# The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein

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.

37 Chapters |
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## Introduction. The Subject Matter, Thesis, and Structure of This Study |
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This study is concerned with the origination of the logic of symbolic mathematics as investigated by Edmund Husserl and Jacob Klein. The ‘logic’ of symbolic mathematics at issue here is that which allows everyone—from barely literate school children to master mathematicians—to employ sense-perceptible letter signs, without a second thought, in a “mathematical” manner. The content of mathematics, like the content of its logic, is immaterial to its topic, which is how it has come about that such signs are self-evidently perceived to represent an “indeterminate” conceptual content as readily and unproblematically as, for example, the perception of the color and shape of this book. What is responsible for this topic is uncontroversially referred to as ‘formalization’. What formalization is, however, is controversial. At one extreme, formalization is understood as the employment of letter signs or other marks to, at the very least, “stand for” or “symbolize” any arbitrary object or content—“whatever”—belonging to a certain “domain.” Let ‘3’ stand for the number of any arbitrary objects whatever; let ‘X’ stand for any arbitrary number whatever; let ‘S’ stand for any arbitrary subject member of any proposition whatever—all these expressions are examples of formalization, and when “interpreted” in a manner that finds nothing especially problematic to speak of here, these examples illustrate pretty much all that is needed—or the minimum needed—to begin formalization. At the other extreme is the view that formalization is the fulcrum for an unprecedented transformation in how the science of the so-called West forms its concepts, a transformation that is as all-encompassing as it is invisible to this day—especially to those who study the history of this science or are engaged in scientific inquiry. |
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## Chapter One. Klein’s and Husserl’s Investigations of the Origination of Mathematical Physics |
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Some seventy years have passed since the first publication of two fragmentary texts on history and phenomenology that Husserl wrote in his last years, |
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## Chapter Two. Klein’s Account of the Essential Connection between Intentional and Actual History |
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Klein’s interpretation of Husserl’s articulation of the phenomenological problem of history in his last writings capitalizes on Husserl’s lifelong concern with “the problems of origin” ( |
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## Chapter Three. The Liberation of the Problem of Origin from Its Naturalistic Distortion: The Phenomenological Problem of Constitution |
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Turning now to Klein’s account of the problem of history in Husserl’s early work, we find him maintaining that in “attacking ‘psychologism,’ Husserl was in fact facing the problem of ‘history’ ” (65). Indeed, Klein maintains “that Husserl in criticizing the attitude of historicism [in ‘Philosophy as Rigorous Science’] puts it on the same level with psychologism. In fact, the former is but an extension and amplification of the latter” (68). Thus: Any “naturalistic” psychological explanation of human knowledge will inevitably be the history of human development with all its contingencies. For in such an account any “idea” is deduced from earlier experiences out of which that idea “originated.” In this view, the explanation of an idea becomes a kind of historical legend, a piece of anthropology. The |
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## Chapter Four. The Essential Connection between Intentional and Actual History |
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Klein unpacks Husserl’s account of how it is that the “intrinsic possibility” of an object’s intentional unity “contains the ‘sedimented history’ of its ‘constitution’ ” (72) in view of two “limits.” These emerge in the analysis of the “universal eidetic ‘form’ of the intentional genesis” of each such object’s unity, that is, in the analysis of “internal temporality” (72–73). |
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## Chapter Five. The Historicity of the Intelligibility of Ideal Significations and the Possibility of Actual History |
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In the foregoing discussion we have indicated that for Klein the transcendental inquiry into the problem of the “intentional history” of the categorial formations of the significance making up an object’s identity “may reveal the essential necessity of its being subject to a history in the usual sense of the term.” That is to say, the transcendental inquiry into the intentional history of an object’s categorial unity may disclose an essential connection between the origin of this unity and its historical development within natural time. For Klein, “[h]istory, in the usual sense of the term, is not a matter-of-course attitude. The origin of history is itself a non-historical problem” ( |
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## Chapter Six. Sedimentation and the Link between Intentional History and the Constitution of a Historical Tradition |
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According to Klein, then, it is Husserl’s phenomenological inquiry into the transcendental constitution of the origins of the ideal formations proper to mathematical and scientific objects that reveals that the “ ‘evidence’ of all the ‘significant formations’ belonging to a science such as geometry” presupposes “the link between ‘intentional history’ and actual history” ( since the product, in the case of geometry, is an ideal product, “anticipation” and the corresponding “accomplishment,” as acts of the subject . . . , are founded upon the “work” of transcendental subjectivity: the ideal formations of geometry are products of the “intentionality at work.” “Anticipation” and “accomplishment” |
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## Chapter Seven. Klein’s Departure from the Content but Not the Method of Husserl’s Intentional-Historical Analysis of Modern Science |
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Klein’s discussion in “Phenomenology and the History of Science” does not “follow Husserl’s pattern” ( |
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## Chapter Eight. Klein’s Historical-Mathematical Investigations in the Context of Husserl’s Phenomenology of Science |
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Part One of our study explored Klein’s interpretation of Husserl’s turn to the problem of history in his last works. We argued that Klein, alone among Husserl’s commentators, recognized that this turn is in harmony with Husserl’s lifelong investigation of the phenomenological origins of the ideal meaning formations that make both philosophy and science possible. We also argued that prior to Husserl’s account (in the final phase of his work) of the essential connection between historical inquiry and the quest for the epistemological foundations of scientific knowledge, Klein’s own investigations of the history of mathematics recognized the same essential connection. We showed, however, that the priority and thus independence of Klein’s investigations in relation to Husserl’s is a complicated affair. To begin with, Klein does not hesitate—though necessarily after the fact—to situate his own mathematical investigations in terms of Husserl’s articulation of the phenomenological problem of the sedimentation of significance. As we have seen, this problem concerns forgetting the original evidence belonging to the origination of the meaning formations that make a given science (e.g., geometry) possible. Klein accepts Husserl’s argument that sedimentation is inseparable from both the primal establishment of science and the historicity of its phenomenal status as a tradition. We have also seen that Husserl characterizes the method of historical reflection that reactivates the forgotten original evidence as involving a back-and-forth or zigzag movement. Beginning with what for Husserl was the present crisis situation of the sciences, reflection strives to uncover the original accomplishments that gave and had to give their formations meaning. As for the crisis itself, we singled out Husserl’s account of the role that the unintelligibility of the epistemological foundations belonging to the meaning formations that make science possible played in the breakdown situation of his time. Finally, we advanced the thesis—but not yet supported it—that the |
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## Chapter Nine. The Basic Problem and Method of Klein’s Mathematical Investigations |
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In |
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## Chapter Ten. Husserl’s Formulation of the Nature and Roots of the Crisis of European Sciences |
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We shall now highlight Klein’s uncanny anticipation in his Husserl’s investigation of the origin of the intrinsic possibility belonging to the objective unity of any meaning formation was shown in Part One to extend to the a priori structure of its genesis as an intentional unity. It was also shown that the latter holds the key to the insight that Husserl’s turn to history in his last writings is the consistent outcome of the phenomenological project of investigating the radical beginnings proper to the things themselves. Far from representing a significant departure from his early rejection of the ability of psychologism and historicism to account for these beginnings, Husserl’s late turn to history is motivated by his realization that the investigation of the origins of certain things themselves is not exhausted by uncovering the sedimented history of their genesis in the stream of consciousness. The backward reference ( |
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## Chapter Eleven. The “Zigzag” Movement Implicit in Klein’s Mathematical Investigations |
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We shall now set about demonstrating the implicit “zigzag” movement belonging to the method of historical reflection operative in Klein’s desedimentation of the formalized meaning formations that characterize modern mathematics. This movement can best be seen by way of an overview of the structure of his investigations in |
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## Chapter Twelve. Husserl and Klein on the Logic of Symbolic Mathematics |
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As Klein notes, the logic of symbolic mathematics was Husserl’s first philosophical problem. Husserl’s investigations in |
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## Chapter Thirteen. Authentic and Symbolic Numbers in Husserl’s Philosophy of Arithmetic |
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Husserl’s |
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## Chapter Fourteen. Klein’s Desedimentation of the Origin of Algebra and Husserl’s Failure to Ground Symbolic Calculation in Authentic Numbers |
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Our discussion of the concepts of authentic and symbolic number operative in Husserl’s analyses in Husserl’s analyses clearly show, on the one hand, that the authentic concept of number refers directly to determinate amounts (i.e., the answer to the question ‘How many?’) of a multitude of determinate objects. Moreover, the latter have the status of generically undetermined—which is to say, physically and “metaphysically” empty and therefore neutral—units or ones (these two concepts being equivalent). On Husserl’s view, the authentic number concepts are manifestly not “abstracta,” since each one involves the “universal form appertaining to the multitude at hand” (82), that is, one and one; one, one, and one, etc. All of this, on the other hand, is in the sharpest possible contrast with the signitively symbolic number concepts, which refer to neither a determinate multitude of units or ones nor to the universal form of their amount. Rather, signitively symbolic numbers, or, more properly, signitively symbolic number signs, indirectly determine “number” through the calculational rules—in the manner of the “rules of the game”—for their combination and transformation. |
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## Chapter Fifteen. Logistic and Arithmetic in Neoplatonic Mathematics and in Plato |
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For Klein “Neoplatonic mathematics is governed by a fundamental distinction which is, indeed, inherent in Greek science in general, but is here most strongly formulated” (23/10). This distinction is between “that which is in no way subject to change, or to becoming and passing away” and that which is subject to change. Thus, one branch of mathematics “contemplates that which is always such as it is and which alone is capable of being known: for that which is known in the act of knowing, being a communicable and teachable possession, must be something which is once and for all fixed.” Klein notes that “whatever pertains to the questions: How large? How many?” belongs to “a certain territory” within the realm of being that has this character of being, and thus is something that can be known. Consequently, “Insofar as the objects of mathematics fulfill the conditions set by the Greeks for objects of knowledge, they are not objects of the senses (ασθητ)” (23–24/10), which “are subject to change, or to becoming and passing away” (23/10), “but only objects of thought (νοητ).” These “mathematical νοητ fall into two classes,” namely: “(1) the ‘continuous’ magnitudes—lines, areas, solids; (2) the ‘discrete’ amounts—two, three, four, etc.” As a result, two parts of the noetic branch of mathematics are distinguished, which correspond to these two classes: geometry and arithmetic. |
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## Chapter Sixteen. Theoretical Logistic and the Problem of Fractions |
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Thus far Klein’s desedimentation of the Neoplatonic background of Diophantus’s Klein also contends that the Neoplatonic texts lack clarity on the precise nature of the underlying material (λη) of ριθμο, insofar as the changing status of this material is held to be responsible for the distinction between 1) the treatment of definite amounts both in themselves |
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## Chapter Seventeen. The Concept of ’Aριθμς |
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Klein’s desedimentation of the presuppositions of the Neoplatonic mathematical background of Diophantus’s |

Details

- Title
- The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein
- Authors
- Burt C. Hopkins, Burt C Hopkins
- Isbn
- 9780253356710
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- Indiana University Press
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- 41.99
- Street date
- September 07, 2011

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