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

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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.

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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.

 

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,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.”

 

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” (PHS, 65) in order to argue that there is an “essential connection, as Husserl understood it,” between “the approach to the ‘true beginnings’ ” formulated in his earlier writings and his adumbration of “the aims which should control research in the history of science” in his last works. Klein locates this essential connection precisely in Husserl’s concern, from beginning to end, with the constitutive problems of origin (true beginnings). According to Klein, Husserl’s phenomenological preoccupation with the “ιζματα πντων, ‘roots’ of all things” (69), traced a continuous path from his early rejection of historicism as a means of accounting for the origin of logical, mathematical, and scientific propositions, to his late formulation of “the historicity (as the ‘historical apriori’) which makes intelligible not only the eternity or supertemporality of the ideal significant formations but the possibility of actual19 history within natural time as well, at least of the historical development and tradition of a science” (74–75). Thus, in marked contrast to later commentators who see in Husserl’s Crisis and “The Origin of Geometry” “the conflict between transcendental philosophy and historicism,”20 Klein aims to show that in these works “Husserl actually confronted the two greatest powers of modern life, mathematical physics and history, and pushed through to their common ‘root’ ” (74).

 

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 Logical Investigations showed irrefutably that logical, mathematical, and scientific propositions could never be fundamentally and necessarily determined by this sort of explanation. (65–66)

 

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).31 The first limit concerns the “general substratum of consciousness” that the “continuous modification of the retentional consciousness approaches and beyond which the ‘prominence’ of the object flows away” (73). The second limit concerns the “ ‘past history’ of the original ‘presentation’ of the object.” Both limits point to sedimented meanings that can be “awakened” (FTL, 280) such that the “intentional genesis” of the meaning in question is “reproduced” as the “history” of its constitution, a history that, “of course, did not take place within ‘natural time’ ” (PHS, 72). For Klein, then, two “histories” are initially at issue in Husserl’s phenomenological account of the intrinsic possibility of an object’s intentional unity. The first history concerns the possibility of such an object’s retaining its unity as an enduring “presence” once it has been presented to consciousness. This history concerns the object’s intentional genesis as an objective “prominence,” its persistence throughout the temporality that is the essential characteristic of the experience of that object. The second history concerns the possibility of the object’s “original presentation” to consciousness. This possibility is more fundamental than that of its persistence as a “prominence,” for what is at issue here is its presentation to consciousness prior to any modification in accordance with consciousness’s structure of internal temporality. It is important to note, however, that neither of the possibilities or “histories” at issue here concern the “natural” existence of objects and their histories. This has been precluded by the phenomenological reduction’s removal of the “index of existence” from the experience of both the object and its history. As a result, what is at issue in the priority of the original presentation of the intentional object to consciousness is decidedly not its being experienced “first” in a supposed natural succession of awareness. Rather, the priority involved here is methodological, in the sense—to be discussed in detail below—that the evidence uncovered by these analyses discloses an “indication” that points to a more original “possibility” belonging to the object than its enduring presence in experience.

 

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” (PHS, 72). This is the case because history in its usual sense is “the ‘story’ of a given ‘fact.’ ” Thus, the telling of any such story about the “fact” of the historical attitude will of necessity presuppose, rather than account for, the “historical attitude” that gives rise to the “telling of the telling” of the story of the historical origin of this attitude: “Whatever historical research might be required to solve it [i.e., the origin of history], it leads ultimately to a kind of inquiry which is beyond the scope of a historian.” Such research “may, indeed, lead back to the problem of inquiry, the problem of στορα as such, that is, to the very problem underlying Husserl’s concept of an ‘intentional history.’ ”34

 

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” (PHS, 76). For Klein, this link is established by Husserl on the basis of the following considerations: i) the “ideal ‘intentional units’ ” at issue in these significant formations are the product (das Erwirkte) of an “accomplishment” (gelingende Ausführung; Verwirklichung) that arises in their “anticipation” (Vorhabe)41—not in their “retention”; ii) “ ‘accomplishment or [sic]42 what is anticipated means evidence to the active subject: herein the product shows itself originally as itself’ ”; iii)

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” translate into terms of “reality” what actually takes place within the realm of “transcendental subjectivity”;

 

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” (PHS, 79) in his last works of providing an “ ‘intentional-historical’ analysis of the origin of mathematical physics,” an analysis that for Klein, “although not based upon actual historical research, is on the whole an amazing piece of historical ‘empathy.’ ” Thus, rather than follow Husserl and analyze the foundations of Galileo’s physics by treating “Galileo’s name [as] somewhat of a collective noun, covering a vast and complex historical situation,” Klein tries “to give a general outline of that actual historical development” of mathematical physics. In so doing, he situates his account of this development within the context of his articulation of the significance of Husserl’s late confrontation with the problem of “the relation between intentional history and actual history” (74), and thus within the context of Husserl’s analysis of the “increase of ‘sedimentation’ [that] follows closely the establishment of the new science of nature, as conceived by Galileo and Descartes” (79). By proceeding in this manner, Klein operates on the assumption that Husserl’s phenomenological analysis of the problem of “true beginnings” has “adumbrated the aims which should control research in the history of science” (65), an assumption that discloses the understanding of history here as being inseparable from philosophy itself. In addition, Klein’s outline of the actual historical development of mathematical physics also operates on the assumption of the aptness of Husserl’s characterization of the method of historical reflection in the Crisis. This is because, for Husserl, the problem of “sedimentation” emerges from out of the “unique situation” (78) that held sway for him at the time he wrote the Crisis, the situation in which both a particular science and science in general “appear almost devoid of ‘significance.’ ” This method characterizes historical reflection as involving “the ‘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 which leads to the “breakdown” of modern mathematical physics.

 

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 operative method of Klein’s mathematical investigations is captured by Husserl’s articulation of the peculiar zigzag movement characteristic of the method of historical reflection. And we suggested that it is precisely this method that permits—explicitly in Husserl’s case and implicitly in Klein’s—their historical investigations to overcome the traditional opposition between the epistemological investigation and the historical explanation of science, which is to say, to overcome the problem of historicism.

 

Chapter Nine. The Basic Problem and Method of Klein’s Mathematical Investigations

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In Greek Mathematical Thought and the Origin of Algebra, Klein situates his historical investigation of mathematics in terms of the “fact that it is impossible, and has always been impossible, to grasp the meaning of what we nowadays call physics independently of its mathematical form” (GMTOA, 19/4). He maintains that this is the case because, “[a]fter three centuries of intensive development, it has finally become impossible to separate the content of mathematical physics from its form” (18/3). This state of affairs is the result of the “intimate connection of the formal mathematical language with the content of mathematical physics,” a connection that stems from “the special kind of conceptualization which is a concomitant of modern science and which was of fundamental importance in its formation” (19/4). Klein argues that because of this connection, any discussion of the problems faced by contemporary mathematical physics must have as its necessary propaedeutic “the limited task of recovering to some degree the sources, today almost completely hidden from view, of our modern symbolic mathematics.” Thus, his study completely bypasses “the fundamental question concerning the inner relations between mathematics and physics, of ‘theory’ and ‘experiment,’ of ‘systematic’ and ‘empirical’ procedure within mathematical physics”—though, “[h]owever far afield it may run, its formulation will throughout be determined by this as its ultimate theme.”

 

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 Greek Mathematical Thought and the Origin of Algebra of Husserl’s formulation of the nature and roots of the crisis of European sciences, together with the method of historical reflection, by elaborating Husserl’s thought on these matters. We shall discuss the fragmentary nature of Husserl’s desedimentation of the origins of modern mathematical physics with a view to showing that Klein’s account of the genesis of modern algebra “desediments” precisely those aspects of Husserl’s historical analyses of the origins of modern mathematics that remain fragmentary in his Crisis.

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 (Rückbeziehung) to the “past history” belonging to the original presentation of the ideal meaning formations proper to mathematical and scientific objects proves unable to account for their possibility so long as the genesis of this history is “reactivated” in accord with immanent time’s a priori form. The original presentation of such meaning formations therefore transcends the limit of their temporal genesis. This limit is the general substratum of consciousness and it is uncovered in the a priori form of the continuous retentional modifications in which their objective prominence as meaning formations is maintained. Thus, the genesis of these meaning formations transcends the a priori temporal form of the individual stream of consciousness.

 

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 Greek Mathematical Thought and the Origin of Algebra. Rather than begin his investigation with a consideration of what he argues is the proximate origin of modern mathematics, namely, Vieta’s assimilation and transformation of Diophantus’s Arithmetic, Klein begins with a consideration of the Neoplatonic literature “which forms its [i.e., the Arithmetic’s] background.” His rationale for this is that “we must first of all see the work of Diophantus from the point of view of its own presuppositions” (20/6). Klein discloses these in terms of the categories of Neoplatonic mathematics and its classification of mathematical sciences, and shows that these “go back to the corresponding formulations in Plato.” Yet because the Neoplatonic categories and classifications “were such as to prevent the integration of the Arithmetic into this literature,” and because the corresponding formulations in Plato are not “identical with them,” Klein’s task of seeing Diophantus’s work “from the point of view of its own presuppositions” must first desediment the Neoplatonic and Platonic background of Diophantus’s Arithmetic.

 

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 Philosophy of Arithmetic seek to establish the foundation of symbolic mathematics, which he also calls ‘universal arithmetic’, on the authentic concept of cardinal number (Anzahl).10 To anticipate the results of our own investigation of Husserl’s treatment of this problem in the next chapter, it begins with the assumption of the logical equivalence, in the sense of the identity of their object, of the contents of the authentic and symbolic concepts of number: Husserl initially presents each as referring to the determinate unity of a determinate multitude of units, albeit directly in the case of the authentic concept of cardinal number and indirectly in the case of the symbolic concept of number. Husserl’s investigation seeks to show that the foundation of symbolic mathematics, and thus its logic as well, lies in the authentic concept of Anzahl.

 

Chapter Thirteen. Authentic and Symbolic Numbers in Husserl’s Philosophy of Arithmetic

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Husserl’s Philosophy of Arithmetic is an attempt to establish the foundation of arithmetic by means of a psychological account of what he refers to as “cardinal number [Anzahl]1 in the true and authentic sense of the word” (PA, 116).2 His critical self-understanding of the failure of this attempt has met with general acceptance in the literature. There is, however, no consensus regarding exactly why Husserl’s attempt failed. Gottlob Frege’s critical review of Philosophy of Arithmetic,3 in which he took Husserl to task for “the influx of psychology into logic” (Frege, 324/332)—to the mutual detriment of each—along with certain of Husserl’s remarks have lent credence to a widely held view that Husserl’s main dissatisfaction with Philosophy of Arithmetic can be traced to the work’s psychologism, that is, to its reduction of both the objects of logical concepts and the objectivity of these concepts themselves to psychological presentations. On this view, Husserl’s statement in the Foreword to the first edition of the Logical Investigations about his “doubts of principle, as to how to reconcile the objectivity of mathematics, and of all science in general, with a psychological foundation for logic,”4 along with his remark recorded by W. R. Boyce Gibson “that Frege’s criticism of the Philosophy of Arithmetic . . . hit the nail on the head,”5 are interpreted as endorsing Frege’s criticism. However, as Dallas Willard has noted, “one searches in vain for passages in [Husserl’s] earlier writings where he advocated such a psychologistic logic.”6 In addition, J. N. Mohanty has shown that Husserl did not acquire the distinction between ‘object’, ‘concept’, and ‘presentation’ from Frege.7

 

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 Philosophy of Arithmetic has shown that he was eventually forced to abandon his initial thesis of their logical equivalence. Specifically, we have demonstrated that rather than substantiate the view that the symbolic presentation of number “can [act as a] surrogate, to the furthest extent, for the corresponding authentic presentation” (PA, 194) of number, Husserl’s analyses conclude by substantiating, in effect, the opposite view. That is, they substantiate the view that the signitively symbolic numbers at issue in arithmetical calculation, which are presented by sensible number signs, do not refer to the same object as authentic numbers.

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.

 

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.

 

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 Arithmetic has disclosed in the Neoplatonic texts an attempt to maintain a systematic distinction between arithmetic, understood as the pure theory of definite amounts in themselves, and the theory of proportions, understood as the theory of the relations that definite amounts have to one another. That is, he has disclosed their attempt to maintain a theoretical distinction between the treatment of ριθμο καθ’ ατ and that of πρς λλο. Moreover, he has shown that this attempt is beset by obscurities and inconsistencies that are rooted in an arithmetical theory that is sedimented in their text, which has its source in Plato and has been passed on according to tradition. This theory aims to investigate the pure relations proper to definite amounts, an investigation that is “intended to stand beside the theory of definite amounts as such, i.e., of their different kinds” (47/39).

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 and according to their kind (κατ’ εδος) and 2) their treatment in relation to one another. Their accounts vacillate between locating this changing constituent, on the one hand, in the realm of ασθητ, and, on the other hand, in the pure λη of the multitude (πλθος) proper to the ριθμο themselves. Klein shows that this lack of clarity about the nature of the material of ριθμο is the source of the inconsistency of the Neoplatonic attempts to determine the precise status of λογιστικ. Thus, on the one hand, they want to assign λογιστικ—in accord with their systematizing inclinations—to the pure theory of proportions, while, on the other hand, as a consequence of their vacillation regarding whether the realm proper to the πρς λλο is constituted noetically or aisthetically, they sometimes opt for the latter and characterize it as the practical application of pure arithmetic.

 

Chapter Seventeen. The Concept of ’Aριθμς

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Klein’s desedimentation of the presuppositions of the Neoplatonic mathematical background of Diophantus’s Arithmetic shows that they are informed by two interrelated strata of presuppositions. Klein reactivates the first of these, which belong to the Neoplatonic stratum proper, by first articulating it and then tracing it to its roots in Plato’s philosophy of mathematics. As we have seen, the Neoplatonic stratum is characterized by the peculiarity that theoretical arithmetic does not deal directly with ριθμο but with their kinds (εδη). Likewise, logistic is characterized by the peculiarity that, while the nature of the material (λη) proper to the ριθμο with which it deals is rendered inconsistently, a dominant view nevertheless emerges that this material is sensible and that, therefore, logistic is not a science (πιστμη) but an art (τέχνη). We have also seen that Klein traces the roots of this stratum to the absence in Plato of any reference either to ριθμς or to ριθμο in the definitions of arithmetic and logistic. It is Klein’s thesis that the definitions proper to each, as having to do with the εδη of the odd and the even, point to the fact that “their formulation [in Plato] presupposes a theoretical point of view” (63/59). However, “the rigor of these definitions consists precisely in the fact that they articulate only one of the two characteristics of the ριθμς,” that is, their kinds, while they “avoid the indefiniteness which attends the term ‘ριθμς’ insofar as by itself it does not reveal the sort of definite objects it is a definite amount of, i.e., of what the definite amount is meant to be a definite amount of.” Consequently, even though these definitions presuppose a theoretical interest, they do not presuppose that the ριθμο themselves are theoretical, that is, that they are ριθμο of “pure” units. Thus, the definitions hold irrespective of whether sensible or noetic material (λη) is understood to underlie counting and calculation and therefore arithmetic and logistic. For Klein, however, owing to the fact that only sensible “units” “are amenable to the partitioning which exactitude of calculation requires” (64/60), the Neoplatonic mathematicians Olympiodorus and the Gorgias scholiast “are forced from the very beginning to regard the ‘hylic’ monads, i.e., the monads which form the λη of the definite amounts,” as sensible.

 

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