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Chapter Seventeen. The Concept of ’Aριθμς

Burt C. Hopkins Indiana University Press ePub

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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Chapter Twenty-nine. Husserl’s Critique of Symbolic Calculation in his Schröder Review

Burt C. Hopkins Indiana University Press ePub

Husserl’s discussion of symbolic calculation in the Schröder review is guided by his critique of the author’s presentation of an algebraic calculus as logic, and in fact as an “ ‘exact’ logic” (Schröder, 20/68). Nevertheless, it is clear that Husserl considers his critique of Schröder’s mistaken, and indeed self-deceptive61 approach to the relationship between logical thinking and calculational technique to be applicable, mutatis mutandis, to the relationship between mathematical thinking and calculational technique in mathematics. Husserl buttresses his main critical point here as follows: “The logical calculus is, thus, a calculus of pure deduction; but it is not its logic. In it we have its logic as little as the arithmetica universalis, which spans the whole domain of numbers, is a logic of that domain” (8/57). When considering the cognitive value of the logical calculus itself, Husserl argues that the failure of its practitioners to properly understand its scope and limits is no more reason to reject it than there is reason to reject universal arithmetic, for “the most gifted of its [universal arithmetic’s] representatives are, and always have been, far removed from a deeper grasp of its fundamental principles” (22/70). Indeed, he then points to a parallel between logical and mathematical calculation:

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Chapter Nineteen. Klein’s Reactivation of Plato’s Theory of ’Aριθμο Eδητικο

Burt C. Hopkins Indiana University Press ePub

Klein’s interpretation of Plato’s dialectical method is guided by the question he has shown was raised, but not answered, by Greek mathematical thought. It is guided therefore by the question of the mode of being proper to the mathematical objects that such thought, in its theoretical guise, cannot help but encounter, a question that it also cannot help but be unable to answer—“for all time” (83/85)—so long as it remains strictly mathematical. The question here, which concerns both the mode of being proper to the “pure” ριθμο as well as to their εδη, is what guides Klein’s desedimentation and reactivation of Plato’s thought of “[t]he Platonic theory of the ριθμο εδητικο [eidetic definite amounts]” (88/91), a theory that “is known to us . . . only from the Aristotelian polemic against it (cf., above all, Metaphysics M 6–8).” Klein writes:

Only the ριθμο εδητικο make something at all like “definite amount” possible in this our world. They provide the foundation for all counting and reckoning, first in virtue of their particular nature which is responsible for the differences of genus and species in things so that they may be comprehended under a definite amount, and, beyond this, by being responsible for the unlimited variety of things, which comes about through a “distorted” imitation of ontological methexis [participation].69 (GMTOA, 89/92–93)

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Chapter Thirty-two. Husserl’s Attempt in the Logical Investigations to Establish a Relationship between “Mere” Thought and the “In Itself” of Pure Logical Validity by Appealing to Concrete, Universal, and Formalizing Modes of Abstraction and Categorial In

Burt C. Hopkins Indiana University Press ePub

Husserl’s distinction in the Logical Investigations between ‘significance categories’ and ‘formal objective categories’ addresses the relation between thinking and the pure “in itself” of ideality that emerges as a result of his break with psychologism. This distinction represents Husserl’s initial attempt to account for how it is that the “in itself” of ideality enters into a relationship with thinking by distinguishing between the “mere” thinking of such ideality and the state of affairs wherein the “in itself” of ideality is intuitively rendered present as a logically pure objectivity.

Mere thinking, which occurs “in symbolic acts” (LI, 566/694), acts whose symbolic quality Husserl maintains is the “significational essence of expressive acts,” manifests what he calls an “intentional relation” (381/555). As intentional, these acts are “directed toward” something either sensibly or categorially objective in a manner that has “a priori precedence over empirical, psychological facticity.” This precedence for Husserl is exhibited by what he calls the “unity of the descriptive genus ‘intention’ (‘act-character’),” where ‘descriptive’ points to the heart of his conception of ‘pure phenomenology’ in the Investigations. Husserl explains that pure phenomenology, in contrast to empirical, psychological facticity, does not apprehend the “contingency, temporality and transience of our [psychic] acts” (175/181), but rather grasps, “in a purely descriptive understanding” (382/55), the “essential determination of ‘psychical phenomena’ or ‘acts.’ ” Descriptive understanding occurs in what Husserl calls “ideation,”73 which is a methodical procedure that compares examples of the immediate experience of something (in the case at hand, symbolic acts of thinking) in order to apprehend “the pure, phenomenologically generic idea” of these experiences as such. Such apprehension prescinds from the empirical content of the experiential examples in a manner that allows it to grasp that which, subsequent to this prescission, is inseparable from the compared experiences as such.

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Chapter Twenty-three. Klein’s Account of the Concept of Number and the Number Concepts in Stevin, Descartes, and Wallis

Burt C. Hopkins Indiana University Press ePub

In contrast to Vieta, whose conservatism regarding the tradition, as we have seen, bound him to the traditional ριθμς-concept (i.e., a determinate amount of monads), even as his logistice speciosa invented the modern concept of number (i.e., the concept of ‘amount in general’), Klein writes that Simon Stevin (1548–1620) “decidedly prefers novel approaches and unusual theses” (195/186). Stevin thus “lightly pushes aside the science of the schools and has little respect even for the authority of the Greeks” (198/190). Indeed, regarding the latter, “he always draws invidious comparisons between Arabic and Greek science” (199/190) on the basis of “the Arabic ciphers and positional system,” which “appears to him to be immeasurably superior to the Greek notation.” Notwithstanding his harsh judgment of the Greeks, however, Stevin nevertheless “is possessed by the idea of a ‘renewal’ ” (196/186) of a “ ‘wise age,’ the ‘siècle sage,’ which once existed and which must be brought back” (196/186–87). Klein quotes at length Stevin’s account of this age:

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