C.W. Spinks
Semiosis, Marginal Signs and Trickster
McMillan 1991

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Nothing makes the process of signing development as clear as what has happened in numbering signs, for their particular float from concreteness to abstract symbolisation parallels the float of signs from object reference to interpretant generation.
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The beginning of numbering starts with a simple recording of a sequence of events by a repetition of marks eventually recognised as a periodicity, but what is most important about this series of marks is that it is a series, some observational division of stuff has already taken place in order to determine that there is repetition and periodicity, and some communicational and mechanical capability is present to provide the need and ability to record the observations. Nevertheless the sign-generated object is observed to repeat its occurrence or presence and is observed to repeat with some kind of regularity. Not only is there an event, but it is the first event of a series of events - thus providing the need for something as cardinal or ordinal concepts of numbering so reflected in the semantics of most peoples.
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To factor time and space is to signify the periodicity of the world, and numbers are ideal for periodicity. Without them prediction for astro-religious or architectural purposes would have been impossible, but periodicity is itself a complex sign operation. Much cognitive activity takes place in noting periodic or quantitative changes in physical objects. There must first be objects and they must be “enumerable". The perceiver must hypostasise a one-to-one correspondence in objects and numbering sign. Then, one must, minimally, hold in mind two enumerations, compare them, evaluate their differences and, perhaps, communicate the results.
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Numbers are generally regarded as a special class of signs, and that the qualities of that special class of signs are particularly relevant to understanding the nature of signs and the problems of the object mystique. Numbers are the most semantically neutral of signs; their semantic carrying capacity is seen as limited to some narrow concept of amount, number or quantity.
Mathematics prides itself on its precision, specificity and semantic mono-valance in signs, and the history of numbering suggests such an objectcentred origin since numbering records were usually inventories. More importantly, numbering is tied directly to the concreteness of our own handedness with base Five, base Ten, or base Twenty being the most common basis for cultural counters.
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Numbering signs, however, go quickly beyond the concreteness of handedness. Lancelot Hogben in Mathematics for the Millions (1983) draws a basic mathematical distinction between “flock numbers” used for counting objects and “field numbers” used for measuring areas in order to distinguish between the two conceptually different mathematical operations. Thus numbering signs are also bifacial, for it is a vastly different thing to record and to count objects in a particular area than it is to measure and predict the general amount associated with a particular slice of time or space. The difference is telling, for Hogben and others point out how important field numbers have been in the development of contemporary mathematics. The close relation between field numbering and survey, geometry and trigonometry, allow astounding leaps in the use of numbering signs. Field numbering is not just a matter of recording a quantity, it is also a matter of predicting and utilising a certain ratio for conceptual purposes; that is, a relation of relations (between objects) used at some point other than the present. The major aspect of the power of numbering signs lies directly in this predictive quality, but also part of the limitation of numbering lies in slavish attachments to concrete objects.
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The evolution of mathematics in as a process of thinking has been a process of finding ways of developing the concept of quantification into more and more inclusive statements about objects and series of objects in periodic relations.
However this object intensity is not something lost in the past, it is still prevalent in contemporary thinking - not so much in the intellectual snobbish sense of a pedestrian understanding of quantification, but in the sense that even at the intellectual frontier, mathematical proofs of newly creations are seen as rearranging the slices of objects known as reality. For example, Newton’s inverse ratio, Einstein's relativity equation, Heisenberg's quantum calculations, the discovery of the still unobserved black holes, all the reiterative formula of fractals are all mathematical operations that do seem to “alter reality”.
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It is this mystique of objectivity and awe at referential potentiality that makes numbering signs most interesting to the problem of the nature of signs, for it highlights the fascinating ambivalence of the object mystique. Numbering signs divert from their very concrete handed origin as tally sequences along the two conceptual paths of the sign, one sensory and abstractive and one intelligible and abstractive.
The right-hand path continues the quantification process and becomes eventually the kind of mathematical thinking we have today.
The left-hand path is the apparent cul-de-sac of Sacred, or Mystical, Geometry which has followed the path of ocultic lore even until modern times, although it is taken somewhat less seriously now all then during the classical age.
The first path leads to the complex symbolisation techniques that produce the mathematical description of physics, systems theory, information theory and a host of other complex contemporary fields, whereas the second path tend to have echoes only and popularised und occultic mysteries of number.
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The mystical geometers were more interested in portraying the eternal and mythic truths of their worlds. They found both the mathematics of shape and celestial movement suited to their purposes of “symbolizing” spiritual concepts like unity, division and growth, and their use of numbering was more in search of archetype than type.
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The semantic neutrality of numbers is a rich field for archetypal mining, and the mathematical mythemes, just like the semantic ones, can be used for either purpose. The nature of the sign is to be a tool turned to a purpose, and what is fundamental to understanding it is its purpose and use. Understanding purpose is, of course, very near to understanding intention, and we are back at the door of a black box, but as Wittgenstein argued, returning to tool use is a way of avoiding the dangers of the homunculus in the box. By focusing on the tool and its use, we can observe what happens physically and thereby hypothesise the rules of operation. So let us return to the idea of the ratio, which informed so many of these ambivalent uses. The search for ratio, as the etymology of related terms suggests, is primarily a search for rationality, and that search is, in part at least, an attempt to explain the periodicity is of sign-objects. Whether one is searching for pi or the squared circle, seeking the Prime Mover or the measurement of infinitesimals, or searching the heavens for God's birth or a stellar fix, one is looking for a rule that governs a series; that is, in Peirce's terms, an interpretant, the law on the sign.
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The search for the rule of periodicity is a search for a rule of feedback, and although the bootstrap problem of informational systems is never very far away, or although the final interpretant is always just beyond our reach, we still believe that we can find a rule of periodicity, for the practical application of our ratios drive us on.
The magic of mathematical mythemes: In fact, because signs are related to physical sensation, the concrete path of quantification’s mystique is a tempting one, and most of us follow it in one form or another. It has all the certainty that our emotional need for validation, intellectual laziness or conceptualising instinct wont to desire. The development of an abstractive capacity does not necessarily remove that temptation…
The capacity of numbers to model the regularities of periodicity and to highlight irregularities for further control is too useful to be ignored very long. ..However, the pragmatists represent only one part of applicability, and trade secrets are not so remote from religious mysteries. As quantum mechanics has taught us, our numbering, whether for business and science, is a process of approximation for particular purposes (course in credit, that the destination, or cash-and-carry), and we're naive if we forget the process of approximation that is operating. We are engaged in activities of ratio, we are examining proportion, we are utilisers of relations, and those very facts mean that we are dividers of the world stuff by our little marks in wood, bones, sand and air. Our perspective is all proportion driven by the Interpretant machine of the sign system.

The magic of numbers: What we tend to forget is the sheer magic of numbering. The setting of #still objects, the manipulation of the signs and the realise changes in objects are a continual source of magic.
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So it is not surprising that the Pythagoreans were tempted by geometry to find the sequence Celestial Harmonies or that the Greeks were tempted to find the Internal and Eternal Forms. It is not surprising that the Babylonians, the Egyptians, the Incas, the Mayas, the Arabs and others found the mathematics of astronomy the basis of an astrological projection of spiritual truth into the human sphere. Numbers are connected with the gods not only because they are always gifts or plunder from the gods, but also because they reflect the periodicities of the heavens and of life. Whether they simply count or predict, numbering signs seem to their human users imminently connected with the order of things; that is probably because the order of things is made up of the signing capacity of the users as much as it is made up of things directed by God's will or entropy’s chaos.
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Mystical Math and Archetypal Numbers: It is instructive to examine the mystical aspects of numbering, for by so doing one can see how easily object centred signs like numbering can be utilised for nonrational processes and how they can be skewed far from any hint of numeric practicality. Early in the history of signing and numbering, the conceptual quality of numbering sign apparently presented itself with much more insistence than now when we easily trafficking numeric abstractions.
First, there were icons of periodicity such as a sequence of lines, the zigzag pattern, or the wave pattern, and such icons were as much a part of practicality as they were of worship of art, for these were media specific, derived in part from the bone or stone to be worked with early tools.

Boe: A most impressive exemple for such icons I saw at Pha Taem in Tailand. These pictures are dated to about 4000 to 6000 BP.



Secondly, there is obviously a concept of discovery here, for the marks represented the periodicity of celestial movements.
But thirdly, one also begins to find archetypal concepts like Unity, Duality, Trinity and Quaternity expressed in icons and graphics.
Fourthly,numbering concepts, because of their semantic neutrality, make perfect archetypes and can supply their conceptionality to a host of human problems.
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As Peirce suggests about the Categorical numbers, what numbers represent are relations, and relations between objects, signs or people of the real fodder of archetypal thinking...It is not difficult to see how quickly numbering can move into spiritual and philosophical concepts when one contemplates the nature of space and time.

Zero: The ultimate abstract concept of spatial number is zero because specifically there is no such thing as “nothingness” in nature. As Wilden in System and Structure (1980) argues, the zero, the nought, the not, the negative are all the results of the digital sign system, and apparent live in the Hindus first developed the concept, they viewed it as a spiritual one. Zero, or sunya, was used to represent a spiritual discipline, but pragmatists took the concept and emphasised its quantitative value so it could be utilised to move computational operations from the counting board to the head. It thereby provided European mathematics with a trajectory to modern scientific and mathic thinking, where periodicity has skewed even calculus into new avenues of thought..
The original intent of the zero was to represent the mystical concept of “emptiness”, which the more pragmatic West still has trouble with. However the important point to all this is that these mathematical and spatial concepts can quickly be complicated in their meaning by moving them from one sense of object reference to another.
The literal moving of the concrete object also ought not to be disregarded, as Piaget’s experiments argue or as attempts at modelling demonstrate. The new sense of objects in periodicity is a second-order concept more complex than simple existence in nature. The handling of an object in space and time and moving it, or its parts, around the concept sure way of operating of signs “torqued” into a hypothetical dimension.
It is an abductive way of developing operational signs from pre-concrete and concrete intuitions and of then critically examining the relationships. Part of the magic of numbering is its concreteness and its predictability, and when one begins to play with a Platonic solids an interesting area of hypothetical thinking is opened that is in direct line with the conceptual spacing done by primitive artists.
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The attempts to square the circle, which may appear to us as useless, really are productive - both in the light of producing the transcendental number pi, but also in the light of the problems for number theory and set theory presented by irrational lumbers like the square root of two, three and five. This geometric playing leads to the development not only of the conceptual use of ratio, processes of trigonometry, problems of route, functions of logarithms, articulation of calculus, but also to the golden ratio of phi the Archimedean Spiral, the problem of magic squares, geometric and arithmetic series, the Fibonacci sequences, and so on.
It is an amazingly fertile process. Anyone who has played with the mathematical concept of infinity knows how quickly one can hear the booming coal of the Spiritual, “for the infinite has turned out to be the hiding place of much that is strange and paradoxical”( Davis and Hersh, 1981). We will still scribble in the Archimedean sand, for we are engaged in the process of abstraction that was started by and maintained by the signing process. It is the interpretant machine in operation, and despite the mystery of the magic numbers, mystical geometry and its objects centredness raise some serious questions about the operations and usefulness of mathematics even in its more complex, mysterious modern versions.
But what is important here is that the object mystique raises the same questions about any sign system because “a sign is something standing for something else”, and that “something else”need not be an external object. It can as easily be a signed object ruled by an Interpretant rule and subject to a host of previous semiotic operations.

Archimedian Spiral