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A pluridisciplinary analysis of the fractal complexity in Mwalimu Harold Isaacs' magnum opus, United States-Mexican Relations during the Gonzalez Administration, 1880-1884.


A review of Mwalimu/Honorable Teacher Harold Isaacs' curriculum vita (1) reveals that he received his Bachelor of Science (BS), Master of Arts (MA) and Philosophy Doctorate (PhD) degrees from the University of Alabama in 1958, 1960, and 1968, respectively. His MA thesis is titled Cocaine in the Relations between the United States and Peru, 1888-1958 and his PhD dissertation is titled United States-Mexican Relations during the Gonza1ez Administration, 1880-1884, (2) which is the focus of this essay.

In addition to Mwalimu Isaacs being the founder of the Association of Third World Studies (ATWS), Inc. and Editor-in-Chief of the organization's Journal of Third World Studies (JTWS), he has served as a graduate teaching fellow and assistant, associate, and full professor, and is now professor emeritus of History. He has taught undergraduate and graduate courses on World Civilization, Historiography, American, Latin American, Black American, African and Third World History. He is a member of three national honor societies; has won 27 major fellowships, grants and honors; has developed 21 individual seminars and 128 seminar series dealing with Third World issues, including revolutions, nationalism, superpower relations, cultural influences, the United States, the post-Cold War world, democracy, economic development, human rights, and Western influences; has organized eight annual Third World Studies conferences, four Black History Month programs, and eight annual ATWS meetings; has engaged in 51 scholarly activities since 1981; has written eight effulgent essays that appear in top-notch scholarly journals and edited 56 issues of the JTWS, and counting; is a member of only one scholarly organization--the ATWS; and has participated in 73 other professional activities. But what struck me about Mwalimu Isaacs' curriculum vita is his extraordinary commitment toward Blacks and other peoples of the Third World, having developed a Black Studies minor in 1974, a Third World Studies minor in 1980 and a Black Leaders Seminar Series in 1981, all at Georgia Southwestern State University. This propelled me to search for the reason of this commitment--one that will be embedded in a deep structure and, thus, his magnum opus: United States-Mexican Relations during the Gonzalez Administration, 1880-1884.

Divided into eight chapters, five appendices, and a bibliography, the 247-page text, United States-Mexican Relations, is about the significant events that took place between the two countries from 1880 to 1884 and the impact on their relations. But before discussing the relations during the Gonzalez Administration, Mwalimu Isaacs began with a survey of the relations between the two countries during the first Porfirio Diaz Administration, covering from 1876 to 1880. (3)

In analyzing United States-Mexican Relations, I employ the mathematical concept of Fractal Dimension and Complexity Theory to explore the idea of spectrum progressing from more orderly to less orderly or to pure disorder in the text, since other works on the topic had characterized Mexico as having been extremely chaotic and backward (more on this later). This called for the utilization of the Pluridisciplinary approach that helped me to mix linguistics and mathematical approaches--more precisely, Linguistic Presupposition and Fractal Methodology. The results generated after the MATLAB computer runs suggest that the combination of negative and positive feedback loops, which form the basis of several African knowledge systems, also form a key mechanism of general self-organizing systems discussed in United States-Mexican Relations. Before discussing the results generated from the MATLAB computer runs, it makes sense to begin with an expose of the research methodology employed in this essay.


The major challenge for me was how to transform the linguistic pragmatic or deep-level meanings in Mwalimu Isaacs's literary text for mathematical modeling. As I stated earlier, this called for the utilization of a pluridisciplianry approach that helped me to mix linguistics and mathematical approaches: more precisely, Linguistic Presupposition and Fractal Methodology. The following is a discussion of these techniques. It behooves me to state here that discussions of this methodology also appear in the following works: Abdul Karim Bangua, "A Mathematical Exploration of Fractal Complexity among the Axioms on the African State" in the Journal of Third World Studies. From John Mukum Mbaku to Pare Badru, Journal of Third World Studies, Vol XXIX, No. 2, Fall 2012, pp. 11-64; Abdul Karim Bangura, "Fractal Complexity in Cheikh Anta Diop's Precolonial Black Africa. A Pluridisciplinary Analysis," CODESRIA Bulletin, Nos. 1 and 2, 2012, pp. 10-19; and Abdul Karim Bangura, "Fractal Complexity in Mwalimu Chinua Achebe's Things Fall Apart: A Mathematical Exploration," Critical Interventions, Number 9/10, Spring 2012, pp. 106-121.

Pluridiciplinary Methodology

Pluridisciplinary Methodology can be generally defined as the systematic utilization of two or more disciplines or branches of learning to investigate a phenomenon, thereby in turn contributing to those disciplines. Noting that Diop had called on African-centered researchers to become pluridisciplinarians, Clyde Ahmed Winters states that a pluridisciplinary specialist is a person who is qualified to employ more than one discipline--for example, history, linguistics, etc.--when researching aspects of African history and Africology in general. (4)

The history of the Pluridisciplinary Methodology can be traced back to the mid-1950s with the works of Diop and Jean Vercoutter. The approach was concretized by Alain Anselin and Clyde Ahmad Winters in the 1980s and early 1990s. A brief history of this development with brief backgrounds of these four pioneers is retold in the rest of this section.

G. Mokhtar in his book, Ancient Civilizations of Africa, traces the development of Pluridisciplinary Methodology to the works of Diop and Vercoutter. Diop was born in Senegal on December 29, 1923 and died on February 7, 1986. He was a historian, anthropologist, physicist, and politician who investigated the origins of the human races and pre-colonial African culture. His education included African history, Egyptology, linguistics, anthropology, economics, and sociology. He is considered one of the greatest African intellectuals of the 20th Century. Jean Vercoutter was born in France on January 6, 1911 and died on July 6, 2000. He was a French Egyptologist. (5)

According to Mokhtar, Diop and Vercoutter were in total agreement on the point that it is necessary to study as much detail as possible all the genes bordering on the Nile Valley which were likely to provide fresh information. Mokhtar notes that Vercoutter considered it necessary to give due weight to the palaeoecology of the Delta and to the vast region which had been termed by other researchers the Fertile African Crescent. Mokhtar points out that Diop advocated tracing the paths taken by peoples who migrated westwards from Darfur, reaching the Atlantic seaboard by separate routes, to the south along the Zaire Valley and to the north towards Senegal, on either side of the Yoruba. He adds that Diop also pointed out how worthwhile it might be to study Egypt's relations with the rest of Africa in greater detail than had been done, and Diop further mentioned the discovery, in the province of Shaba, of a statuette of Osiris dating from the 7th Century before the Christian era. Similarly, argues Mokhtar, a general study might be made of the working hypothesis that the major events which affected the Nile, such as the sacking of Thebes by the Syrians, or the Persian invasion of -522, had far reaching repercussions on the African continent as a whole. (6)

Furthermore, according to Winters, two major scholars who have advanced the pluridisciplinary approach by combining anthropological, historical and linguistic methods to explain the heritage of African people, constituting a third school of Africancentric researchers (the first and second schools being the African American and the French-speaking African and African Caribbean, respectively), are Anselin and himself. (7) Anselin teaches ancient Egyptian linguistics at the University of Guyana Antilles. He is an anthropologist and also the founder of the Journal of Caribbean Egyptology. Winters is a lecturer at Governors State University at University Park in Illinois where he teaches curriculum design and research methods courses. He also is a 28-year teaching veteran of the Chicago Public Schools system.

Anselin is the author of three important pluridisciplinary Africancentric books--(1) Samba, (2) La Question Puele, and (3) Le Mythe d'Europe--and numerous articles. In Samba, Anselin demonstrates how the corpus of Egyptian hieroglyphics explains both the Egyptian civilization and the entire world of the Paleo-Africans. He also makes it clear that Kemetic civilization originated in the Fertile African Crescent and that Black African and Kemetic civilization at its origination was unified from its foundations in the Sahara up to its contemporary manifestations in the languages and culture of Black Africans. In La Question Puele, Anselin examines the unity for Egyptian, West African and Dravidian languages, political traditions and culture. He also provides a detailed discussion of the "Black Ageans." The findings comprise a thorough representation of the affinities between the Agean and Dravidian civilizations. (8)

Winters is the only African American that attempts to confirm Diop's theories in relation to the genetic unity of the Egyptian, Black African, Elamite, Sumerian and Dravidian languages. Winters is mainly concerned with the unity of the ancient and new worlds' Black civilizations and the decipherment of ancient Black writing systems used by these Africans. This interest had led him to learn many languages, including French, Tamil, Malinke/Bambara, Chinese, Arabic, Otomi, and more. (9)

Winters had used Diop's genetic model in his research by combining anthropological, linguistic and historical methods to confirm that the center for the rise of the originators of the Egyptian and Manding civilizations, the Magyar or Hungarian civilization, the Dravidian civilization, and the Sumerian and Elamite civilizations was the Fertile Crescent of the highland regions of Middle/Saharan Africa. He also explains how Blacks founded civilizations in the Americas and East and Southeast Asia. A major finding from Winters' work is that the ancestors of the Dravidian and Manding-speaking people seem to have left Africa at the same time around 2600 BC, and that these people founded civilizations in Europe, Elam, India and ancient China. (10)

Like Diop before him, Winters also discusses the African sub-stream in European languages, the conflict between African people and Indo-European-speaking people, and the loss of early African settlements in Europe to the contemporary European people due to natural catastrophes and wars around 1000 BC. Winters provides valuable source material for the elaboration of the African influence on European languages and those of East and Central Asia. (11)

Winters had discovered that the Proto-Saharan people used a common writing system. He also was able to read the ancient inscriptions left by these people in the Sahara dating to 3000 BC. He was able to confirm this development by comparing the Manding and the Elamite languages, and the Sumerian and Dravidian languages. The evidence of a genetic relationship between the Manding languages, which Winters used to decipher the earliest Proto-Saharan writings and other languages spoken by the founders of civilization in India and Mesopotamia, led him to hypothesize that the writing systems used by these ancient founders of civilization could be deciphered. The utilization of Diop's linguistic constancy theory allowed Winters to confirm his own hypothesis and read the common signs used to write the Harapant, Minoan and Olmec scripts. (12)

Winters' most significant finding is the cognate language of Meroitic. By employing the evidence presented by the Classical sources that the Kushites ruled empires in Africa and Asia, Winters is able to show that the cognate language of Meroitic was the Tokharian language spoken by the Kushana people of Central Asia. He has been able to decipher many Meroitic inscriptions by using the Kushana/Tokharian language. (13)

According to Dani Nabudere, Pluridisciplinary Methodology involves the use of open and resource-based techniques available in an actual situation. Thus, it has to draw upon the indigenous knowledge materials available in the locality and make maximum use of them. Indigenous languages are therefore at the center of the effective use of this methodology. (14)

What all this suggests, according to Nabudere, is that the researcher must revisit the indigenous techniques that take into consideration the epistemological, cosmological and methodological challenges. The researcher must be culture-specific and knowledge-source-specific in his/her orientation. Thus, the process of redefining the boundaries between the different disciplines in our thought process is the same as that of reclaiming, reordering and, in some cases, reconnecting those ways of knowing, which were submerged, subverted, hidden or driven underground by colonialism and slavery. The research should therefore reflect the daily dealings of society and the challenges of the daily lives of the people. (15)

Towards this end, following Nabudere, at least the following six major questions should guide pluridisciplinary research: (16)

(1) How can the research increase indigenous knowledge in the general body of global human development?

(2) How can the research create linkages between the sources of indigenous knowledge and the centers of learning on the continent and in the Diaspora?

(3) How can centers of research in the communities ensure that these communities become "research societies"?

(4) How can the research be linked to the production needs of the communities?

(5) How can the research help to ensure that science and technology are generated in relevant ways to address problems of the rural communities where the majority of the people live and that this is done in indigenous languages?

(6) How can the research help to reduce the gap between the elite and the communities from which they come by ensuring that the research results are available to everyone and that such knowledge is drawn from the communities?

The truism that indigenous knowledge is critical to Africa's development prompted a workshop titled "Indigenous Knowledge Systems and Intellectual Property in the Twenty-First Century: Perspectives from Southern Africa" convened at the University of Botswana from November 26 to 28, 2003 which culminated into a book with the same title published in 2007 by the Council for the Development of Social Science Research in Africa (CODESRIA) based in Dakar, Senegal. The tenor of the workshop and subsequent book is that the twin themes of indigenous knowledge systems and intellectual property rights have moved to the center of academic discourse within the context of innovation and the commercialization of knowledge. This is because wealth is no longer reckoned in terms of physical assets alone. Unfortunately, the traditional imbalance between the North and the South, which has for long manifested itself mainly through trade, is replicated even in tapping intellectual property given to residents of the developing world who remain largely unable to define their property rights. Once again, the West exploits Africa and the rest of the developing world by expropriating indigenous knowledge systems and patenting them in the West. (17)

Various scholars have suggested many major concepts to underlie the Pluridisciplinary Methodology, but it is Dan Nabudere who has provided the most succinct definitions and discussions for most of these concepts. They are as follows:

(a) African Spirituality refers to those aspects of people that have enabled them to survive as a human community throughout the centuries. It transcends European classical humanism with its class, socioeconomic and geographical limitations based on Greece and the Athenian City-State, which is based on a system of slavery. African Spirituality leads to enlarged humanities and recaptures the original meaning of humanity which Western scholars, beginning with Plato, in their hollow and lopsided search for material progress, have abandoned. (18)

(b) Contemporary African Philosophy is a critique of the Eurocentric "idea" and "general philosophy" in its metaphysical perception that European humanism is superior to that of the African people. This falsehood, which has been perpetuated by Europe to this day, hinges upon the belief that the rest of humanity has to be forced to believe like Europe in order to be "humanized" into a singular humanity. Contemporary African Philosophy seeks to "de-structure" this European pretext and emphasize humankind's "shared humanity." (19)

(c) The African Renaissance is the initiative to recapture the basic elements of African humanism (ubuntu, eternal life, and immanent moral justice) as the path to a new humanistic universalism. This initiative, according to Chancellor Williams, "is the spiritual and moral element, actualized in good will among men (and women), which Africa itself has preserved and can give to the world." (20)

(d) The Pan-Afrikan University does not begin in a vacuum, for it has a deep heritage of culture and "civilizational" values that must inform its recreation (e.g., the Sankore University in Timbuktu). These institutions are to be found within Africa's ancient achievements. They must be unearthed and reclaimed. If the Pan-Afrikan University is to respond to this historic challenge and be a part of the correction of its historical distortion and theft of African heritages, it has to provide deeply thought out and well-conceived vision and mission, with a well articulated strategy to achieve its objectives. For it to be successful, it must be a part of the creation of a counter-hegemonic discourse which can enable the "triple agenda of deconstruction, reconstruction, and regeneration" to be undertaken at the same time. Consequently, the Pan-Afrikan University must develop the University as a new institution of higher education, which can help in reshaping the direction of education on the continent toward a more culture-specific and culturally relevant curriculum and pedagogy of liberation. It must draw from those heritages and provide the students, adult learners and the communities with a space in which they can learn as well as carry out their research and be trained by their teachers, community experts, and consultants at the University campuses as well as in the community knowledge sites. Essentially, the Pan-Afrikan University must be people-centered and community-based in which everyone enjoys the freedom to learn and speak. (21)

(e) African Epistemology and Cosmology imply the development of an all-inclusive approach which recognizes all sources of human knowledge as valid within their own contexts. This calls for the adoption of hermeneutic philosophy in its African essence. This African-based epistemologicai and cosmological foundation is the prerequisite for the production and development of knowledge. (22)

(f) African Humanism/Ubuntu is a concept from the Southern African Nguni language family (IsiNdebele, IsiSwati/IsiSwazi, IsiXhosa and IsiZulu) meaning humanity or fellow feeling; kindness. Ubuntu serves as the spiritual foundation of African societies. It is a unifying vision or worldview enshrined in the maxim umuntu ngumuntu ngabantu: i.e. "a person is a person through other persons." This traditional African aphorism, which can be found in every corner of the continent, articulates a basic respect and compassion for others. It can be interpreted as both a factual description and a rule of conduct or social ethic. It both describes the human being as "being-with-others" and prescribes what that should be. (23)

(g) African Languages are at the center of developing the Pan-Afrikan University at all knowledge sites. Language, as Amilcar Cabral correctly pointed out, is at the center of articulating a people's culture. He stated that the African revolution would have been impossible without Africans resorting to their cultures to resist domination. Thus, culture is a revolutionary force in society. It is because language has remained an "unresolved issue" in Africa's development that present day education has remained an alien system. As Frantz Fanon put it, "to speak a language is to assume its world and carry the weight of its civilization." Kwesi K. Prah has argued consistently that the absence of African languages in the curriculum has been the "key missing link" in the continent's development. Consequently, the Pan-Afrikan University must build its curriculum on the basis of promoting African languages at the sites of knowledge and at the same time try to build libraries at those sites in the languages of the people living there. They must be promoted as languages of science and technology. This calls for the complete revamping of the epistemological and cosmological worldview of the current discourse. It also calls for the application of different methodological and pedagogical approaches to learning and research in African conditions. (24)

(h) New Humanities is to serve as the core department in the division of the Pan-Afrikan University concerned with research and advanced studies. In the words of Chancellor Williams, the New Humanities "will have the task of enlisting the services of the world's best thinkers of the work of developing a science of humanity through studies expressly aimed at better human relations. It is to be at the heart of the entire education system and, therefore, the nation." Williams believes that the central idea in this philosophy is life. He argues that since neither Western science nor religion has provided satisfactory answers to three questions (From where do we come? Why? And where are we bound?), it is imperative for the Pan-Afrikan University to provide the space for discussing these eternal questions. This approach calls for the reorganization of the disciplines of the social and human sciences as well as the natural sciences into a holistic learning process. The reorganization should lead to a breaking down of the over-compartmentalization and over-fragmentation of faculties, departments, and branches of knowledge. It should explore the reunification of allied disciplines (which have been subdivided into sub-disciplines) into unified fields of study. (25)

(i) Hermeneutic Philosophy recognizes the basic unity of human endeavor through "discourse" that expresses "the intelligibility of Being-in-the world." (26)

(j) Integrated and Synthesized Knowledge is based on the notion that privileging African-centered curriculum must transcend a narrow conception of what is purely African to include such knowledge within the wider synthesized framework of global knowledge. (27)

(k) Afrikan-based Pedagogy draws inspiration and materials for learning from real life situations of the African people, especially in the rural areas, by adopting those pedagogical methods and techniques that inform their philosophy of life, their worldview, and their lived experiences and practices. The key to developing an Afrikan-based Pedagogy hinges upon the knowledge specific-sites where African experts of different branches of knowledge are located. These sites will inform both the content and the pedagogy. The pedagogy will incorporate "oracy," which contains forms of art and techniques to which they give expression, which is essential for adult learning. By mainstreaming this form of expression, its agents gain visibility and recognition in knowledge creation and production. This will enable indigenous tales, stories, proverbs, legends, myths, symbols and epics to be resuscitated, for these forms of knowledge incorporate people's philosophies of life, norms, values in a kind of "moving" and "living library." (28)

(l) Life Long Learning, which has recently become a mantra of many developed countries and international organizations as a novel approach to learning in the 21st Century, is deeply embedded within African culture and epistemology. Learning and "culturalization" in African societies were considered continuing processes that "took place from birth until death with the family unit, extended family, the village and the entire community participating." (29) Life Long Learning will bring adult learners to formal institutions of learning and remove the division between informal, non-formal, and formal education in line with African traditions and culture. It will also provide for the cooperation in research between the Pan-Afrikan University and the communities, in addition to providing for the recognition of learning outcomes gained through their own contexts outside the formal education system. (30)

(m) Kemetic Civilization is a Black African civilization whose origination in the Fertile African Crescent was unified from its foundations in the Sahara up to its contemporary manifestations in the languages and culture of Black Africans. (31)

The favored methodological approach for pluridisciplinary studies is Hermeneutics, an open-ended approach that permits cross-cultural communication and exchange of ideas and opinions to promote understanding between all knowledge systems in their diversities. This African philosophical-pedagogic approach hinges upon the acceptance of pluralism and cultural diversity. It stresses the need for the "fusion of historical horizons" as the best way of transmitting understanding between different lived histories or experiences of different communities as the basis of their existence. It insists on both the cultural context and the historical contingencies of events as necessities for a true comprehension of the different lived experiences. Furthermore, the approach has its roots in the African/Egyptian mythical figure of Hermes, the messenger of knowledge from the gods to mortals and the interpreter of the divine message to humankind, and that is why Hermeneutics is named after Hermes. (32)

Hermeneutics is to be employed on the premises that encourage self-directed learning, which engages with the knowledge, interests, and real life situations that learners bring to their learning situations. This notion of site-specific knowledge attempts to offer a corrective to the Eurocentric tendency of universalizing knowledge around Occidental centers and sites of knowledge which are privileged to the disadvantage of others, claiming to be the only sites of "rationality" and "scientific knowledge." Recognizing the other sites and centers leads to a truly multi-polar world of global knowledge culled from all sources of human endeavor. (33)

Linguistic Presupposition as the Unit of Analysis

As stated earlier, the unit of analysis for the present essay is linguistic presupposition, which can be defined as an implicit assumption about the world or background belief upon which the truth of a statement hinges. The linguistic presuppositions for this study are drawn out of the writer's/Mwalimu Isaacs's topics in the text examined. The writer's topics here are the a priori features, such as the clear and unquestionable change of subject focus, for defining types of linguistic presuppositions found in the text examined. While there are many other formulations of 'topic' from which to chose, the writer's topics are employed for this essay because it is the writer who had topics, not the text. The other formulations of 'topic' include sentential topics, discourse topics, presuppositional pools, relevance and speaking topically, topic boundary markers, paragraphs, paratones, representation of discourse content, position-based discourse content, and story. Thus, the notion of 'topic' in the present essay is considered as one related to representations of discourse content.

In choosing the writer's topic as the recording unit, the ease of identifying topics and correspondence between them and the content categories were seriously considered. Guiding this choice was the awareness that if the recording unit is too small, such as a word, each case will be unlikely to possess any of the content categories. Furthermore, small recording units may obscure the context in which a particular content appears. On the other hand, a large recording unit, such as a stanza, will make it difficult to isolate the single category of a content that it possesses. For the current essay, two methods were appropriate. First, there is the clear and uncontestable change of subject focus. Second, topicalization was found to have been used to introduce new characters, ideas, events, objects, etc.

Finally, in order to ascertain the reliability of the coding unit employed for the essay, attempts were made to show inter-coder reliability: that is, two or more analysts, using the same procedures and definitions, agree on the content categories applied to the material analyzed. Two individuals, who had extensive training in discourse analysis and especially topic identification, were given copies of the text studied to identify what they perceived as topics, or more specifically, where one topic ends and another begins. Although there were no differences between the two individuals and I, the identified topics and the texts were also given to a linguist who has done a great deal of work on topic analysis for comments and suggestions. This approach was quite useful for increasing my confidence that the meaning of the content is not heavily dependent on his analysis alone.

After identifying the presuppositions in the text studied in terms of the topics identified, these propositions were placed into two categories (order versus disorder) based on the bottom-up processing approach common in linguistic analysis for further examination. This involved working out the meanings of the propositions already processed and building up composite meanings for them.

Because the text examined is a representation of discourse in text, the level of analysis is naturally the written text. Text is used here as a technical term--in Gillian Brown and George Yule's conceptualization, "the verbal record of a communicative act." (34)

In order to ascertain the presuppositions and in the text examined, the test known as Constancy under Negation Rule was employed. This test is important because, following Gottlob Frege (35) and Peter Strawson, (36) presuppositions are preserved in negative statements or sentences. A researcher can therefore simply take a sentence, negate it, and see what inferences survive: that is, are shared by both positive and negative forms of the sentence. But because, as Stephen Levinson is quite correct in pointing out, "constancy under negation is not in fact a rich enough definition to pick out a coherent, homogenous set of inferences," the tests for presuppositional defeasibility (the notion that presuppositions are liable to evaporate in certain contexts) and the projection problem of presuppositions (i.e. the behavior of presuppositions in complex sentences) were also employed. (37)

Consequently, in order not to necessarily presume the conclusions to be drawn, cues to the intent of the author of the text examined are 'deconstructed.' How, then, are these cues mapped out for the present essay? According to Herbert Paul Grice's characterization of meaning or non-natural meaning (which is equivalent to the notion of intentional communication), intent is achieved or satisfied by being recognized. A sender's communicative intent becomes mutual knowledge to sender and receiver: that is, S knows that H knows that S knows that H knows (and so ad infinitum) that S has this particular intention. (38) So following Roger Shuy, it is necessary to begin by asking "What did the writers (here, Mwalimu Isaacs) do"? (39) Thus, it is clearly necessary to look at specific topics developed by the author of the text analyzed. This is particularly true because, according to Wallace Chaf (40) and Carol Kates, (41) the structure of intentions can neither be' defined by the grammatical relations of the terms, nor the semantic structure of a text. Therefore, mapping out the cues to the intent of the author contained in the text analyzed called for: (a) identifying communicative functions, (b) using general socio-cultural knowledge, and (c) determining the inferences made.

Fractal Methodology

It is only logical to begin any discussion of Fractal Methodology with a definition of what a fractal is. As I state in my book, Chaos Theory and African Fractals, the concept of fractal remains inexplicably defined. (42) This shortcoming is pointed out by Philip Davis as follows, albeit he himself does not provide and explicit definition: "I consulted three books on fractals. Though there were pictures, there was no definition" (43) The following is a small sample of the various ways the concept of fractal has been described as provided by Lynn Steen:

   The concept of fractional dimension, or fractals, was
   developed in order to describe the shapes of natural
   objects.... An interesting property of fractal objects is that as
   we magnify a figure, more details appear but the basic shape
   of the figure remains intact. (44)

In addition, according to Steen,

   The word fractal--coined by (Benoit B.) Mandelbrot--is
   related to the Latin verb frangere, which means "to break."
   The ancient Romans who used frangere may have been
   thinking about the breaking of a stone, since the adjective
   derived from this action combines the two most obvious
   properties of broken stones--irregularity and fragmentation.
   The adjectival form is fractus, which Mandelbrot says led
   him to fractal. (45)

Furthermore, as Steen points out, "Fractal dimension (is) a measurement of the jaggedness of an object." (46)

Keith Weeks (in Hargittai and Pickover, 1992) states:

   [J. E.] Hutchinson laid the foundations of a certain concept
   of self-similarity, the basic notion being that of the object
   made up of a number of smaller images of the original
   object, and so on ad infinitum, typically resulting in detail at
   all levels of magnification, a trait commonly associated with
   objects referred to as fractals. (47)

From the preceding descriptions, I venture to offer a general definition of a fractal as a self-similar pattern: that is, a pattern that repeats itself on an ever diminishing scale.

As for Fractal Methodology, more popularly referred to as Fractal Analysis, itself, with its applications in the social sciences, Clifford Brown and Larry Liebovitch in their recent work appropriately titled Fractal Analysis published as part of the Sage Publications Quantitative Analysis of the Social Sciences series have a succinct expose on the subject. (48) The rest of the discussion in this section is based on their work.

Brown and Liebovitch begin by stating that several early applications of fractal mathematics emerged in the social sciences. These works include Vilfredo Pareto's 1897 study of the distribution of wealth; Lewis Fry Richardson's 1948 and 1960, but published posthumously, study of the intensity of wars; and George Zipf's 1949 studies of the distributions of word frequencies and city sizes. Brown and Liebovitch argue that while these ideas were known by experts in the field, they were isolated, quirky concepts until Mandeibrot developed the unifying idea of fractals in the 1970s and 1980s. Since that time, however, in spite of the fact that Zipfand Pareto distributions represent fractal distribution, social scientists have lagged behind the physical and natural sciences in utilizing fractal mathematics in their works. (49)

Brown and Liebovitch observe, however, that in recent years, the application of fractal mathematics by social scientists in their studies has grown exponentially. Their variety, they note, has expanded as rapidly as their numbers. They cite the examples that fractal analysis had been employed by criminologists to investigate the timing of calls for assistance to police, by sociologists to investigate gender divisions in the labor force, and by actuaries to study disasters. The surprising range of fractal phenomena in the social sciences led Brown and Liebovitch to call for a comprehensive survey that would investigate the common threads that unite them, thereby leading to a broader understanding of their causes and occurrences. (50)

According to Brown and Liebovitch, if a researcher has rough data, strongly nonlinear data, irregular data, or data that display complex patterns that seem to defy conventional statistical analysis, then fractal analysis might be the solution to the researcher. They posit that the non-normal and irregularity of so much of social science data apparently are the result of the complexity of social dynamics. Thus, for them, fractal analysis offers an approach for analyzing many of these awkward data sets. And more important, they note, the method also offers a rational and parsimonious explanation for the irregularity and complexity of such data. They insist that the data are not behaving badly; instead, they are simply obeying unexpected but common rules of which we are unaware. (51)

Brown and Lieboviteh go on to conceptualize fractals as "'sets defined by the three related principles of self-similarity, scale invariance, and power law relations." They postulate that when these principles converge, fractal patterns form. They note that the statistic called fractal dimension is employed to capture the essential characteristics of fractal patterns. They add that much empirical work in fractal analysis focuses on two tasks: (1) showing that fractal characteristics are present in a particular data set and (2) estimating the fractal dimension of the data set. They also mention that there are various techniques for implementing these two tasks (Brown and Liebovitch, 2010:2), the discussion of which is beyond the scope of the present essay. Nonetheless, it is necessary to provide brief definitions of the preceding five italicized concepts based on Brown and Liebovitch's work for the sake of clarity. The significant fact about sets is that almost all data sets can be fractal: that is, points, lines, surfaces, multidimensional data, and time series. Since fractals occur in different types of sets, various procedures are required to identify and analyze them, with the approach hinging upon the kind of data. (52)

Brown and Liebovitch define self-similarity as a characteristic of an object when it is composed of smaller copies of itself, and each of the smaller copies in turn are made up of yet smaller copies of the whole, and so on, ad infinitum. The word similar connotes a geometrical meaning: that is, objects that have the same form but may be different in size. (53)

Scale invariance for Brown and Liebovitch refers to a thing that has the same characteristics at every scale of observation. Thus, when one zooms on a fractal object, observing it at ever-increasing scale of magnification, it will still look the same. (54)

According to Brown and Liebovitch, power law relations denote the rule that for a set to achieve the complexity and irregularity of a fractal, the number of self-similar pieces must be related to their size by a power law. Power law distributions are scale invariant because the shape of the function is the same at every magnitude. (55)

Finally, Brown and Liebovitch characterize fractal dimension as the invariant parameter that characterizes a fractal set. An analyst uses the fractal dimension to describe the distribution of the data. It is akin to having a "normal" set of data and using the mean and variance to describe the location and dispersion of the data. (56)


Before engaging in the fractal analysis of the data generated from Mwalimu Isaacs's text, I will begin with a discussion of the descriptive and inferential statistics employed to analyze them first. Before computing the univariate and bivariate statistics to do the descriptive analyses of the data teased out of Mwalimu Isaacs's text, a two-dimensional ad hoc classificatory system was developed within which the data were categorized. The first of these categories entails the presuppositions of order: that is, presuppositions that suggest a condition of logical or comprehensible arrangement among the separate elements of a group. This type of presupposition is triggered by presuppositional discourse stretches such as "Diaz considered Gonzalez to be his most trusted and loyal subordinate, a factor which contributed greatly toward Gonzalez' becoming President in 1880," "The years between 1880 and 1884 marked the first period of any significant length when relations between the two nations (i.e. Mexico and the United States) were cordial," and "Mexico exhibited a friendlier attitude toward the United States during the French intervention." The second category encompasses presuppositions of disorder: that is, presuppositions that suggest a condition of confusion, mess, disturbance, disarray, or muddle. This type of presupposition is triggered by presuppositional discourse stretches such as "According to many writers, including Henry Bamford Parkes, James Creelmen, Daniel James, Ernest Gruening, and Wilfrid Callcott, Gonzalez was a scandalous and corrupt individual whose sole significance was in making the Mexican people realize that only Diaz could rule Mexico in a successful manner," "Previous to 1867 the principal disruptive factor in the relations of the United States with Mexico was American determination to carry out what she considered her 'manifest destiny' to round out the continental boundaries of the United States; the object required the acquisition of Mexico's northern territories which were absorbed into the American union as a result of the annexation of Texas and the Mexican War," and "Unfortunately, after the withdrawal of the French, border troubles plagued relations for a number of years."

After the descriptive and inferential statistics were computed, the data were then plotted for oscillations between order and disorder in the book. This technique made it possible to show visually the attractor reconstruction for the text.

As shown in Table 1, a total of 1,437 topic entries were teased out of Mwalimu Isaacs's text. Of these, 1 categorize 890 or 62 percent as presuppositions of order and 547 or 38 percent as presuppositions of disorder. There are more presuppositions of order than presuppositions of disorder for all but one chapter in the text. The mean for the order category is about 111 presuppositions, with a standard deviation of approximately 51 presuppositions; the mean for the disorder category is about 68 presuppositions, with a standard deviation of approximately 25 propositions. The range for the order category is 145 presuppositions and that for the disorder category is 88 presuppositions, while the variance for order is about 2,608 presuppositions and that for disorder is approximately 603. This means that there are more (in fact, almost twice as many) and statistically significant topic entries for presuppositions of order than there are of those for disorder. But given the fact that the number of presuppositions of disorder is significantly large, it would be farfetched to assert that Mwalimu Isaacs was romanticizing the Mexican society. Moreover there are significant variations among the chapters for each category in terms of topic entries, as can be gleaned from the ranges.

From Table 2, it can be seen that there is a statistically significant difference between the topic entries for order and those for disorder at the 0.01 level. It is also evident that there is a moderately or substantially positive correlation between the two dimensions, albeit not statistically significant at the 0.05 level. In essence, even though the order dimension is dominant, but as it increases, so does the disorder dimension, even though not at the same rate.

As can be seen from Figure 1, a log-log plot (or log-log graph) was employed to represent the observed units described by the two-dimensional variable encompassing order (y) and disorder (x) as a scatter plot/graph. The two axes display the logarithm of values of the two dimensions, not the values themselves. If the relationship between x and y is described by a power law,

y = [x.sup.a];

then the (x, y) points on the log-log plot form a line with the slope equal to a. Log-log plots are widely used to represent data that are expected to be scale-invariant or fractal because, as stated before, fractal data usually follow a power law.

A logarithm is an exponent. It is illustrated in the following definition:

For b>0, b [not equal to] 1 and for x>0,

y = [log.sub.b] x if and only if [b.sup.y] = x

Thus, since a logarithm is an exponent, it is easy to use exponent laws to establish mathematical generalizations.

Figure 1 illustrates the fractal dimension of the two-dimensionality of the variable. The binary logistic statistics reveal that the relationship between the two dimensions is moderate or substantial, albeit not statistically significant at the 0.05 level. In sum, Mwalimu Isaacs's text moves halfway across the spectrum--it typically moves from periodic fractal, rather than stretching all the way to pure order or disorder. In essence, the results generated after the MATLAB computer runs suggest that the combination of negative and positive feedback loops, which form the basis of several African knowledge systems, as Ron Eglash (57) suggests, also form a key mechanism of general self-organizing systems discussed in United States-Mexican Relations.

As I mentioned in the introduction of this essay, other works on the topic had characterized Mexico as having been extremely chaotic and backward--racist postulates that are disputed by the preceding findings in Mwalimu Isaacs' text. These other works' perspectives, which have been discussed by Gilbert G. Gonzalez in his article tilted "The Ideology and Practice of Empire: The U.S., Mexico, and the Education of Mexican Immigrants," (58) can be classified into three categories. The first category is the "peaceful conquest": the belief that Mexico is so extremely chaotic and backward that "any progress in Mexico must come through colonization by some higher and more progressive race (i.e. "the Anglo-Saxon--or rather the Anglo-American"), or by the introduction of capital in large amounts to develop her natural resources by the aid of native races." (59) The second category is the "Oriental": the perception that Mexicans are "exotic, poor, strange, appealing, possibly loathsome, and for sure a subordinate people, practicing an impenetrable culture." (60) And the third category is the "sleepy peons and sad eyed burros": the belief that a Mexican is "someone who walks rather than rides a horse (a caballero)," replacing the old notion of the "standby greaser," and a "Southern darkey." (61)

Indeed, the preceding findings make it plausible to include Mwalimu Isaacs into the school of thought I characterize as atenu in Ancient Egyptian/Hieroglyphics or Mapinduzi in Kiswahili or Revolutionary in English, as opposed to either the atenu m 'ten in Ancient Egyptian/Hieroglyphics or Mapinduzi ya Malazi in Kiswahili or Revolutionary-Accommodationist in English, or the kheperu in Ancient Egyptian/Hieroglyphics or Kubadilisha in Kiswahili or Reformist in English, in my paper titled "Pan-Blackist Conceptualizations of the Black Power Paradigm: From Cheikh Anta Diop to Ali Al'amin Mazrui." I define these concepts in the essay as follows: the term atenu was employed by Ancient Egyptians to describe revolutionaries, rebels or fiends who wanted radical change. Such people were perceived as Mesti, the divine parents of the God of Sun or Day Ra; Mesu, the gods who begat their own fathers or divine beings; and Mesut, children of God Osiris or divine beings. The concept atenu m 'ten was employed by Ancient Egyptians to refer to those who wanted change but would accept things, listened to, obeyed, or be content with things as long as their burdens were assuaged. The word kheperu for Ancient Egyptians described those who sought change in form, manifestation, shape, similitude, or image. (62)

The justification for my placing of Mwalimu Isaacs in the atenu or Mapinduzi or Revolutionary school of thought is further glaringly evident in the first sentence of his concluding chapter when he emphasizes justice as follows: "This study has showed that the historians' neglect of United States-Mexican relations during the Gonzalez administration has not been justified." (63) As Joshua Goldstein and Jon Pevehouse postulate, scholars with a revolutionary worldview tend to value justice, while those with a conservative worldview value order and those with a liberal worldview value freedom. (64)


The data gleaned from Mwalimu Isaacs's United States-Mexican Relations made it possible to explore fractal patterns embedded in two dimensions: (1) order and (2) disorder. The substantive findings, as stated earlier, reveal that Mwalimu Isaacs's text moves halfway across the spectrum--it typically moves from periodic fractal, rather than stretching all the way to pure disorder. It can therefore be suggested that the combination of negative and positive feedback loops, which form the basis of several African knowledge systems, as Ron Eglash (65) suggests, also form a key mechanism of general self-organizing systems discussed in United States-Mexican Relations. in essence, Mwalimu Isaacs's framing of his characters' experiences is reminiscent of African ways; despite the challenges and hardship, their thought processes never became extremely chaotic.

Indeed, as Kofi Nyidevu Awoonor (66) and also Bangura (67) posit, the African life concept is holistic--that is, it is based on an integrative world view. All life to the African is total; all haman activities are closely interrelated. This has as its underlying principle the sanctity of the person, her/his spirituality and essentiality. This essentialist view of the person confers value to her/his personhood. All else--her/his labor and achievements--flow from this value system. Even personal shortcomings cannot invalidate it.

In addition, Awoonor and also Bangura (68) point out that for Africans, politics defines duties and responsibilities alongside obligations and rights. All these relate to the various activities that have to do with survival. The survival concept is continuing, dynamic and dialectical. The fundamental principle that is at the basis of this conception is a moral one. Moreover, the African moral order never defined rigid frontiers of good and evil. Good and evil exist in the same continuum. Whatever is good, by the very nature of its goodness, harbors a grain of evil. This is a guarantee against any exaggerated sense of moral superiority which goodness by itself may entail. The notion of perfection, therefore, is alien to African thought. Perfection in itself constitutes a temptation to danger, an invitation to arrogance and self-glorification. The principle of balance defines the relationship between good and evil. As life operates in a dialectics of struggle, so also does good balance evil and vice versa.

Thus, the essence of an African-centered approach is that it is imperative and urgent for Africans to be concerned about broader development as well as approaches to development that are undergirded by humanity or fellow feeling toward others. When African-centeredness is considered along with the idea of the socialization effects of developmental environments and the possibilities of a reinforcement of these notions and contexts, the implications for an African development process appear vital.

Although compassion, warmth, understanding, caring, sharing, humanness, etc. are underscored by all the major world orientations, African-centered thought serves as a distinctly African rationale for these ways of relating to others. African-centeredness gives a distinctly African meaning to, and a reason or motivation for, a positive attitude towards the other. In light of the calls for an African Renaissance, African-centeredness urges Africans to be true to their promotion of good governance, democracy, peaceful relations and conflict resolution, educational and other developmental aspirations.

We ought never to falsify the cultural reality (life, art, literature) which is the goal of African-centeredness. Thus, we would have to oppose all sorts of simplified or supposedly simplified approaches and stress instead the methods which will achieve the best possible access to real life, language and philosophy.

Final Words:

And if I may borrow his favorite verbum (or word) he uses to extol the effulgence of others, it is only fitting to say that Mwalimu Harold Isaacs is simply a SUPERLATIVE Afrikan!


(1.) Harold Isaacs, Curriculum Vitae of Harold Isaacs, retrieved on January 29, 2012 from 2010/08/HIVITA2010.pdf

(2.) Harold Isaacs, United States-Mexican Relations during the Gonzalez Administration, 1880-1884 (doctoral dissertation, University of Alabama, Tuscaloosa, Alabama, 1968).

(3.) Ibid, p. 6.

(4.) Clyde Ahmad Winters, "The Afrocentric Historical and Linguistic Methods," The Western Journal of Black Studies, (1998), Issue 22. Consult also the following: Cheikh Anta Diop, Precolonial Black Africa: A Comparative Study of the Political and Social Systems of Europe and Black Africa, from Antiquity to the Formation of Modern States, translated from the French by Harold Salemson (Chicago, IL: Lawrence Hill Books, 1960/trans. 1987); Cheikh Anta Diop, The Cultural Unity of Negro Africa (Paris, France: Presence Africaine, 1962); Cheikh Anta Diop, Black Africa: The Economic Basis for a Federated State (Chicago, IL: Lawrence Hill/Africa World Press, 1974); Cheikh Anta Diop, The African Origin of Civilization: Myth and Reality (Chicago, IL: Lawrence Hill/Africa World Press, 1974); Cheikh Anta Diop, Civilization or Barbarism: An Authentic Anthropology (Brooklyn, NY: Lawrence Hill Books, 1981/trans. 1991).

(5.) G. Mokhtar, Ancient Civilizations of Africa (Berkeley, CA: University of California Press, 1990).

(6.) Ibid

(7.) Clyde Ahmad Winters, "The Afrocentric Historical and Linguistic Methods.

(8.) Ibid.

(9.) Ibid.

(10.) Ibid.

(11.) Ibid

(12.) Ibid.

(13.) Ibid.

(14.) Dani W. Nabudere, "Towards the Establishment of a Pan-African University: A Strategic Concept Paper," African Journal of Political Science, Vol. 8, No. 1 (2003), pp. 1-30.

(15.) Ibid.

(16.) Ibid.

(17.) Isaac Mazonde and Pradip Thomas (eds.), Indigenous Knowledge Systems and Intellectual Property in the Twenty-First Century: Perspectives from Southern Africa (Dakar, Senegal: Council for the Development of Social Science Research in Africa [CODESRIA] Publications, 2007).

(18.) Dani W. Nabudere, "Towards the Establishment of a Pan-African University: A Strategic Concept Paper."

(19.) Ibid.

(20.) Ibid.

(21.) Ibid.

(22.) Ibid.

(23.) Abdul Karim Bangura, "Ubuntugogy: An African Educational Paradigm that Transcends Pedagogy, Andragogy, Ergonagy, and Heutagogy," Journal of Third World Studies, Vol. xxii, No. 2, (2005), pp. 13-53; Abdul Karim Bangura, African Peace Paradigms (Dubuque, IA: Kendall Hunt Publishing Company, 2008).

(24.) Dani W. Nabudere, "Towards the Establishment of a Pan-African University: A Strategic Concept Paper."

(25.) Ibid.

(26.) Ibid.

(27.) Ibid.

(28.) Ibid.

(29.) Ibid.

(30.) Ibid.

(31.) Clyde Ahmad Winters, "The Afrocentric Historical and Linguistic Methods."

(32.) Dani W. Nabudere, "Towards the Establishment of a Pan-African University: A Strategic Concept Paper."

(33.) Ibid.

(34.) Gillian Brown and George Yule, Discourse Analysis (Cambridge, UK: Cambridge University Press, 1983), p. 6.

(35.) Gottlob Frege, "On Sense and Reference," in P. T. Geach and M. Black (eds.), Translations from the Philosophic Writings of Gottlob Frege (Oxford, England: Blackwell, 1892/1952).

(36.) Peter F. Strawson, Introduction to Logical Theory (London, England: Methuen, 1952).

(37.) Stephen Levinson, Pragmatics (Cambridge, UK: Cambridge University Press, 1983), p. 185.

(38.) Herbert Paul Grice, "Logic and Conversation," in P. Cole (ed.), Syntax and Semantics: Vol. 3: Speech Acts (New York, NY: Academic Press, 1975).

(39.) Roger W. Shuy, "Topic as the Unit of Analysis in a Criminal Law Case," in Deborah Tannen (ed.), Analyzing Discourse: Text and Talk (Washington, DC: Georgetown University Press, 1982).

(40.) Wallace L. Chafe, "Discourse Structure and Human Knowledge," in R. O. Freedle and J. B. Carroll (eds.), Language Comprehension and the Acquisition of Knowledge (Washington, DC: V. M. Winston, 1972).

(41.) Carol A. Kates, Pragmatics and Semantics: An Empiricist Theory (Ithaca, New York: Cornell University Press, 1980).

(42.) Abdul Karim Bangura, Chaos Theory and African Fractals (Washington, DC: The African Institution, 2000), p. 6.

(43.) Philip J. Davis, Spirals: From Thoedorus to Chaos (Wellesley, MA: A. K. Peters, 1993), p. 22.

(44.) Lynn A. Steen (ed.), Introduction to Contemporary Mathematics (New York, NY: W. H. Freeman and Company, 1988), p. 409.

(45.) Ibid., p. 420.

(46.) Ibid., p. 413.

(47.) Istvan Hargittai and Cliff A. Pickover (eds.), Spiral Symmetry (Singapore, Malaysia: World Scientific Publishing Company, 1992), p.107.

(48.) Clifford Brown and Larry Liebovitch, Fractal Analysis (Thousand Oaks, CA: Sage Publications, 2010).

(49.) Ibid., p. ix.

(50.) Ibid

(51.) Ibid, p. 1.

(52.) Ibid., pp. 2-3.

(53.) Ibid., p. 3.

(54.) Ibid, p. 5.

(55.) Ibid.

(56.) Ibid, p. 15.

(57.) Ron Eglash, African Fractals: Modern Computing and Indigenous Design (New Brunswick, NJ: Rutgers University Press, 1999), pp. 173-4.

(58.) Gilbert G. Gonzalez, "The Ideology and Practice of Empire: The U.S., Mexico, and the Education of Mexican Immigrants," Cultural Logic, Vol. 4, No. 1 (Fall 2000/2001).

(59.) Gonzalez, 2000/2001; F. E. Prendergast, "Railroads in Mexico," Harper's New Monthly Magazine, Vol. lxiii (1881), p. 276; Nevin O. Winter, Mexico and Her People Today, (Boston, MA: L. C. Page, 1907), p. 53; Alan Knight, U.S.-Mexican Relations, 1990-1940 (San Diego, CA: University of California Press, 1987), p. 26.

(60.) Gonzalez, 2000/2001; George B. Winton, Mexico Today: Social, Religious, and Political Conditions (New York, NY: Missionary Education Movement, 1913), p. 2; Mary Elizabeth Blake, "Picturesque Mexico," in Mary Elizabeth Blake and Margaret F. Sullivan (eds.), Mexico (New York, NY: Lee and Shepherd Publishers, 1888), p. 39; Mary Barton, Impressions of Mexico with Brush and Pen (London, England: Metheun and Company, 1911), p. 20.

(61.) Gonzalez, 2000/2001; Percy F. Martin, Mexico of the Twentieth Century (London, England: Edward Arnold, 1907), p. x; Frederick Simpich, "The Little Brown Brother Treks North," The Independent, Vol. 116, No. 3 (1926), p. 238; Alan H. Rogers, "Character and Habits of Mexican Miners," Engineering and Mining Journal, Vol. 85, No. 14 (April 14, 1908), p. 700; "The Riots at Cananea," Engineering and Mining Journal, Vol. 81, No. 23 (June 9, 1906), p. 1099; Eva Frank, "The Mexican 'Just Won't Work'," The Nation, Vol. 125, No. 3241 (July 17, 1927), p. 156; J. Nelson Nevius, "Letter to the Editor," Engineering and Mining Journal, Vol. 81, No. 23 (June 9, 1906), p. 213; Frank H. Probert, "The Treasure Chest of Ancient Mexico," The National Geographic, Vol. 30, No. 1 (July 1916), p. 43.

(62.) Abdul Karim Bangura, "Pan-Blackist Conceptualizations of the Black Power Paradigm: From Cheikh Anta Diop to Ali Al'amin Mazrui," Proceedings of The International Black Power Conference on Reflections, Relevance and Continuity: Caribbean and Global Perspectives of Black Power (convened by the University of West Indies St. Augustine Campus, Trinidad and Tobago, September 18-19, 2010), pp. 1-66.

(63.) Harold Isaacs, United States-Mexican Relations during the Gonzalez Administration, 1880-1884, p. 217.

(64.) Joshua S. Goldstein and Jon C. Pevehouse, International Relations 7th ed. (New York, NY: Pearson Longman, 2004), pp. 8-9.

(65.) Eglash, 1999.

(66.) Kofi Nyidevu Awoonor, Ghana: A Political History (Accra, Ghana: Sedco Publishing Ltd. and Woeli Publishing Services, 1990).

(67.) Abdul Karim Bangura, Sojourner-Douglass College's Philosophy in Action: An African-Centered Creed (San Jose, California: Writers Club Press, 2002).

(68.) Kofi Nyidevu Awoonor, Ghana: A Political History; Abdul Karim Bangura, Sojourner-Douglass College's Philosophy in Action: An African-Centered Creed.

By Abdul Karim Bangura *

* Abdul Karim Bangura is professor of Research Methodology and Political Science at Howard University. He also is researcher-in-residence of Abrahamic Connections and Islamic Peace Studies at the Center for Global Peace in the School of International Service at American University. He holds a PhD in Political Science, a PhD in Development Economics, a PhD in Linguistics, a PhD in Computer Science, and a PhD in Mathematics. He is the author of 68 books and more than 600 scholarly articles. He is fluent in about a dozen African and six European languages, and studying to increase his proficiency in Arabic, Hebrew, and Hieroglyphics. He is the recipient of many teaching and other scholarly and community service awards. He also is a member of many scholarly organizations and has served as President and then United Nations Ambassador of the Association of Third World Studies.

Table 1: Univariate Statistics by Types of Presuppositions in the Text

                         Topic Entries for     Topic Entries
                         Presuppositions of    Presuppositions
                               Order            of Disorder

Chapter                  N              %      N                  %

1: Diplomatic            111            62     67                 38
Relations between
United States and
Mexico 1876-1880

2: Mexican-Guatemalan    99             61     62                 39
Boundary Dispute and
the Role of the
United States

3: Obstacles to          55             42     76                 58
Effective Border
Control and
Diplomatic Problems
Attendant to the A
rehension of Outlaw

4: Indian                180            63     107                37
Depredations During
the Gonzalez

5: Miscellaneous         113            58     82                 42
Border Problems
during the Gonzalez

6: Economic              177            73     67                 27
Communication and

7: Economic              120            64     67                 36
Relations: Mining and
Agricultural Holdings

8: Conclusions           35             65     19                 35

Totals Scores & Mean     890            62     547                38
Percents =
1,437 or 100%

Mean                     111.25                68.38

Standard Deviation       51.071                24.553

Range                    145                   88

Variance                 2,608.214             602.839

Source: Self-generated data from the text and computed by using MATLAB

Table 2: T-Test: Paired Samples Test and Correlation

Pair 1: Order-Disorder                       Paired Differences

Mean                                         42.88
Standard Deviation                           38.720
Standard Error Mean                          13.690
95% Confidence Interval of the Difference

Lower                                        10.5
Upper                                        75.25

t-Static                                     3.132
Degrees of Freedom                           7
Significance 2-tailed                        0.017

Pair 1: Order and Disorder                   Paired Samples

Correlation                                  0.683
Significance                                 0.062

Source: Self-generated data from the text and computed
by using MATLAB
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Date:Mar 22, 2013
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