CRITIQUE-OF-AR.NET/MONOGRAPHS/WYNN/EUCLIDEAN.HTM – Version 1, June 2017

Critique of Archaeological Reason
8. Monographs: Wynn

Euclidean concepts

Giorgio Buccellati, October 2013


Euclidean concepts
Measurement
Parallel axes
Bilateral symmetry

Euclidean concepts

Definition p. 39      the development of general constructions of space, spatial notions that transcend individual objects and their relations and organize space into a framework in which objects occupy positions. A euclidean space is one of positions, not of objects. It is defined by an arbitrary framework, a set of coordinates, for example. In a sense, one empties space of objects and organizes what is left by means of a reference system that consists of all of the potential positions objects may hold.
Units of consistent spatial quantity p. 39      Such a space requires intervals, which are units of consistent spatial quantity. Amount of space is irrelevant to topological and projective space, but it is one of the key concepts around which euclidean space is built. Spatial intervals act as an independent and constant reference against which objects can be compared and located in space.
A space of positions p. 39      A space of positions is the space in which modern adults habitually act and think. Whether or not it is consciously thought of as a threedimensional coordinate grid, the everyday world (the life-world of phenomenologists) is acted upon as if space were a given and immutable entity in which objects occupied positions. However, "... it would be a complete mistake to imagine that human beings have some innate or psychologically precocious knowledge of the spatial surround organized in a two- or three-dimensional reference frame" (Piaget and Inhelder 1967:416).
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Measurement

Definition p. 40      any notion of a relatively constant amount of space used as a reference of some sort. This need not be a formally defined idea of meters or inches, but could be such readily available references as hand breadth and literal feet. Of course, it would be difficult to document the use of such intervals in prehistory. However, there are two constant intervals that appear to have been used fairly early in the manufacture of some stone tools – the radius and diameter. [...] The interval of a radius is not general, but is specific to a particular tool. Nevertheless, it is an interval that consists, however briefly, of a constant quantity of space.
The notion of interval
p. 40      There are two constant intervals that appear to have been used fairly early in the manufacture of some stone tools – the radius and diameter. [...] The interval of a radius is not general, but is specific to a particular tool. Nevertheless, it is an interval that consists, however briefly, of a constant quantity of space.
p. 45 I do not wish to overemphasize the importance of a concept of interval. It is not in and of itself evidence for a competence in euclidean space. However, the notion of a quantity of space is one of the prerequisites to euclidean concepts, and here we have it, in rudimentary form perhaps, in the regular diameters of upper Bed IIdiscoids and spheroids.
Example:
discoids
p. 40
     Figure 20. The amount of trimming on this discoid suggests that the final shape was probably intentional. The rudimentary spatial notions of order and separation are insufficient to explain this artifact; the knapper
must have employed some additional spatial concept. At the minimum, he must have used some concept of radius or diameter, that is some notion of a constant amount of space separating all of the edges. Alternatively, he might have used some idea of a regular curve employing a reference system of chords and arcs. But this is even more complex than a radius. We may conclude that the knapper of this discoid used some notion of interval, either radius or diameter, while making this artifact. Again, I do not mean to argue that the knapper was a geometrician and reflected upon such concepts, only that he used a simple notion of interval in his spatial repertoire.
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Parallel axes

Parallel axes and "affine" geometry p. 45      A second spatial notion that is an element of euclidean space is that of parallel axes, which are essential to the formal definition of a coordinate grid. Actually, parallels constitute a geometry all their own, termed "affine" geometry, with its own set of axioms and theorems. While it is possible to define parallel lines using euclidean notions of angles and measurement, it is not necessary to do so. Any two lines on a plane that never meet are parallel. This definition assumes a concept of plane, but in practice is well within the competence of an intuitive geometry. This does not mean, however, that it must exist as an independent notion in an intuitive conception of space, nor that it must precede a truly euclidean geometry in a developmental sequence.
p. 48      This argument about affine geometry may appear, I admit, rather like sophistry. But it does bear on two important aspects of our problem – the antecedents of euclidean concepts and the relation between developmental and "logical" sequences. The lack of an affine "stage" will turn out to be quite important.
Example p. 45
     Figure 22 (Isimila). The lateral edges of this cleaver are extensively trimmed, and it is fair to assume that the overall shape was intentional. The parallel here is about as good as one could hope to achieve by bifacial trimming. This artifact actually approaches being a regular quadrilateral. But does it exemplify a notion of parallel, or is the parallel just a side effect of bilateral symmetry, which is not an affine concept?
Caveats p. 47      Based on artifacts such as these, an argument could be made that the Isimila hominids had some concept of parallel in their spatial repertoire. However, there need not have been a concept of parallel that was separate from euclidean notions of space since, as we shall see, there are euclidean notions such as congruency that were used by the Isimila hominids. To document a separate use of affine spatial relations, we would need an assemblage in which affine but not euclidean relations were required. To my knowledge no such assemblage exists.
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Bilateral symmetry

Definition p. 49      Bilateral symmetry in its strict sense is a euclidean concept. "Two points are said to be symmetrical with respect to a point, P, if P bisects the line segment joining the two points. Two figures can be considered symmetrical with respect to a point (line) if each point in one figure has a symmetrical point in the other drawing" [...] In the case of symmetry with respect to a line (bilateral symmetry), the bisecting line must be perpendicular to the segment joining any two symmetrical points. The notions of "bisect" and "perpendicular" require the quantitative concepts of angle and distance (concepts of rotational and linear measurement). A simpler intuitive notion of symmetry consists of mirror images, one an exact but reversed duplicate of the other. Yet even here the idea of an exact duplicate usually means a congruency, which is of course a euclidean notion requiring conserved amounts of space and angle.
Reversal and congruency p. 50      In this section I will examine the development of bilateral symmetry by citing examples of artifacts that required a reversal about a perpendicular bisecting line (which I will simply term a midline) as well as some notion of symmetrical congruency, that is, the theoretical infinity of symmetrical points that constitute symmetrical figures.
Example p. 50f
     An even more powerful argument for a concept of symmetry can be based on minimally trimmed artifacts. Figure 26 (Isimila). The trimming at A especially suggests attention to mirroringa congruent shape. The bit has been "pulled in" for the same distance on the retouched margin as it was naturally done on the opposite margin. The subsequent constriction extends for the same length along each side. The knapper also trimmed the butt into
an even curve, giving the ardfact an excellent symmetry in plan with minimal effort. The biface discussed earlier in Figure 7 presents a similar approach to achieving symmetry. Both of these artifacts argue for a concept of the relation between whole and parts. They also suggest that the conceived "whole" was in large part defined by a concept of symmetry. In both cases, some notion of mirroring and of congruency of shape must have been part of the knapper's spatial repertoire.
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