The Origin of the Logic of Symbolic Mathematics

Edmund Husserl and Jacob Klein

Burt C. Hopkins

• Publisher: Indiana University Press
• Year: 2024
• Print ISBN: 9780253356710
• eBook ISBN: 9780253005274
• Topic: Philosophy
• Subtopic: Logic in Philosophy

Burt C. Hopkins presents the first in-depth study of the work of Edmund Husserl and Jacob Klein on the philosophical foundations of the logic of modern symbolic mathematics. Accounts of the philosophical origins of formalized concepts—especially mathematical concepts and the process of mathematical abstraction that generates them—have been paramount to the development of phenomenology. Both Husserl and Klein independently concluded that it is impossible to separate the historical origin of the thought that generates the basic concepts of mathematics from their philosophical meanings. Hopkins explores how Husserl and Klein arrived at their conclusion and its philosophical implications for the modern project of formalizing all knowledge.

Part One

Klein on Husserl’s Phenomenology and the History of Science

Chapter One

Klein’s and Husserl’s Investigations of the Origination of Mathematical Physics

§ 1. The Problem of History in Husserl’s Last Writings

Some seventy years have passed since the first publication of two fragmentary texts on history and phenomenology that Husserl wrote in his last years,¹ texts that unmistakably link the meaning of both the crowning achievement of the Enlightenment (the new science of mathematical physics) and that of his own life’s work (the rigorous science of transcendental phenomenology) to the problem of their historical origination. It is striking that in the years following the original publication of these works and their republication in 1954 in Walter Biemel’s Husserliana edition of the Crisis, commentary on them has, with one significant exception, passed over what Husserl articulated as the specifically phenomenological nature of the problem of history. It has been ignored in favor of mostly critical discussions of Husserl’s putative attempt to accommodate his earlier “idealistic” formulations of transcendental phenomenology to the so-called “problem of history.”

As it is typically understood, this problem begins with the notion of a contingent sequence of events that shape cultural formations and human experience in a manner that defies rational calculation. History in this sense becomes a problem when its contingency is understood to condition the intellectual content of cultural formations, such as philosophy and science. The problem here concerns the influence of the historically conditioned heritage of taken-for-granted ideas, meanings, and attitudes on the knowledge claims made by philosophy and science. When the intellectual content of the latter is understood to have as its insuperable limit the particular historical situation of the philosopher and the scientist, as well as of philosophy and science, the knowledge claims of both are correspondingly understood to be incapable of ever achieving “universality.” Formulated in this manner, the “problem of history” assumes, as is well known, the guise of what since the beginning of the twentieth century has been called ‘historicism’.

The reception of Husserl’s last works has been preoccupied with the story of their departure from his own early rejection of historicism and his late attempts to establish what many have deemed oxymoronic and therefore impossible: a phenomenology of the apriori proper to the historical origination of meaning. Motivated by the goal of establishing phenomenology as a presuppositionless universal science of a priori meanings, Husserl’s early thought had identified the “facticity” of history as among those presuppositions standing in the way of a “pure” phenomenology. Husserl’s late turn to the problem of history has therefore led many to suspect that pure phenomenology and the historical preoccupation of his last texts are intrinsically incompatible.

§ 2. The Priority of Klein’s Research on the Historical Origination of the Meaning of Mathematical Physics over Husserl’s

Part One of this study is concerned with the major exception to the trend in the literature to overlook the significance assigned to history in Husserl’s Crisis alluded to above, namely, the work of Jacob Klein.² Its twofold aim is to elaborate Klein’s understanding of the phenomenological problem of history sketched by Husserl in his last works³ and to introduce Klein’s own contribution to the understanding of the problem of the historical origination of the meaning of mathematical physics. The latter’s contribution occurs in his little known but remarkable works on Greek mathematics and the origin of algebra.⁴ On the assumption that Klein’s contribution to that understanding came after his appropriation of Husserl’s formulation of the phenomenological problem of history, the execution of this twofold aim would seem to be a fairly straightforward matter. One would need only to show how the method and content of Husserl’s path-breaking investigations influenced or otherwise provided the context for Klein’s own research. However, Klein’s work on the historical origination of the meaning of mathematical physics actually preceded Husserl’s work on this same issue by a number of years.⁵ Thus, Hiram Caton’s felicitous characterization—in another context, and one that will be taken up shortly—of Klein’s relationship to Husserl as “a scholarly curiosity”⁶ proves apt here as well, since Klein’s work on the history of mathematics represents an uncanny anticipation of Husserl’s own work.

In 1959 Leo Strauss characterized Klein’s magnum opus, “Die griechische Logistik und die Entstehung der Algebra,” then still untranslated, as a work that is “much more than a historical study.”⁷ Strauss continued: “But even if we take it as purely a historical work, there is not, in my opinion, a contemporary work in the history of philosophy or science or in ‘the history of ideas’ generally speaking which in intrinsic worth comes within hailing distance of it. Not indeed a proof but a sign of this is the fact that less than half a dozen people seem to have read it, if the inference from the number of references to it is valid.” Strauss’s characterization of this work as “much more than a historical study,” along with his comparison of it—without limiting it—to both the “history of philosophy” and the “history of ideas,” is instructive here. For while it claims that Klein’s treatment of his topic is of unparalleled historical import, the cryptic suggestion that its true significance transcends contemporary studies in the history of philosophy or science, as well as studies in the history of ideas generally, gives occasion to formulate a major thesis of the present study: that both the methodology and the content of Klein’s mathematical studies fall outside the traditionally distinct methodological approaches to the likewise traditionally distinct domains staked out, respectively, by the history and the philosophy of science. Before developing this thesis within what here will be argued to be the proper context for considering both the method and the content of Klein’s mathematical studies, it is necessary to digress briefly so as to situate this context in relation to how the methods and the contents of the history of science and the philosophy of science are typically understood to differ. The goal thereby is to provide a context in contrast to which the radicality of Klein’s approach to both historical and systematic issues in his mathematical studies can be demonstrated.

With respect to method the difference in question here concerns the traditional contrast between the “empirical” approach to science characteristic of the history of science and the “epistemological” approach characteristic of the philosophy of science. Accordingly, the history of science is usually defined by its investigation of the contingent series of mathematical, scientific, and philosophical theories involved in the formation and development of a given science. By contrast, the philosophy of science is usually defined by its investigation of the cognitive status of the philosophical problems posed by the employment of logic, mathematics, and metaphysics in the knowledge claims advanced by the systematic sciences. Corresponding to these methodological differences are the differences in content of the domains typically treated by the historical and the philosophical investigations of science. Thus, the content of the history of science reflects the changes over time that mark the development of a science, whereas the content of the philosophy of science reflects the temporal stability that defines scientific knowledge.

§ 3. The Importance of Husserl’s Last Writings for Understanding Klein’s Nontraditional Investigations of the History and Philosophy of Science

Rather than work within the context of this traditional understanding of the difference and indeed opposition between these methods and their domains, Klein’s mathematical studies are characterized by a method—albeit one that largely remains implicit—that overcomes the opposition between historical explanation and epistemological investigation in the study of science. His studies are thus historical without being limited to empirical contingencies and epistemological without being cut off from the historical development of scientific knowledge. In other words, Klein’s work overcomes the problem of history that leads to historicism by showing, in effect, that the disclosure of the “historicity” of scientific knowledge does not lead to an opposition between the contingency of history and the universality of knowledge. His work shows this by uncovering the heritage of ideas, meanings, and attitudes that underlie the basic concepts of the modern mathematics that makes mathematical physics possible; that is, he uncovers aspects of what Husserl will refer to as the “historical apriori” (Origin, K380/C375) of modern physics. Yet it is Husserl who in his last works was the first to articulate explicitly the methodological issues involved in overcoming the opposition in question here. The assessment of both i) the scope and limits of Klein’s implicit method and ii) the cogency of its results must take Husserl’s reflections on this methodology as its point of departure. Husserl’s later articulation of the “theory of knowledge . . . as a peculiarly historical task” (F220/C370), a task he assigns to his final formulation of transcendental phenomenology and its now defining goal of overcoming “[t]he ruling dogma of the principial separation between epistemological elucidation and historical explanation” (ibid.), provides the proper perspective from which to assess Klein’s work. It is Husserl’s formulation of the “universal apriori of history” (K380/C371) as “nothing other than the vital movement of the coexistence and the interweaving of original formations and sedimentations of meaning” (F221/C371) that serves as the “guiding clue” for overcoming the “ruling dogma” in question. The methodology that discloses this “vital movement” thus is indispensable for taking the measure of Klein’s investigations, and it is to be found in Husserl’s sketch of phenomenologically historical reflection. Husserl characterizes such reflection in terms of a “ ‘zigzag’ back and forth” from the “ ‘breakdown’ situation of our time, with its ‘breakdown of science’ itself,” to the historical “beginnings” of both the original meaning of science itself (i.e., philosophy) and the development of its meaning leading up to the “breakdown” of modern mathematical physics (see Crisis, 59/58).

§ 4. Klein’s Commentary on Husserl’s Investigation of the History of the Origin of Modern Science

Klein himself provides the warrant for this account of the significance of Husserl’s methodology for understanding his own mathematical studies in his article “Phenomenology and the History of Science” from 1940. After first explicating Husserl’s articulation of the phenomenological problem of history in the original published versions of the Crisis and “The Origin of Geometry,” Klein goes on to outline “[t]he problem of the origin of modern science” (PHS, 82) in a manner that corresponds to Husserl’s formulation of the problem, save for one significant deviation. There Klein adds a third task to the two tasks that Husserl articulates in connection with this problem. Whereas for Husserl the problem of the origin of modern science involves the “reactivation of the origin of geometry” (83) and “the rediscovery of the prescientific world and its true origins,” (84) according to Klein there is yet another aspect to this problem. He articulates this aspect in terms of “a reactivation of the process of symbolic abstraction” (83) whose “ ‘sedimented’ understanding of numbers is superposed upon the first stratum of ‘sedimented’ geometrical ‘evidences’ ” (83–84). Klein therefore positions this additional task between the twin tasks that Husserl articulates in the Crisis.⁸

Klein’s introduction of this third task is significant for a number of reasons, all of which will be taken up here in due course. At this point, however, only one requires comment, namely that the task of the “reactivation of the process of symbolic abstraction” had in fact already been undertaken and indeed completed by Klein himself in “Die griechische Logistik und die Entstehung der Algebra.” There can be no mistake about this. In the final section of his “Phenomenology and the History of Science” (see 79–83), Klein presents a synopsis of the development of the symbolic transformation of the traditional Greek theory of ratios and proportions, as well as of the ancient Greek “concept” and science of number, into François Vieta’s “ ‘algebraic’ art of equations” (80). In addition, he discusses the “formalization” of Greek mathematics that was prepared for with the “anticipation” of an exact geometrical nature by Galileo and his predecessors and realized with the symbolic transformation of Euclidean geometry into Descartes’s analytic geometry—the latter being made possible by Vieta’s “invention” of modern mathematics. The formalization of Greek mathematics, upon which are “laid the foundations of mathematical physics” (82), is said by Klein to “have already lost the original intuition” (81) of the Greek mathematics underlying it. He traces this loss to modern mathematics’ technique of operating with symbols. As a result of this, the “reactivation of the process of symbolic abstraction” (84) that makes possible the formalization of the mathematics that prepares the way for mathematical physics is held by Klein to involve, “by implication, the rediscovery of the original arithmetical evidences.” For him these original evidences concern “the original ‘ideal’ concept of number, developed by the Greeks out of the immediate experience of ‘things’ and their prescientific articulation” (81).

What Klein lays out in this synopsis amounts to a précis of the “argument” of his work on Greek mathematics and the origin of algebra from 1934–36. This fact calls attention to a second “scholarly curiosity” characteristic of Klein’s relationship to Husserl, namely, his failure to provide any reference to that work in an ...

Table of contents

• Cover
• Half title
• Title
• Copyright
• Contents
• Preface
• Acknowledgments
• List of Abbreviations
• Introduction. The Subject Matter, Thesis, and Structure of This Study
• Part One. Klein on Husserl’s Phenomenology and the History of Science
• Part Two. Husserl and Klein on the Method and Task of Desedimenting the Mathematization of Nature
• Part Three. Non-symbolic and Symbolic Numbers in Husserl and Klein
• Part Four. Husserl and Klein on the Origination of the Logic of Symbolic Mathematics
• Glossary of Greek and German Terms
• Bibliography
• Index of Names
• Index of Subjects