Abstract
This paper explores some of the constructive dimensions and specifics of human theoretic cognition, combining perspectives from (Husserlian) genetic phenomenology and distributed cognition approaches. I further consult recent psychological research concerning spatial and numerical cognition. The focus is on the nexus between the theoretic development of abstract, idealized geometrical and mathematical notions of space and the development and effective use of environmental cognitive support systems. In my discussion, I show that the evolution of the theoretic cognition of space apparently follows two opposing, but in truth, intrinsically aligned trajectories. On the epistemic plane, which is the main focus of Husserl’s genetic phenomenological investigations, theoretic conceptions of space are progressively constituted by way of an idealizing emancipation of spatial cognition from the concrete, embodied intentionality underlying the human organism’s perception of space. As a result of this emancipation, it ultimately becomes possible for the human mind to theoretically conceive of and posit space as an ideal entity that is universally geometrical and mathematical. At the same time, by synthesizing a range of literature on spatial and mathematical cognition, I illustrate that for the theoretic mind to undertake precisely this emancipating process successfully, and further, for an ideal and objective notion of geometrical and mathematical space to first of all become fully scientifically operative, the cognitive support provided by a range of specific symbolic technologies is central. These include lettered diagrams, notation systems, and more generally, the technique of formalization and require for their functioning various cognitively efficacious types of embodiment. Ultimately, this paper endeavors to understand the specific symbolic-technological dimensions that have been instrumental to major shifts in the development of idealized, scientific conceptions of space. The epistemic characteristics of these shifts have been previously discussed in genetic phenomenology, but without devoting sufficient attention to the constructive role of symbolic technologies. At the same time, this paper identifies some of the irreducible phenomenological and epistemic dimensions that characterize the functioning of the historically situated, embodied and distributed theoretic mind.
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Notes
More recent research suggests that some animals, great apes for instance, are also capable of higher-order forms of symbolic cognition and communication (Savage-Rumbaugh and Lewin 1994). However, in the case of (nonhuman) primate symbolic cognition and communication, certain limitations still apply, for instance when it comes to acquiring and using a comprehensive vocabulary.
The sequence of the evolutionary stages that Donald identifies here, and that of their distinctive symbolic means, is roughly mirroring that of the decisive stages that can be observed on the level of ontogenetic development see my discussion in the “Symbolic technologies as cognitive tools: from gesture to notation” section.
In many behavioral contexts, the animal organism cognitively structures its spatial environment by way of direct reference to the place of its own, perceiving and moving body. Take the case of the human organism’s embodied cognition of space, which is generally characterized by an ego-centric frame of reference. In such a frame of reference, the perceiving individual’s own living body, the cognitively indispensable “medium of all perception” (Husserl 1989, p. 61), functions, as both phenomenologists and psychologists have traditionally stressed, as a “relational center of all spatial orientations” (Husserl 1997, p. 109; see also Piaget and Inhelder 1956; Ströker 1987). This is due to the fact that this body, in the first-person perception of space, constitutes an “ever-abiding point of reference” that gives rise to the appearance of spatial directions such as left and right, front and back, and above and below (Husserl 1997, p. 66). At the same time, more recent psychological and linguistic research has shown that in various developmental, biological and cultural contexts, (some of) the structures governing the organism's cognition of space, and the organism's spatial orientation in particular, have their basis in allocentric frames of reference (see, e.g., Gentner 2007; Haun et al. 2006; Levinson 2003; Sinha and Jensen de López 2000; see also Acredolo 1978). In this case, a salient and stable feature of the environment, existing independent of the organism’s own living body, may serve as the dominant source of reference for orientation in space. Allocentric frames of spatial reference, which can be observed already among human infants and also among non-human primates (Acredolo 1978; Gentner 2007; Haun et al. 2006), nevertheless continue to rely indirectly on the organism’s own, living body. This is because in an organism’s active spatial comportment, an allocentric frame of reference becomes practically useful for orientational purposes only if the absolute referent it also effectively related to place of the individual organism’s own living and moving body.
For a brief discussion of the history, benefits, and limitations of measures that were directly modeled on the human body see Kula (1986, chapter 5).
This is despite the impressive mathematical and geometrical knowledge that was accumulated and applied in the institutions of both these states. In Egypt, it has been claimed, the domain of geometrical and mathematical techniques generally remained limited to the status of “merely applied arithmetic”; i.e., bound to practical applications that were relevant for the smooth functioning of the socio-economic apparatus (van der Waerden 1961, p. 31; but see Gillings 1972; Joseph 2011, chapter 3). Besides the aforementioned surveying and calculation of areas of farmland, elaborate geometrical and mathematical methods were, for example, used for structural and area calculations in the domain of architectural construction, as well as for a range of calculations necessary for taxation purposes (Kline 1972, p. 21). Similarly, it is commonly claimed that Babylonian geometry generally remained of an applied nature. The only notable exception is that in the context of the emergence of a more or less systematized form of astronomy in Babylonia, knowledge about the calculation of the radius and the area of circles existed that apparently did not serve any immediate practical purposes (Kline 1972, pp. 10–11).
For instance, in his Timaeus Plato (1997) argues that the divine creator of the cosmos used ideal geometrical forms (lines, surfaces, and triangles) as the first grounds and principles to order the pre-existent world of chaos. In the Republic, the general priority Plato gives to the realm of ideas and ideal forms explicitly includes also that of number; both calculation and geometry are regarded as useful and spiritual exercises in so far as they concern the “numbers themselves, never permitting anyone to propose for discussion numbers attached to visible or tangible bodies” (1997, p. 1142).
This is, of course, not to say that there were no further important innovations and departures made in geometry and mathematical conceptions of space alike. Of significance in this regard are, e.g., the development of a sophisticated relational conception of space by Leibniz (which was later elaborated upon by Einstein), the formulation of Non-Euclidean geometries by Riemann and others, and the development of modern mathematical topological notions of space.
Interestingly, in his writings Husserl himself occasionally comes to question, at least tentatively, the notion that the theoretic construction of ideal–objective geometrical and mathematical formations is exclusively a product of “pure” thinking. For instance, in his famous essay on the origins of geometrical thinking, Husserl notes that the constitution of theoretic formations of ideal objectivity, such as they can be found in their purest form in geometry and mathematics, as well as the “mental manipulation” of mathematical significations (Husserl 1970, p. 27), can only be accomplished by way of these formations’ symbolic embodiment (Husserl 1970, p. 358). Furthermore, such symbolic embodiment, Husserl notes, particularly if it involves a durable, material medium such as writing, is essential as it allows for the retaining of sense; including that of the original ideal constructs of human geometrical and mathematical thinking (Husserl 1970, p. 366). Despite these intriguing insights, however, Husserl ultimately falls short to give proper due to the constructive role that symbolic technologies play in the constitution of theoretic knowledge and that of a rational-scientific type of intentionality. This applies mainly in two respects. First, Husserl never came to fully recognize the extent to which symbolic technologies such as writing may fulfil a both creative and constructive cognitive function. Specifically, Husserl never fully admits that these technologies, and their material dimensions, are instrumental, indeed decisive, in bringing new domains of theoretic thinking into being. Second, in his discussions Husserl does not distinguish between different types of symbolic technologies (e.g., representations of geometrical figures, diagrams, different types of notations, etc.) and specify their respective contribution to the theoretic achievement of the constitution of ideal objectivity.
See for a related discussion of these and other external media assisting numerical thinking, e.g., De Cruz (2008).
Notably, there also exists a considerable range of historical material documenting a cultural practice or art that involved the use of the fingers to assist in processes of counting and ultimately also calculation and computation (see for an instructive historical overview Menninger 1969, pp. 201–220). The existence of sophisticated finger-based counting and manual computation practices is particularly well documented for Antiquity (see the discussion by Williams and Williams 1995) as well as for Medieval times (Kusukawa 2001). Apparently, such manual techniques were still widely used relatively recently, e.g., among some traders in the Middle East (Menninger 1969, p. 201). Some of the historical examples of techniques of finger counting and manual computation were astonishingly developed; the Romans, for instance, were able to represent, and distinguish, numbers from 1 to 10.000 just by using the fingers (digiti) of both of their hands (Menninger 1969, p. 201).
For a detailed and still unrivaled, comprehensive historical overview of mathematical notations systems, see the classic work by Cajori (1952).
This is because sophisticated mathematical notations allow the human mind to quickly infer the exact magnitude of large sums from their corresponding numerical representation. For instance, the numerically literate individual is capable of almost immediately recognizing the exact value that the numeral 543 represents. This is not possible with more ‘primitive’ notational means—it would take considerable time, for instance, to count 543 tally marks and thus grasp the exact number represented.
For instance, cross-cultural studies have shown the directionality of the prevailing writing systems shapes the orientational structure of the mathematical mind’s internal, spatial representations (Zebian 2005; also Dehaene 1997, pp. 81–83). Similarly, the repeated use of the abacus as a cognitive tool is capable of significantly informing the structure and cognitive function of one’s own, mental representations (Hatano et al. 1977; Stigler 1984).
In may thus not come as a surprise to learn that the Romans, similarly to the Greeks who for the most part employed alphabetic letters as mathematical notations, commonly did not yet use their numerical representations as a means and medium for calculation. Instead, they resorted to the abacus to assist their numerical thinking, and thus to a systematically spatially organized cognitive tool, the functioning of which does not involve notations (see on this point Cajori 1952, vol. 1, chapter 2). In relation to the cognitive efficacy of the Hindu–Arabic notation system, it is likewise of importance to note that this system, in contrast to the Roman system, for instance, and also in contrast to other historical examples of notation systems that already used place-value notation such as the Babylonian, has a symbol notation for zero. The notational symbol zero came to fill the void in the positional place-value number system, where the void indicates an absent unit in a number comprising several digits. This in turn made it possible to unambiguously represent magnitudes (Danzig 1954, pp. 30–31). In the numeral 3,003, for instance, the first digit 3 unambiguously means 3 × 1,000. By comparison, the traditional practice of leaving spaces blank to indicate the absent units (as in 3 3) is a likely source of confusion. Dantzig has argued that in addition to this representational efficacy, the invention of a symbol for “an empty class, a symbol for nothing”, ultimately also made it possible for the theoretic mind to conceive of this symbolically denoted void as a number (Danzig 1954, p. 31). This theoretic realization, he claims, in turn opened up the way for new, formerly unthinkable ways of calculation, and ultimately, for modern arithmetic thinking (see for a comprehensive argumentation Danzig 1954).
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This research was completed with the support of the Australian Research Council’s Discovery Projects funding scheme (project number DP110102466).
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Woelert, P. Idealization and external symbolic storage: the epistemic and technical dimensions of theoretic cognition. Phenom Cogn Sci 11, 335–366 (2012). https://doi.org/10.1007/s11097-011-9245-8
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DOI: https://doi.org/10.1007/s11097-011-9245-8