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Excerpts from Wynn 1989 Evolution
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.
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Methodology
Qualitative analysis  p. 3f.  Such a qualitative technique may appear sacrilegious in this age of indices and computer analysis, but there is a sound reason for it: quantification would accomplish nothing. 
Concentrating on exceptional pieces  p. 32  [...] such artifacts are by no means the most common. Almost all biface assemblages, even very late ones, have crude bifaces with irregular cross sections. In many assemblages these predominate. Nevertheless, in a study such as this one, which is concerned with the evolution of a competence, it is fair, I think, to focus on the exceptional pieces. 
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Developmental sequences
Gaps in sequence  p. 33  As was the case with the projective straight line, the developmental sequence of the notion of regular cross section is far from complete. 
No intermediate stage for straight edge  p. 38f  What is not clear is the development of these projective abilities. No argument can be made for projective relations during the Oldowan. No straight edges exist, and there are no intentionally regular cross sections. The West Natron artifacts are more equivocal. Some are extensively trimmed, some have vaguely straight sides, and some of the bifaces have somewhat regular cross sections. [...] In other words the "intermediate" position of West Natron artifacts in regard to topological relations is not corroborated by their position in regard to projective relations. These appear to have been a relatively late addition to the hominid repertoire. 
Late date for euclidean concepts  p. 39f  The idea of space as a general framework of positions is one that develops during ontogeny, and, as I hope to show in this section, it appeared in the behavior of early humans by 300,000 years ago. [...] there is early evidence for a concept of measurement but not for one of parallel axes, a fact that has interesting implications for the evolution of euclidean space. 
Seriation problems  p. 54f  If we limit our consideration of symmetry to euclidean concepts of congruency, perpendicular and bisected lines, and so on, then the developmental sequence presented by stone tools consists of only two stages  early stone tools for which an argument for symmetry cannot be made, and later artifacts, like those from Isimila, for which it must be made. This is not much of a sequence, and, in fact, it tells us nothing about the antecedents of euclidean notions, which is just what we would like to know. 
Overview  p. 58  It seems that the development of a general frame of spatial reference did not include an "affine" stage of competence in parallels, but did include the relatively early appearance of notions of interval, shape, and mirroring. 
Apelike stage for Olduwan  p. 63  Oldowan hominids attended to the nature of the edge  probably tied to a specific task at hand  and paid no attention to the overall shape of their tools. These edges required, as we have seen, only rudimentary spatial concepts. Indeed, in many respects these tools resemble tools made by modern chimpanzees. Both have simple modifications tied to an immediate task, and both are spatially simple. [...] in terms of spatial concepts, Oldowan tools look very apelike. 
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Nature of progress in the Palaeolithic
No cumulation  p. 62f.  It is my contention that spatial concepts did not accrue in additive fashion, like the reading of a geometry text, but were acquired in constellations of concepts tied, ultimately, to single overriding idea or conceptual breakthroughs. These breakthroughs were unlikely to have resulted from conscious theorizing on the part of individual hominids. They were, rather, developments within the repertoire of daytoday spatial strategies that yielded more pleasing or desirable results. Nevertheless, these developments opened up many new possibilities for artifact form. 
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Spatial thinking
Nature of spatial thinking  p. 1  While spatial thinking may not be the key to the evolution of human thinking, it may be a window through which we can glimpse the evolution of the hominid mind. 
Correlation ancient thinking ~ modern analysis  p. 2  The analysis [...] is organized according to types of spatial relations. The division is one of convenience; I do not mean to suggest that hominids compartmentalized their spatial thinking in such a fashion. Topology, projective geometry, and euclidean geometry are all fields of formalized geometry and as such are at least vaguely familiar to readers. 
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Spatial competence
Minimum spatial competence  pp. 45  ...the problem of jniruffluriy necessary competence. [...] . Is it not possible that hominids used their most sophisticated spatial concepts in realms other than stone knapping? It is difficult to get around this problem. Indeed, we can never logically eliminate the possibility that a twomillionyearold Euclid made crude stone tools while drawing triangles in the sand. 
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Overall shape
Disregard for overall shape  p. 23f.  the knapper had to position the trimming blows in a rudimentary series in which sequential blows were spatially related to several preceding blows. This is a simple concept of order. All of these Oldowan concepts of space are simple, and, interestingly, all appear to be focused on the configuration of edges. There appear to have been no ideas about appropriate overall shape, nor any spatial concepts used solely to control overall shape. [...] Notions of overall artifact shape appear after the Oldowan. [...] The minimally trimmed bifaces from Isimila [300,000 years ago] represent the most sophisticated concepts of topological space. Here the knapper directed blows to achieve an overall shape in a very economical manner. 
p. 60  There appears to have been little attention to the overall shape of Oldowan tools. It was the configuration of edges that was the goal of these stone knappers. [...] The hominids appear to have made no attempt to alter the overall shape of the tools, for whatever reason. Such attention to overall shape is a characteristic of succeeding technologies.  
Appearance 1,200,000 years ago  p. 61  Both the notion of interval and the notion of symmetry contributed to the development of the idea of the artifact as a whole. No longer did hominids focus attention only on the edge; they attempted to attain certain overall shapes. Discoids, spheroids, and bifaces all required some notion of overall design. 
Awareness of intended results  p. 62  Like earlier hominids they had an idea of final shape, but, unlike their earlier counterparts, they had a sophisticated array of spatial concepts for conceiving and attaining the desired result. 
A landmark (1,200,000 years ago)  p. 63  Oldowan hominids attended to the nature of the edge  probably tied to a specific task at hand  and paid no attention to the overall shape of their tools. These edges required, as we have seen, only rudimentary spatial concepts. [...] The acquisition that first takes hominid tools out of range of ape spatial concepts is the notion of overall shape, or, to put it a bit differently, some conception of the tool as a whole. While the edgeoriented technology of the Oldowan appears to have been tied to immediate tasks at hand (an ad hoc technology), the presence of tools with repeated, overall shape suggests that in these cases, an idea of the whole tool existed previously in the mind of the knapper. [...] We do not know why these 1.2millionyearold hominids strived to attain this particular shape, but we do know some of the shape's conceptual requirements. With bifaces, the whole tool, more or less, had to be conceived ahead of time. 
Repetitive patterns  p. 64  While the idea of artifact or objectasawhole may seem painfully rudimentary, it represents a much more comprehensive organization of space than that used in the Oldowan. Simple spatial notions like order and separation had now become elements arranged into higherlevel patterns. Moreover, these patterns could be repeated again and again because they existed as specific intentions. Technology could exist apart from a specific task at hand, and, indeed, had come to occupy a conceptual realm of its own. 
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External perspective
Importance of such ability  p. 38  The ability to distinguish and coordinate independent perspectives is an important component of modern adult concepts of space. It is commonplace for us to imagine how an object or scene will appear from another perspective (we even use similar skills to measure intelligence on standard tests). It is an efficient and useful way to apprehend the world. Without this ability, the world would constantly present unfamiliar scenes. Even wellknown objects would, if approached or seen from a new angle, appear different because of the inability to imagine how they would appear from the other, familiar angle. [...] projective notions [...] are important spatial notions. Awareness of independent points of view implies that an individual recognizes that both he and the object exist within a space of changing relative positions. 
Children know only their viewpoint  p. 38  Young children do not possess this concept, and its absence yields an interesting notion of space. For example, when presented objects arranged into a scene, young children consider the relative spatial position to be permanent; that is, they assume that the leftright and frontback order that is visible from their perspective is the same for all other observers. When asked to predict the shape of a shadow on a screen, they describe the object as they see it from their own position. In other words, young children do not consider viewpoints other than their own (Piaget and Inhelder 1967). 
A landmark  p. 64  The second constellation of spatial concepts includes euclidean and projective notions, along with a more sophisticated understanding of the relation or whole to parts. The acquisition of this constellation appears to have hinged on a single breakthrough in spatial thinking  the invention or discovery of perspective. This projective notion was the key that allowed the extension of internal spatial frames that were anchored to specific objects into general constructions of space that organized possible as well as real positions. 
A conceptual feat  p. 64  As we can see from the early bifaces from West Natron and Olduvai, a notion of interval was in use fairly early, but there is no evidence of general frames of reference. Intervals are, by themselves, insufficient. What is required "as well is a constant orientation, and such constant orientation is impossible without some notion of viewpoints that exist independent of that of ego. One must be able to step away from the tyranny imposed by direct perception and construct alternative views. This is quite a conceptual feat, requiring complex substitutions and restructuring of shapes. Moreover, it requires the subjugation of what one actually sees to what one thinks. 
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Intentionality
Secondary trimming as indicative of intentionality  p. 3  Much that is regular about stone tools is purely accidental. [...] Intention does not require some sophisticated image or mental template. [...] The analyses in this book concern, therefore, the patterns of trimming found on tools. Occasionally they also consider the overall shape of the artifact, but only in cases where it is clearly the result of extensive trimming. 
p. 37  Without extensive trimming it is difficult to argue for any kind of intention.  
p. 39  It is the amount of trimming that makes me cautious; the shape just might be fortuitous.  
p. 41  Moreover, neither artifact is extensively trimmed, making it hard to argue for an intentional shape. [...] The disc shape is probably fortuitous.  
Repetitive examples  p. 4  I did try to avoid unique examples. Even though a remarkable pattern or configuration seemed, at least to me, certainly intentional, it remained just possible that it was an accident. However, if this pattern was repeated on another artifact, I decided that the possibility of its being an accident was reduced to an acceptably low level, and felt comfortable in including it in the sample. There are only two or three such unusual patterns in the study. 
Symmetry  p. 18  For objects such as the choppers and scrapers of previous examples it is impossible to make judgments 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. 
Quick planning  p. 19  [...] we need not envision the hominid in agonizing contemplation, but even quick, onthespot planning required a notion of whole and part. 
Caution  p. 32  These examples are cautionary and I include them to emphasize the care that must be taken in determining the necessary competence of hominid stone knappers. Regularity in stone tools can be accidental, and, moreover, even extensively trimmed artifacts can be irregular. 
p. 45  The same caveats hold when one examines stone tools for parallel edges that hold when one inspects them for projective relations. It is especially hard to prove intention.  
Simple notions  p. 40  The amount of trimming on this discoid suggests that the final shape was probably intentional. [...] 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. [...] 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. 
Insufficient evidence  p. 52f.  [...] there is nothing about the location of trimming that suggests that the shape of the original cobble was significantly altered. [...] In this case, the symmetry is in the heads of the archaeologists. 
No verbalization  p. 57  Again I must emphasize that the knapper need not have been able to describe or to verbalize his concepts in nice mathematical terms. 
No conscious theorizing  p. 63  Like earlier hominids they had an idea of final shape, but, unlike their earlier counterparts, they had a sophisticated array of spatial concepts for conceiving and attaining the desired result. 
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Symmetry
Regularity of plan  p. 33  Neither artifact's cross section shows the symmetrical regularity occasionally seen on the Isimila bifaces. However, the plan shape of each of these artifacts is at least roughly symmetrical. I will treat the symmetry later; what is important here is that there is attention to regularity in plan, which can be directly checked, but no clear attention to cross section, which would have required imaginary perspectives. 
Relationship to natural shapes  p. 44  Is it possible that spheroids resulted not from some spatial concept of interval but simply from the copying of a model in nature – a fruit, for example? This is a knotty problem to which I will return later in the discussion of symmetry. 
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Structure
There is no specific discus sion of structure in the book, but the central notion is implied by the following remarks.
No analysis of structure per se  p. 3  The analyses in this book concern, therefore, the patterns of trimming found on tools. Occasionally they also consider the overall shape of the artifact, but only in cases where it is clearly the result of extensive trimming. 
Sequential complexity [as a clue to structure]  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. 
Non random sequence of blows  p. 13  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. 
Relation of single constituents to the whole  p. 15  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. 
p. 18  The final topological notion of use in analyzing stone tools is that of continuity  the relationship of parts to a whole. A line is, topoIogically, 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.  
p. 19  [...] we need not envision the hominid in agonizing contemplation, but even quick, onthespot planning required a notion of whole and part.  
Linear order  p. 17  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. 
Potential elements of the whole  p. 19  In order to have done this the knapper needed some notion of the shape broken down into potential constituent 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. 
Whole to parts, 300,000 years ago  p. 62  the notion of wholepart relations, the understanding of the spatial relationship of a whole to its constituent parts. The minimally trimmed bifaces required an idea of final shape (which could include the euclidean notions discussed above) and also an awareness of the minimum modifications required to achieve that shape. The knapper had to understand how each possible modification would affect the result, and choose accordingly. Trialanderror knapping, even with constant checking, would be insufficient. 
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Bracing
The term and the notion as such are missing in the book, but the ability to distinguish and to unite non contiguous elements (paraperception) is implied by the following.
Analysis and synthesis  p. 18  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 wholepart 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. It is obviously impossible to see an infinitely small point or to inquire about understandings of lines. However, it is possible to examine the results of the process of analysis and synthesis, the ability to break a whole into constituent elements and put it back together again. 
Coordinating viewpoints  p. 28  As is true of the topological concept of analysis and synthesis, there is good evidence for consideration of perspective in later Acheulean bifaces. This competence included not only the ability to operate from one stable viewpoint but the ability to coordinate viewpoints. This conclusion is corroborated by the evidence from cross sections. 
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Typology and taxonomy
No standard modern typology  p. 4  The analysis does not use any standard archaeological typology. I do use some archaeological terms such as "scraper" and "handaxe" but only as convenient terms of reference. The implied function in such terms is irrelevant. Indeed, the purpose for which the hominid made the tools is not at issue. [...] A hominid could have made two tools with very different tasks in mind, but if he used the same spatial notions to conceive both of them, then, from the point of view of this analysis, the tools are equivalent. There is a typology of sorts inherent in the analysis. It consists solely of spatial relationships identified on the tools  proximity and symmetry, for example. 
No isolated assemblages of items that are based only on proximity  p. 11  ... there is no known artifact assemblage that consists exclusively of unmodified flakes and singleblow cores. 
p. 12  ... there is no known assemblage that consists entirely of artifacts of such simplicity. However, it is at least conceivable that an assemblage of such artifacts did in fact constitute the earliest recognizable (though as yet unrecognized) set of stone tools. 
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About archaeology
Archaeology is "optimistic"  p. vii  Archaeology is by nature an optimistic discipline that strives to build an understanding of the past out of small and fragmented pieces. 
Not fossilized theories  p. 57  The artifacts themselves are not fossilized theories but they do supply clues to the antecedents of such a general frame of reference 
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Topological concepts
##### 1. Proximity {#proximity}
Placing of blows on rough stone provides a firm clue to intention  p. 10f  In order to use topological concepts to analyze stone tools we must ask what spatial notions (there had to be some!) the hominid knapper used while striking one stone onto another. We cannot simply use the geometry of the finished product, because almost every stone tool is topologically equivalent – a threedimensional solid equivalent to a solid ball (the exceptions are stone rings found in some late assemblages). As a consequence we must step down and look at the arrangement of elements on the tool itself. Certain topological relations can be apprehended from the way in which the knapper positioned trimming flakes during the manufacture of an artifact. In a very real sense, the placing of trimming blows, particularly as they are positioned relative to previous ones, reflects concepts used by the knapper. Three topological notions are especially evident in the trimming patterns of stone tools: proximity, order, and continuity. These represent increasingly complex spatial coordinations. 
Significance of contiguity  p. 13  The result of contiguous placing of trimming blows is a single prominent edge. One could argue that the flake scars of the previous examples of "proximity only" artifacts (Figures 1 and 2) were also placed contiguously and hence also represent a rudimentary concept of order. This is possible. However, there is nothing about these more primitive artifacts, such as one prominent edge, that required a concept more sophisticated than simple proximity, and, to reiterate a crucial point, we can only argue about minimum competence. 
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 singleblow 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 following 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 {#orderpairing}
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, I am
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 more complicated 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 makes this 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 
##### 3. Order – ordered sequence {#ordersequence}
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. 1417 
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. 
##### 4. Order – reversed sequence {#orderreversal}
The notion of reversed sequence (symmetry)  p. 17  [...] there are some kinds of artifacts that required a reversal, though the spatial elements were more abstract than simple trimming flakes. I am referring, of course, to bifaces, whose bilateral symmetry requires the reversal of order around a midline. ABC/CBA is a symmetry. 
#### Continuity {#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 wholepart 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 make judgments 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 

#### Projective concepts {#projective}
Definition: constancy of items in spite of change in perspective 
p. 24  Projective geometry is a geometry of viewpoints or perspectives. It is concerned with qualities of figures that remain constant when the source of a projection changes. A simple way to explore projective relations is to study changes in shadows cast by a point source such as a candle or flashlight. One can easily transform the shadow of a square piece of cardboard into a rhombus or a trapezoid, but it is impossible to change the shadow into a triangle or a circle. Some qualities of the figure remain constant regardless of its relation to the light source. 
Must be argued from within the object  p. 25  We are interested in intuitive notions of space and, in particular, in how these intuitive notions govern the manufacture of stone tools. The intuitive notion of perspective is therefore of some interest. Do any patterns displayed by stone tools require concepts that include some consideration of perspective, that is, that consider the relative orientation of object and observer? [...] We are left again with patterns internal to stone tools, and only a few of these suggest specific projective notions. The most notable are intentional straight edges and regular cross sections. 
Must be argued from within the object  p. 28  As is true of the topological concept of analysis and synthesis, there is good evidence for consideration of perspective in later Acheulean bifaces. This competence included not only the ability to operate from one stable viewpoint but the ability to coordinate viewpoints. This conclusion is corroborated by the evidence from cross sections. 
MInimum necessary competence  p. 33  Unless the artifact is extensively trimmed and extremely regular, it is difficult, if not impossible, to document projective relations. The minimum necessary competence does not, therefore, include projective notions. 
#### Straight line {#straight}
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 

#### Cross section {#cross}
Definition  p. 28  The cross section of an object is the twodimensional 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 twodimensional 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 


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 device. 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 threedimensional 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.  
Threedimensional 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 {#euclidean}
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 lifeworld 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 threedimensional reference frame" (Piaget and Inhelder 1967:416). 
#### Measurements {#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 

#### Parallel axes {#parallel}
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 


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 {#bilateral}
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 

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