Methodology

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Nature and limits

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.
Ape-like 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 ape-like.

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Nature of progress in the Paleolithic

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 day-to-day spatial strategies that yielded more pleasing or desirable results. Nevertheless, these developments opened up many new possibilities for artifact form.

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Structure

There is no specific discussion 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, on-the-spot 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 whole-part 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. Trial-and-error 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 (para-perception) 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 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. 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 single-blow 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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