1. Introduction

African philosophy and philosophy in Africa have come a long way in the past four decades. Philosophy in Africa obviously is easier to comprehend than African philosophy. We imagine that the difficulties surrounding African philosophy as a discipline might have compelled the philosopher P. O. Bodunrin to prefer the title: Philosophy in African to a book he edited and which was devoted to African philosophy. Indeed a colleague once sent a journal article to the United States of America on African philosophy. An angry reviewer lambasted him for using the adjective “African” to qualify philosophy.

We shall discuss this over-flogged problem of the nature and meaning of African philosophy briefly. This work will be divided into three parts. Part one will focus on issues surrounding the notion and substance of African philosophy. Part two will examine some issues in mathematics and Physics in connection with the struggle for African survival. Part three will focus on culture and ethno-philosophy.

2. Part One: The Notion and Substance of African Philosophy

We read about the Chinese philosophy; we stumble on books having titles like German Ideology, French socialists, British empiricists: Indian philosophy, etc. These titles and academic courses with such titles do not generate the kind of intellectual heat you experience in some quarters when it comes to African philosophy. The reason is not unconnected with the jaundiced perception of the African as not being as full-fledged human being. Some prominent philosophers who share the view that African philosophy does not exist or that African philosophy exists in a very rudimentary form include Odera Oruka; Placide Tempels; Sedar Senghor; John Mbiti; Alexis Kagame; Peter Bodunrin and Kwasi Wirendu; Joseph Omoregbe (in Boidunrin 1975) and C. S. Momoh (in Momoh, 1989) have each adequately responded to the positions of those eminent philosophers with doubt the existence of African philosophy.

Omoregbe has criticized Wirendu for maintaining that traditional African society lacks an understanding of logic – he has shown that there are other types of logic besides what is called Aristotelian. It is pertinent to note that during Aristotle’s time, there were other competing methods of doing logic. The Megarian School of Logic established by Euclides and further developed by Eubikides. Stilpo, Diodorus Cronus, Appollonius Crunus and Chrysippus to mention a few, was reputed to be more advanced than Aristotelian logic in the solution of numerous everyday problems (Kneale & Kneale, 1962: 112). We should always bear in mind that these Greek scholars were plagianists and that the position of Aristotle was enhanced by the status accorded him by Alexander the Great. Reading African history from an Afro-centric perspective, it will be noted that whatever ancient Egypt had acquired or developed existed in other parts of Africa, south of Egypt. This point of view has been properly argued for and documented by Chancellor Williams (1986), Cheikh Anta Diop (1974: 1978) and Andah, (1996).

Still examining the place of logic or absence of it in African philosophy, it will be useful to remind ourselves that the traditional laws of thought in logic fail to hold firmly in dialectical logic, quantum logic, fuzzy logic which is related to fuzzy set theory in mathematics in fuzzy logic; you will recall that we have truth values like: True; more or less true; Rather true; Not very true; Not very false; more or less false; rather false…False (Alozie, 2003).

Logic continues to develop. This kind of situation makes it difficult to understand the brand of logic these our “African logical neo-positivists” (as our late dear C. S. Momoh styled them) are referring to. The lesson is that every apprehended phenomenon has a history. Logic, which is human endeavour, has its own history. Such history did not start with our contact with the European and Arab invaders. P. O. Bodunrin, who belongs to this group of African logical neo-positivists have made a statement with which I completely agree. The statement avers that philosophers should not give the impression that Africans are radically different from the rest of mankind in their conceptual system Bodunrin is to give an African intellectual stamp to racism against the Africans.

Colonialism, neo-colonialism, years of slavery and living in slave prison condition have made Africans think they are different from the rest of mankind. African peoples have been terrorized traumatized and policed for the benefit of current league of economically powerful G8 Nations. The terrorization, traumatisation and policing are done by the indigenous “Diverse”, neo-colonial governments that would not bother to destroy whole towns if their expatriate and neo-colonial masters so demand. The difference alluded to is the result of years of myth planting into the consciousness of the black population; cultural invasion which places the conquering culture on top of the conquered peoples’ culture mis-education and media war against blacks. We become “nice guys” or what Fela Kuty will call “Zombies”, who can be manipulated at will. This is probably why Malcolm X lamented the docility of the blacks in the United States of America in the faces of brazen state dehumanization. Examine the situation in Ireland; Middle East; Far East, Asia. You will observe that force is greeted with counterforce. Such people are respected. The docile members of the planet earth are entities for elimination. What Malcolm “X” says of the African Americans equally applies to Black Africans on the African continent. Instead of talking about the black person in this or that country, we shall be talking of the black person anywhere in the world. This in fact would satisfy Malcolm X’s desire to internationalize racial issues in America.

The point has been made by some scholars that Africa is not homogenous culturally. Even an aspect concerning gender has been given almost three different approaches by three Nigerian scholars and philosophers (Amadiume, 1098; Nzegwu 1985; Onyewuenyi, 1997). Amadiume and Nzegwu are Igbo yet their interpretations of the gender issue differ. Nkeonye Otakpor an Igbo male probably unaware of these three female scholars on gender issues has used concrete legal examples to show that Amadiume’s positions are tenable. As if he was reading Nzegwu’s criticism of Amadiume, Otakpor says “same sex marriage particularly of the female kind is part of our history as a people” (p. 43). From the Yoruba end, Onyewuenyi’s position can be seen as a very sophisticated level as being close to Amadiume’s and Otakpor’s of Igbo extraction. And there are other cultural traits that link up the Igbo and Yoruba. But for the past forty years, the two ethnic groups are programmed by the political establishments to regard each other as mortal enemies who also differ from each other greatly. We shall be returning to the issue of culture later. The point that must be noted here is that despite cultural variations, there are some bonds that hold the Africans together. People of the same ethnic group, or indeed identical twins brought up, separately, in divergent socio-cultural environments will necessarily differ culturally. What is very paramount is the colour of the skin, texture of hair, and the person’s physiognomy. A number of mulattoes will be in trouble, given this “mirror test” as Chinweizu will put it. Every mulatto will have to prove, through action and words on which side of the divide such a person belongs. It so happens that those who often torment the black-skinned persons are the off-spring of black skinned and white-skinned persons. In making this racial suggestion, we are conscious that there is nothing like a pure racial form. The works of J. A. Rogers and others have amply illustrated that (Rogers, 1978). In a war situation, this is an instant identification criterion. The black-skinned who take the sides of the perfidious oppressors of their fellow black-skinned persons will be found out by their action and dealt with accordingly. You may call this reverse “racism”. That is needed for the world to understand that racism could be a two-way traffic.

The argument against oral tradition and presumed lack of writing has been dealt with by S. B. Oluwole in her Philosophy and Oral Tradition (Oluwole 1997). She battled successfully in our opinion, against philosophers like Henry Oruka, Peter Bodunrin, Kwasi Wirendu, Paulin Hountondji, etc., who anchored philosophy on written text, and who maintained that writing in Africa is a recent phenomenon. It is however false to state that writing is a recent phenomenon in Africa. Some communities in Africa, today, cannot read or write. The same is true of some communities in Britain, United State of America, France, etc. Circumstances beyond those non-literate communities in the twenty-first century have forced them to persist in their illiteracy. We cannot generalize that the whole of America, Britain, and France do not have written tradition. Similarly, it is fallacious to maintain that Africans are new to the art of writing.

We could excuse scholars who believe that Africans have not developed writing skill if there are no references to what Ancient Africans of black-skin, thick lips, woolly hair have done in the sphere of writing. But there are abundant classical materials on these authored by white-skinned acholars attesting to the fact that the blacks of ancient Egypt, Ethiopia, Meroe, Zimbabwe developed writings. Cheikh Anta Diop labored fruitfully to make modern scholars be aware of the immense contribution of black Africans in the field of writing, science, and technology in antiquity.

In the Cross River state of Nigeria, Nsibidi is a type of writing that has subsisted for many centuries, but transmitted through some fraternities of secret cults. Secret societies like Ekpe, Okonko in the Eastern parts of Nigeria have their writings similar to hieroglyphics but intelligible to members. Centuries of despoliation, terrorization and dehumanization of the African peoples have made them lose most of their sterling qualities. The business of the African philosopher is to think out ways of making these dehumanized Africans regain their humanity.

The major attack on African philosophy by Paulin Hountondji is his insistence that Africans are not grounded in science. We agree that the “metaphysical problems of origins of the world, the meaning of life, the wherefore of death; human destiny, the reality of the beyond, the existence of God, and all those insoluble problems which belong to mythology, yet are the usual fodder of philosophical rumination” should not be the concern of African philosophy during this period when the people are faced with the real threat of extermination (Hountondji, 1976:99). Problems of science, technology, economics, politics, war and peace should engage the Africans and their philosophy. But this agreement on the need for science does not imply that science is absent in Africa’s antiquity or that philosophy did not exist in Africa before contact with the Europeans. Cheikh Anta Diop had done tremendous work unearthing the hidden, and often distorted fact that Western science originated from Black Africa. Diop assembled evidence from archeology, linguistics, chemistry, mathematics, ancient records and writings and from ancient history to demonstrate that Western civilization, including science, were the bye-products of black African civilization. Indeed, Cheikh Anta Diop could be regarded as a one-man intellectual military brigade against Western and imperialist falsifiers of Africa’s history.

Hountondji stated that philosophy is a theoretical discipline and therefore belongs to the same genus as algebra, geometry, mechanics, linguistics, etc. (P.47). Hountondji is still immersed in the outdated mindset of having a dividing wall between “facts” and “theories”. Facts and theories go hand in hand. We re-call the period when some philosophers, notably the positivists, logical positivists and critical rationalists, attempted to demarcate science from non-science. Verification, confirmation, and falsification approaches were tried. These approaches could not provide the test for all purposes in science. What is more, it was found that metaphysics, myth and so-called non-scientific enterprises had some of the attributes of science. Mathematics (including algebra and geometry) is saturated with assumptions and unprovable components. We use the word unprovable from the foray of Kurt Godel who demonstrated that the various respected logical approaches to mathematics will never enable us prove the consistency of mathematical theories. This position shows that there are mathematical theorems that are neither provable nor disprovable.

In philosophy of mathematics, we are aware of the various shades of intuitionism. The intuitionists in general do not hold on to the dogged assignment of “true” or “false” to mathematical statements. There is the third assignment of “undecidable” to mathematical statements from the point of view of intuitionist mathematical philosophers. Our dear Hountondji stated that it is not philosophy but science that Africa needs first (p. 98). Elsewhere he goes on to state that, in the last analysis, philosophy has always been a reflection on science (p.155). We shall, however, use concrete examples in physics and mathematics to show that mathematics and physics could legitimately be known as philosophy.

3. Part 2: Physics

The mathematical formulations of physical theories lend themselves to the procedures of formal proof. Although we are interested here in physical theories, we should remember that mathematics had its own theories too. Mathematical theories, whether they are about infinite series, linear differential equations, partial differential equations, algebra and geometry of vectors, field theory, probability, or numerical analyses, have some common features. These mathematical theories are presumably devoid of any fixed “empirical reality”. This is probably designed to make mathematics a rigorous way of reasoning.

It has been argued, in some quarters, that mathematics is an abstraction of reality. This is partly correct. But the total picture of the origin of mathematics must also be traced to imagination and creative thinking. In some sense, mathematics can be likened to the creative arts with concepts of elegance, form and beauty. Whatever way we perceive mathematical theories, however, we will observe that these theories attempt to mirror the real world. The abstracted, mirrored world is put in axiomatic forms. The axiom systems are supposed to be self-evident truths that do not require further investigations. We now know that there is not such concept or truth that cannot be further investigated with possible useful results. Every mathematical theory contains some axioms and undefined forms, as well as symbols that may be constants or variables as in mathematical logic. The theory will stipulate the various steps to be taken in order to arrive at what is known as proof. The method of proof is stipulated in advance. Physical theories however differ from mathematical theories in that physical theories inject real data and real world or universal physical entities into theories. Sometimes, the assumed physical entities in physical theories do not exist. An example is the quantity of entity found in the equation of displacement current.

This quantity is said not to be a true current, although it were the toga of reality in Maxwell’s modification lf Ampere’s law. Mathematical tools could be used to serve the interest of chemistry, physics, geology, economics, politics or any other aspect of human endeavour. Even in physics a particular symbol could mean different things depending on the topic or aspect of physical reality that is being considered. For instance, “E” could be assigned the meaning “energy|” or could stand for the term, “electricity”. Similarly, “C” could stand for speed of light or for Coulomb’s measurement of electric charges. What should be clear in our minds is that mathematical theories are not necessarily synonymous with physical theories. They both complement each other in the quest for truth and knowledge. A mathematical proof is not necessarily equivalent to everyday truth in reality. Physical truths and mathematical truths could tally if the conditions for assessment are properly specified in advance.

4. Micro-Physics (Quantum Physics)

Microphysics could be said to have been inaugurated in the year 1900 when Max Planck offered a solution to the problem of black body radiation. A black body is a body that absorbs and emits wavelength and temperature. Such a body would appear black when viewed from outside. Radiation in physics is the dissemination of energy from a sosurce. The energy is disseminated as the inverse square of the distance from the source in the absence of absorption. The term, ‘radiation’ is applied to assorted kinds of particles like protons, neutrons, etc., to electromagnetic waves, x-rays, radio waves, ultraviolet, infrared-rays, etc. Radiation of electromagnetic waves is experienced by all bodies, whether hot or cold. Animal and human beings are also sources of radiation. Very hot bodies radiate electromagnetic waves that are visible with our naked eyes whereas very cold bodies radiate electromagnetic waves that could be detected with specialized instruments.

5. The Copenhagen Interpretation of Quantum theory

The Copenhagen interpretation of quantum theory revolves principally round Heisenberg and Bohr. Heisenberg developed the uncertainty relations in which we cannot simultaneously establish both the position and momentum of a particle in motion. The uncertainty relations of Heisenberg are equivalent to Bohr, on his own part, developed the concept of complementarity. Complementarity was used by Bohr for the encouragement of the admission of the contradictory positions in quantum theory. For instance, the wave and particular picture of light. Furthermore, we are aware that the uncertainty relation specifies that we cannot simultaneously know the position or velocity of a particle. For Bohr, however, the knowledge of the position of a particle is complementary to the uncertainty relations and both constitute the Copenhagen interpretation of quantum theory. The name “Copenhagen” is used because Heisenberg, Bohr, and Pauli and a few others exhausted themselves in the town of Copenhagen trying to find a suitable interpretation of quantum theory.
We can summarize the Copenhagen interpretation of Bohr and Heisenberg by saying that quantum theory of relativity is statistical and opposed to determinism and objectivity, as we know it. The Copenhagen interpretation makes the point that it is meaningless to talk about the physical properties of quantum entities without precisely specifying the experimental arrangement by which we intend to measure them. Quantum reality is in part an observer-created reality, and human intention influence the structure of the physical world. In the words of Max Born,

the generation to which Einstein, Bohr, and I belong was taught that there exists an objective physical world, which unfolds itself according to immutable laws independent of us; we are watching this process as the audience watches a play in a theatre. Einstein still believes that this should be relation between the scientific observer and his subject. (Pagels, 1982)

Despite the difficulties and controversies over the interpretation of quantum theory, a lot of new grounds have been broken, using the quantum theory as the foundation. These new fields, which have been developed on the basic of quantum theory, are in the areas of the theory of electrical conductivity; quantum theory of solids, band theory of solids, quantum chemistry, molecular biology and many more areas. The scientific enterprise appears with popular logic, metaphysics, epistemology and axiology. Newtonian physics was very useful in technological advancement despite the fact that there were competing parallel approaches to logic, and to other areas of philosophy and physics. The same can be said of the relativistic physics of Einstein. Modern physics has raised the question about the truth of physics in terms of the nature of reality. It has also raised the issue of how to unite all these schools of physics, which have helped solve numerous problems for technology.
For the purpose of Africa’s survival, it is necessary to note that the entity or quantity found in the equation of displacement current does not exist or is said not to be a true current, although in Maxwell’s modification of Ampere’s law, the current or equality appears to exist. There is need to idealize various life and death situation for Africans. Deliberating constantly on how to get out of the approaching storm, it is very likely a way out will emerge. The development of physics shows that logic alone, at least in the conventional sense, could not have led to discoveries in the field. Just as physical theories are built on models, Africans have to get viable models to use. We should bear in mind that Africa, from socio-political, geo-political and historical perspectives, would necessarily design a unique approach to her liberation. The other lesson from physics is that scientists made discoveries in science. Africans, despite divisions, as you have in science should liberate themselves, and by all means possible. You could be told that black people have never been under one government. This may be true glaring fact. Max Planck held on to it, Stefgan, Boltzman, Wein, Einstein, Compton held onto the idea. Many discoveries were made.

The black body in physics featured in classical physics and in quantum physics under different covers. There is some doggedness in science. Africans focusing on the battle for survival will have to be dogged on this singular issue. You will be surprised the number of workable theories that could emerge. When there is need to violate rule in the science games that could be done in order to advance. This violation of rules and procedure occurred severally in the development of physics. It does happen that Aristotle’s (or rather plagiarized African) physics, Newtonian physics, Einsteinian relativistic physics and quantum physics could exclude each other in real scientific and technological terms; these approaches are used depending on the problem that is to be solved. Africans could try this approach of evaluating the various theories of liberation that could emerge or are already available, and these approaches as the need arises.

6. Mathematics

The battle of ideas over nature of reality in physics was also replicated in mathematics. The debate concerning the relationship between mathematics and physics even became very crucial with the formulation of the special theory of relativity of Albert Einstein, a physicist, and Henri Poincare, a mathematician, almost at the same time. Poincare’s several achievements in physics seem to suggest that physical discoveries could be made through mathematics. It must be quickly added that there are physicists who differentiate Poincare’s approach to the special theory of relativity from that of Einstein. This differentiation, in my view, is only relatively justifiable. It is partially justifiable at the level of the special theory of relativity. When we move into the atomic or sub-atomic level, the differentiation takes a more complicated form.

According to Silvio Bergia, Poincare had been working on various aspects of the theory of relativity between 1895 and 1904 (French, 1979). These aspects included the criticism and final rejection of the concept of ether, the concept of space-time and his conclusion that absolute motion for the Earth is impossible: and the principle of relativity which stipulates that “…the laws of physical phenomena must be the same for a ‘fixed’ observer or for an observer who has a uniform motion of translation relative to him so that we have not, and cannot possibly have any means of discerning whether we are, or are not, carried along in such a motion” (Whitaker). Poincare is said to have made these predictions in a paper he delivered to an international congress of arts and sciences in St. Louis. The paper predicted that no velocity will exceed the speed of light; the relativity of time and also the consistency and appropriateness of Michelson-Morley result and Fitzgerald-Lorentz contradiction (French, 80).

From the above account, it is easy to conclude that Poincare’s principle of relativity is almost identical with that of Albert Einstein. The major criticism of Poincare’s principle of relativity is that he made references to “fixed” and “moving” systems. If this is a fault in terms of theory of relativity in terms of quantum mechanics. Einstein, as is well known, was aware of the issue in quantum theory and indeed was one of the founders to a significant extent. Yet, Einstein had co-ordinates, clocks, clocks in the box and deterministic categories in his physical orientation. From the point of view of quantum mechanics and kits Copenhagen interpretation by Neils Bohr and Heisenberg and from that of Stephen Hawking who aims at joining Einstein’s theory of relativity, quantum mechanics and electro-thermodynamics into a unified theory, Einstein’s approach will definitely appear inadequate. What is being suggested here is that neither Poincare nor Einstein could be said to have arrived at the final truth about physical reality.

The apparent difference between the paths pursued by Poincare and that of Einstein was further magnified by Einstein himself in the article, “Geometry and Experience”. In that article, Einstein made a summary of what he considered to be Poincare’s conception of geometry. According to that summary, “geometry (G) predicates nothing about the behaviour of real things, but only geometry together with totality (P) of physical laws do so. Using symbols, we may say that the sum of (G) + (P) is subject to experimental verification. Thus (G) may be chosen arbitrarily and also part of (P); all these laws are conventions. All that is necessary to avoid contradiction is to choose the remainder of (P) so that (G) and the whole of (P) are together in accord with experience. Envisaged in this way, axiomatic geometry and the part of natural law which has been given a conventional status appear as epistemologically equivalent”, (Einstein, 1921, MCMLIV). In the same breath, while applying the above characterization of geometry of Reimannian metric of the four-dimensional space-time continuum, Einstein is of the view that the question of whether this continuum has a Euclidean, Reimannian or any other structure is a question of physics proper which must be answered by experience. Interestingly, Einstein realized that his interpretation of geometry breaks down when applied.

The insistence on experience by Einstein for the assessment of mathematical postulates is useful for physics. The little difficulty here is that experience could be either at the macro-level or micro-level of physical existence, are number-oriented. Even when infinities are discussed at that level, the orientation is number-oriented. Probabilities also follow the same numerical pattern. At the micro-level of physical existence, a resort could be made to matrices.

It may be necessary to emphasize that the problems in philosophy of physics are very much present in philosophy of mathematics. As a result, there are numerous schools in philosophy of mathematics. These schools include logicism, formalism, intuitionism, Platonism, and conceptualism. These schools could be divided into two groups depending on the assumed origin of mathematical truth and mathematical entities. The first group are those who are of the view that mathematical truth and mathematical entities are discovered but not created or invented. The second group is the direct opposite of the above group and do believe that mathematical truth, entities, proofs, etc., are all human creations and are also a part of our cultural heritage (Barow, 1988).

Platonism broadly represents the school which believes that there exist mathematical concepts and truths beyond the human level. The business of mathematical concepts and truths beyond the human level. The business of mathematicians is to discover such truths and concepts. This view is sometimes linked with the popular view that says, “God is the supreme and ultimate mathematician”. It does happen that schools which thought to be far from Platonism make use of Platonism at some critical points. Thus, logicism believes that all mathematics is reducible to logic in this sense, deals with the rules for correct reasoning and for drawing appropriate inferences. The meaning of the word “correct” and “appropriate” has never been satisfactorily arrived at. Although logic is supposed to be the creation of human beings, some mathematicians of the logicist school claim that since human beings are created by God, all human creation are directly of indirectly creations of God. The intricacies, uncertainties and problems of such claims and counter –claims about the divine origin of mathematics and logic are beyond the scope of this essay. In the same vein, schools like “informalism” and “intuitionalism” sometimes are interpreted in Platonic undertones.

The other broad school is conceptualism; it states that all that we find in mathematics is human creation and culturally derived. Just as in Platonism, you would discover that some adherents of conceptualism share some view with logicim, intuitionism and formalism. Formalism in the philosophy of mathematics is associated with the mathematician, David Hilbert. It is however, more appropriate to state that names like Gottlob Frege and Bertrand Russel were also involved in the establishment of formalism in some respects. Formalism makes use of formal language and formal system. Formal languages are used in logic mathematics, computer designs and computational devices. A formal language has what is known as primitive symbols together with what is known as formation rules. The formation rules tells us what can be regarded as a well-formed formula. An examination of formalism as a philosophy of mathematics has to grapple with the fact that there are indeed two principal type of formal systems of logic. These are a) axiomatic system and b) the natural deductive system.

The axiomatic system, as had been earlier stated, is built on the assumption that there are self-evident truths that need no further proofs. It is later to be established that such blind belief in the truth of axioms need not be so. However, the axiomatic system is a sub-set of the well-formed formulae. The elements of this sub-set are the axioms and the theorems of the system. Given the assumed infallibility of axioms, they are specified at the outset. On the other hand, theorems are derived from axioms following transformation rules provided by the system. There is an obvious difficulty in a further assumption that an axiom can also be called a theorem, since it can be derived from itself. This difficulty was to be highlighted by Bertrand Russell’s paradox on the set theory.

The axiom system is generally perceived as a set of meaningless symbols and strings or symbols that provide specific rules concerning how to formulate new strings from old ones. Interpretation of system in formal logic is done by assigning meaning to the hitherto meaningless symbols in the well-formed formulae. According to Gottlob Frege (1884 – 1925), formal systems have an intended interpretation such that their theorems are considered as necessarily true statements.

Frege made the bold move of stating that the concepts of mathematics had to be defined in terms of logic, and that the theorems of mathematics are indeed truths of logic. This appears to follow the footpath of another German mathematician and philosopher like himself known as Gottfried Leibniz (1946 – 1716). Leibniz was an encyclopaedic, eradite, deep thinker known by philosophers as a representative of nationalism. His writings covered such diverse fields as physics, medicine, mathematics, politics, linguistics, history, and many more disciplines. Leibniz tried to reduce everyday language and logic into symbolic forms. This eventually was construed to mean that move, although we are aware that his programme was unrealizable for the very simple reason that there are issues and notions that are beyond the realms of mathematics in the sense that mathematics is understood by many mathematicians.

Frege, who borrowed a lot from Leibniz, proceeded to reduce arithmetic to logic. He did not seem to include geometry in his reductionist programme. That had to wait for David Hilbert (1862 – 1943) in his Foundation of Geometry in which he laid the foundations of an axiomatized formal system. It was Bertrand Russell and Alfred North Whitehead who systematically tried to show that all mathematics can be reduced to logic. This they did in their joint work: Principia Mathematics (1910 -43). It is interesting to observe that Bertrand Russel, who helped to popularize the work of Frege to the English speaking audience, was also the person who pointed out the inconsistency in Frege’s logic. Frege’s logic makes the assumption that every predicate determines a class. Russell in 1902 pointed out to him that the assumption is inconsistent, since it leads to the paradox concerning the class of classes that do not contain themselves as elements. This terrific blow on Frege’s programme by Russell’s did not prevent Frege’s programme from forming an important cornerstone for the philosophy language.

The natural deductive system as distinct from the axiomatic formal system discussed above, is again made up of primitive symbolic formation and basic rules of inferences. Axioms are clearly ruled out of this system in the interpretation of the natural deductive system, the rules of interpretation can be understood to be valid inferences, and conclusions sthat emerge and they do not depend on assumptions that are necessarily true statements found in the axiomatic system. When cracks, contradictions or paradoxes were spotted in mathematics – the language of science – David Hilbert worked tirelessly to expunge those contradictions or paradoxes from mathematics. The results of such effort appeared in his book, Foundations of Geometry, (1899). Formalism as a philosophical approach to mathematics was thus firmly established by Hilbert, as had been stated.

Hilbert Foundations of Geometry, Russell and Whitehead Principia Mathematica received a devastating shock from another mathematician known as Kart Godel (1906 – 78) in an article entitled “On Formally Undecided Proposition in Principia Mathematica and related system”. According to the theorem in the article, every formal arithmetic is incomplete in the sense that there exist a sentence (in the language of the first-order predicted calculus) which expresses an arithmetical truth and yet is not provable within the system. As we can recall, a formal system is made up of a set of axioms in a formal way without reference to meaning. What is needed in the formal system is the effective definition of the axioms sets and the corresponding rules of derivation. The rules must include a mechanical method for deciding membership of the set and corresponding method for deciding, in any particular case, whether the method and rules have been appropriately applied. Given the rules and definite steps to be taken within the system, it means that the class of formulae is enumerable or counted. Computers make use of this formalist system to generate all and only the derivable formulae, any one of them in a finite amount of time.

Kart Godel demonstrated that given any consistent system (like the one used by the computer) an arithmetical true sentence can be formed which is nsot derived from the system. David Hilbert’s formalist plan, to exchange arithmetical truths with that of derivability in a formal system, was shattered by Kurt Godel’s undecidability theorem. We could observe some elements of conventionalism in Poincare’s and Einstein’s sense in the formalist stratagem – conventionalism in the sense that there are laid down rules which are agreed upon for decisions of matters in mathematics.

A study of various philosophies of mathematics will reveal that those philosophies often cross their boundaries to the ones they criticize. This happens because mathematics, like physics, is a human creation. Mathematics changes with time and in places, depending upon numerous phenomenon. Given the cultural background of mathematics, absolute truths and absolute objectives may not exist even in mathematics.

All these have implications for the philosophy of survival for Black Africans. Our survival, which will have political and economic components, will be viewed as events. Imagination, creativity and commitment to the success of the African project should be weapons for Africans wherever they may be in the battle for survival.

In some institutions of higher learning, mathematics as we have noticed so far, is philosophy, poetry, mysticism, material, logical and much more. Remember that Poincare and Einstein arrived at the door of the theory of relativity using different routes and theoretical frameworks. Just as in physics, different routes in human affairs could achieve definite results. Paradoxes could emerge in the bid to get Africa liberated. This should not bother us. Observe that mathematical paradoxes abound. I am not sure that Bertrand Russell’s attempt to solve the paradox of set theory can be regarded as very successful. Set theory is still in use. To complicate matters, there is the theory of chaos. What is important is for Africans to realize where they are now; attempt to discover why they are where they are now; and decide on the direction of movement. When a viable direction has been chosen, mathematical and political theories which will land the people in the chosen direction and a destination will emerge. There are aspects of mathematics that tend to be mystical. Russell as much wrote about that. In physics, the book, Toa of Physics, by Fright Capra, shows some relationship between modern physics, Buhddist physics, and Buhddist religion. Religion should not be ruled out in the battle for survival. Christian religion suffered terribly in Russia because its adherents tried to obstruct the course of the evolution. If Christianity will take the form some catholic priests and churches in Latin American countries promote, then religion could play some socio-political transformative roles in the interest of the masses. Religion is not to be construed to be just. Western or Eastern type of religion. All shades of opinion that can lead to the expected end should be accommodated.

7. Part 3: Culture, Ethno-philosophy and Africa’s Survival Philosophy

Sekou Toure defined culture thus:
By culture we mean all the material and immaterial works of art and science, plus knowledge, manners, education, a mode of thought, behaviour and attitudes accumulated by the people both through and by virtueof their struggle for freedom from the hold and domination of nature, we also include the result of their efforts to destroy the deviationist politics-social systems of dominion and exploration-through the productive process of social life. (Toure, 1974:60)

We agree largely with this definition.
In this section, we shall explore the relationship between culture, ethno-philosophy and the major project of ensuring the survival of black, Africans. Toure (1974) made the important observation that the early European adventures denied the existence of culture to the African. He pointed out that contrary to what these invaders did to Asians culturally, the European invaders of Africa, did not “exterminate the men with the sole idea of seizing their treasures”, but treated them “as beasts to be sold into slavery and, as they pleased, sold on the spot or exported to America, or even killed when their capabilities and selling prices no longer assured an inadequate profit”. To generate the success of such a venture a preliminary step was necessary – that of easting one’s conscience and reassuring oneself by thinking that it was a question of dealing with beasts and not human beings. Hence, the alleged barbarity of Africans and the denial of their culture and civilization gave birth to their estrangement from the human race – (Toure, 1974).

This picture of the African by the dominating power persists up to this moment. We should ignore the cosmetic approaches that tend to hide this fact. Forget that the black person was at a point in time, the Secretary General of the United Nations and that a black person is the Secretary of State in the United States of America. These cosmetic appearances of black persons at the world of stage remind us of Franz Fanon’s Black Skin, White Masks. Those black-skinned government or international agency officials are actually representing white-skinned people. The point made by Toure had also been expressed by Franz Fanon in The Wretched of the Earth, (Fanon, 1967, 170). There is no pretence of being respectable or human to the native African. If the creature is a wild beast or a cannibal, then it follows that such a creature should not have a territorial claim and could be hunted like a beast or kept in a zoo garden at best. In this respect, Teresa Okure’s assertion that Africa is a huge Refugee Camp is a gross understatement (Okure, 1993). When Macre Massoni discusses intercultural philosophy as it concerns Africans and in the light of Theophilus Okere’s Hermeneutics, the assumption is that the African has been admitted into human society by the others. This is a false assusmption (Massoni, 2005). Okere makes the assumption when he adopts Herbert Macruse’s pacifist definition of culture (Okere, 2005: 39)

There is need to respond to the denial of the quality of being human and having any culture to Africans. Like Franz Fannon had pointed out severally, colonialism’s treatment of the African is continental in scale, the imperialists have no need for national imperialism as such, although we have what looks like Nation-States, Pan-Africanism, as an organization, is one response to this onslaught on the African. But it has been criticized for ignoring the class nature of black people of the world (Reed Jr., 1974). The organization of the Pan-African Party in the United States and their kind of programme justifies the criticism leveled by Reed (Amiri Baraka, 1974).

Sekou Toure’s approach takes care of that criticism. According to Toure, culture has to be considered from a dialectical perspective. He warns, I think correctly too, that “one should not mistake this expression for a form of neo-idealism: by ideological classes, we are referring to classes which are by no means born from a simple economic and social stratification, we are faced here with a fundamental choice between two possibilities which are mutually exclusive, that is, between

1) The ideology of domination, and prostration under domination;
2) The ideology of struggle against any kind of domination and of the complete sovereignty of the people, power being exercised by and for the people”-(Toure, 60-61).

Toure argues that culture is a more effective weapon of struggle than guns. Culture for the people is the nightmare of the capitalist. Indeed, there may be no culture without class differentiation. The oppressors use forms of culture to oppress. The masses use forms of popular culture to liberate themselves from tyranny, dehumanization and from planned genocide by imperialist powers.
The experiments and work of “brain washing” experts in the United States clearly show the importance of a person’s culture to behaviour. Richard Restalk in Premeditated Man quotes Edgar Schein as saying:

In order to produce marked change of behaviour and/or attitude, it is necessary to weaken undermine, or remove the supports to the old patterns of behaviour and the old attitudes”, breaking emotional ties by severing all contacts between the prisoner and those whom he cares about and establishing instead a ‘total’ environment which inflexibly provides reward and punishments only in terms of the new behaviour and attitudes to be obtained. (Resak, 1977: 40)

Any development or plan for change which fails to incorporate the generality of the populace would definitely collapse.

George F. Mclean, an European, agrees with our earlier position on culture and method of social change. Just as I made the illustration of the amoeba using Persig’s novel. Mclean asserts that “the animal seeks out its sustenance and defends its life, even ferociously when necessary”. Human beings add consciousness and determination to these animal qualities in self defense. Mclean states the obvious, that culture is under continued re-evaluation by each generation which must decide what to pass on to its children and how to adapt it in order that it be life giving for them. He marshals out three aspects of cultural tradition with which we totally agree. The first is that culture is fundamentally a creative work of freedom. The second is that it is the only real possibility available to a person or people for a life of meaning and dignity for themselves and their children, nothing will be defended more rightly or more fiercely when necessary. Thirdly, culture is the place where the spirit of a people dwell (Mclean, 2005: 300). In the face of pluralism and the convergence of civilizations he makes rooms for paradoxes as veritable vehicles of emasculating the weak nations of the world. Collective conscious action by the oppressed of the world will halt the negative trend in globalization. It is in the light of the role of culture in society that we commend the insight of Oguejiofor in his critique of Okere’s brand of hermeneutics and Okere’s criticism of ethno-philosophy (Oguejiofor, 2005). When we consider what science and technology have done to the planet earth, it will be difficult to claim that the kingdom of scientific progress has come. These include: world wars, invention of aids virus as a biological weapon, colonialism, neo-colonialism depletion of the ozone layer, dumping of toxic waste in cities of under-developed countries of Africa like Ivory Coast and so on.

From the foregoing, we could appreciate the enormity of wickedness, violence, and deliberate effort to exterminate a race based on skin colour. Given this degree of crime against humanity by the dominating powers and their local agents, it becomes crystal clear why discussing metaphysics, logic, epistemology and other philosophies without connecting same with burning issues about the very existence and survival of our race becomes diversionary. The wealthy and wretched have been bracketed for elimination. What those who are about to be eliminated do under the circumstance is their private and collective choice. For those of them sojourning in enemy territories, they may be well informed that some of them will be human guinea pigs for various laboratory and medical tests; thus, they need to fight for Africa. It is in this context that the initial quotation about fighting for Africa our motherland is made.