Thomas Wynn

Excerpts are particularly extensive because of the great significance of this book, and its particular relevance to the central argument of CAR. The quotes provide an exemplary model of inferential reasoning.

Methodology

Developmental sequences

Nature of progress in the Paleolithic

Spatial thinking

Spatial competence

Overall shape

External perspective

Intentionality

Symmetry

Structure

There is no specific discussion of structure in the book, but the central notion is implied by the following remarks.

Bracing

The term and the notion as such are missing in the book, but the ability to distinguish and to unite non contiguous elements (para-perception) is implied by the following.

Typology and taxonomy

About archaeology

Topological concepts

     1. Proximity

Applying the notion of proximity	p. 10f	Perhaps the most rudimentary spatial concept is that of proximity – the perceiving or placing of elements in the same spatial field. It is simply a notion of "nearness" or "byness," and is so basic that virtually every trimmed artifact is equivalent to every other trimmed artifact. Each trimming blow is in some sense "near" the others on the artifact. The only exceptions are unmodified flakes and cores that have had only one trimming blow or, perhaps, two unrelated trimming blows, though it would be difficult if not impossible to prove that the second trimming flake was not placed somehow near the first. From the perspective of this simple spatial notion all trimmed stone artifacts, from polyhedrons to fine bifacial points, are equivalent because multiple trimming flakes have been placed in proximity to one another. Now there is no known artifact assemblage that consists exclusively of unmodified flakes and single-blow cores.      Competence in proximity is, therefore, a kind of base point from which we can begin to describe the evolution of spatial concepts. As it turns out, there are artifacts whose knapper need only have used a notion of proximity. The fol lowing examples are from Bed I at Olduvai.
Specific examples	p. 10f	Figure 1 (FLK N). To have made this artifact, the stone knapper need only have struck the cobble repeatedly in the same general vicinity. It was not necessary that he place one blow in some precise relationship to a previous one, only that the blows be more or less near one another. The result was a number of sharp projections and edges that would probably have been useful for several kinds of tasks.      Figure 2 (DK). At first glance this artifact may appear somewhat more sophisticated than the preceding. There are more trimming flakes, removed by blows from several different angles. Simple bashing would not suffice. Because we can only argue about minimum competence, though, we must conclude that this artifact could have been manufactured by someone who simply placed his trimming blows near other trimming blows.                            Figure 1                                                            Figure 2

     2. Order – pairing

Applying the notion of pair	p. 12	Separation is the notion of distinguishing or segregating the elements in a spatial field so that each can have a specific, rather than general, relation to the others. Order coordinates separation with proximity to create such concepts as the pair,the sequence, and, most complex, the reversible sequence. In a pair, proximity and separation are coordinated but there is no need for a constant direction of movement. The elements in a pair must be distinct yet contiguous, hence the coordination of separation and proximity.
Separation	p. 23	If the hominid wanted to make a tool with a single edge, however, proximity alone would not suffice. He would need a concept of separation, in which the edge acts as a kind of boundary or reference separating faces of the tool. He would have to direct his blows with regard to that boundary; this concept is more complex than simple proximity.
Specific examples	p. 12f	Figure 3. [...] Four trimming blows have produced a single sharp edge along about one third of the circumference of a cobble. Trimming flakes A and B preceded trimming flakes C and D. In effect, the artifact is the result of two pairs of trimming blows. The order of the blows within each pair is irrelevant, as is the order of the pairs to one another. Again, Iam emphasizing how the knapper had to place trimming blows to achieve the result. Bashing in a general vicinity would not have been sufficient. The knapper directed three specific blows (dissociation, separation) to a position adjacent to preceding blows (proximity). Indeed, preceding negative scars appear to have supplied specific striking platforms for subsequent blows. The result is not a random collection of projections and edges but a single working edge. The minimum necessary concept is that of the pair.      Figure 5. This chopper has at least eleven trimming flakes and, at first glance, appears more sophisticated than the preceding two examples. In addition to simple pairs, the single edge on this chopper required that the concept of separation incorporate several elements. This pattern is intuitively morecomplicated than that of the simple pair. In mathematical topology, separation is often exemplified by points positioned on opposite sides of a boundary of some sort (inside or outside a circle, for example). On the chopper in Figure 5, the knapper produced a sinuous edge by trimming onto two faces of a cobble. True, the placing of each blow required the idea of a pair in space. But the notion of a single edge also acted as a kind of boundary separating the blows from each face of the tool. What makesthis more sophisticated than a simple pair is that the boundary—the edge—had to have been maintained for several trimming blows. This boundary was a spatial reference used by the knapper to orient his blows. The concept of a simple pair was not quite sufficient                          Figure 3                                                            Figure 5

     2. Order – ordered sequence

The notion of ordered sequence	p. 14f.	The notion of an ordered sequence requires the coordination of proximity and separation with the addition of a constant direction of movementor orientation. A pair is a pair regardless of the directional orientation of the second element to the first. But if we want to create a series of elements, such as a line of posts, we must employ some concept in addition to separation and nearness. We must place the third element and all subsequent elements in some specific spatial relation to all of the preceding elements. We cannot simply place the third post near the second; we must also consider the position of the first. This is most easily done by maintaining a constant direction of movement. This coordination is more complex than that of the pair because several distinct elements (separation) must be related through some concept of arrangement, even one as simple as "move to the left.
Specific examples	p. 14-17	Figure 6a (FLK N). In the view shown here, the lower edge of the artifact consists of a series of six unifacial flake scars. On close examination one can see that the knapper did not strike off the trimming flakes one after another in exact sequence (at least one is out of place; see arrow). Consequently, any comparison to a strict sequence such as ABCDEF would be spurious. Nevertheless, the end result is the same—one flake scar following another. The violation of a strict order should not be disturbing. It is unlikely that these early hominids played logical games with themselves, at least using stone tools.      The coordination of proximity and order in this artifact is certainly more complex than that of a pair. Each flake is placed in relation not just to another one but to all of the other flake scars. The knapper achieved this result by restricting successive trimming blows to a single direction. This constant direction of movement is the third notion in a concept of order.      Figure 6b. This example differs from earlier ones in that it is a flake tool; that is, the blank itself was a flake rather than a cobble or a chunk of stone. This implies two major manufacturing steps—the removal of the flake from a core and the subsequent trimming of the flake. While this increase in number of steps reflects a greater complexity of sorts, it is of no direct relevance to a discussion of spatial concepts, especially since we have no way of knowing whether the whole sequence was conceived at the outset or whether the knapper simply selected a previously struck flake as most appropriate for the task at hand. The spatial concepts necessary for trimming this flake are identical to those used on the cobble in the preceding example.      Figure 6c. This small scraper was also made on a flake. The trimming flakes are related to one another in the same manner as those of the two preceding examples. What is important here is the small size of the trimming flakes; the knapper needed a fairly fine motor control to make this artifact. The "primitiveness" of Oldowan tools cannot, therefore, be explained in terms of poor motor coordination on the part of the hominids. In makingthese three artifacts, Oldowan hominids employed a spatial concept more sophisticated than that of a simple pair. Put abstractly, we can argue for a concept of linear order. Each element, in this case trimming flakes, was placed in relation to several other elements, the result being a sequence. Though their order was not as elegant as a purely mathematical series such as ABCDEF, it is still clear that these hominids were capable of arranging a sequence of elements.

     3. Order – reversed sequence

Continuity

Breaking down and putting back together	p. 18	[ ] the relationship of parts to a whole. A line is, topologically, an infinite series of indiscernably small points. In order to conceive of a line in this fashion, it is necessary to be able to subdivide the line infinitely and then recreate it in thought from the resulting elements. Stated in another way, one must analyze a whole (the line) into its constituent elements (the points) and resynthesize these elements into the whole. It is a breaking down and putting back together which, in terms of simpler topological relations, requires the coordination of proximity, separation, and order. Such whole-part relations are among the most sophisticated of topological notions and, if we can get at them, should be quite important in characterizing the spatial competence of early hominids. [A relatively late development: the book discusses at this point the question of possible transitional items.]
intention	p. 18	For objects such as the choppers and scrapers of previous examples it is impossible to makejudgments about minimal trimming because we have little idea of the knapper's intention concerning overall shape, if in fact he had one. But for artifacts with symmetry or regularity of any arbitrary kind we can make such judgments, keeping in mind that they are, in fact, judgments
Example	p. 19	The symmetry of this particular biface resulted from four short sec¬ tions of trimming (A, B, C, and D) that are, for the most part, unconnected with each other. In order to have done this the knapper needed some notion of the shape broken down into potentialconstituent elements, in this case trimming flakes, and of their combination into the finished whole—in other words, a fairly sophisticated idea of the spatial relation of parts to the whole. Only by reducing the shape to potential trimming flakes could he have determined which were necessary and which were not. Again, we need not envision the hominid in agonizing contemplation, but even quick, on-the-spot planning required a notion of whole and part.

Projective concepts

     Straight line

Definition	p. 25f.	When the position of the observer is taken into account, certain spatial qualities are added to the simpler topological ones. A straight line viewed from the proper position becomes a point (the terminal point of a straight line is capable of masking all of the other points), but there is no position from which a curved line can be seen as a point. The ability to create a straight line presupposes an awareness of this stable viewpoint, either by actual sighting or by imagination. [...] What is true of projective straight lines must also be true of straight aspects of artifacts, but only if the straightness is clearly the result of intentional modification. [...] Much that is straight on stone tools is a result of manufacture on flakes, many of whose natural edges approach straightness.
Example:	p. 26	The trimmed lateral edge of this cleaver is remarkably straight, and, more important, the extent of the trimming suggests that the original shape of the edge was considerably altered. The edge is also straight in profile. This required the knapper to control two viewpoints or perspectives at the same time. Competence in the basic topological notions discussed in the previous chapter would not have been sufficient. The knapper had to have related the trimming of the edge to a constant point of view. Moreover, because the edge is also straight in profile, the knapper had to have considered a point of view located on another plane. Even if the knapper continually checked the edge by actual sighting, he had to be aware that the shape varied according to the viewpoint, that what was straight from one sighting point was perhaps not straight from another. In other words, some notion of perspective must have been present in the knapper's spatial repertoire, and a fairly sophisticated one at that since there are two coordinated perspectives evident here.

     Cross section

Definition	p. 28	The cross section of an object is the two-dimensional shape produced by a plane intersecting the solid. For example, the cross section of a sphere is a circle; of a cylinder, a circle or an ellipse; of a pyramid, a rectangle. The cross section of a biface is the two-dimensional shape produced by an imaginary plane intersecting the artifact. We say that a biface has a "regular" cross section if it presents a recognizable twodimensional figure such as a rhombus or a triangle or a lens shape. Symmetry is usually the most telling quality of regularity.
Imagination	p. 28f.	The ability to conceive of and create a regular cross section requires a sophisticated coordination of viewpoints or perspectives. First, it is necessary that the observer – in our case the knapper – be aware of a viewpoint that is not actually available to him. One cannot sight into a solid. Without the aid of some assisting apparatus such as a template the image of the cross section must be imagined. Second, the observer must coordinate a virtual infinity of cross sections or perspectives.
Example	p. 29	The cross section was taken with a template at the point of maximum width. As can be seen from Figure 13, the artifact is extensively trimmed, and it is fair to conclude that the original shape of the blank, including the cross section, has been changed. Could the knapper have controlled the shape of this cross section by directly sighting from a perspective to the rear of the artifact? Such a direct sighting would actually combine (and confound) the dimensions of maximum width and maximum thickness, which do not fall at the same point on the length of the artifact (maximum thickness is farther toward the tip). In other words, the knapper would see a "view" that was a composite of two separate cross sections.      There is, in fact, no position from which the cross section of the artifact in Figure 13 can be directly observed. It must be created in the imagination or, as Ihave done, measured with a device. Moreover,a planar intersection taken virtually anywhere on this artifact, even at angles not parallel to the major axes, would yield a regular cross section. The knapper constructed not just one imaginary viewpoint but a virtual infinity of them. His control in this regard was quite remarkable.
Reversal and congruency	p. 51	[The item in Figure 13] is not only bilaterally symmetrical in plan, it is symmetrical in profile and in all of its cross sections. Furthermore it is extensively trimmed, and the shapes are certainly intentional. The symmetry is extremely regular, and the mirrored shapes are congruent. One could improve little on the result if one were to use a measuring de¬ vice. The symmetries in this artifact come very close to fulfilling the formal definition of symmetry.
End point of developmental sequence	p. 52	The handaxe in Figure 13 also suggests something even more remarkable than simple bilateral symmetry. The artifact demonstrates symmetry across not just a midline in plan, but also a midline in profile and a midline for all of the cross sections. These lines intersect to define a three-dimensional space, intuitively equivalent to a space of Cartesian coordinates. It is difficult to imagine how a more sophisticated conception of space could be employed in stone knapping. Artifacts like this one constitute the endpoint of our developmental sequence.
Three-dimensional frame of reference	p. 57	The fine bifaces, like those in example 13, have bilateral symmetry in three dimensions (as we would define them). The knapper had to be able to invert congruent shapes across an imaginary midline for three separate dimensions and to coordinate them. This required a frame of reference of considerable complexity, and, while I suppose it possible that this frame was applied only to spatial relations on or within artifacts, this seems to me very unlikely. The abstract nature of such a spatial framework suggests that it was a general one, extendable to all of space. Again I must emphasize that the knapper need not have been able to describe or to verbalize his concepts in nice mathematical terms. That he employed such a framework, however, is, I think, undeniable.

Euclidean concepts

Measurement

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.

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.

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.