Preface to the International Union of Anthropological of Anthropological Sciences and Ethnological Sciences, Commission on the Anthropology of Mathematics
Paul W. Dixon
University of Hawaii
Inaugural address: International Union of Anthropological
and Ethnological Sciences Inter-Session Tokyo 2002
Quaero non pono, nihil hic determino dictans
Coniicio, conor, confero, tento, rogo
(I inquire, I do not assert; I do not here determine anything with final assurance; I conjecture, try compare, attempt, ask…)
Motto to Christian Knorr von Rosenroth,
Adumbratio Kabbalae Christianae
The neurophysiological basis of our number sense is found in the angular gyrus, Broadman's Area 39. It is here where the effects of cerebrovascular accidents, i.e., strokes, give rise to acalculia, an impairment of the ability to calculate. This is an analagous condition to those speech impairments found in Broca's and Wernicke's Aphasia with injuries to the speech centers in the left cerebral hemisphere. (Dehaens, 1997) .) From this evidence, we may conclude that a neurophysiological basis exists for our mathematical ability which parallels our development of speech in its dependence on specialized neurophysiological centers.
It may be postulated, that the mathematical centers in both left and right cerebral hemisphere are sculpted in a holographic sense by experience. By neurophysiological analogy, we may compare these structure with those neurophysiological structures which mediate place localization in the visual sense. In this model the neural structures have equal latencies of 0.1 milliseconds and their ability to localize sound sources is based on the effect of experience on innate neurological structures. In this way, precise inhibition is essential for microsecond interaural time difference coding, (Brand, Behrend, Marquardt, McAlpire, Grothe, 2002)
We may postulate by analogy that the number line which forms the basis of our mathematical ability is likewise, in large measure, based on the effect of experience on the innate structure of the number line which gives rise to our ability to compare orders of magnitude.
In a conceptual sense, therefore, we may posit that there must arise innately the ability to categorize in an array which provides for an organizing principle for perceptual experience. The angular gyrus where Broadman's Area 39 lies may be thought of as the organizing of organizing areas of the brain. The ubiquity of a numerical descriptive array for the organization of perception is clearly evident which provides for the utility of innate perception of this universal organizational principle.
Thus, for example, the intrinsic relationship found between number and geometry has long been apparent in mankind's conceptual history. We may cite in this connection the conceptual framework of Pythagorous whose work in geometry shown in the Pythgorean Theorm of, a2 + b2 = c2 , was supported by the observation that, "Nature speaks in numbers." The central role of perception in forming the basis of numerosity may then be derived from evolutionay adaptation within our native habitat.
Numerosity, from an evolutionary standpoint, offers a reliable index for perceptual comparison. This may provide for a more consistent basis of comparison between different perceptions both within and between different modalities such as vision and audition. Training in musical performance and composition which are structured in a linear mathematical meter could then, under this postulation, provide for instruction in the foundation of perception within the realm of numerosity.
In examining comparative data from other species, some evidence indicates that wolves may be able to count in the sense that they can compare different sizes for packs of wolves. To do this the animal has to recognize that each object of a set corresponds to a single number. Also, that the last number in a sequence then represents the total number of objects.
Dogs, as related to wolves, have been shown to have numerosity when tested experimentally. Dogs paid little attention when one plus one treats equalled two treats. However, in experimental testing, when one treat plus one treat equalled three treats, they attended to the stimulus configuration for a much longer period of time. (Mateo, 2002)
The universal perception of aesthetic qualities in our species, in other words, to perceive what is beautiful and what is not, may indicate the ability to perceive a certain mathematical ratio. This has been termed the golden number ( v 5 - 1) : 2. This may easily be produced by dividing a segment in two parts so that the smaller part is to the larger part as the larger part is to the entire segment. This has proven to be an important concept in ancient and modern artistic and architectural design. (The New Encyclopedia Britannica, Fibonnacci Numbers, 15th Edition)
We may, therefore, postulate with the common experiences of childhood as well as those innate constituents of Broadman's Area 39 we may support the developmental theory of Jean Piaget where he states that the child acquires the basic formulae of symbolic logic during the first 12 years of life. We may also modify the original position of Piaget from a purely innatist position towards one where the acquisition of mathematical concept's progression in the child's mind is influenced by experience and is not purely innate in origin.
On the other hand, within the province of innate ideas, it could be said that we know of mathematical knowledge yet are unaware of it. Culture then plays an important role in determining what is allowable in experience. (Bursent, 1986)
We may postulate, therefore, that the interplay between our innate
capacity for mathematics and experience is in large measure determined within
a diversity of cultural milieu. We may again draw an analogy between language
development and mathematical development. This may be derived from a quote form
the works of Walter Goldschmidt in Man's Way where he states, "The capacity
for language is a capacity not only to respond to symbols but to create them
- for language is symbolic behavior ."
(Swisher, 1989)
It may be observed, from the foregoing, that in the sense of a Darwinian evolutionary adaptation, we possess a number line and a sense of proportion in psychophysical perception. This kind of experiential/neurophysiological interface arises within the diverse panorama of human culture. As we are symbol users, and also symbol creators we then apply this capacity for symbolization to these perceptual elements. Similarly, to the acquisition of language, the slow and laborious repetition of times tables, rules of multiplication and division , subtraction and division fill the lives of school children around the world.
The general value or meaning of a symbol as an encoding of a perceptual stimulus is acquired through repetition as best exemplified in language learning. Also, the parent carries out a similar operation when the child learns to count. Thus counting steps, objects such as buttons and marbles, through sensor--motor action provides ready evidence of the effectiveness for this avenue towards early learning of mathematical concepts.
The quantum jump which the child experiences toward the abstract principles of arithmetic, calculation of volume, etc., then algebra and latter differential and intergral equations in the calculus would seem to be those barriers towards numerosity which create those vast numbers of mathematically phobic members of modern civilization. If it would be possible to use the basis of theoretical anthropology/ethnolinguistic understanding to improve instruction in mathematical skills and understanding, this would be of great service internationally! This kind of work would also serve to cross another barrier - which is to form an acquaintance with computer based instruction, computer use and programming skills as well.
One of the commonest faults in a learning environment is to free the learning from passion and involvement. In this way, laughter and joy are replaced by grim effortfullness and memorization. Certainly, this is more true in mathematical instruction than elsewhere. Knowledge acquired through schooling does not come fully formed either from genius or revelation. Nor can we say that in the learning process that it from the outset exists abstractly before it has been expressed in some other form. In the process of learning, we may observe that knowledge is produced by intimate and tempestuous relations with organized forms such as you have in mathematical instruction which are the necessary limits for organized acquisition of knowledge. (Game, Metcalfe, 1996)
At the very outset, we may recommend that mathematical instruction be made a part of active participation in music and art. Laughter, pattern and rhythmic interaction may capture the child's interest and turn the effort of learning into an entrancing game of skill and personal development.. These activities are also part of nature and through geometrical illustration as well as patternment in time.
At the very least, we may conclude that instruction in mathematics incorporate dance and song infused with laughter amid positive social interactions. Those phobic reactions now associated with mathematical learning and performance may be dispelled at least in part by this teaching methodology. Within the context of learning theory, this teaching process may represent the substitution of the phobic responses toward mathematical stimuli with more positive learning oriented response sets.
Mnemonic devices may be defined as method or system, physical activity, verbal formula, graphic indication or material contrivance which aids in the process of memorization or remembering. (Leach, Fried, 1972) The ethnological usage in the preliterate societies without mathematical or musical notation is seen where these mnemonic devices may be used in record keeping, calculation of time, the reckoning of accounts, also the preservation of historical and genealogical records, mythical and musical and ritual lore and for the prompting of singers and for the recollection and recitation of tribal memorabilia.
These kinds of mnemonic devices as shown in the literature are: 1.) Some form of counter often used to convey more than numerical information, 2.) Some form of graphic symbol often combined with counters of various types to represent a broad background of meaning. 3.) Oral recitation with or without other methods of recollection.
An example of one of these mnemonic devices was the quipu of the Incas, a tallying device based on knotted cords depending on the main cord with subsidiary strings tied to the secondary ones. Thus numbers and sums were recorded in a decimal system, and the subject of the reckoning was distinguished by the color of the string and the relative importance of the parts was shown by position.
The quipu was thus not a calculating device but an aid to memory for trained interpreters. Similarly the wampum of the North American Indians, was a mnemonic device. These were wampum collars, belts and strings used as a form of record and communication.
Even for those with written languages and exact methods of for writing mathematical and musical notation, mnemonic devices are often used. Diagrammatic systems , may be used and hand charts as well as rhythm, rime, melody, word plays, as well as systems of alphabetical organization. Rhythm as a form of repetition with its repetition of stress is thus a reliable method of assisting memory.
In nature also, we have exemplar of these repetitive patterns of a mathematical nature. Our mathematical ability is thus in this sense a reflection of nature. (Dixon, 1998) We note the phenomenon of phylotaxis, from the Greek (phyllon - leaf, taxis - order). Interestingly, the spiral arrangement of branches on a stem, or the number of petals, sepals and spirals in flowerheads during the flowering of the plant represent successive numbers in the series of the famous Italian mathematician Fibonacci in which each number is the sum of the preceeding two numbers, e.g., 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.. (Klar, 2002)
Encoding these kinds of progressions within the quipu of the Incas would provide a ready means of remembering a given sequence and also a mnemonic device providing ready method for understanding the propagation of the Fibonacci series. Following this example, mnemonic devices may be developed which encode the basic operations of symbolic logic as well as set theory.
Also in the modern world, these mnemonic devices may be translated into computer based methods of instruction. Thus computer based iconic presentation may be adapted to assist in the encoding of mathematical semantics thus increasing recollection and improving problem solving.
More generally, the power of oral recitation can be used to encode mathematical notation and structure into comprehensible syntax within any given ethnolinguistic milieu. Thus a system of translation of mathematical notation into the appropriate syntactic and semantic (idiomatic) translation by native speakers may serve as the sine qua non for mathematical instruction internationally. Clearly, if the mathematical text is left in its original form, none of the foregoing principles of instruction will have been met and the end result must be the continuance of mathematical illiteracy under this system of instruction.
Since innate structures exist for mathematical operations, we postuatle that mathematical eidolon are indeed present within the mentatlese of the human cerebral functionalities. Composite structures such as are found in Broca's Area and also Wernicke's Area for phonemes and syntax as well as semantic categories may be found for mathematical understanding as well. Just as the phoneme is considered an ideal in modern linguistics so also in this analysis number corresponds to the platonic ideal of number. Form and figure (number) as platonic ideals may then be ascribed to innate neurophysiolgical processes since form, i.e., geometric shape, in the realm of mathematics is the formal basis for mathematical understanding.
In mathematical instruction, it may be then hypothesized that we bring together number within symmetric diverse geometric shapes as vividly illustrated by nature in the phenomenon of phylotaxis, as mentioned before.
Thus, per example, interdisciplinary classroom instruction may incorporate diverse plant specimen to illustrate mathematical progressions. Mathematical progressions in this exemplar are well illustrated and are also countable within a laboratory setting. In the most general sense, therefore, the instructional methodology matches the basic Darwinian principle found in the Anthropology of Mathematics that symmetry in nature and number are through natural selection encoded in our Broadman's Area 39. Rather than imposing a pre-set code of memorized number relationships or pre-set categories upon the mathematical learner it may be recommended, based on first principles, that an inductive knowledge of mathematics be formed through example in the classroom.
What is present as an innate structure in Broadman's area 39 may be thought of as providing a Quantitative Recognition Device (QRD) analagously with Noam Chomsky's Language Recognition Device (LAD). The process of quantitization would then be due to innate neurophysiological structures under this postulation. Counting which proceeds from, one, two, three … to many, as well as the basic operations of addition and subtraction, similarly to the assigned functions of phoneme coding and encoding, may then be analyzable from those functions of the innate number line of Homo sapiens.
Once these fundamental understandings have been established, then within a given ethnolinguistic milieu mathematical problems may be treated as puzzles, games and mental teasers rather than as sources of failure and frustration. Children, and adults as well, enjoy mental challenges and solve puzzles with alacrity. The introduction of higher level mathematical concepts within this format, it is here hypothesized, will go a long way toward easing the transition towards problem solving with more complex equations.
The essential stimulus properties, from a Darwinian perspective, may exist as successions of discriminable conditions. Transfer between different stimulus modalities, going for example, from size to shape to colour may provide a means for the learner to encode the essential conception of number. It would also be necessary to provide for active sensori-motor manipulation of these dimensionalities to assure rapid and accurate learning.
What stimulus fields may be appropriate for the learner would be discernable to the instructor within the cultural milieu of the of the learner. Linguistic and semantic analysis and instruction appropriate to the mental age of the learner are essential for the encoding of a numerical sequence and those other operations of symbolic logic as theorized by Jean Piaget.
Philosophically, where Plato has ascribed to a higher realm both form and figure (i.e., number) as ideals, (eidolon) this higher realm in the analysis would consist of those cortical regions in Broadman's area 39 whose function it is to quantitize those regularities found in the evolutionary environment for Homo sapiens. In a sense also, analogously with Broca's and Wernicke's and Area in the right cerebral hemisphere, the linguistic referent is acquired. It is then learned through ethnolinguistic experience and is not present a priori. The basis for competence would, however, exist innately for mathematical understanding under this postulation.
Mathematics, it may be hypothesized, can be instilled into the child's developing intelligence in the same way as linguistic knowledge. Cultural enrichment within a familial context based on early experience with number via counting, mathematical problems presented as intriguing puzzles, where philosophical and mathematical understanding are seen a conceptual framework for human reason would then be necessary constituents for the development of mathematical excellence. So also, the school environment may partake of this culture of mathematics. How this may best be instituted within diverse cultural settings is the essential understanding established as programmatic goal and value under the IUAES Commission on the Anthropology of Mathematics.
As a general principle of mathematical understanding, we find the progression towards more advanced forms of mathematics from algebra to differential and integral equations in the calculus and beyond requires increased abstraction and simplification. The higher forms of mathematics are then simpler and more readily understood and practiced once an adequate foundation is established necessary for their comprehension. The higher forms of mathematics provide a simpler means of calculation even though, for example, conceptually the use of the infinitesimal in the calculus may be among the most difficult mathematical values to visualize.
Since computer languages are the most abstract form of mathematics, it would also be recommended that the modern elementary and junior-high school curricula include, at the introductory level, instruction in computer languages. Thus, as it is generally recognized, that introduction to a foreign language both ancient and modern be accomplished at the elementary and junior-high school level of instruction so also it would be recommended that basic programming languages be introduced early on.
An acquaintance with mathematics provides for an introduction to metaphysical relationships that are perfect within the axiomatic system from which they are derived and demonstrated as theorems and lemmas. From this basic work, the underpinnings of modern science, particularly in physics as in work in fluid dynamics, electrodynamic and plasma physics are derived. It is, therefore, with some degree of justification that mathematical understanding is found to be a prerequisite for progress into college level curricula or even going from middle school into high-school.
Modern mathematics, with its use of algorithms, mathematical topography, set theory and Boolean algebra have transformed our modern civilization into a mathematical universe. The esoteric subjects of a few years ago have now become the most valued knowledge on our planet. Those introverted, sometimes awkward, chess players have now taken center stage in our civilization. These people have become the, "titans of industry, social lions, cultural icons." (Krauthamer, 2002)
Knowing, in the sense of mathematical knowledge, has both cultural and neurophysiological sources in accordance with these postulations. The fundamental utility of mathematical understanding is clearly illustrated by it presence in all human cultures. This ubiquity also supports the more general Darwinian conception of mathematical coding abilities positively influencing survival potentialities for our species. Let us, therefore, implement both theoretical and applied anthropological understandings to comprehend, implement and educate within this line of endeavour.
Historically, the value of empirical research has been heralded as the triumph of science over pure philosophy. There is, however, a general reluctance to accept progress in science unless it is supported by a mathematical derivation. Now, with the advance of the Information Age, with cybernetics in the from of modern computer as well as the internet - we note an even greater worth being placed on mathematical understanding. Hence, this IUAES Commission on the Anthropology of Mathematics may serve this pressing need in the 21st Century.