Some remarks on Husserl’s Doppelvortrag

Abstract

We provide an interpretation of Husserl’s 1901 Doppelvortrag in Göttingen from the viewpoint of the modernist transformation of mathematics. We emphasise the dialectical aspects of the Doppelvortrag, and especially the underlying conflict between abstraction and intuition, which often resurfaces in Husserl’s philosophy. We focus on three key aspects: (1) the relation between Husserl’s idea of pure logic and Hilbert’s axiomatic method; (2) the concept of Definitheit and the early developments of model theory; (3) the nature of imaginary numbers and their justification in arithmetic and algebra. We stress that, in contrast to Hilbert’s Completeness Axiom from his Grundlagen, Husserl viewed the Definitheit as a statement requiring a proof, and he believed that one could be provided in the cases of arithmetic and geometry.

1 Introduction

In the September of 1901, Edmund Husserl was appointed nicht-etatmäßig Extraordinarius at the Faculty of Philosophy in Göttingen.¹ This was essentially a promotion, as he was previously a Privatdozent at the University of Halle. It seems that Husserl was not received well by the Faculty of Philosophy, where he encountered the hostility of Julius Baumann and of the psychologist Georg Elias Müller.² Husserl was better welcomed by mathematicians, as both Felix Klein and David Hilbert saw him as a potential collaborator.³ As testified by his notes, Husserl attended the meeting of the Göttingen Mathematische Gesellschaft of November 5th, where Hilbert held a talk Über die Grundlagen der Geometrie und Arithmetik.⁴ Later in the meetings of November 26th and December 10th, Husserl gave two lectures which came to be known as his Doppelvortrag. The title of his first lecture was The Passage through the Impossible and the Completeness of an Axiom System [Der Durchgang durch das Unmögliche und die Vollständigkeit eines Axiomensystems], and in the second talk, he continued by discussing the notions of relative and absolute definiteness.⁵ The overall topic of these two lectures is what Husserl calls “the problem of the imaginary”, but they also touch upon many aspects of Husserl’s views of logic and of his philosophy of mathematics.

The goal of this article is to provide an interpretation of Husserl’s lecture from the viewpoint of the so-called modernist transformation of mathematics.⁶ On the one hand, we find in Husserl’s conference (and in his other writings from the same period) all the fundamental elements of modern mathematics. Husserl develops a theory of pure logic as a Mannigfaltigkeitslehre which has its core in the notion of abstract form, and which fully embraces the formal turn of mathematics towards arbitrary systems of objects, ultimately justified by their internal consistency. And yet, on the other hand, Husserl often seeks the ultimate grounding of these abstract systems in those very entities from which the modernist transformation was breaking away from—namely, the natural numbers, the epitome of the traditional conception of mathematics as the science of quantities. The problem of the passage through the imaginaries—the central topic of the Doppelvortrag—is an extremely good witness of this twofold attitude. If mathematics deals with purely abstract forms, what justifies its concrete calculations and deductions? More broadly, if mathematics deals with abstractions, what does it have to do with reality? We take these very questions as witnessing a transitional point in time. Husserl agrees with Hilbert’s fundamental tenet that the consistency of an axiom system coincides with its truth. But while the modern mathematician views groups, rings, Banach spaces, manifolds, simplices, etc., as interesting in themselves and does not call them into question, Husserl often seeks a further philosophical justification which crucially lies outside of them—and which brings back into the fore the privileged and foundational role of the natural numbers. This twofold tendency makes Husserl’s text especially interesting to our present look. Even more so, we believe this dialectic between intuition and abstraction is very well representative of Husserl’s whole philosophy, spanning from the Philosophie der Arithmetik to the Krisis. While his phenomenology attributes to the Gegebenheit the ultimate justification of knowledge, it also recognises the autonomy of the realm of concepts—of the sciences and of the purely abstract and formal knowledge provided by mathematics. In a sense, while in later works such as Erfahrung und Urteil, the dialectical relationship between experience and judgement is approached from the bottom up, moving from the analysis of experience to the genetic justification of concepts, the Doppelvortrag is an instance of the opposite viewpoint, where this conflict is examined from the side of the abstract sciences.⁷

In this article, we shall try to analyse the Doppelvortrag and emphasise its dialectical aspects—between natural numbers and formal systems, between intuition and abstraction, between meaning and meaningless. In doing so, we will depart at times from other interpretations of the Doppelvortrag, which have often focussed on the problem of providing a rational reconstruction of Husserl’s notion of Definitheit from the viewpoint of modern mathematical logic.⁸ We do not wish to resolve the dialectic underlying Husserl’s text, but merely to indicate it and comprehend it. Also, we do not plan to clarify what we think is necessarily opaque—we believe that the Doppelvortrag is interesting exactly because of its conflicting character, and we shall not try to disentangle concepts and meanings that are essentially bundled together in the text. Still, we emphasise that our interpretation builds upon, and is very largely indebted to, the prior scholarly research on the Doppelvortrag and on Husserl’s philosophy of mathematics, and in particular to the work of Centrone (2010, 2011, 2013) and Hartimo (2003a, 2007a, 2007b, 2017, 2018, 2021).

We shall especially focus on three key directions. First, Husserl’s lectures at the Göttingen Mathematische Gesellschaft can be read in the light of the diachronic development of Husserl’s own phenomenology, and of its interplay with the axiomatic method independently developed by Hilbert. The Doppelvortrag displays more concretely the idea of pure logic that Husserl had just exposed in his Logische Untersuchungen, and on which he had lectured in 1896 in Halle.⁹ More specifically, we will focus in Sect. 2 on the relationship between the philosophical side of Husserl’s pure logic—the Wissenschaftslehre of the Logische Untersuchungen—and its mathematical side—which is fully expressed by his concept of Mannigfaltigkeitslehre and which has several points of convergence with Hilbert’s axiomatic method.

Second, we believe that the Doppelvortrag should also be read in the context of the early developments of model theory around the beginning of the twentieth century. A large part of Husserl’s conference was devoted to introduce the notion of Definitheit of a theory—a concept which contemporary scholars have associated now to completeness, now to categoricity, now to some other technical notions. We believe that there is a strong risk of projecting our contemporary categories to a period where these categories were still in the making, and that one should rather try to understand what Husserl meant by Definitheit in his own terms, without necessarily trying to rephrase his concept with the toolkits of modern mathematical logic. As testified by the topic of Hilbert’s lecture on the 5th of November and by his contemporary text Über den Zahlbegriff, notions such as that of completeness were “in the air” and scholars were intensively discussing them at the time. Following this attitude, we shall try in Sect. 3 to emphasise three different aspects of the notions of Definitheit, and simply notice how they relate to quite different concepts from contemporary model theory. Our conclusion is simply the following: what today is disambiguated was not yet clear at the time in which Husserl was writing, and the Doppelvortrag is itself one key chapter in the development of these ideas. Interestingly, one of Husserl’s key contribution with the notion of Definitheit is perhaps the fact that, differently from Hilbert, he did not view the Definitheit as an axiom, but rather as a statement requiring a proof. We shall emphasise this aspect later, by comparing Husserl’s text with Hilbert’s consideration on the “axiom of completeness”.

Finally, in Sect. 4, we will examine the very topic of the Doppelvortrag, i.e. the passage through the imaginary. As a matter of fact, one should keep in mind that for Husserl the Definitheit was an essentially instrumental property. As we read in the Notizen of the Deutsche Mathematische Gesellschaft, mentioning Husserl’s talk, “in definite systems, and only in these, the passage through the impossible is permitted”.¹⁰ The Definitheit of a system of axioms ensures that we can perform such a “passage through the impossible”. More concretely, it is the Definitheit of arithmetic which guarantees that we can use the complex numbers to solve polynomial equations with integer coefficients. As we shall further emphasise later, the centrality that Husserl attributes to the problem of the imaginary numbers also characterises Husserl’s attitude in the foundations of mathematics, and it goes hand in hand with the privileged role that Husserl had attributed to the natural numbers already in his Philosophie der Arithmetik. We thus believe that the Doppelvortrag must be interpreted both mathematically and philosophically, and we agree with Majer (1997, p. 43) that it should be read (also) in the light of the problems underpinning his Philosophie der Arithmetik, and especially the relationship between “the proper and symbolic apprehension of numbers”.¹¹

We conclude this introduction with some brief philological remarks. There are nowadays two versions of Husserl’s Doppelvortrag: the Husserliana edition compiled by Lothar Eley and the more recent and extended version compiled by Elisabeth and Karl Schuhmann.¹² While these two editions agree on the content of the first lecture, they present different materials belonging to the lecture of the 10th of December. In the following, we will not give a summary of Husserl’s lecture, but rather examine the topics which we emphasised above. We will thus maintain a rather neutral attitude with respect to the exact content of the second lecture and refer to both versions of the Doppelvortrag. The texts that Elisabeth and Karl Schuhmann added to Eley’s version of the Doppelvortrag comprise material that Husserl used to prepare his second lecture, though we cannot know for sure how close their reconstruction is to the actual presentation, as there is no written record of it. In the following three sections, we will thus analyse the Doppelvortrag along the three main threads which we emphasised above, and we will base our reading on both versions of the Doppelvortrag.

2 Husserl’s pure logic and Hilbert’s axiomatic method

Husserl indicates the nature of logic as its subject of inquiry already at the very beginning of the Doppelvortrag.¹³ However, he had already considered this topic in several of his works, most importantly in the Prolegomena to the Logische Untersuchungen, which had just been published in 1899. In this work, Husserl embarks in a lengthy critique of several theories of logic, and then moves on to describe his own view on the argument. On the negative side, Husserl articulates in the Prolegomena his famous critique of psychologism, and he deems as inconsistent the theories of logic that authors such as Theodor Lipps, Christoph von Sigwart, John Stuart Mill and Wilhelm Wundt had advanced in their recent works.¹⁴ On the positive side, he proposes in the Prolegomena his own idea of pure logic as Wissenschaftslehre, i.e. an autonomous theoretical discipline which does not rely on psychology or any other empirical science. Husserl’s development of a theory of science is at the same time philosophical and mathematical, and it brings together a multiplicity of different influences and traditions coming from the 19th century.¹⁵ In this section, we will outline Husserl’s idea of pure logic, we will explain how it comprises both philosophical and mathematical elements, and we will relate it to Hilbert’s developments of the axiomatic method.

First, on the more philosophical side, let us focus on the idea of logic as a theory of science. Husserl’s source of influence is here clearly Bernhard Bolzano, who had described the theory of science as “the science which instructs us on how we should present the sciences in appropriate textbooks”.¹⁶ Even if Husserl expresses several concerns with Bolzano’s definition,¹⁷ he agrees with the idea of a second-order discipline which takes as its subject the very notion of science. However, Husserl was also concerned with Bolzano’s methodology, which he deemed insufficient from a philosophical standpoint as it lacked an analysis of intentionality.¹⁸ This relates to a second aspect of Husserl’s logic, namely to a transcendental perspective which is not (immediately) possible to find in Bolzano’s elaboration. In the Prolegomena, in fact, Husserl largely elaborates on the topic of the conditions of possibility of scientific theories, thereby relating to a more Kantian tradition.¹⁹ These two aspects of the theory of science are largely developed in the Prolegomena and also call for many of the (phenomenological) analysis that Husserl carries out in the six following investigations.²⁰ In particular, in the Prolegomena Husserl outlines his analysis of the notion of Begründung, he distinguishes different notions of sciences and, finally, he introduces the problem of the a priori conditions of possibilities of the scientific theories.²¹

These aspects of the theory of science find a final exposition in the last sections of the Prolegomena, where Husserl outlines the three tasks of his pure logic. Central to the project of a Wissenschaftslehre are the first two tasks of logic, which he describes in Sects. §67 and §68 of the Prolegomena. On the one hand, logic must determine the fundamental categories of meaning, and make explicit how complex units of meaning are built upon simpler ones. On the other hand, logic must make explicit the formal laws that distinguish possible and impossible complexes of meanings—that is, in contemporary terms, laws that distinguish consistent and inconsistent propositions. At the same time, these laws correspond to the very deductive rules that are used in formal deductions, one key example being the law of noncontradiction.²² As Husserl remarks, these two levels of pure logic are sufficient to capture the scope of a science of the conditions of possibility, so that the very idea of a Wissenschaftslehre is fully tied to these two levels of analysis.²³ Most interestingly, these two aspects of logic are naturally related also to other considerations that Husserl makes later in the Logische Untersuchungen—for instance in the Third and Fourth Investigation on parts and wholes—and they also make explicit how the concept of meaning is the essential bridge from the Prolegomena to the six following investigations.²⁴

These first considerations mainly relate to the idea of logic as Wissenschaftslehre and to problems which are closely tied to the philosophical developments of Husserl’s philosophy in the Logische Untersuchungen. It is also possible, however, to switch the order of priority between logic and theory of science and put the spotlight on the Wissenschaftslehre as logic. This is a justified theoretical move, as the theory of science is for Husserl not only a philosophical theory, but also a discipline with a mathematical content. After introducing the first two tasks of pure logic, Husserl writes

Once all these investigations are concluded, the idea of a science of the conditions of possibility of theories in general is sufficiently realized. At the same time, we also see that this science hints beyond itself to a complementary one, which deals a priori with the essential types (forms) of theories and their respective relational laws.²⁵

The first two tasks of logic are closely connected to the problem of the a priori possibility of a theory in general. The third task of pure logic takes into considerations the different possible forms of theory that one can obtain. While the former two aspects point in the direction of a transcendental philosophy—broadly intended as a Wissenschaftslehre—the latter relates Husserl’s considerations to his views of mathematics. As he makes fully explicit both in the final sections of the Prolegomena, as well as in the Doppelvortrag, mathematics cannot be considered anymore as the science of quantity, but it is in its essence a formalen Theorienlehre, “the most general science of the possible deductive systems in general”.²⁶

Following Gray (2008, pp. 203–209), it is natural to qualify as modernist the conception of mathematics that Husserl outlines here. Mathematics is not considered anymore as the science of quantities, but rather as a discipline characterised by its abstract and formal nature.²⁷ In fact, such a Theorienlehre must abstract away from the specific content of all concrete theories and characterise them in purely formal terms. It is thus exactly the process of abstraction which makes it possible to move from the concrete to the formal theories, which can in turn be analysed from the viewpoint of their reciprocal relationships. In fact, as Husserl says, “these different forms are not unrelated to each other”,²⁸ and it is possible to classify them in a systematic way.²⁹ In the Doppelvortrag, Husserl considers the example of Euclidean geometry to clarify these two orthogonal levels of analysis.

The Euclidean geometry is a concrete theory that, once formalised, yields the theory form which we call the theory of the tridimensional euclidean manifold. And again, this is only a single instance of the systematically interconnected class of manifolds with variable curvature.³⁰

On the one hand, there is the process of formalisation, which abstracts away from the concrete subject matter of a theory and allows us to grasp its underlying form. On the other hand, there are multiple relations between theories considered in formal terms. The problem of the relationship between different formal theories comes up very naturally once one embraces the axiomatic approach that Husserl suggests.

A systematically developed theory in this sense is defined by a collection of formal axioms, i.e., by a limited number of purely formal principles, consistent with each other and independent from one another. The systematic deduction yields, purely logically, namely, according to the law of noncontradiction, the dependent statements and thus the entire collection of statements that belong to the defined theory.³¹

Formal theories have an axiomatic presentation and are closed under those rules of deduction that Husserl had considered in the second task of his pure logic. This notion of formal theory is thus reminiscent of the concept of nomological theory that Husserl had introduced in the Prolegomena, and that according to him exemplifies the very ideal of science.³² However, while the nomological theories discussed in Sects. §64–§66 of the Prolegomena are real theories, with a material and concrete content, the Theorienlehre deals with purely formal theories, that is, with the abstract and logical scaffolding of the real theories.

In other words, Husserl’s third task of logic is first a Theorienlehre, a pure mathematics dealing with formal theories and their relationships. However, this description does not fully capture Husserl’s idea of mathematics and his view of the theory of science as logic. In Sect. §62 of the Prolegomena, Husserl also points out that there is always a correlation between the level of theories and the level of reality.³³ The “interconnection of truths” and the “interconnection of things” are such that “one and the other are given a priori with each other, and they are inseparable from one another”.³⁴ Such a duality between meaning and reality is constant in Husserl’s thought, and it finds its final justification in the correspondentist theory of truth of the Sixth Logical Investigation.³⁵ Most importantly for us, this duality between an apophantic and an ontological level applies also to the three tasks of logic that we have taken into consideration before: theories [Theorien] correspond to manifolds [Mannigfaltigkeiten], so that the Theorienlehre becomes also a Mannigfaltigkeitslehre.

However, the domain of objects is defined by the axioms in the following sense, that it is delimited in general as the sphere of objects, independently of whether they are real or ideal, for which fundamental principles of this and that form are valid. We call a domain of objects defined in this way a determined, but formally defined, manifold.³⁶

Let us first clarify the terminology. In Husserl’s context, the word “Mannigfaltigkeit” should be interpreted as referring to some sort of mathematical structure: a manifold is a collection of elements which satisfy a list of properties and that are related by specific forms of compositions.³⁷ A formal theory determines a corresponding manifold and, moreover, a manifold is always specified by a formal theory. There is thus a one-to-one correspondence between theories and manifolds. It is quite natural for the contemporary reader to associate Husserl’s manifolds to models in the sense of contemporary model theory. Model theory deals exactly with the duality between syntax and semantics, and especially with the interplay between theories and structures. There are however two key disclaimers which one should keep in mind. On the one hand, we should be careful not to project concepts and notions which were developed only (some) years later. Model theory as we know it was made precise by Tarski and other logicians around the ’30 s and it did not exist as a discipline at the time when Husserl wrote the material for the Doppelvortrag.³⁸ In particular, Husserl does not work with the precise notions of formal language, derivation and (first-order) structure which we are now accustomed to from modern mathematical logic. On the other hand, as Hodges (2018, p. 440) remarks, “one feature of the early work on models of axioms was the looseness of some of the formulations”. Husserl is no exemption to this attitude. The views expressed above fully belong to these early developments of model theory, where the precise formulation of its concepts was still in the making. The notion of Definitheit which we will consider in the next section is possibly the clearest example of the transitionary nature of Husserl’s work.

These terminological specifications being made, we can surely state that the introduction of the Mannigfaltigkeitslehre expresses Husserl’s own stance on the mathematical changes at the turn of the century. Especially under the influence of the non-Euclidean geometries, Husserl argues for a view of mathematics as a formal discipline, dealing with theories on one side and manifolds on the other.³⁹ Such a discipline is thus fully theoretical, but at the same time has an important connection to possible applications: “the formal mathematics wants to be the instrument of concrete mathematical discoveries”.⁴⁰ The connection to the practical sphere is not a specificity of formal mathematics, but applies more generally to the Wissenschaftslehre. The theory of science is a theoretical discipline, but it has a normative role which becomes in turn an instrument to evaluate real disciplines in concrete circumstances. In the Prolegomena, Husserl speaks therefore of a theoretical Wissenschaftslehre, a normative Wissenschaftslehre and a practical Kunstlehre von der Wissenschaft.⁴¹ It seems therefore correct to intend the Mannigfaltigkeitslehre in similar terms and view it as a formal discipline whose utility will be eventually proved in practice. As Husserl says, in fact, this new formal mathematics “wants to provide a method of incomparably greater generality and power, which will make all methodical work of concrete mathematical nature unnecessary”.⁴² In virtue of its formal nature, the Mannigfaltigkeitslehre can provide mathematicians with general methods which apply to every area of mathematics, and that could in principle even replace the manual activity of calculating.⁴³

From the viewpoint of our reconstruction, the perspective advocated by Husserl turns out to be compatible, if not convergent, with the axiomatic method that in the very same period Hilbert was applying to geometry and arithmetic. In fact, while the first two tasks of the pure logic pave the way for the transcendental and philosophical developments of Husserl’s phenomenology, with the notion of Mannigfaltigkeitslehre Husserl was developing a view of mathematics very close to the one advocated by Hilbert in the same years. A text which clearly represents Hilbert’s axiomatic approach around the 1900 is without any doubt his Grundlagen der Geometrie.

Hilbert introduces in this work a system of axioms for the Euclidean geometry, and he discusses the problems of their independence and consistency. In the context of arithmetic, a similar role is played by Hilbert’s article Über den Zahlbegriff. Both in the context of geometry and arithmetic, Hilbert was replacing the so-called genetic method—which aims to clarify the foundations of mathematics by constructing its objects from simpler ones, taken as granted—with his own axiomatic method.

My opinion is this: despite the high pedagogical and heuristic value of the genetic method, the axiomatic method should be preferred for what concerns the conclusive presentation and the full logical verification of the content of our knowledge.⁴⁴

The axiomatic method makes it possible to clarify the logical relations between statements, and in turn to justify our mathematical knowledge. Crucially, this view was not new in itself. A very recent example of application of the axiomatic method in the setting of arithmetic had been recently provided by Frege, who in the Grundlagen der Arithmetik from 1884, and later in the Grundgesetze der Arithmetik from 1893 set as his own goal to ground the entirety of arithmetic on a few limited number of principles [Grundgesetze]. However, the very comparison with Frege displays the radical difference of Hilbert’s approach (which rather builds on Pasch’s influence, as highlighted by Eder and Schiemer (2018)). The main novelty of Hilbert’s axiomatic lies in fact in his specific way to interpret the epistemological foundation of the axioms. In the traditional axiomatic approach all deductive knowledge rests upon the axioms, but the ultimate justification of these first principles lays outside the deductive science.⁴⁵ The problem of the validity of the first axioms is therefore viewed as external to mathematics, and it is ascribed now to one, now to another specific discipline—to physics, psychology, metaphysics, etc.—depending on the philosophical inclination of the author in question. In Frege, the justification of the principles of arithmetic is guaranteed by means of logical truths, whose ultimate foundations rests in their (alleged) immediate evidence. In the case of geometry, the ultimate justification goes back to the fundamental intuition of space. In his review of Hilbert’s Grundlagen der Geometrie, Frege expresses the traditional view of the axiomatic method: the justification of the first axioms of geometry does not proceed by deduction, and it does not belong to the realm of mathematics.

Since ancient times, people have called axiom a thought which is hold for true, but that cannot however be proven by a logical chain of derivations. Also the logical laws are of this kind. Some, however, would perhaps be inclined not to use the qualification of “axiom” for these general principles, but to reserve it for the fundamental principles of a restricted field, such as those geometry. But this is a question of smaller significance. The question of the justification of the truth of the axioms will not be addressed here. For the axioms of geometry, the intuition is usually seen as the source [of their justification].⁴⁶

In stark contrast to the traditional viewpoint, Hilbert’s solutions does not look for an external foundation of the axioms, but rather for an internal justification.⁴⁷ As his exchange of letters with Frege makes clear, for Hilbert the axioms are true because they are consistent, and not vice versa. In his letter from December 27th 1899, Frege outlines to Hilbert his perplexities concerning the Grundlagen der Geometrie.

I call axioms propositions that are true, but that cannot be proven, because their knowledge comes from a source quite different from logic, one which can be called spatial intuition. From the truth of the axioms it follows that they cannot contradict each other.⁴⁸

Frege articulates here the classical axiomatic conception of geometry, essentially in its Kantian version: it is the spatial intuition which justifies the truth of the axioms, and which eventually ensures their consistency and reciprocal compatibility.⁴⁹ Hilbert’s reply to Frege’s objection is quite telling.

You write “I call axioms propositions.... From the truth of the axioms it follows that they cannot contradict each other”. I found exactly this sentence of yours very interesting, because personally, as long as I think, write and lecture on such things, I always say the exact opposite. If arbitrarily stipulated axioms do not contradict each other with all their consequences, then they are true, then the things defined by the axioms exist. This is for me the criterion of truth and existence. The proposition “every equation has a root” is true, or the existence of roots is proven, as soon as the axiom “every equation has a root” can be added to the remaining arithmetical axioms without thereby ever creating a contradiction with any adequate conclusions.⁵⁰

According to Hilbert, a system of axioms is true exactly because it is consistent. Hilbert’s modernism rests exactly on this overturning of the traditional order of explanation. The justification of mathematical knowledge is not extrinsic, but rather internal to mathematics. Most specifically, it would be provided by a proof of the fact that a certain family of axioms, such as those of arithmetic or geometry, is consistent. Still, one should be cautious and not immediately think about the so-called Hilbert’s Programm, and the related problem of the purely syntactic consistency proof which Hilbert will later set as his own (unreachable) goal. In this text, in fact, Hilbert does not yet endorse those ideas, which he will only later formulate with the program of a finitistic Beweistheorie.⁵¹ As we mentioned earlier, at this point in time, the syntax and semantics divide is not yet fixed, and the notion of consistency also oscillates between these two sides. Actually, what Hilbert has in mind in his letters with Frege is probably a notion of semantic consistency. The consistency proofs presented in the Grundlagen der Geometrie (and mentioned also in Über den Zahlbegriff) rely on the explicit description of concrete structures witnessing the validity of the axioms.⁵² For instance, to prove that the negation of Desargues’ Theorem is consistent with the axioms of Euclidean geometry (Hilbert 1899, Thm. 33) describes a geometry which satisfies all the axioms of plane geometry but which also fails to verify Desargues’ property. In contemporary terms, he provides a model of the theory of plane geometry together with the negation of the Desargues’ property.

Later on, Hilbert will reject this semantic approach to consistency, and he will advocate for a finitistic (and proof-theoretic) foundation of arithmetic.⁵³ However, independently from this later developments, this early Hilbertian axiomatic phase essentially overlaps with Husserl’s developments of a pure Mannigfaltigkeitslehre. The affinity between Husserl and Hilbert on this topic is manifest in Husserl’s own comments to the letters between Frege and Hilbert.

I notice the following. Frege does not understand the meaning of Hilbert’s “axiomatic” foundation of geometry, namely that it is a purely formal system of conventions, which overlaps with the Euclidean for its theory form.⁵⁴

Husserl shares with Hilbert the same (modernist) axiomatic view. The task of logic is to investigate the abstract mathematical systems of axioms, and their mutual relationship. And yet, the process of abstraction that turns a concrete theory into its formal counterpart also modifies its sense—it empties out its material meaning and provides it with a purely logical and formal meaning, which Husserl at points also qualifies in operational terms as a Spielbedeutung. Introducing this notion in the First Logical Investigation, Husserl criticises the idea that arithmetical expressions are mere signs, and he clarifies them as follows:⁵⁵

Thus, in the spheres of symbolic-arithmetic thinking and calculations, one does not operate with meaningless signs. It is not the “mere” signs in their physical sense, emptied out of all meaning, which replace the original signs with their arithmetical meaning. Instead, it is the very same signs, but taken in a specific operational- or game-meaning, that replace the signs with arithmetical meaning.⁵⁶

As Husserl further says in Sect. §20 of the First Logical Investigation, the operational use of symbolic expressions calls for a logical justification—“one must ground the logical justification of such a procedure and reliably determine its limits”.⁵⁷ The problem of this justification brings us closer to the topic of the Doppelvortrag, which essentially focuses on the use of imaginary numbers in arithmetic. We notice that, by attributing an operational Spielbedeutung to arithmetic expressions, Husserl is not saying that such expressions are also void of any symbolic content. As he puts it in Sect. §21 of the First Logical Investigation “also the expression with a symbolic function still means something, and this is nothing else than what can be intuitively clarified”.⁵⁸ The notion of Spielbedeutung does not exhaust the meaning of symbolic expressions, which also bear a conceptual content that can be clarified by the (categorical) intuition.⁵⁹

The interaction between the two spheres just described—the concrete meaning of material theories on one side, and the operational and conceptual meaning of their formal counterpart on the other—is again representative of Husserl’s modernism, and it relates to a family of problems which is at the same time philosophical and mathematical. The problem of the imaginaries, namely the central topic of the Doppelvortrag, emerges exactly in this context. The natural numbers are regarded by Husserl to be closer to intuition, while the imaginary numbers (a broad term, which encompasses the negative, the rationals, the reals, and the actual imaginary complex numbers) are essentially more abstract objects.⁶⁰ Their Spielbedeutung calls for a mathematical justification, while their conceptual content for a phenomenological investigation of the categorical intuition. While this latter goal is pursued by Husserl in the Logische Untersuchungen, and in several of his later phenomenological texts, the Doppelvortrag deals exactly with the former issue. From this philosophical perspective, the Doppelvortrag can be read as an investigation into two levels of meaning—the concrete, material, and denotative notion of meaning, and the Spielbedeutung of logical expression.⁶¹ The notion of Definitheit crystallises Husserl’s own solution to this problem.

3 The notion of Definitheit

The concept of Definitheit is introduced by Husserl in the second lecture of his Doppelvortrag, where he uses it as a criterion to justify the “passage through the imaginary”. Husserl provides several characterisations of the Definitheit of a system of axioms, and he also further distinguishes between a relative and an absolute Definitheit. The right interpretation of this notion has been object of a lively debate in the secondary literature (Aranda 2022; Centrone 2010, 2011; Da Silva 2000, 2016; Hartimo 2007b, 2018; Hartimo and Okada 2015; Hill 1995, 2002; Majer 1997; Okada 2013). Here we will argue for an interpretation which is mainly based on what Husserl writes in the manuscripts (K I 26/36) and (K I 26/90), where he provides a coherent enumeration of three properties which characterise the notion of Definitheit.⁶² The interpretation which we are proposing does not aim to find a single way in which one can express the Definitheit of an axiom system in modern terminology, nor to phrase Husserl’s concept in formal mathematical terms. We simply wish to identify the multiple layers of meaning which Husserl associates to this notion, and to put it in its mathematical and philosophical context. In particular, we do not think that Husserl’s Definitheit is fully captured in terms of (versions of) syntactic completeness (Centrone 2010; Da Silva 2000, 2016) or categoricity (Hartimo 2007b, 2018), nor that Husserl foresaw the problem of establishing the semantic completeness of a calculus (Majer 1997).⁶³ We rather think that the notion of Definitheit is better clarified by emphasising the fact that it is, in Husserl, still an informal concept, which partly does, partly does not, overlap with our formal notions of syntactic completeness, categoricity, etc. We thus especially agree with Aranda (2022, pp. 58–59) that there is a gap “between the intuitive notion of completeness introduced by Husserl and the formal concepts of syntactic completeness and categoricity”.

In the light of these considerations, we follow in our interpretation a pluralistic and informal approach, and seek to identify what are the multiple elements that converge in Husserl’s notion of Definitheit (which, for the moment, coincides for us with what Husserl calls relative Definitheit). This multiplicity has been often recognised, e.g. by Hartimo (2018, pp. 1523–1524), who stresses that “Definiteness, for Husserl, thus embraces in the end three ideals of completeness: ‘pureness’ and non-extensibility as captured by categoricity, syntactic completeness, and often also a kind of computational completeness to aspire for richer determination of the ‘existential domain’”. We shall clarify these three ideals in a slightly different way, and argue that the notion of Definitheit relates to three different properties that a mathematical theory should fulfil. A definit system of axioms is such that: (i) it cannot be extended, (ii) it is complete, and (iii) it fully describes its corresponding manifold. By considering these three properties, it will then become clear that Husserl took definit theories to provide a full (syntactical) description of the corresponding (semantic) manifold, so that the correspondence between theories and manifolds is made more precise via the correspondence between manifolds and definite theories.

First, Husserl says that a system of axioms is definite if it is not possible to add new axioms to it, provided they refer to the same objects of the original system. We read in the materials to the second Vortrag:

The additional question: would such a system be definite? It would be definite if no new axiom is possible for the restricted sphere of existence, for the given individuals and for the not given individuals.⁶⁴

Similarly,

An axiom system is definite if it delimits a domain of objects as existing, and in such a way that for the objects of this domain no new axiom (which is deductively independent from the axiom system) is possible. For this domain: I keep fixed the axiom system and I do not add any new existential determination, I do not extend the domain, so that it is not possible to add any new axiom deductively without contradiction. Clearly, this pertains however only axioms that builds on the concepts already defined, and which are thus meaningful propositions.⁶⁵

One can find similar definitions also in the manuscripts (K I 26/36) and (K I 26/90).⁶⁶ According to this first definition, a system of axiom is definit if it cannot be extended with new axioms having the same concepts. We stress here two aspects. First, as the second quote above makes explicit, Husserl is also well aware of the fact that every (consistent) system of axioms can be extended to one which is inconsistent, and he therefore remarks that the resulting extension has itself to be consistent.⁶⁷ Second, and more importantly, Husserl explicitly remarks that the notion of Definitheit does not exclude arbitrary extensions to new axiom systems, but only extensions which keep fixed the domain of objects (namely, the Mannigfaltigkeit) which the axioms refer to.

Following Centrone (2010, pp. 167–176), this aspect of the notion of Definitheit can be interpreted in terms of the maximality of an axiom system, namely the property that a system of axioms $\Gamma$ has whenever, for every sentence $\phi$ of the underlying language $\mathcal {L}$ with $\phi \notin \Gamma$, the set $\Gamma \cup \{\phi \}$ is inconsistent. One way to phrase the first aspect of the Definitheit is thus by saying that an axiom system $\Gamma$ is definit if it is maximally consistent. Crucially, we remark that under this interpretation $\phi$ ranges over sentences in the same language $\mathcal {L}$ of $\Gamma$, as Husserl does not exclude extensions by axioms involving new concepts, and referring to new entities. As it is well-known, maximal consistency is equivalent, in classical logic, to the syntactic completeness of a theory.⁶⁸ In general, we say that a system of axiom $\Gamma$ in the language $\mathcal {L}$ is syntactically complete if, for every sentence $\phi$ of $\mathcal {L}$, either $\Gamma \vdash \phi$ or $\Gamma \vdash \lnot \phi$, i.e. $\Gamma$ proves either $\phi$ or its negation $\lnot \phi$. It is important to distinguish syntactic completeness from its semantic counterpart. A system of axioms $\Gamma$ in the language $\mathcal {L}$ is semantically complete if, for every sentence $\phi$ of $\mathcal {L}$, either $\Gamma \models \phi$ or $\Gamma \models \lnot \phi$, i.e. $\Gamma$ entails either $\phi$ or its negation $\lnot \phi$. While the former notion refers to the possibility of deducing $\phi$ or $\lnot \phi$ from $\Gamma$, the latter says that either $\phi$ holds in all the models where $\Gamma$ holds, or $\lnot \phi$ does. This conceptual difference is nowadays taken for granted in mathematical logic, but was not obvious at the time in which Husserl was writing.⁶⁹ Syntactic and semantic completeness are for us different properties, which can sometimes be equivalent, as it happens in first-order logic, or which are essentially different, as in the case of second-order logic. Interestingly, Husserl seems to be aware of the relation between maximality and completeness, but he does not really distinguish between the syntactic and the semantical aspects of this latter notion. Consider the following passage from the manuscript (K I 26/34):

I can however say: an axiom system is definite when it formally defines a domain of object in such a way that every meaningful question on this domain of objects finds its answer through the axiom system, or that every meaningful proposition given the axioms, if we restrict exclusively to the objects whose existence is grounded in the axioms, either follows from the axioms or it contradicts them.⁷⁰

Similarly, in (K I 26/36):

An axiom system that delimits a domain is called definite if every proposition understandable on the basis of the axiom system, intended as a proposition for the domain, is either true or false on the basis of the axioms. Said otherwise, if there are only the following two possibilities: either the proposition follows from the axioms, or it contradicts them.⁷¹

As these quotes make clear, Husserl continuously oscillates between syntactic and semantic notions. First he says that a system of axioms is definite if it either proves or disproves every proposition, then he argues that a system of axioms is such if every proposition is either true or false given the axioms. In other terms, Husserl does not feel the need of proving the soundness and the completeness of a proof system and “takes the correspondence between consistent theory and structure for granted” (Centrone 2010, p. 180).⁷² The Definitheit as a notion of maximality is considered by Husserl to be equivalent to the completeness of a system of axioms, both in its syntactic and semantic version. This characterisation of the Definitheit is clear from the quotes above, but appears also in other passages—where the focus is sometimes on the syntactical, sometimes on the semantic side. For instance, Husserl writes that

An axiom system is definite if every proposition meaningful in virtue of the axiom (every understandable proposition) is also true or false on the basis of the axioms.⁷³

Or, as we read in (K I 26/82), an axiom system is definit “if every proposition meaningful in it is decided with respect to its domain”.⁷⁴” The Definitheit of a system of axioms is thus not only a maximality criterion, but also a completeness property of theories which is at the same time a syntactic and semantic requirement. A system of axioms $\Gamma$ is definit if, for every sentence $\phi$ from the same language, it proves either $\phi$ or its negation. Moreover, and this is equivalent according to Husserl to the previous statement, a system of axioms $\Gamma$ is definit if, for every sentence $\phi$ from the same language, it makes true either $\phi$ it or its negation.

Finally, as one can already see from the equivalence between syntactic and semantic completeness, Husserl does not take the Definitheit to be only a property of axiom systems, but also a characterisation of their corresponding manifolds. In the secondary literature this aspect has been especially stressed by Hartimo. For example, (Hartimo 2018, p. 1517) remarks that with the notion of Definitheit “Husserl also intended to capture a full characterisation of the domain in question”. Husserl is quite explicit on this additional aspect of the Definitheit. In (K I 26/36) he writes

An axiom system with a domain is definite if it does not leave open, it does not leave undecided, any meaningful question (with respect to the axiom system) relative to this domain.⁷⁵

We read analogously in (K I 26/90):

An axiom system is definite when it does not leave open any question for the formal objects of its domain, or when it does not leave open any relation on the basis of its definition (no operation relation or connection).⁷⁶

Finally, we find in (K I 26/91):

The axiom system is definite if it does not leave open any question for the domain that it delimits. All the objects in the domain are determined by the definitions.⁷⁷

According to this third group of definitions, a system of axioms is definite if it fully captures its corresponding manifolds. Indeed, if the completeness of an axiom system is interpreted semantically, and not only syntactically, it is then natural to take it also as a property of the structure which corresponds to the axioms. How to express in formal terms this aspects of the concept of Definitheit has been largely discussed in the secondary literature. While Centrone (2010, pp. 194, 199) appears to relate this feature of Husserl’s definiteness to completeness in second-order logic, Hartimo (2018) relates it to categoricity.⁷⁸ Independently from how we choose to frame Husserl’s text in the light of our present-day concepts, it is clear that he thinks that the Definitheit of a theory guarantees that it fully describes a manifold and its elements. Definite theories correspond to manifolds which are fully determined [bestimmt], namely manifolds in which every object can be unequivocally characterised.

What is the “determinateness”? Any arbitrary object of the domain, any arbitrary operation, general or specific, is equivalent to a univocal result of the operations from the specific class.⁷⁹

Husserl is not always consistent about the relationship between Bestimmbarkeit and Definitheit. While generally he seems to view them as equivalent, sometimes he also seems to regard the former concept as stronger, as it implies some kind of “equational normal form” in which every formula can be written out, as he believes to be the case for mathematical theories.⁸⁰ However, the general idea that the Definitheit of a theory allows to fully characterise a manifold is surely present in Husserl’s text and it makes more precise the correspondence between theories and manifolds that we have outlined in the previous section. Since in Husserl we do not find any precise notion of elementary equivalence, second-order equivalence, or isomorphism, we do not think that one can provide his concept with a fully formal mathematical definition. After all, we should not try to make Husserl’s concept so precise that it loses its original meaning. The Definitheit of a theory expresses the fact that it fully captures its corresponding manifold, whatever this means. To our eyes, this idea simply did not yet have a clear-cut mathematical definition at the time in which Husserl was writing.

Let us recapitulate what we have said so far. In the Doppelvortrag Husserl characterises the notion of Definitheit in three different ways. A theory is definit if it is maximally consistent, if it is (syntactically and semantically) complete, and if it fully describes its corresponding manifold. However, in the former analysis, we have not yet focussed on one important distinction that Husserl draws and that sheds some further light on his Definitheit. In the Doppelvortrag Husserl distinguishes between two notions of Definitheit, that is, a relative and an absolute Definitheit. What we have said so far applies exclusively to the notion of relative Definitheit, which is the concept originally introduced by Husserl. Differently, by absolute Definitheit Husserl essentially refers to Hilbert’s notion of completeness (Vollständigkeit).⁸¹ There are at least three references to the Vollständigkeit of an axiom systems in the works of Hilbert from this period.⁸² First, Hilbert introduces an axiom of line completeness in his second edition of the Grundlagen der Geometrie from 1903, but the same axiom appears already in the French translation of the Grundlagen from 1900.⁸³

V2 (Completeness Axiom). The elements of geometry (points, lines, planes) constitute a system of objects which, if we keep fixed the totality of all said axioms, does not admit any extension, i.e., it is not possible to add to the system of points, lines, planes any other system of objects so that the resulting system satisfies all the listed axioms I—IV, V1.⁸⁴

As Hilbert (1903, pp. 16–17) emphasises, this axiom makes sure that one cannot add elements to a system in such a way that the resulting extension is still governed by the same original axioms. This completeness axiom is essentially a maximality requirement which applies to a system of geometrical objects—completeness is viewed here as a property of the geometrical space itself, and not of its axiom system.⁸⁵ We find a similar conception of completeness in Hilbert’s article Über den Zahlbegriff from 1900:

IV 2. (Completeness Axiom) It is not possible to add to the system of numbers another system of objects, so that also in the system resulting from their composition all the axioms I, II, III, IV 1 are satisfied. Or, more briefly, the numbers constitute a system of objects which does not admit any extension, if we keep fixed all the axioms.⁸⁶

Completeness is viewed again as a maximality condition of a system of elements—here a system of numbers—and not directly of a system of axioms. Husserl rephrases this Hilbertian notion of completeness with a focus on systems of axioms. In Husserl’s own notes from Hilbert’s conference at the Mathematische Gesellshaft in Göttingen from the 5th of November 1901 we read that:

An axiom system is closed (unfortunately I do not remember anymore Hilbert’s expression), if it determines its domain of conceptual objects in such a way that no new object (of new type) can be added to the domain, that this axiom system now governs also the extended domain (it cannot be added—without thereby giving rise to a contradiction).⁸⁷

We now find a version of completeness stated in terms of axiom systems and their corresponding manifolds. A system of axioms is complete if it characterises a domain of objects which cannot be extended to a new system consistent with these axioms. In particular, we stress that here the maximality of a system of axioms does not refer only to the axioms pertaining the original domain of objects, but also excludes any possible extension by new objects and their associated axioms. We thus arrive at the concept of completeness which Husserl attributed to Hilbert at the time of the Doppelvortrag. Completeness, or as Husserl calls it absolute Definitheit, refers to the maximality property of an axiom system ensured by some “closure axiom” [abschließende Axiom], like those of the Grundlangen or Über den Zahlbegriff.⁸⁸

The first key difference between Husserl’s (relative) Definitheit and Hilbert’s Völlständigkeit is in the way these two notions force their corresponding manifold not to be extensible.

An axiom system is relatively definite when every proposition meaningful in it is also decided relatively to its domain. An axiom system is absolutely definite when every proposition meaningful in it is decided in general. Therefore, absolute definiteness = completeness in Hilbert’s sense.⁸⁹

And we find similarly in (K I 26/95):

If a manifold is relatively definite, then there is no additional axiom for its objects that can be added to the defined axioms. If a manifold is absolutely defined, then there is no additional axiom in general that can be added to the axioms.⁹⁰

The absolute Definitheit refers to axiom systems which cannot be consistently extended to any other system. The domain of a theory which is complete in Hilbert’s sense cannot be extended tout court to a larger one which still satisfies the same axioms. Differently, the kind of maximality that Husserl has in mind with his notion of relative Definitheit applies only to those sentences which have a meaning with respect to the original domain, and not to every sentence in general. According to Husserl, examples of theories which are complete in Hilbert’s sense are the real and the complex numbers, while examples of theories which are relatively definite are the natural numbers, the integers, the rationals and the Gaussian rationals.⁹¹ In particular, Husserl’s definite theories can be extended to new theories in an expanded language, and with new operations relative to these newly added elements. As we shall see in the next section, the possibility of constructing extensions of an axiom system is fundamental for Husserl’s solution to the problem of the imaginary.

We should also remark another crucial difference between Hilbert’s and Husserl’s notion of definiteness and completeness. While Hilbert’s completeness is obtained by means of a closure axiom, which forces a domain not to be extendible, Husserl views his notion of relative definiteness essentially as a property which must be proved for a given axiom system.

In my view the completeness is never an axiom, but rather a theorem for the definite axiom systems and manifolds.⁹²

Differently from Hilbert, Husserl considers the Definitheit of arithmetic essentially as a theorem which needs to be demonstrated. Husserl is quite explicit on this matter in the Doppelvortrag, and he clearly believes that it is possible to prove that the different systems of arithmetic are definit. For example, just before the previous quote Husserl remarks the following:

Once the axiom system is so devised and complete, that every point of the space (when we express everything in terms of relation between points) can be grasped as a specific result of the operations with respect to the original individual values of the operations, then completeness is also proved. It is proved that the axioms allow to deduce every geometrical proposition.⁹³

Although the quote above is admittedly obscure, and does not provide a proof of the completeness of geometry, the key point is that Husserl believes it possible to show that a system of axioms is definite, i.e. by expressing all its elements as the result of the operations over some original domain. In the light of this view, Husserl considers proved the Definitheit in the context of Hilbert’s geometry. His considerations in the context of arithmetic are quite similar—in the Doppelvortrag we find the following excerpt (cited in Centrone (2010, p. 173)):

A formal axiom system which does not contain any extrinsic closure axiom is said to be definite if every proposition that has a meaning according to the axioms, eo ipso falls under the axiom system, either as a consequence or as a contradiction. And this is the case whenever, on the basis of the axioms, it is possible to reduce every object of the domain to the group of “numerical objects”, for which every true relationship is satisfied identically, and every other is thus false. For example, when every defined proposition can be reduced to an equality, or to <, > between numerical objects, then the axiom system is definite.⁹⁴

Husserl believes that we can rephrase every arithmetical proposition into either an equation or an inequality, and therefore that every arithmetical manifold is definite. As remarked by Centrone (2010, p. 172) (and studied also in Okada (2013) and Hartimo and Okada (2015)), Husserl’s rationale is broadly the following: both in the context of Euclidean geometry and in arithmetic, every element t of the underlying manifold can be expressed by a “normal form” $t'$—so that to establish an equality $t=s$ it suffices to first compute the normal forms of t and s, and then compare them. As Centrone (2010, p. 174) puts it, “the arithmetics are syntactically complete because every expression $\alpha$ can be reduced to an equation and the axioms of the theory ‘calculate’ all equations”. But how do these two steps concretely work? Although Centrone’s analogy with the theory of real closed fields $T_{\textrm{RCF}}$ is surely interesting (cf. Centrone (2010, p. 172))—as $T_{\textrm{RCF}}$ is the first-order theory of $(\mathbb {R},0,1,+,\times ,<)$ and it is complete, decidable, and has quantifier elimination, it seems to us that it suggests a reading which is a little too charitable with Husserl.

In particular, concerning the first step, Husserl does not really explain how an arbitrary proposition can be rephrased in terms of equations (i.e. he does not foresee the topic of quantifier elimination). And, even if he considers the possibility that “only propositions of a specific form have this property with certainty”,⁹⁵ he either (implicitly) excludes more complex arithmetical sentences, or he (implicitly) derives the Definitheit of the whole theory of arithmetic from the Definitheit of its equational fragment. Additionally, concerning the second step of Husserl’s argument, it is also not clear how one would compare two numerical expressions in the case of the real numbers (which he believes to be both relatively and absolutely definite). In fact, while in the context of the natural numbers one can compare them by rephrasing every $n\in \mathbb {N}$ as a sum $1+1+\dots +1$ of n-many terms equal to 1 (a possibility which Husserl emphasises also in the Sixth Logical Investigation, cf. Husserl (1901, VI, §18)), it is entirely unclear how one could extend this procedure to the real numbers. This point has been rightly stressed by Okada (2013, p. 14): “Husserl believed that his completeness/definiteness works not only for the theories of natural numbers, of integers, of rationals, but also for the theories of real numbers and of complex numbers in the uniform way (although as we now know that it does not work with real number theory as the decision of the primitive numerical equality-relation is not decidable anymore”.

In other words, there is no natural rephrasing of Husserl’s notion of Definitheit in modern terms that captures his view that both the natural numbers and the reals are definite. On the one hand, the theory of the ordered reals $T_{\textrm{RCF}}$ is complete, decidable, and has quantifier elimination, but the equalities between any two specific real numbers cannot be determined by rewriting procedures as in the setting of the natural numbers. In a way, the decidability of $T_{\textrm{RCF}}$ is a consequence of its lack of expressibility (at least compared to Husserl’s ambitious goals)—definable subsets of models of $T_{\textrm{RCF}}$ are simply intervals with algebraic endpoints (cf. Marker (2002, Thm. 3.3.31)), whence $T_{\textrm{RCF}}$ cannot really settle equalities concerning transcendental numbers such as $\pi$ or e. On the other hand, even if natural numbers admit the normal-form reduction studied in Okada (2013) and Hartimo and Okada (2015), their first-order theory does not have quantifier elimination and it is neither decidable nor categorical in any cardinal. Husserl is simply oblivious with respect to these key difficulties, which relate to the developments of mathematical logic in the 20th century and also to the problems which led to Gödel’s Incompleteness Theorem. Husserl naively believes that the Definitheit of (any) arithmetic is proved immediately from the fact that “every defined operational entity is a natural number, and the relation of every natural number to each other can be determined on the basis of the axioms”.⁹⁶ It seems to us that Husserl’s confusion on the matter is a result of the informal nature of his concepts, and of the fact that his intuitions had not yet been translated into formal, mathematically rigorous terms. In the light of these considerations, it seems to us that Husserl’s contribution to the development of the mathematical notion of completeness does not rest on his attempts to prove it, but rather in his key intuition that the Definitheit should be a theorem, and not a definition.

As a matter of fact, the conceptual difficulties that we highlighted, as well as the terminological and philosophical differences between Hilbert and Husserl, are all witness of Husserl’s struggle to establish a new vocabulary for logic and model theory. While nowadays we take concepts such as completeness for granted, the texts of Husserl (and Hilbert) put us in a context where they were still in the making, and where they had not yet been sharply demarcated from one another. In particular, we have seen how in Husserl the concept of Definitheit expresses three main different properties. A definite theory is maximal (relative to its domain), it is (syntactically and semantically) complete and, finally, it perfectly characterises its corresponding manifold. Because of these properties, Husserl will later consider the notion of Definitheit as the full expression of the ideal of theoretical sciences which he had already talked about in the Prolegomena.⁹⁷ In fact, even if the original role of the notion of Definitheit in the Doppelvortrag is to tackle the problem of imaginary, Husserl will come back to this concept in later texts—in the Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie, in Formale und Traszendentale Logik and in Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie. In these works the concept of definiteness is considered as the full realisation of the deductive sciences, and as the fundamental subject matter of his theory of science. As Husserl writes “the theory of manifolds in its distinguished sense is the universal science of the definite manifolds”.⁹⁸ Here, however, we shall not explore the late developments of the notion of Definitheit, but rather consider its conceptual role in the Doppelvortrag itself.

4 The passage through the imaginary

The central topic of Husserl’s Doppelvortrag is the problem of the imaginary numbers. Husserl’s first conference was titled The Passage through the Impossible and the Completeness of an Axiom System [Der Durchgang durch das Unmögliche und die Vollständigkeit eines Axiomensystems], and the same topic is expanded in his second lecture.⁹⁹ We find a clear exposition of the problem that Husserl had in mind in the first pages of the Doppelvortrag:

Problem: let there be a domain of object in which, because of the specific nature of the objects, the forms of connections and relations are determined, and they are expressed in a specific axiom system A. On the basis of this system, whence on the basis of the specific nature of the objects, some specific forms of connection have no real meaning, i.e. they are contradictory forms of connection. What does justify the use of contradictions in calculations? What does justify the use of contradictions in the deductive thinking, as if they were consistent? How do we explain that it is possible to operate with contradictions according to rules, and in such a way that, once the contradictions fall our from the propositions, the resulting propositions are correct?¹⁰⁰

Consider any domain of objects, then there are expressions which make sense with respect to this domain and, also, expressions which are void of any meaning in it. Expressions of this latter kind cannot be said to be true or false, as they do not even make sense in the domain and do not refer to any object in it—they rather belong, as Husserl says in an older manuscript, to the “sphere of objectlessness”.¹⁰¹ However, in some cases, such expressions can be used in formal calculations concerning the objects of the original domain—they can be intermediate steps in derivations whose first and last steps are perfectly meaningful statements. The problem of the imaginary is thus the following: are we justified in using such expressions in our deductions? Can meaningless expressions be used in derivations and provide us with true statements for the objects of the original domain?¹⁰²

This formulation of the problem of the imaginary is very general, and it allows us to see the relationship between Unmögliche and Imaginären. In fact, even if the title of Husserl’s conference talks about a “passage through the impossible [Durchgang durch das Unmögliche]”, in the text of the two conferences we do not find this expression, but rather that of “passage through the imaginary [Durchgang durch das Imaginäre]”.¹⁰³ This apparent “Diskrepanz” between the title and the content of the lectures, as Elisabeth and Karl Schuhmann call it, can be solved if we understand that the problem of impossible objects, which Husserl was dealing with in his manuscripts from the ’90 s, and the problem of imaginary numbers, which is the subject of the Doppelvortrag, are two sides of the same philosophical issue. In both cases Husserl’s concern is the same: under what conditions can concepts which are meaningless and do not have a reference provide us with meaningful knowledge of real objects? We find a clear formulation of this problem in the manuscript (K I 26/73b), which Ierna (2011, p. 218) dates between 1890 and 1894.

Let us now take precisely this case as the starting point for our further observations and pose the question: May we use the domain of objectlessness as bridge to obtain knowledge for the domain of objectivity? And when is this the case?¹⁰⁴

In the transition from this original formulation of the problem and the later one from the lectures in Göttingen, one can see that the difference is mainly that in the Doppelvortrag Husserl expresses the problem in starkly more mathematical terms. The issue whether objectless concepts can provide us with meaningful knowledge is now phrased in terms of axiom systems and formal calculations. The problem of the Durchgang durch das Unmögliche is introduced by Husserl in the Doppelvortrag as one of the main difficulties resulting from the application of the Mannigfaltigkeitslehre to the concrete mathematical practice. As Husserl says, “the problem of the imaginary appeared in the first historical form of pure mathematics, inside arithmetic, especially in the form of arithmetical algebra”.¹⁰⁵

By focussing on the mathematical features of the problem of objectless concepts we see the reason of the shift from the original question in terms of Unmöglichkeit to the more technical issue of the Imaginärität.¹⁰⁶ In fact, in the context of mathematics the problem of the imaginary can also be phrased as follows: What is the justification of the use of imaginary numbers in the arithmetic of the natural numbers? Even if in his statement of the problem Husserl generalises the problem of the imaginary to an arbitrary domain and its “contradictory forms of connection”—and he regards as imaginary also the negative, the rational and the real numbers—what he has in mind is clearly the problem of the application of complex numbers in algebraic calculations concerning the integers.¹⁰⁷ In this form, the problem considered by Husserl can be traced back at least to the Italian algebraists from the 16th century, and to the problem of finding the solutions of arbitrary third-degree polynomial equations (over the integers).¹⁰⁸ Given an arbitrary third-degree monic polynomial

$$\begin{aligned} x^3+ax^2+bx+c, \end{aligned}$$

it is possible to solve it by first changing the variable and letting $y=x+\frac{a}{3}$, so that one obtains the depressed cubic polynomial

$$\begin{aligned} y^3+py+q=0, \end{aligned}$$

where $p=\frac{-a^2+3b}{3}$ and $q=\frac{2a^3-9ab+27c}{27}$.¹⁰⁹ Once in this form, the so-called Cardano’s formula provides us with the following solutions, namely

$$\begin{aligned} y = \root 3 \of {-\frac{q}{2}+ \sqrt{\frac{q^2}{4}+ \frac{p^3}{27}} } + \root 3 \of {-\frac{q}{2}- \sqrt{\frac{q^2}{4}+ \frac{p^3}{27}} } \end{aligned}$$

whence,

$$\begin{aligned} x = -\frac{a}{3} + \root 3 \of {-\frac{q}{2}+ \sqrt{\frac{q^2}{4}+ \frac{p^3}{27}} } + \root 3 \of {-\frac{q}{2}- \sqrt{\frac{q^2}{4}+ \frac{p^3}{27}} }. \end{aligned}$$

We thus see that, in order to find the roots of an arbitrary third-degree polynomial with integer coefficients, it is necessary to use algebraic operations which bring us outside the scope of well-defined quantities (i.e. outside of $\mathbb {Z}$, $\mathbb {Q}$ and even outside of $\mathbb {R}$). In particular, one sees from the formula above that it may be necessary to calculate the square root of a negative number, which does not have any value in the reals, let alone in the integers. Crucially, this may happen even if we are just interested in calculating only the positive roots of a polynomial with integer coefficient, as one may verify by applying the previous formula to the polynomial $x^3-15x-4$, or in general to any depressed cubic polynomial (with integer coefficients) where $\frac{q^2}{4}+ \frac{p^3}{27}<0$.¹¹⁰ In passing, we remark that a polynomial like $x^3-15x-4$ is well defined not only over the integers, but in a sense also over the natural numbers, since a root for $x^3-15x-4$ is exactly a solution of the equality $x^3=15x+4$ (this is exactly how the algebraists from the Reinassance were dealing with such expressions, as they wanted to avoid reference to the “fictitious” negative numbers).¹¹¹

The solution formula of the third-degree polynomials appeared the first time in print in Cardano’s Ars Magna.¹¹² Even if one may consider the problem of imaginary already implicit in the very nature of the negative numbers (as, in a way, did these Renaissance algebraists), it is the problem of calculating the square roots of the negative numbers that most puzzled mathematicians. Consider the following quote from Cardano’s Ars Magna, cited in Gavagna (2022, p. 258):

And notice that $\sqrt{+9}$ is either $+3$ or $-3$, in fact plus [times plus] and minus times minus are plus. Therefore $\sqrt{-9}$ is neither $+3$ or $-3$ but something of a third unknown nature.¹¹³

The problem of computing the square root of negative numbers, which Cardano tellingly calls radices sophisticae, led Cardano to various attempts in his De regula aliza libellus to find an alternative solution formula that did not contain such expressions.¹¹⁴ The problem of the radices sophisticae would resurface shortly after in the work of Bombelli, who addressed it by introducing new signs, which he called più di meno (plus of minus) and meno di meno (minus of minus) in his Algebra.¹¹⁵ Bombelli’s work did not find great echo in the mathematical community, but was nonetheless one of the first key steps in the development of the complex numbers. Some decades after, the idea of imaginary quantities would appear again in the history of algebra, now in the context of “the problem of computing the number of roots” (Gavagna 2014, p. 186), and of the early modern formulations of the Fundamental Theorem of Algebra by Girard and Descartes.¹¹⁶

Moving now aside from the historical developments of algebra, we can summarise the key philosophical problem underlying the role of imaginary quantities as follows. Cardano’s formula provides a method to find the solutions of an arbitrary third-degree polynomial. However, in order to compute the solutions of these polynomials, it is necessary to employ imaginary expressions such as $\sqrt{-9}$ in concrete calculations. Most strikingly, these seemingly nonsensical expressions appear also in the calculation of the solutions of polynomials with integer coefficients—a situation which had been identified by Cardano as the casus irreducibilis in the Ars Magna.

Addressing this problem in the Doppelvortrag, Husserl considers five different theories which had been previously advocated by mathematicians. The first two theories are just mentioned without further discussions, as Husserl probably felt that they were not serious attempts to answer the problem of the imaginary.¹¹⁷ The first theory is attributed by Husserl to Bain and Baumann, and it says that the imaginary numbers are justified empirically by induction. The second theory is attributed by Husserl to Boole, and it states that imaginary numbers are immediately justified by their a priori self-evidence. Husserl discusses more extensively a third and a fourth proposal. The third theory which he considers is essentially the one proposed by Dedekind in Stetigkeit und irrationale Zahlen, which Husserl explicitly quotes in his text.¹¹⁸ According to Dedekind, numbers are always the outcome of a mental act of creation, so that the problem of imaginary numbers finds its solution with the introduction of new definitions, which would allow us to proceed beyond a given sphere of numbers and create new kinds of numbers. Husserl disagrees with this creative view of definitions: according to him a definition is not a mental act of creation, but simply “an arbitrary determination of the meaning of a word”.¹¹⁹ A fourth theory, often argued for by those who believe that definitions have a creative power, says that imaginaries are justified by their concrete applications outside of pure mathematics.¹²⁰ The problem of the imaginary numbers is thus purely apparent, and it vanishes once we take into account the many usages of mathematics. However, according to Husserl, the difficulty of this fourth theory relies exactly on its exclusive interest in the applications of mathematics. The definition of number takes place inside pure mathematics, and the question of its possible extensions is a theoretical matter, which cannot be solved by appealing to the concrete use of mathematical concepts.¹²¹

Husserl seems to consider both the third and the fourth solution to the problem of imaginary numbers as serious misunderstandings of the relation between different arithmetical theories. According to Husserl, in fact, every arithmetical theory is delimited in scope by an underlying defining concept.¹²² For instance, the arithmetic of the natural numbers is determined by the definition of number as a possible reply to the question “How many? [Wieviel?]”, i.e. by the concept of number as a positive integer.¹²³ However, it is not possible to extend the range of a concept via arbitrary definitions. By using a terminology which is slightly foreign to Husserl here, but that nonetheless fits well, we could say that it is the intension of a concept which determines its extension. Once we consider numbers to be quantities determined by the concept of Wieviel, we cannot artificially stipulate what falls under such concept—we cannot simply add new objects to its extension. In fact, even if Husserl has in mind a stepwise extension of the natural numbers to the other numerical domains, which is very reminiscent of Dedekind’s approach, he also believes in a very different justification of such procedure. One should sharply distinguish between two different aspects of arithmetic. On the one hand, every arithmetical system is associated to an underlying concept of number, and to a corresponding domain of numbers which fall under such concept. On the other hand, to every arithmetical system also belong specific operations and algorithmic procedures, which one can use to perform calculations and deduce results about the objects of the system.¹²⁴

The different arithmetics do not have common parts, they have rather quite different spheres but an analogue structure, they have partially the same forms of operations, although different concept of operations.¹²⁵

Different arithmetical systems have fully different Operationsbegriffe, as their operations refer to different objects, namely to different systems of numbers which are specified by different concepts. However, the algorithmic procedures and the operations that pertain to these different domains partially overlap. For example, the operations of sum and product between two natural numbers are the same both if we view them as two integers, or also if we see them as real numbers. Even if the number systems are different, sum and product agree on a specific subset of elements, and they satisfy the same associativity and distributivity laws. Clearly, this does not hold true of any arbitrary operation, as there are operations which make sense with respect to one manifold but not in another. As Husserl says, “forms of operation that are meaningful in one conceptual domain, are however contradictory in another”.¹²⁶ Here one can think of a simple subtraction like 7–9, which makes full sense in the ring of integers $\mathbb {Z}$, but that is meaningless in the semiring of the natural numbers $\mathbb {N}$. Different concepts of number put forward different limitations of the algorithmic procedures. Therefore, according to Husserl, the stepwise extension of the number domain does not concern the concept of number itself, but rather the scope of application of the arithmetic operations. While Dedekind viewed the stepwise extensions of the domain of numbers from the naturals to the reals as a mental creation of new numbers, for Husserl we are rather considering, at each step, “a purely formal new concept”,¹²⁷ which determines a system of arithmetic that partially overlaps with the previous ones.¹²⁸

A similar focus on the algorithmic aspects of the stepwise extension of the natural numbers to new number systems is present also in the fifth theory that Husserl considers in his first lecture. This solution to the problem of the imaginary relies on the so-called principle of permanence—a principle that Husserl takes over from the mathematician Hermann Hankel.¹²⁹ In his Theorie der complexen Zahlensysteme of 1867, Hankel introduces it as follows:

The guiding principle contained here can be named the principle of the permanence of formal laws and it consists of the following: If two forms expressed in the general signs of the arithmetica universalis are equal to each other, then they should remain equal to each other also when the symbols do not refer anymore to simple quantities, and thus also the operations obtain another arbitrary meaning.¹³⁰

By arithmetica universalis, Hankel refers here to the formal or symbolic arithmetic, where one can apply the algorithmic procedures of algebra in their full generality and without any restriction induced by the underlying domain of interpretation. The principle of permanence thus states that the equalities obtained in the arithmetica universalis remain valid once they are provided an intepretation relative to a specific domain of objects. In particular, this principle places some constraints on the possible extensions of an operational system: it requires that the operations from the extended arithmetical system should be equivalent to the original operations when we restrict ourselves to the objects of the narrower arithmetical system.¹³¹ According to this fifth theory presented by Husserl, the principle of permanence actually justifies the passage through the imaginary. In fact, if the extended arithmetical system is consistent, all results obtained in it should be also consistent with the original restricted system. For example, coming back to our go-to example of third-degree polynomials, if using Cardano’s formula and calculating with imaginary numbers we find an integer root $r\in \mathbb {Z}$ of a polynomial $p(x)=x^3+ax^2+bx+c$, then the equality $p(r)=0$ remains true even when we interpret it in the narrower domain of the integers.

Although Husserl does not agree with this fifth proposal, it constitutes however the basis on which he structures his own solution to the problem of the imaginary. More precisely, Husserl’s position could be regarded as an attempt to make the principle of permanence more formal and precise. Differently from Hankel, Husserl stresses that the extended arithmetical system is consistent, and he interprets this fifth approach to the problem of the imaginaries as the statement that consistency entails conservativity.¹³² Let $T_{\textrm{AR}}$ be a restricted arithmetical system, for example the arithmetic of the integers, and let $T_{\textrm{AU}}$ be the extended arithmetica universalis, i.e. the general arithmetical system where the algorithmic rules are not constrained by limitations due to the specific objects which we are dealing with. The fifth theory can be then formally reconstructed as follows. If $T_{\textrm{AR}}$ and $T_{\textrm{AU}}$ are consistent with each other, then if $T_{\textrm{AU}}\vdash \phi$ and $\phi$ is in the same language of $T_{\textrm{AR}}$, it follows that $T_{\textrm{AR}}\vdash \phi$.¹³³ The consistency of $T_{\textrm{AU}}$ over $T_{\textrm{AR}}$ entails, via the principle of permanence, the conservativity of $T_{\textrm{AU}}$ over $T_{\textrm{AR}}$. At the end of the first lecture, Husserl provides his main argument against Hankel’s theory.

However, this inference is questionable. First, it is surely correct that no derived proposition that includes imaginary complexes contains an inconsistency, that it cannot conflict neither with the extended nor with the original and narrower axioms. But how do we know that what is consistent is also true? Or, as it should be put here: how do we know that a proposition, which pertains exclusively the concepts occurring in the narrower domain, and that are defined there, and which does not contradict the axioms of the restricted domain, that such a proposition is valid for the narrower domain?¹³⁴

Husserl agrees with the fact that, since the newly introduced axioms concern imaginary numbers that do not belong to the original manifold, the consistency of the original arithmetic entails the consistency of the extended system. However, for Husserl, this argument only concludes that the restricted system $T_{\textrm{AR}}$ is not inconsistent with the consequences of the extended system $T_{\textrm{AU}}$, but it does not yet establish that $T_{\textrm{AR}}$ proves them. In modern notation, from the fact that $T_{\textrm{AU}}\vdash \phi$ and $T_{\textrm{AR}}\nvdash \lnot \phi$, we cannot conclude that $T_{\textrm{AR}}\vdash \phi$. The application of the principle of permanence relies on a unwarranted step from consistency to provability. More concretely, even if the statement $p(r)=0$ (for some integer $r\in \mathbb {Z}$ and some third-degree polynomial $p\in \mathbb {Z}[x]$) is consistent with the arithmetic of the integers $T_{\textrm{AR}}$, this does not mean that $p(r)=0$ is actually provable in $T_{\textrm{AR}}$ (or, equivalently for Husserl, it does not follow that it is true in $T_{\textrm{AR}}$). It follows that also this fifth theory does not suffice, according to Husserl, to solve the problem of the imaginary.

After criticising the five theories presented above in his first lecture, Husserl introduces in the second lecture of his Doppelvortrag his own solution to the problem of the imaginary. This can be seen as a refinement of the fifth theory which we have just considered, and it consists essentially in the specification of the precise cases in which the appeal to the principle of permanence is justified. As a matter of fact, the notion of (relative) Definitheit that we have analysed in the previous section of this article is introduced by Husserl exactly to identify when the passage through the imaginary is licit. As Husserl clarifies in the second lecture and in other preparatory manuscripts, “the passage through the imaginary is thus connected with the condition of definiteness, and in fact the partial definiteness already suffices for this”.¹³⁵ Of the various aspects of the notion of Definitheit that we have highlighted in Sect. 3, it is certainly the definiteness as completeness (syntactic or semantic) which makes the passage through the imaginary possible.

The use of a larger system to find propositions of the narrower is allowed only when we have any characteristic from which we can recognize that every proposition, which has a meaning in the narrower domain, is also decided in it, thus that it must also be a consequence or a contradiction of it.¹³⁶

If the restricted arithmetical system is definite then every sentence concerning the objects of its corresponding manifold is either true or false, and (equivalently for Husserl) it either follows from the axioms or it contradicts them. Husserl’s idea is thus to restrict the argument based on the principle of permanence to those theories which are (relatively) definite. Following Centrone (2010, p. 170), we can reconstruct Husserl’s argument as follows. Suppose the extended arithmetical system $T_{\textrm{AU}}$ proves a sentence $\phi$, that is, $T_{\textrm{AU}}\vdash \phi$. Since $T_{\textrm{AU}}$ is a consistent extension of $T_{\textrm{AR}}$, it clearly cannot be the case that the restricted arithmetic $T_{\textrm{AR}}$ contradicts $\phi$, i.e. that $T_{\textrm{AR}}\vdash \lnot \phi$. Thus, it follows that $T_{\textrm{AR}}\nvdash \lnot \phi$. Now, since $T_{\textrm{AR}}$ is definit, it follows that either $T_{\textrm{AR}}\vdash \phi$ or $T_{\textrm{AR}}\vdash \lnot \phi$, and therefore, we conclude that $T_{\textrm{AR}}\vdash \phi$. Therefore, while the fifth theory considered by Husserl in the first lecture simply argues that consistency entails conservativity, Husserl additionally identifies the Definitheit of the restricted system as a necessary requirement for this step.¹³⁷ The passage through the imaginary is therefore possible if the following two conditions are met: (i) if the extended arithmetical system $T_{\textrm{AU}}$ is consistent, and (ii) if the restricted arithmetical system $T_{\textrm{AR}}$ is definit.

The passage through the imaginary is the main topic of Husserl’s Doppelvortrag, and the main motivation behind Husserl’s introduction of the concept of definiteness. In this section, we have tried to describe what exactly was the problem that Husserl wanted to address, and to present his own solution in the light of the historical developments of algebra. However, we see how the modern reader may find Husserl’s approach quite puzzling, as an explanation of obscurum per obscurius. After all, Husserl explains something which appears quite justified to us—the use of complex numbers to solve polynomial equations—via something which is largely more subtle and contentious—the consistency and the completeness of arithmetic. We believe that to make justice of Husserl’s approach to this issue one needs some kind of historical Verstehen.

First, on the technical side of the problem, Husserl seems to think that the consistency and the completeness of arithmetical systems were not particularly hard mathematical properties to establish: he seems to take the consistency of arithmetic for granted, and he suggests a proof of the Definitheit of arithmetic (cf. Sect. 3 above).¹³⁸ One is tempted to regard as naive his approach, especially in the light of Gödel’s incompleteness theorem and the developments of mathematical logic in the past century. However, we believe that one should try to appreciate the fact that Husserl’s problems in the Doppelvortrag—as well as many mathematicians’ problem around this time—are not immediately our own problems. The role of the imaginary numbers that Husserl discusses in his two lectures in Göttingen was already being discussed by the mathematical community, and was possibly also one key influence behind the debate on the foundations of mathematics.¹³⁹ It is thus interesting to notice that the very notion of completeness of an axiom system, now so central in mathematical logic, was originally elaborated and discussed in this quite diverse historical context—as a means to justify the use of the imaginary numbers in algebra and arithmetic.

Second, from a more philosophical perspective, one should keep in mind the somewhat privileged status of the natural numbers, which appears to be shared by many mathematicians of the time. Dedekind’s genetic approach to the foundations of mathematics can be seen as one example of this philosophical inclination, which is epitomised by Kronecker’s famous dictum, that “the integers were made by God, everything else is human work”.¹⁴⁰ In the Philosophie der Arithmetik, Husserl had defined the concept of number in terms of Vielheit, thereby assigning a central role to the naturals in his philosophy of mathematics. There is indeed a sense in which, according to Husserl, numbers are first the natural numbers, and only second, via the introduction of new concepts of number and with the extension of the associated algorithmic procedures, also the integers, the rationals, the reals and the complex numbers. From this perspective, the Doppelvortrag can be seen as a rare instance of a foundational program which brings together the two approaches that Hilbert had identified in Über den Zahlbegriff, namely the genetic and the axiomatic method.¹⁴¹

5 Conclusion

The Doppelvortrag is a point of intersection of several different lines—a moment which possibly belongs to alternative narratives. In this article, we have tried to select three levels of conceptual and historical interpretation. In Sect. 2, we have described Husserl’s proposal of a theory of science in its two declinations of a philosophical Wissenschaftslehre and a mathematical Mannigfaltigkeitslehre, and we have related Husserl’s approach to the parallel developments of Hilbert’s axiomatic method. In Sect. 3, we provided an interpretation of Husserl’s notion of Definitheit in his Doppelvortrag. We tried to place Husserl’s Definitheit in the historical context of the early developments of model theory, and we especially stressed how Husserl noticed the need to prove the definiteness of an axiom system, even if his attempts to do so were essentially flawed. Finally, in Sect. 4, we focussed on the very problem of the passage through the imaginary—the main topic of the Doppelvortrag—and we outlined Husserl’s own solution to it. We hope this article helps in providing a conceptual and historical understanding of the Doppelvortrag—of its place in Husserl’s philosophy and its role in the modernisation of mathematics.