On Mathematical Towers of Babel and “Translation” as an Epistemic Category

In his manuscript “The Origin of Geometry,” written in 1936 but published posthumously in 1939, Edmund Husserl notes the following:

The Pythagorean theorem, [indeed] all of geometry, exists only once, no matter how often or even in what language it may be expressed. It is identically the same in the “original language” of Euclid and in all “translations”; and within each language it is again the same, no matter how many times it has been sensibly uttered, from the original expression and writing down to the innumerable oral utterances or written and other documentations” (Husserl 1970, 357).¹

Husserl, the founder of the school of phenomenology, was no stranger to mathematics. He obtained his PhD in mathematics in 1883 with a thesis “Contributions to the Calculus of Variations” [Beiträge zur Theorie der Variationsrechnung]. Hence, assuming that Husserl encountered (at least) two different proofs of the Pythagorean theorem in his mathematical studies, one has to wonder, when looking at this theorem, whether the “original” proof of Euclid and its “translations” are indeed the same. It is clear that Husserl here conflates not only the numerous proofs of the Pythagorean theorem that were already known back then, but also the dependence of those proofs on the cultural milieu, where and how those proofs were discovered and developed: taking into account the class of “all scientific constructions and the sciences themselves,” he notes that “[w]orks of this class do not, like tools … or like architectural and other such products, have a repeatability in many like exemplars” – since they exist only once (ibid., 356–357); that is, these works have what Husserl calls an “‘ideal’ objectivity”² (ibid., 356).

Although I will not discuss in this paper Husserl’s philosophy of geometry, one must note that from today’s perspective of the philosophy (and the history) of mathematics, and especially of the philosophy of mathematical practices, not only the use of the term “translation,” but also the activity itself of translating, into, for example, a (more) formal language, is considered anachronistic, not to say misleading and distorting. As I will survey briefly at the end of this article, Thomas S. Kuhn and Peter Galison, to give two examples, criticize translation from the perspective of the philosophy of science, though their solutions were different. To focus on the philosophy of mathematics, Brendan Larvor indicates that

in examining the logic of a proof, we should look at the proof in the form and the idiom in which it is used, rather than first translating it into some other idiom. It may be that the proof employs inferential moves that are possible only using precisely that kind of representation, governed by precisely those rules (Larvor 2019, 2723).

Moreover, there is sometimes, according to Larvor, a “non-universality of [the inferential] moves available in some notational or diagrammatic practice,” which resists “translation into formal logic” (ibid., 2725).

Nevertheless, the question that arises is whether and how the term “translation” was actually used by mathematicians during the nineteenth and twentieth centuries. Does Husserl’s view, however misleading it may be regarding how mathematics is practiced in “real life,” reflect in any way the opinions of other mathematicians? Did other mathematicians employ the term “translation” as an actor’s term, thus understanding it as an epistemic category to describe relationships between mathematical idioms or domains, and if so, what did they mean by it? To answer the last question, I will present three case studies from the nineteenth and twentieth centuries of mathematicians (or communities of them) who used the term “translation.” The examples are taken mostly from the history of geometry and its interaction with algebra, where the term “geometry” is taken in its broader meaning, including, for example, algebraic geometry. This, let me emphasize, does not mean that there do not exist other mathematicians who used this term with respect to other areas of mathematics. The goal of this article, as will be shown in the second part, will be to explicate what was meant by the mathematicians who were using this term: were they following Husserl’s view, or did they have a more restricted and local notion of translation, in the meaning of Galison’s “trading zones,” as I will argue at the end of this paper?

“Translation” as an Actor’s Term: Three Case Studies

I will begin by considering three case studies in which the term “translation” was indeed used by mathematicians: the first from the beginning of the nineteenth century, concentrating on the rise of descriptive geometry in France; the second, which deals with how hyperbolic geometry was treated at the end of the nineteenth century; and the third, how the metaphor of the “Tower of Babel” was employed in considering twentieth-century mathematics in general and formalism in particular. I chose these case studies because they all show communities of mathematicians who were using the term in a similar fashion.³

Case Study I: Descriptive Geometry: Monge, Laplace, Lacroix, and Biot

Our first example deals with the relationships between algebra and geometry at the beginning of the nineteenth century, when the methods of algebra and analysis slowly were able to prove new results that were not predicated by geometry. The rise of descriptive geometry and the extensive use of analytic geometry signify one of these new developments. What prompted these new developments was the founding of two novel institutes in Paris: the École Polytechnique, providing training for civil and military engineers, and the École normale de l’an III, which was devoted to advancing teacher education. The country’s best mathematicians, for example Monge, Laplace, and Lagrange, taught in both schools. The lessons given as well as the new results that were proved were often accompanied by reflections on the nature of the mathematical language.

I would like to begin with Monge, and especially how he defined the objects of descriptive geometry.⁴ In his teaching at the beginning of 1795, Monge emphasized in his lesson program at the École normale de l’an III the geometric visualization of mathematical and physical problems and defined the aim and objectives thus:

The first [objective] is to accurately represent, with drawings that have only two dimensions, objects that have three and are amenable to rigorous definition.

From this point of view, it [descriptive geometry] is a necessary language for the man of genius who designs a project, for those who must direct its execution, and finally for the artists who must themselves execute its various parts.

The second objective of descriptive geometry is to deduce from the exact description of the [geometric] bodies everything that necessarily follows from their respective shapes and positions. In this sense, it is a way of seeking the truth (Dhombres 1992, 305–306).⁵

Taking this statement into consideration, and especially the “necessary [geometric] language” for the introduction of new, possibly abstract, concepts, Monge described either the concrete situations or the possible applications in which a concept appears (Sakarovitch 1998, 214). This was done through drawing and modeling. Indeed, Monge was one of the first mathematicians at the École polytechnique to build three-dimensional models of mathematical surfaces, a tradition that was further developed by his student Théodore Olivier.⁶ Moreover, Monge emphasized the connections to practical concerns (such as the needs of the army) as well as the methods of stone carvers. He also stressed the importance of the relationships between analysis, analytic geometry, and descriptive geometry in his book Géométrie descriptive at the end of the eighteenth century:

It is not without an object that we make a comparison here between descriptive geometry and algebra …. There is no construction in descriptive geometry that cannot be translated into analysis; and if the problems have no more than three unknowns, then every analytic operation can be regarded as the script for a spectacle in geometry (Monge 1799, 16).⁷

It is clear that Monge thought about geometry and algebra as languages. This way of presenting the relationships between algebra and geometry, as a “translation” between languages, is also to be seen in the writings of other French mathematicians, such as Biot and Lacroix. Biot noted in 1802 that

to show how one can write down in analysis the questions of geometry, and conversely, how one can translate into geometry the results of analysis: this is the goal of the application of algebra to geometry (Biot 1802, 1).⁸

Three years earlier, Lacroix called for a much stronger separation between the two languages, writing:

but I would not wish to mix … geometric considerations with algebraic calculations; it would be better … that the results of each illuminate the other, corresponding to each other, so to speak, as the text of a book and its translation (Lacroix 1797, xxvi).⁹

Indeed, while Lacroix attempted to point out an equivalence between the two languages as separate but functioning as equivalent “books” that illuminate each other,¹⁰ Monge noted that this image of “translation” does not imply that it is a perfect translation, that is, the translation employed does not necessarily point toward a full equivalence between the languages: in order to translate into the geometric language, thus Monge, one must have “no more than three unknowns.”¹¹

This image of “translation” as either what may constitute a partial equivalence or as what points toward a hierarchy of languages in the sense of a hierarchy of generality is to be seen also with other teachers at the École normale de l’an III, namely Lagrange and Laplace. On the one hand, Lagrange claims “without metaphor” [sans métaphore] that arithmetic and geometry are the “foundation and essence of all the sciences that treat magnitudes”¹² and that one must “translate” arithmetic results into geometric ones or conversely to see how these languages complete each other (in Dhombres 1992, 211).

One may assume that by the expression “without metaphor” Lagrange means that this “translation” between the two foundations is what should be done in practice. But nevertheless, it is an equivalence that is restricted to the sciences “that treat magnitudes.” On the other hand, for Laplace, “translation” occurs only into the “algebraic language”—and both expressions, “translation” and “algebraic language,” are used often side by side (see, for example, Dhombres 1992, 46, 60, 61, 67, 103). However, the expression “geometric language” does not appear at all in his lecture notes, hence implicitly underlining a hierarchy between the languages—which one is more general—or even an implicit understanding of what may even be considered a language.

Case Study II: Poincaré and the “Dictionary” Between Languages

As we saw with the first example, at the beginning of the nineteenth century, several mathematicians—Lacroix, Laplace, Monge, and Biot—considered the relationship between presenting the same problem or the same theorem in both geometric and algebraic language as one of translation, though some of them noted that one language may be more powerful than the other, or that such translation can be only partial (obviously, this discussion did not end at the beginning of the nineteenth century, but seems rather to persist through descriptions of analytic geometry). That is, these mathematicians did not look at different objects and translate between them; rather, it is the same object that is considered, presented in different languages.

A similar situation occurred at the end of the nineteenth century with the rise of different “interpretations” of non-Euclidean geometries. To recall, during the first decades of the nineteenth century, the mathematicians János Bolyai and Nikolai Ivanovich Lobachevsky developed non-Euclidean geometries, in particular hyperbolic geometry. They did so by developing a geometry in which a revised parallel postulate was introduced, according to which more than one line can be drawn through a given point parallel to a given line. Though neither of them proved that this new geometry is consistent, the novelty from their side (and also from Gauss, who did not publish his results) was a strong belief that other geometries were possible in addition to the Euclidean one. To make this belief concrete and more intuitive, the question was whether one could find a surface on which those geometries could be illustrated.¹³

In 1868 and 1869, Eugenio Beltrami published his two seminal papers “Essay on the Interpretation of Non-Euclidean Geometry” and “Fundamental Theory of Spaces with Constant Curvature.” Beltrami asked whether there are two-dimensional surfaces, embedded in Euclidean space (and having another metric), that satisfy the revised parallel postulate. To answer the question in the affirmative, he constructed several surfaces of revolution (or partial such surfaces) embedded in three-dimensional Euclidean space on which the geometric theorems of the Bolyai–Lobachevsky hyperbolic plane could be easily interpreted and verified. As a result of the construction, these were surfaces of constant curvature, which Beltrami called pseudospherical. Moreover, at the beginning of the “Essay,” Beltrami noted that his goal was to find a “concrete interpretation” (Beltrami, 1868, 381), that is, “to find a real substrate for this theory [of the hyperbolic plane] before admitting the need for a new order of entities and concepts to support it” (ibid., 375).

The “real substrate” is, for example, an infinite surface of revolution embedded in a three-dimensional space, but it was considered—as Moritz Epple showed—a product of “concrete” imagination, one that went beyond the traditional world of geometric illustration and visualization (Epple 2016, 23). This concretization was also materialized in the form of a physical material model, constructed out of paper (see Figure 1 and also Friedman 2018, 142–150). The material model, being an object in the physical world and hence a “concrete” object, is also what can (and should) be imagined—exactly since what is modeled is only a part of the entire surface.

Figure 1

Left: Bonola’s drawing of the pseudosphere and the tractrix (Bonola 1906, 123). Right: photo of one of Beltrami’s models, indicating that it is to be found in the model collection of the mathematical institute in Pavia (ibid., 124).

For Beltrami, the use of the term “interpretation” was still rooted in the tradition of Anschaulichkeit, to be seen either in his “concrete imagination” or in his construction of material models. However, one may ask whether the term itself was considered as what may also to be found in the semantic field of the word “text” as one might expect from expressions such as “interpretation of a text.” This shift in meaning is indeed to be seen in 1891 with Henri Poincaré, who coupled the term “interpretation” with other terms: “translation” and “dictionary.”¹⁴

Poincaré discusses in his paper “Les géométries non euclidiennes” (1891) the different models of hyperbolic geometries, calling them “interpretations.” Though he mentions Beltrami’s surface of revolution explicitly, he does not note his material models (or any other material models; see Poincaré 1891, 770–771). He then claims, in the section “Interpretation of Non-Euclidean Geometries”:

It would be easy to extend Beltrami’s reasoning to three-dimensional geometries, and minds that do not recoil before a space of four dimensions will see no difficulty in it … . I prefer to proceed otherwise. Let us consider a certain plane, which I shall call the fundamental plane, and let us construct a kind of dictionary by making a double series of terms written in two columns, and corresponding each to each, just as in ordinary dictionaries the words in two languages that have the same meaning correspond to each another … [we can obtain theorems from one interpretation within the second one] as we would translate a German text with the aid of a German–French dictionary (ibid., 771).¹⁵

Poincaré provides an excerpt from just such a dictionary, as can be seen in Figure 2.

Figure 2

Poincaré’s translation table (Poincaré 1891, 771). The use of a table that transfers terms from one “language” to another was not new. An obvious example from the beginning of the nineteenth century is Gergonne’s two-column method to introduce duality between concepts in the framework of projective geometry.

While Poincaré refers to Beltrami’s choice of words when he notes that “Lobachevsky’s geometry, being susceptible of a concrete interpretation, ceases to be a useless logical exercise (ibid.),” it is clear that with the use of the “dictionary” between two languages, a shift of meaning of the expression “interpretation” occurred. For Poincaré, the main point was to show that hyperbolic geometry is consistent, based on the assumed consistency of Euclidean geometry. The way he formulates the following claims underlines how the terms “dictionary” and “translation” functioned together in his discourse:

if two of Lobachevsky’s theorems were contradictory, the translations of those two theorems made with the aid of our dictionary would be contradictory as well. But those translations are theorems of ordinary geometry, and no one doubts that ordinary geometry is exempt from contradiction¹⁶ (ibid.).

It is essential to note that for Poincaré, this “interpretation is not unique, and several dictionaries may be constructed analogous to [the] above” dictionary. This nonuniqueness is reflected in the fact that for Poincaré, the axioms of geometry are conventions and are neither empirical nor a priori synthetic judgments (ibid., 773). Maria de Paz stresses that this concept might have been transferred from the field of jurisprudence, and that Poincaré’s use of this concept “implies … the existence of alternative conventions” (de Paz 2018, 246).¹⁷ But it is exactly these alternative conventions that reflect the nonuniqueness of the dictionary. Just as there are several possible conventions, there are several possible dictionaries—each dictionary is between two different languages.

Poincaré’s “dictionary” and the possibility of using several alternative such dictionaries did not go unnoticed; it was certainly a part of the elaborate and profound mathematical discussion during the 1890s regarding the nature of the axioms of geometry and the relationships between the theory and the different “systems” satisfying it. Indeed, the term “system” was employed to denote a collection of axioms that satisfy a theory.¹⁸ Though a discussion of the different concepts and terms that were used by various mathematicians dealing with the foundations of geometry at the end of the nineteenth century (Hermann Wiener, Moritz Pasch, David Hilbert, to name only a few) is beyond the scope of this paper, it is illuminating to give one example from this period. Hilbert, in a letter to Frege on December 29, 1899, wrote the following:

every theory is only a scaffolding [Fachwerk] or schema of concepts together with their necessary relations to one another, and … the basic elements can be thought of in any way one likes. If in speaking of my points I think of some system of things, e.g. the system: love, law, chimneysweep … and then assume all my axioms as relations between these things, then my propositions, e.g. Pythagoras’ theorem, are also valid for these things. In other words: any theory can always be applied to infinitely many systems of basic elements. One only needs to apply a reversible one-one transformation [Transformation] and lay it down that the axioms shall be correspondingly the same for the transformed things (in Frege 1980, 40–41).¹⁹

How Hilbert refers to the Pythagorean theorem certainly reminds us of how Husserl, almost thirty years later, would refer to it, as we saw in the introduction; this is perhaps not surprising, given the fact that from their acquaintance in 1901, Hilbert and Husserl were close friends and knew each other’s works well (though there were differences in their approaches).²⁰ And although Hilbert does not talk about “translation” in 1899 as Poincaré does in 1891, he does talk about “transformation,” and one may assume that for Hilbert, in a way resembling Poincaré’s dictionary, there was a dictionary transforming the system of points, lines, and planes into the system of love, law, chimneysweep. But whether it is (or may be considered) a “dictionary” or “transformation” from a theory to a system or only from one particular system to another remains somewhat unclear in the letter from 1899.

Poincaré’s concepts of “dictionary” and “translation” were also used during the first two decades of the twentieth century. In 1906, the Italian mathematician Roberto Bonola published his influential book La geometria non euclidea: Esposizione storico critica del suo sviluppo.²¹ In that book, Bonola, in the section “Geometry on a surface,” first discusses surfaces of constant curvature: positive, zero, or negative. He then presents two drawings: one of the tractrix (together with its equation) and one of the surface of revolution obtained from it, called the “pseudosphere.” The section ends with a photograph of one of Beltrami’s paper models (see Figure 1 and Bonola 1906, 123–124). Immediately following this discussion, Bonola emphasizes another way of thinking about non-Euclidean geometry:

There is an analogy between the geometry on a surface of constant curvature … and that of a portion of a plane, both taken within suitable boundaries. We can make this analogy clear by translating the fundamental definitions and properties of the one into those of the other (Bonola 1906, 125).

Also here, the term “translation” appears to show the logical relationships between different terms, which Bonola presents in a translation table.

Bonola’s work became quite well known. To give one example, in 1913, Hermann Weyl published his influential book Die Idee der Riemannschen Fläche; but in discussing the non-Euclidean plane, he does not mention any of the material models of Beltrami. He does, however, emphasize a “translation” between basic concepts (points, lines) similar to Bonola’s; after this translation, “the complete Bolyai–Lobachevsky geometry holds for these ‘points’ and ‘straight lines’” (Weyl 1913, 152). As a reference for this “model of plane non-Euclidean geometry” (ibid.), Weyl provides the title of the German translation of Bonola’s book. Essential to note here is that Weyl employs the term “model” in the sense of a system of concepts and axioms and not in the sense of a material one.²² In his book Raum Zeit Materie, Weyl presents Beltrami’s “Euclidean model” (1919, 83) of non-Euclidean plane geometry (which holds on a surface of revolution; hence by Beltrami’s “model” Weyl does not mean the physical one). But he once again principally stresses the relationship between models of Euclidean and non-Euclidean geometry as a kind of translation between two languages: one constructs a “Euclidean model for the non-Euclidean geometry” and “sets a lexicon by which one translates the concepts of Euclidean geometry into a foreign language, that of non-Euclidean geometry” (ibid., 71–72).²³ Again, Weyl completely ignores Beltrami’s paper models and his “interpretations.” The concept of “translation” becomes related to other semantic fields, related to “text” and “language.” While the term “translation” might have indicated previously a translation between a concrete material interpretation and a set of equations, now it was between two abstract models, consisting solely of concept and axioms.

One should note that the use of the term “translation” between two languages, though echoing the former case study, was not used identically to how Monge, for example, employed it. First, Monge (and several other mathematicians mentioned in the first case study) did not use the term “dictionary” explicitly, and second, Monge (but also Laplace implicitly) pointed toward a certain hierarchy between languages. That is certainly not the case here, since for Poincaré, Bonola, and Weyl, the different interpretations are of the same geometry, having always the same strength.²⁴

Case Study III: Bourbaki and the Towers of Babel

The two previous examples focused on specific time periods in the history of mathematics: the beginning of the nineteenth century and the rise of descriptive geometry and the end of the nineteenth century with the various approaches to non-Euclidean geometry. These two examples show that the term “translation” was used as an epistemic category, that is, as a term expressing certain claims regarding the nature of mathematical knowledge. Its use pointed toward a success—to a certain extent—in expressing a mathematical theorem or system in two languages that are different though translatable into each other. However, the term was also employed to point toward a different conception: to indicate when mathematical domains could not communicate with each other.

This view is to be seen in a short note by the mathematician Felix Klein. In 1926, Klein described several approaches to dealing with algebraic functions at the beginning of the twentieth century—the Italian school of algebraic geometry during the first decades of the twentieth century with its “geometric thinking” and the German school with its “arithmetic procedures”—and he notes that there is an unsolvable dissonance: “The situation is like the Tower of Babel, that soon the different languages no longer understand each other”²⁵ (Klein 1926a, 327). Klein was referring to the 1916 edition of the Enzyklopädie der mathematischen Wissenschaften, in which a two-column table was included comparing the different terms used in each language (Figure 3). This kind of table, which was already implicitly called by Poincaré a table of “translation,” was obviously not a novel visual means for showing relationships between different mathematical idioms.²⁶

Figure 3

A table of various concepts used in different theories (the “algebraic-geometric” and the “arithmetic”) for investigating algebraic functions was added to the Enzyklopädie der mathematischen Wissenschaften, at the beginning of the article “Die algebraischen Flächen vom Gesichtspunkte der birationalen Transformationen aus” of Castelnuovo and Enriques. The editors, Meyer and Mohrmann, noted, “Here is a short table of corresponding basic terms” [“Hier ein kurzes Verzeichnis der einander entsprechenden Grundbegriffe”] (in Enzyklopädie der mathematischen Wissenschaften, 1903–1915, vol. 3, second part, first half, 676).

The use of the metaphor of the Tower of Babel was also certainly not new, at least when one considers German philosophical thought from the late eighteenth century to the early twentieth: Kant, Goethe, Hegel, and Benjamin all referred to this architectural metaphor, though having different aims in bringing it to the fore (see Purdy 2011). This is not to imply that Klein was necessarily aware of such philosophical use of this metaphor, but one may assume that he was aware of the use of this architectural metaphor by mathematicians during the late nineteenth century and the beginning of the twentieth (cf. Schlimm 2016). What I would like to focus on, however, is not the architectural metaphor and how it reflected the conception of mathematics as a language constructed by man, but rather how the image of the Tower of Babel was associated with a global task of translation. The intertwining of the two terms is to be found explicitly in the work of Bourbaki.

To recall, Nicolas Bourbaki was the pseudonym of a group of (mainly French) mathematicians who, starting at the end of 1934, met regularly and attempted to reformulate mathematics in a modern and rigorous way, writing self-contained and up-to-date books that, based on set theory, would unify many branches of mathematics. In order to see how the two concepts—that of translation and that of the Tower of Babel—appear in the writings of Bourbaki, I will focus on two of their works: the first, their manifesto of 1948, translated into English in 1950, called “The Architecture of Mathematics”; the second, their 1968 book Theory of Sets.

In “The Architecture of Mathematics,” the term “translation” appears only once, and it is used in a similar sense to how Hilbert described the passage from an abstract theory to a concrete system (“model” in today’s terminology) satisfying the axioms of the theory. In talking about three of the axioms of a group, Bourbaki emphasizes that

one can take the[se] three following [axioms], which we shall express by means of our symbolic notation … which it would be very easy to translate into the particular language of each [example discussed] (Bourbaki 1950, 224).

However, another conception of translation arises implicitly in the manifesto. Bourbaki, like Klein, also evokes the metaphor of the Tower of Babel in regard to the languages of mathematics:

it is legitimate to ask whether this exuberant proliferation [of mathematics] makes for the development of a strongly constructed organism … or whether it is the external manifestation of a tendency towards a progressive splintering, inherent in the very nature of mathematics, whether the domain of mathematics is not becoming a Tower of Babel, in which autonomous disciplines are being more and more widely separated from one another, not only in their aims, but also in their methods and even in their language (ibid., 221).

One may wonder where Bourbaki’s metaphor of Tower of Babel originated. While it is somewhat improbable that it was taken directly from Klein’s remark, one possible source is André Weil’s 1946 book Foundations of Algebraic Geometry. Weil was an early member of the Bourbaki group during the 1930s, and hence one can assume that the other members of the group knew his book. In the preface to that book, Weil (1946, viii) expresses his wish that all mathematicians (or at least those who deal with algebraic geometry) would speak one clear and well-defined language, avoiding the fate of the Tower of Babel:

attention must be and has been given to the language and the definitions. Of course every mathematician has a right to his own language—at the risk of not being understood … [leading to the] same fate … for mathematics as once befell, at Babel … in such a subject as algebraic geometry, where earlier authors left many terms incompletely defined … all terms have to be defined anew, and to attach precise meanings to them is a task not unworthy of our most solicitous attention.

I would like to suggest that Weil and Bourbaki both claim that in order to prevent the danger arising from this “Tower of Babel” (that is, against the dangerous splintering), one should employ a mathematical machinery called “formalism,” by means of which mathematical proofs can be translated, as stated in Bourbaki’s book Theory of Sets.²⁷ As stated in the introduction of the book, the mathematician

[in] general is content to bring the exposition to a point where his experience and mathematical flair tell him that translation into formal language would be no more than an exercise of patience (Bourbaki 1968, 8).

However, while the first conception of translation, proposed in the 1948 manifesto, was more local, in the sense that one transforms a particular theory into concrete examples, here one deals with a global translation: the translation of the entire body of mathematics into the “single formalized language” of set theory (ibid., 9). However, Bourbaki admits that writing down everything in a formalized language would be a “tedious” exercise²⁸ (ibid., 8), almost impossible, and eventually notes that their book would be just like any other book on mathematics. They eventually declare that “[w]e shall therefore very quickly abandon formalized mathematics” (ibid., 11).

“Translation” as an Epistemological Category: Toward Trading Zones between Mathematical Domains

Although in all of the above case studies the term “translation” was considered in the semantic field of other text- and language-related concepts (such as “dictionary,” “language,” “interpretation”), there are clear differences among the three studies. One major difference between the approach of Bourbaki to the term “translation” (and the activity itself) and those presented in the first two case studies concerns their domain of applicability: while the actors in the first two case studies concentrated on local domains of mathematical practices and research (descriptive geometry and the analytic vs. geometric idioms, or non-Euclidean geometry and its different models), Bourbaki’s concept of translation is global: the official aim is to translate the entire body of mathematics into a single language. In this sense, it reflects Husserl’s approach, that geometry has only one language, “and within each language it is again the same,” to cite Husserl again.

To make a necessary detour, it should be emphasized that neither Bourbaki nor Husserl thought that with the formalization into a single language, or its use, mathematical research and development would be either stopped or completed. This was already seen with Bourbaki’s explicit image of incessant splintering and the implicit construction of the Tower of Babel. But Husserl was also aware of that; in a slightly different version of The Crisis of European Sciences and Transcendental Phenomenology,²⁹ he employs the same metaphor of the Tower of Babel, as Bourbaki did several years afterward. Recalling the different mathematical cultures that existed before and along with that of Greek antiquity—the Egyptians, the Babylonians, and the Indians—Husserl notes:

Only Greek philosophy leads in its own development to the science of infinite theories, in which Greek geometry was the dominant example and model for thousands of years. Mathematics, the idea of the infinite, that of infinite tasks, is a Babylonian tower construction that despite its endlessness, remains a sensible task ad infinitum. The correlate of such infinitude is the new man, the man of infinite goals (Husserl 1987, 44, 46).³⁰

That is, Husserl notes that mathematics is an infinite project, echoing the infinite, never-ending construction embodied in the Tower of Babel project.³¹ The task, however, is to ground mathematics in a single language; indeed, “Euclidean geometry, and ancient mathematics in general, knows only finite tasks, a finitely closed a priori”³² (Husserl 1970, 21).

It is interesting to note that one can read Husserl, as Jacques Derrida does,³³ in two almost contradictory ways: on the one hand, Husserl calls for a destruction of the mathematical Tower of Babel, since as the project fails eventually, one needs a univocal language to make “a decisive rupture with spontaneous language” in order “to fix meanings”;³⁴ on the other hand, Husserl calls for the construction of the very same tower, since only in that way would one have access to this univocity, via the comparison between the original and its “translations”—although this access would be “inauthentic and indirect.”³⁵

One has to emphasize yet again that the global translation project into a formalized language was considered in the 1950s to be a theoretical one (i.e., it was never done or even attempted in practice). It can be suggested that the project was announced to give it greater prestige by establishing it with a more rigorous basis, hence prompting a vertical image of translation.³⁶ “Translations” that are local appear, as claimed above, when mathematicians deal in practice with translating, concentrating on a specific restricted field in mathematics.

And while Husserl’s reflections are highly philosophical, they underline, although implicitly, similar reflections that were discussed when various mathematicians dealt with local translations. One may hence also interpret Husserl’s example of the Pythagorean theorem as emphasizing a concrete theorem: his claim for the indirect access to the ideal sense of the mathematical expression due to translations may underline a philosophical approach to the limits of local translations. One may even suggest that Husserl is underlining the limits of local translations, in the sense that if every translation is “inauthentic and indirect,” then also every equivalence relation between mathematical idioms cannot be without discrepancies.

This local versus global approach to “translation” between different idioms of scientific languages has also been present in discussions within the philosophy of science. For example, Thomas Kuhn emphasizes the problematic of using the concept of “translation” with respect to the communication between different scientific practices and languages. In 1970, he noted the use of a universal language that would be able to display all scientific idioms “as one”:

The point-by-point comparison of two successive theories demands a language into which at least the empirical consequences of both can be translated without loss or change. That such a language lies ready to hand has been widely assumed since at least the seventeenth century when philosophers … sought a “universal character,” which would display all languages for expressing them as one (Kuhn 1970, 266).

Kuhn criticized this positivist idea that scientific knowledge can be built upon neutral empirical ground:

Why is translation, whether between theories or languages, so difficult? Because, as has often been remarked, languages cut up the world in different ways, and we have no access to a neutral sub-linguistic means of reporting (ibid., 268).

If one follows Bourbaki and applies Kuhn’s description to mathematics, this neutral language—as the only universal language—is the one to be found at the basis of mathematics, as is implied by the prefix “sub.” What is here presented is a pursuit of a universally accepted single language, but which is—according to Kuhn—absent and impossible to find.³⁷

One may quite rightly argue that Kuhn’s description refers to scientists who research the same object with different scientific languages, and this is not the case with mathematicians, who may consider different idioms and translate between them. While this critique might be applied to Bourbaki’s description of a global translation from (in theory) any mathematical idiom and practice into a formalized language, in most of the examples we examined above, mathematicians were actually looking at the same set of objects and theorems (curves and surfaces in descriptive geometry or models of hyperbolic geometry), described in (at least) two different ways. They did not make any claims of being global, and certainly did not express a holistic point of view regarding mathematics, but rather expressed relationships of “translation” between more local domains of mathematics. It is useful to think of this relation in terms of Peter Galison’s “trading zones.” According to Galison, a trading zone is “an intermediate domain in which procedures could be coordinated locally even where broader meanings clashed” (Galison 1997, 46). Since trading zones express “different subcultures of theorizing” (ibid.) and how the interaction between the different subcultures develops, it takes history into account. In this sense, seeing “translation” between local but different mathematical idioms as a “trading zone” concentrates on the local coordination between those idioms, and certainly not on their homogenization by establishing a “universal language” (as is done by the “global translation” project; cf. Galison 1999, 145, 157).

This local mathematical “trading zone” is also to be observed in the last example I would like to present. In 1950, the mathematician Emil Artin published a paper entitled “The Theory of Braids,” a presentation to the general public of his research in braid theory (cf. Friedman 2019). Artin states:

The theory of braids shows the interplay of two disciplines of pure mathematics: topology, used in the definition of braids, and the theory of groups, used in their treatment (Artin 1950).

While several statements are proved only symbolically, other results are described by requiring the reader to imagine the stretching of several threads and by referring to a diagram. Regarding these results, Artin states that one can “describe the translation of our geometric procedure into group theoretical language.” A few paragraphs later, Artin notes that “the problem of classification of closed braids, at least, can be translated into a group theoretical problem” (ibid., 119). One may assume that Artin used the term “translation” to describe the relationship between the two languages: the diagrammatic-–topological and the symbolic–algebraic. However, this translation was restricted to only parts of these idioms, as is to be seen from another paper of Artin’s, “Theory of Braids,” published in 1947. Presenting in this paper the theory of braids in a more algebraic idiom, Artin strongly discourages the reader from either imagining or drawing certain braids, explicitly encouraging working only with algebraic expressions. After presenting an example of a certain normal form of a braid, both algebraically and with the help of a drawing, he emphasizes: “Although it has been proved that every braid can be deformed into a … [specific] normal form, the writer is convinced that any attempt to carry this out on a living person”—that is, drawing it—“would only lead to violent protests and discrimination against mathematics” (Artin 1947, 126). This underlines that for Artin, “translation” is a restricted epistemic category: it may help in understanding theorems and concepts or even prompt the emergence of new knowledge, but it should not be applied to every algebraic proposition in order to visualize its objects or its results.

Seeing the term “translation” in this way—as pointing toward an epistemic activity whose domain of action is restricted and local—helps to frame how the actors considered themselves and their activity while engaged in “translating.” If this activity is between two local domains, which from time to time have to be coordinated and adjusted, this means that a complete dictionary is not given in advance. Hence “translation” is not just a description of an already given procedure, but rather an operative, constitutive action, which enables the emergence of new knowledge, but at the same time may also be restrictive, in the sense that it may be that not all of the theorems, objects, and procedures in the first domain can be and are being translated into the second. This restriction occurs, for example, with Weyl (and also implicitly Poincaré) ignoring Beltrami’s paper models,³⁸ with Laplace’s preference for the “algebraic language,” and with Monge’s explicitly noting that translation would work but only “if the problems have no more than three unknowns,” implying that if a problem had more than three variables, then no translation into the language of geometry would be possible. The last example of Artin’s shows this restrictedness clearly; indeed, Artin dramatically ends his paper with a warning against the “violent protests” that might erupt were such a “translation” to occur. To recall Monge: just “as every analytic operation can be regarded as the script for a spectacle in geometry,” this spectacle itself has its limits, just as “translation” as an epistemic category was employed as a local and at times restricted one, in opposition to the Husserl–Bourbaki impractical dream of a single universal language.