“This much needed book should go a long way both toward correcting the under-appreciation of Jacob Klein’s brilliant work on the nature and historical origin of modern symbolic mathematics, and toward eliciting due attention to the significance of that work for our interpretation of the modern scientific view of the world”
“Burt Hopkins, already our leading scholar on Klein and his relationship to Husserl and phenomenology”
“the central thesis of Klein’s Greek Mathematical Thought and the Origin of Algebra: that modern mathematics is based upon a symbolic reinterpretation of the received Greek understanding of number (arithmos)”
“His discussion has the added virtue of placing Klein’s achievement in the context of the philosophical task set forth in Husserl’s later phenomenology: the “desedimentation,” as it were, of the sense-history of symbolic mathematics (and of modern mathematical science as such insofar as the latter is determined by a symbolic-mathematical representation of the world.)”
“Klein himself affirmed the interpretive significance for his own work of the historical-phenomenological task set forth by Husserl in later writings such as The Origin of Geometry and Crisis of the European Sciences”
“on Hopkins’ view, Klein successfully carried out the phenomenological task defined by, but not satisfactorily completed by, Husserl himself”
“Husserl’s expressed concern over the loss of an “original intuition” to ground symbolic mathematical science, and the consequent breakdown of meaning in that science”
“the history of this breakdown consists of two stages. First is the geometrical idealization of the world via what he terms “Galilean science” (taken as a kind of collective noun). Second is the formalization of that science by means of symbolic algebra, which latter surreptitiously substitutes symbolic mathematical abstractions for the directly intuited realities of the real world (“life-world”)”
“we might consider the concept “force” used in contemporary physics textbooks. Typically, force is defined in such texts in an experientially direct or intuitive manner, as a “push or a pull.” That conception of force soon gives way, however, to the algebraic formula F=ma, in which F (“force”) is understood as the algebraic product of m (mass) and a(acceleration)”
“Husserl’s aim is not to debar science from progressing beyond concepts of direct experience, but rather to demand that symbolic meaning formations laying claim to physical intelligibility account for themselves in terms of an intelligible sense-genesis in intuition or direct experience”
“The result is that empirical meaning finds itself replaced by mere empirical correlation, as experimental predictions are derived from mathematical formulas themselves devoid of intelligibility”
“while Klein endorses both Husserl’s analysis of the breakdown of meaning, and the latter’s historical-phenomenological method of desedimenting the modern mathematization of nature, he departs from the content of Husserl’s analysis”
“Klein characterizes Husserl’s attempt (sketched principally in “The Origin of Geometry” and Crisis and not itself based on actual historical research) as an “amazing piece of historical ‘empathy’” that nevertheless falls short of an adequate rendering of the meaning-intentional structure of modern symbolic mathematical physics”
“Hopkins’ thesis is that for Husserl, but not for Klein, the symbolic formations of modern mathematical physics nevertheless have an intuitive origin in the ontology of individual objects, and are therefore in principle subject to being “cashed in” in the intuition of the life-world”
“By contrast, Klein’s desedimentation, concentrates on the transformation of the Greek arithmosconcept and, on Hopkins’ account at least, involves the rejection of Husserl’s assumption that desedimenting symbolic meaning-formations is tantamount to cashing them in in the intuition of the life-world. That is because for Klein, the modern symbolic number concept bears within itself, in sedimented form, unclarified ontological assumptions of its origins”
“In Husserl’s case, Hopkins focuses on the treatment of number in his Philosophy of Arithmetic (1891, hereafter PA)”
“Husserl’s account of number in PA distinguishes between “authentic” and “symbolic” number concepts, both of which are regarded by Husserl, at least initially, as referring ultimately to the same object, which Husserl calls a “multiplicity.””
“Authentic numbers (cardinal numbers 2, 3, 4, etc.) are species of the concept of multiplicity; they are “authentic” because, according to Husserl, they refer directly to something directly intuitable, i.e., the countable as such”
“number has its origin in an act of abstraction by which we address the object as simply an “anything at all” and unify some particular assemblage of such anythings in what Husserl terms a “collective combination.””
“In the realm of symbolic numbers, by contrast, a sign substitutes for the actual concept of a multiplicity. Husserl starts out by attempting to interpret symbolic number concepts as “inauthentic,” in the sense that the latter still intend the same object as the former even if we cannot “take it in” intuitively (large numbers, for instance)”
“His final verdict on symbolic number concepts is that they are not genuine concepts at all (even “inauthentic” ones), but rather substitutes for concepts, constituted by calculational rules (“rules of the game,” as Hopkins puts it) that take the place of thought”
“The remainder of Part Three is devoted to Klein’s interpretation of the advent of the modern symbolic conception of number. I cannot begin to do justice to either the startling originality of Klein’s analysis nor to the depth and comprehensiveness of Hopkins’ interpretation”
“Klein launches Greek Mathematical Thought with an analysis of the distinction between arithmetic and “logistic” in Greek mathematics”
“The touchstone of his account is that for Greek mathematics in general, a number (arithmos) is always a countable collection of units of a definite kind, and never itself a concept or generalized object. Thus, the hallmark of an arithmos is its complete definiteness – a number is “just so many” of “just this kind” of units (apples, oranges, fruit, or in the case of arithmetical science, pure units accessible only to thought)”
"”Logistic,” or the art of calculation, on the other hand, makes demands on the concept of number that strain its ontological basis in collections of indivisible units or “pure monads.””
“For Aristotle, the “separability” of number from matter arises rather from an act of abstraction by which we disregard everything in countable units except for their being countable. For Aristotle, then, the being of the object of the concept a number is still a definite collection of definite things, but these things (monads) are now the much stripped-down units resulting from our disregard of everything but their countability”
“as he famously puts it in Metaphysics VI, while physics or natural science deals with things that exist inseparable from matter, and metaphysics or first philosophy deals with things that exist separate from matter, mathematics deals with things that cannot exist separate from matter, but regards them qua separate from matter”
“for Aristotle, the indivisibility of the unit of counting derives from its having been taken as the measure of counting, and the unity of a number itself (that is, its collective being as an assemblage of units) derives from its having been counted”
“Klein’s reading of Diophantus’ Arithmetic prepares us for his subsequent interpretation of Vieta’s symbolic algebra as in origin a reinterpretation of Diophantine algebra, but on ontological assumptions foreign to Diophantus himself and Greek mathematics in general”
“the Diophantine notion of the number “species” (eidos). The Diophantine “species” is a class or general category of numbers whose determinate amount (three, four, and so forth) is yet to be determined (or has been left undetermined)”
“the species, represented symbolically by Diophantus (unknown, known, etc.) always signifies, in accord with the general Greek mathematical conception, a determinate amount of units of a determinate kind”
“for Diophantus the algebraic variable or sign designates a general concept, but that general concept does not intend a general object”
“It is on the basis of the Diophantine conception of “species,” but on radically different ontological assumptions, that Vieta will coin (in his Analytical Art of 1591) the modern symbolic concept of number”
“Hopkins proceeds next to Klein’s desedimentation of Vieta’s reinterpretation of the Diophantine procedure (along with subsequent innovations by Stevin, Descartes, and Wallis). This is Klein’s real tour de force in Greek Mathematical Thought. The gist of it is that Vieta transforms the received (Greek) concept of arithmos, determined above all by a direct relation to its object (numbers themselves as collections of countable things in the world), into a symbolic conception the proximate object of which is “number in general,” and whose relation to countable things in the world is therefore indirect”
“the arithmetical intelligibility of symbolic numbers remains parasitic upon the received intelligibility of arithmos as a countable collection. This is because symbolic numbers, which are in the final analysis a certain kind of reified abstraction, are nevertheless subjected to arithmetical calculation as if they were collections of countable units”
“Herein lies the fundamental tension (or incoherence, if you like) uncovered by Klein’s desedimentation: modern symbolic mathematics, the intelligibility of which derives from the received Greek concept of arithmos (as a collection of countable units), nevertheless violates that very intelligibility by setting up as its proximate object a general concept of number”
“At the hands of Descartes, finally, in a fateful development for all subsequent science, geometry itself, the traditional science of figures in space, is reinterpreted as a symbolic representation of generalized quantitative relationships or algebraic “equations” (think “graphing an equation” in your high school course on analytical geometry)”
“Hopkins’ key interpretive insight here, in my view, is his careful unpacking of, and emphasis on, Klein’s uncovering the decisive role of the written sign in the constitution of the being of symbolic number. It is not merely that in Vieta the thing signified by algebraic signs is a generalized object (unlike the object signified by Diophantine algebraic signs), but that the written sign enters in such a way as to render the modern symbolic conception of number self-referential in a very specific sense. As Hopkins puts it, “Vieta’s letter as a sign – in the novel sense of a symbol – of an indeterminate magnitude does not signify anything but itself” (286, n.147). Modern symbolic number, then, is born when Vieta subjects his letter signs to syntactic rules, which symbolically constitute their own object as an object at all, while simultaneously maintaining the object’s numerical character as an “amount” to be calculated upon”
“1. Husserl fails to account for the meaning-structure of symbolic numbers because he assumes that the sense-genesis of symbolic numbers necessarily has its origin in authentic numbers, with the consequence that symbolic numbers at least in principle are susceptible to being “cashed in” intuitively”
“2. Klein succeeds in uncovering the meaning-structure of modern symbolic number and in doing so completes the project defined in Husserl’s Crisis”
“Klein specifically shows that desedimentation does not necessarily yield a “cashing in” or redemption of symbolic meaning formations in intuition, since the object of the symbolic number concept is symbolic number itself”
“I think we must consider more carefully what it would mean to “cash in” the equations of modern mathematical physics intuitively. As we noted, such equations prescribe numerical operations that can be performed solely on symbolic-dimensionless numbers, thereby violating the very intelligibility of physical quantities, which are inherently dimensional. This, of course, raises the question of why algebraic physics works”