wonder what happens to narrative in between those crises. Would you say that it plays a role then as well?

**Plotnitsky**: I would , again, advocate a more multiple and dynamic picture, along the lines suggested above. Many histories happen at the same time, and the “same” event may be a crisis, a radical break, in one history and, in Kuhn’s terms, normal practice, part of a continuous trajectory, in another, either contemporaneously or in retrospect. These types of events may, however, have a great transformative power.

**Alexander**: In thinking of narrative in this context I can see that it has an enormous transformative power. It seems to me that narratives by their nature are polemical, and that they would naturally come to the fore at those moments of crisis. Their function in-between these points is to me an open question.

**Plotnitsky**: That is certainly true because I think narrative is a crucial part of the rhetoric of culture. Let me clarify something, however. I think that almost everyone, even the most rigid Platonists, would subscribe to the theory that mathematics is an exploration of the unknown. But nonclassical epistemology entails a different narrative model, in which, at a certain point, you can no longer continue this type of exploration and move into an unknown territory and gradually make more and more of it known through exploration. Nonclassically, the unknowable is always part of knowledge, which also entails a different type of narratives of knowledge. Now, to return to history, via the question that Peter Galison asked me concerning a possible concept or model of history grounded in nonclassical epistemology, coupled to the irreducible discursive and, especially in this context, specifically narrative multiplicity. Would you agree, as a historian, that it is not an easy model of history to develop or practice?

**Alexander**: Definitely.

**Plotnitsky**: Indeed, even if applied to events that involve encounters with nonclassicality, the models and narrative of history used in my paper or that we have invoked thus far are more or less classical models, coupled as they may be to discursive and narrative multiplicities. That is, however, different from constructing a non-classical model of history and a corresponding narrative, which is hard.

**Alexander**: It is very hard; on the other hand, it is also very clear today that classical narratives of history are sorely deficient. You can tell all the grand narratives you want, and you might capture various things about, say, technology, the march of progress, or politics, or the neo-conservative march of democracy or whatever else you like. You can tell those stories, but whichever narrative you tell, it more than falls short. It is almost an act of wilful desperation for a historian to try to tell a single “classical” narrative.

**Plotnitsky**: That’s quite true. Let me, however, reiterate the distinction between the narrative of history that involves encounters with nonclassicality, in which narrative itself could still be classical, and the role of nonclassical epistemology in the history of mathematics and science, or elsewhere. Dealing with nonclassicality does not in itself imply a change in our thinking concerning the nature of history or our practice of history.

**Alexander**: I perfectly agree.

**Plotnitsky**: However, non-classical narratives of historical events, even apart from any role of non-classical epistemology in this event themselves (say, those of classical physics) is, in principle, possible, and perhaps necessary.

**Alexander**: Yes, that’s what I’m trying to get at. What would such non-classical narratives look like?

**Plotnitsky**: I cannot think of historical studies where this type of history is practiced (perhaps because of the disciplinary constraints), although one might find certain non-classical elements in such studies. We might need to look at literature for examples of enactments of non-classical historical and narrative models—Joyce, Beckett, Woolf, or, closer to history, Faulkner. I do think that Faulkner gives us a non-classical picture of American history. I am not unhappy that I am not a historian and that I don’t need to write non-classical historical accounts, because they would be difficult to construct and, I suspects, even more difficult to publish.

**Alexander**: Very difficult. The examples that you give are of self-contained literary narratives, but to apply these to a historical narrative is very hard. You probably end up as with mathematics – with a multiplicity of fields and a multiplicity of narratives. And that is in fact what you see in actual historical practice – a multiplicity of narratives. You find so many overlapping, and competing stories, so many different ones, and none of them can make a claim to being hegemonic.

**Plotnitsky**: It’s difficult for psychological reasons, it’s difficult for rhetorical and institutional reasons. It is one thing for a writer like Faulkner or Beckett to write a novel of this type; it’s another thing to write a history of mathematics and science that operates in a radical register of discourse.

**Alexander**: But that’s the thing; what separates historical narrative from literary narrative is the claim of the factual truth of actual events, which literary works don’t require. You can never give up this claim as long as you’re doing history.

**Plotnitsky**: You’re quite right. How do you describe a fact or event nonclassically? That could, I think, be done in principle, for example, on a literary model, say, the way certain events, especially conversations, are described in Faulkner. Whether it could be done in practice of your discipline is another question, because a specific narrative of certain events, often key events, such as the discovery of the irrationals would be like a novel, more like Faulkner’s Absalom! Absalom!, Joyce’s Ulysses or Finnegan’s Wake, or Beckett’s The Unnameable, a most fitting title here.

**Alexander**: Yes, it will not be a historian who will write that history. Historians are willing to qualify their truth claims to some extent, to say that they tell stories about the past, that they use imagination, empathy, etc. But ultimately, the only basis, the only reason for the existence of the field is that in some way it’s about “what really happened.” Nevertheless, things have thoroughly changed. Not so long ago it was generally assumed that underlying all of history was a single unified narrative; you might be telling part of it, somebody else might be telling a different part of it, but all ultimately join together into a unified narrative that will tell us what the past really was. So a researcher looking for a topic would search for a topic “that hasn’t been done.” The point was to cover a little bit more of this great unified landscape of history. That is no longer the case. Historians do not pick topics in that way. The current assumption is that there are so many different possible questions, so many different possible stories, so many possible heroes and villains that you could talk about. And this multiplicity is now integrated into the structure of history. Without any acknowledged crisis, the underlying assumptions of the field have been transformed. This strikes me as profoundly related to the non-classic move.

**Plotnitsky**: To what degree did the history of mathematics itself follow this shift? It seems to me, for example, that the heterogeneity or non-Euclideanism of mathematics has not as yet found its way into the multiplicity of narratives about mathematics. It also seems to me that mathematics itself has continued to have and portray a more homogeneous Platonist image of itself than sciences have, especially biology and information sciences and technologies, which recently replaced physics as dominant sciences in culture. It appears that the field of the history of mathematics—I don’t mean the particular historians, but the institution or the field as a whole—has not as yet portrayed this multiplicity, which is always historical (an intersection of events and trajectories) of modern mathematics. If we had adopted a more Aristotelian view of mathematics, which would be related to the world, the history of mathematics could also benefit from it. On the other hand, this divorce of mathematics from reality, from everyday phenomena was also extraordinarily effective and productive.

**Alexander**: Extraordinarily productive, no question… But here’s the paradox of mathematics. I think you make a wonderful case about this multiplicity of mathematics and those overlapping fields and making it almost a prototypical non-Euclidean, non-classical field. On the other hand, mathematics has a cultural role which can be described as “the guardian of modernity,” of the modern hierarchy of knowledge. For example, you are right when you say that information science or technology is the dominant form of knowledge, at the moment. But mathematics has a role here; it may not be at the centre of things, but it guarantees their validity. It guarantees that information technology is rational, that it works and so on. Because of this function there is a widespread sense that mathematics must be kept unified and pristine, insulated from the world.

**Plotnitsky**: The event, to use a narrative category, like this—the conference, the book, the interviews—is significant from this point of view because it tries to bring the world to mathematics, and mathematics to the worlds, through narrative. It might help a greater dissemination of mathematics in culture and, as such, may have an effect upon the history of mathematics. It seems to me that our discussion now arrived to what defines the event of which it is a part: the role of the narrative in mathematics at various levels, from the cultural narrative of the kind you analyse to the kind of epistemologically defined narratives that are subject of my paper to the quest narratives analysed in Apostolos Doxiadis’ and Michael Harris’ papers, to the narratives of journeys of mathematical ideas themselves considered by Barry Mazur, so forth. Or, from the other pole (although there are more than two poles) of this event, that of narrative theory, there is an introduction of certain, let us say, quasi-mathematical models into the theory. This bringing narrative into our understanding of mathematics could, I think, also be productive in reaching the field of history of mathematics and, again, in enriching the relationships between mathematics and culture in general, in part because, while much of modern mathematics is divorced from the world, most of the world in which most of us live, the narrative is far from being so divorced.

**Alexander**: I agree. Mathematical narratives are a kind of “Jacob’s ladder”…

**Plotnitsky**: I think, to return to my earlier metaphor, it’s a ferry that can bring us to the island of mathematics or run aground close enough, so we swim across. I prefer horizontal metaphors.

**Alexander**: I understand why.

**Plotnitsky**: It brings us close enough to see and, at least to some degree, understand, the architecture of mathematics, which, as we discussed here, is more like a city than a building because of its interactive heterogeneity, multiplicity. This, I now realize, is itself a metaphor and a narrative that connects mathematics and the (urban) world, at least, formally, kind of mathematically. But in any event, I think the narrative can help us come closer to mathematics.

**Alexander**: I would definitely like to do that…

**Plotnitsky**: But I think that’s what you are doing here, what we are doing here. And historical narratives, such as those told in your paper, are part of that process, as are also fictional narrative, such as that of Apostolos Doxiadis’ novel. They are, let us say, narrative aspects of the grand phenomenon of mathematics. The narrative could serve to open, to use a metaphor of Gilles Deleuze, new lines of flight, in both directions, between mathematics and the world.

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