[Xmca-l] Re: Objectivity of mathematics

[Martin John Packer]

Let me copy it here:
On Nov 9, 2014, at 10:14 AM, Martin John Packer <mpacker@uniandes.edu.co> wrote:

> In the paper I attached, I also took a shot at a reconstruction of the history of math. 

Classical Mathematics
               The Greeks considered mathematical entities—numbers, shapes, solids—to exist prior to their knowledge of them, in the famous ‘Platonic’ realm. Math was the way of finding these preexisting, underlying forms. The geometry that Euclid (c. 300 BC) presented in his Elements operated within Plato’s ontology of perfect, immaterial, singular forms (such as Triangle), imperfect, multiple, sensible objects, and “mathematicals”—the mathematician’s inscribed figures, such as ABC and XYZ, which were both immaterial and plural (Lachterman, p. 118). Every constructed square, for example, was considered at the same time a unique individual, different from every other square, and yet also representative of the form of squareness, the perfect Square, and so identical with every other square.
               In this classical ethos and ontology, student and teacher were not discovering or enacting something new when they constructed a figure, or divided two numbers. They were rediscovering what had been there all along, “reenacting in time what has already been done all along and thus never for the very first time” (p. 121). Mathematical activity was seen as evoking the perfect geometric and arithmetic forms which were the genuine objects of study (p. 120-121).
 
Modern Mathematics
               The radical moderns, of whom Descartes (1596-1650) was preeminent, viewed math, and knowledge in general, very differently. The 17th century is commonly seen as a time of revolution in mathematics, with the invention of analytic geometry, differential and integral calculus, decimal fractions, and more. Lachterman sees in this revolution a change in ethos and ontology. “Construction” became a central concept in modern mathematics, a concept that was at core ontological. Mathematical entities were now seen as having no existence prior to their human construction. Cartesian math was something active and creative, an example of the mind’s essential power of making. Descartes’ approach to geometry illustrates this: he introduced in La Géométrie (1637) a coordinate system – his famous X & Y coordinates – whereby geometrical problems could be expressed as equations involving variables, known quantities, and unknowns. The solution of the problem (finding a locus, for example) amounted to solving – finding the ‘roots’ of – a polynomial equation. Problems of a specific degree of complexity (number of lines, dimensionality of figures) corresponded to equations of a particular degree. Hence there was an order, a seriality, to this method. It was very general, and powerfully iterative – a solution at one degree of complexity provided the basis for a solution at the next degree. In a real sense this approach turned every geometrical problem into the single problem of finding an equation’s roots. The successes of this form of math fed its confident extension widely through accounts of human thinking and learning.
               This modern, Cartesian math of formulae and constructed proofs differed in both ethos and ontology from the classical Greek math of demonstration and proof of theorems. Whereas the Greeks considered mind the mirror of external things and math a way of finding entities such as numbers that already existed, the Cartesians considered mind active and creative, in its essence a power of making. Math was the outward embodiment of this creative power, made visible in the construction of problems (e.g., positing axes, etc.) and of their solutions (drawing up equations). Modern mathematics dissolved the classical distinction between arithmetic and geometry, and in doing so dissolved the perfect Forms that the Greeks had emphasized. Integers became merely a particular type of rational number; square and cube became merely particular cases of  constructions in multi-dimensional space. Infinity and infinitesimals were manipulated with ease.
 
Problems with Modern Mathematics
               This modern mathematical ethos, though born in the 17th century, is alive today. (Its influence on the constructivism of Piaget will be apparent; cf. Rotman, 1977.) But it is not without its problems, as well as its successes. The conception of mathematical construction straddled an ontological divide – commonly associated with Descartes – between the subjective (intentionality, the conceptual) and the objective (the referent, the sensible). The concept of construction came to the fore in large part, Lachterman argues, because it promised to be the mediator between reason and reality, mind and mindless nature. As Lachterman puts it, Descartes “most conspicuously exploited the power inherent in the view that symbolization frees us to work ingeniously beyond the boundaries apparently fixed by nature as it is sensuously, premethodically given... while at the same time serving to direct those mechanical operations or movements from which outwardly manifest configurations artfully issue” (p. 125). Construction seemed to bridge or dissolve the gap between subjective and objective; it seemed to solve the problem of the relationship of reason to the ‘real’ world, of the ‘true’. The “secret” of modernity, what it struggled to achieve, in Lachterman’s view, was the “willed coincidence” of human making with truth or intelligibility.
               But the promise couldn’t be made good: increasingly this ethos came to deny the reality of the ‘external’ world. To put this another way, Cartesian mathematics, skeptical of perception as a source of genuine knowledge (e.g., Descartes’ criticism of Locke), seemed to offer a powerful illustration of the power of ‘reason’ (individual mental capacity) to construct truths about the world. But a tension developed. On the one hand, true knowledge could not be based directly on the natural world, for this, in the Cartesian ontology, was a realm of contingencies, while true knowledge was unconditional. On the other hand, reason in general, and mathematics in particular, still strove to be about what is ‘real.’ The criterion of adequacy to reality seemed unavoidable, yet indeterminate.
               What had begun as a relatively innocent effort to bring order to the natural world, to “master and possess” it (p. 23), ended with the denial that reality had any existence outside the human mind, a denial that dissolved into relativism and “self-divinization.” For example, when non-Euclidean geometries were developed in the nineteenth century, both these and classical geometry were viewed as abstract systems resting on conventional and arbitrary axioms, rather than either logically necessary or natural postulates. The “reality” of the axioms was deemed irrelevant, and these geometries were considered to offer conjectural models of physical space (e.g., Einstein’s use of Riemannian geometry), not as descriptions of how space really is.

Postmodern Mathematics
               These problems within modern mathematics have motivated several attempts at a new conceptualization of mathematical investigation. Lachterman, for example, can tell his history of ancient and modern math only because he adopts a third position, which one can call postmodern. From this vantage point the problems of modern mathematics become visible.
               Rotman (1993), too, seeks to deconstruct the view that math is a purely formal enterprise. In his account, the formal procedures of mathematics – operating on apparently decontextual and abstract entities – are not self-sufficient, but are sustained by the informal practices of a community of mathematicians. “Mathematics is neither a self-contained linguistic formalism nor an abstract game played entirely within the orbit of its own self-referring rules, conventions, and symbolic protocols” (1993, p. 24). Rather, there is a relationship between mathematical formalism and everyday language which “allows an embodied subject – the corporeal, situated speaker of natural language – to register a presence in and connection to the world of real time, space, and physical process” (1993, p. 25).1 We will return to the character of this “presence” later. Rotman is clear on the need “to demolish the widely held metaphysical belief that mathematical signs point to, refer to, or invoke some world, some supposedly objective eternal domain, other than that of their own human, that is time bound, changeable, subjective and finite, making” (Rotman, 1987, p. 107). Math does refer to a reality, but this is a human reality, itself constructed, neither timeless ideal Forms nor lifeless matter. Mathematical signs circulate within, and help reproduce, a specific sociocultural human context.
               Rotman sees modern mathematics as motivated by an “illusion of mastery” that is literally fantastic. For example, the assumption of infinite iterability that one finds in integral calculus and series expansions presumes a counterfactual capacity: such iteration would take infinite time and infinite energy. In its place Rotman proposes a “non-Euclidean arithmetic” in which the iteration that underlies number and counting is not unbounded and infinite, but closed and finite. Such an arithmetic, which Rotman works out in some detail, is only “locally Euclidian.”