Numerical Logic of the Basotho – Part I

By Zulumathabo Zulu © 2015

The popular myth in South Africa and elsewhere is that the Africans did not have mathematical knowledge until the advent of the European colonizers. Statements like If the White man didn’t come here, the African would still be living like a primitive or a technophobe are common place in one form or another.

This flagrant denial of indigenous mathematical knowledge flies in the face of the indigenous counting system of the Basotho {Lehe – the cosmic egg; Ngwe – one; Pedi – two; Tharo – three to n – 1}. Moreover, Pedi is a cosmogenous principle of genesis that says that in order for something to exist, it must have a baseline of two as in the intersect of the feminine principle and the masculine principle. Thus, the universe was created out of these principles namely Teyateyane (the intersect) of (1) Peo (the primeval seed) and (2) Mokedi (the primordial water) of the cosmic egg. This Teyateyane resulted in Seqhomane sa Mokgubu wa Kganare (a massive explosion) wherein the fragments turned into stars; constellations; galaxies; black holes; carbon existence and large bodies of water.

I still recall my experience in the great city of Ottawa, Canada where a Canadian man posed a question to me while I was waiting for an OC Transpo bus near the University of Ottawa in the few months of my arrival in the 80s: “Do you have TV in Africa?” and I enthusiastically responded “Yes Sir! The TV we have and I even watched Star Trek at a mountain village in QwaQwa powered by a car battery. Are you a trekie?”. The guy was shocked to learn that not only had the African native been exposed to a TV but had actually watched Mr. Spoc and Captain Kirk.

What is also noteworthy here is that the word Trek in Star Trek came from Africa via the Afrikaans language of South Africa. Afrikaans was invented as a Creole language by the African natives in the Dutch colonial settlements in South Africa in the 1700s. In the book Sesotho Dictionary of Mathematics I pay deserved tribute to the Nama people who led the invention of the Afrikaans language. I also pay another tribute to the Nama people as a result of their inspirational role in my other books The Sacred Knowledge of the Desert: African Philosophical Transcendence and A Woman in the Bush.

The gentleman probably liked the click sound of QwaQwa. At least this afforded me an opportunity to dispel some myths. I was not irked by the Canadian because I understood that he was genuinely interested in knowing if there was a TV in Africa. This is what I refer to as a genuine case of ignorance and this was my duty to educate him about Africa.

Nonetheless, these myths persist here in the beautiful country of South Africa among some. In this article, I want to share the knowledge system of the Basotho of South Africa with the people of South Africa and the world. In this case, the focus is on my favourite topic: the numerical logic of the Basotho. I have presented this topic to various conference and social gatherings.

The Basotho, who influenced my philosophical thinking, are mostly found in the Eastern Free State province of South Africa which borders the KwaZulu Natal province on the Eastern side, Lesotho country on the South-Eastern side, and Gauteng on the Northern side. I am genealogically linked to these beautiful people through my Mosotho mother.

The Basotho are great philosophers and their Sesotho language provides the numerical platform to philosophize.

The Basotho knowledge system is a derivative of cosmology. Basotho cosmology is defined as an epistemic framework comprised by three taxa namely (1) cosmogony to refer to genesis; (2) cosmography to locate the star system and describe its features as in Tosa le madinyana a yona (Jupiter and her babies) and (3) indigenisation to refer to the localisation of the knowledge of the cosmos with respect to the terrestrial mileau of the people.

As a matter of necessity, knowledge is linked to the context of its origin wherein context is defined as a spatial source or environment that inspires the genesis of knowledge. The Basotho, like other Africans, trace their origin to the cosmos as described in the book The Sacred Knowledge of the Desert: African Philosophical Transcendence. Thus, in the mathematical science of numerical logic, context must persist as a moral compas of knowledge production.

Teaching mathematical knowledge without context as is the case with Canadian universities and other Western institutions is tantamount to peeling away a lifeblood layer of knowledge so that students are devoid of an epistemic access to the metaphysical essence of context and subtext with respect to the knowledge of numerical logic. It is a treacherous act of miseducation to teach a contextless knowledge. Numerical logic, like any other knowledge discipline, necessitates and must espouse the metaphysics of context.

Numerical logic is a powerful form of reasoning which provides for a robust platform for philosophical thinking. Numerical logic manifests its intellectual and analytical prowess in terms of level headed thinking, strategic knowledge and mental discipline. The Basotho even have a ceremony known as Mokete Wa Lewa (the ceremony of strategic thinking) which they celebrate in their New Year of Phato (Phato refers to the month of August) as part of their sacred calendar. By far the only people I have come across who possess and perform a philosophical ceremony.

The Basotho have made numerical logic an integral part of their language; in the Sesotho language there is no distinction between numerical and linguistic usage. Unlike the English language in which mathematical reasoning and the language are separated, numerical and linguistic domains are closely coupled in the Sesotho language and this is something that consumed me after discovering this amazing fact in this African language.

Having sojourned among many nations of the world in many countries, I have found the Basotho to be unique in their worldview in that their language boasts an intricate system of syntactic rules with a morphological system that is also governed by the rules of the language. As an independent researcher who serves the function of an intellectual historian, I have observed the Basotho in a variety of situations in terms of how they troubleshoot, conceptualize, analyze, strategize, problem-solve and design solutions.

The Basotho have a concept of Numerical Analysis known as Qapollo Ya Dipalo and the concept of Error Analysis known as Mokgero. The concept of Mokgero also encapsulates the concept of standard deviation in statistics. The Sesotho axiom “Haekgera ephekoleha ka dipalo” says that if it (the object) deviates from the baseline, it gets healed by a numerical solution. It is notable that medicinal healing is part of the numerical specification. This is the case because the Basotho have the concept of sacred numbers which they use in their therapeutic ceremonies. Another axiom refers to “Lekgwaba la methati” meaning the bird of graphical gradients. In this philosophical construct, the gradient is conceptualized as a graphical sequence. This is an incredible form of knowledge which forms part of Basotho cosmic knowledge and their transdisciplinary approach to knowledge production.

The word Mothati in everyday use refers to both linear and non-linear gradient. In accordance with Basotho mathematical knowledge, a gradient is a non-linear system and the straight line is merely a special case of that non-linearity. For the Basotho, a gradient is straight, curved or round. For them, a gradient is a graphical system. This numerical knowledge contrasts sharply with the English mathematical convention in which case a gradient is a linear object. The Merriam-Webster Dictionary describes a gradient as “a part sloping upward or downward”. This description confirms the English concept of a gradient as a straight line.

For some of you who are mathematicians or engineers, we can also see the linearity of a gradient in calculus. In vector and multivariable calculus, where we analyze a gradient of a multivariable function as in f(x,y,z), a gradient is described as a vector of that function. Analytically, a gradient of a function at a particular point describes the largest increase of that function with respect to a particular direction. This is what we refer to as the extrema (maxima or minima) of the function at that point. This mathematical analysis, using partial derivatives, still treats a gradient as a linear object.

The Basotho boast expressions like “Hothathika ka thaba” meaning to traverse the small changes of the mountain’s gradient. If the mountain is round then the gradient is also round. This numerical reasoning is an everyday use of the language something that is unique among the Basotho. I explore numerical concepts in my book Sesotho Dictionary of Mathematics.

In addition to numerical reasoning, Sesotho language also provides a high degree of mental discipline owing to its intricate system of syntax. The Sesotho morphological system is governed by the syntax of the language unlike the English language whereby the morphology is not subject to the syntax because the English syntactic system only operates at a sentence level vis-à-vis the Sesotho language whereby syntax operates at both sentence and word levels. Among the Basotho, word formation is a level one abstraction while sentence construction is a level two abstraction.

There are other non-African languages that possess this syntactic-morphological feature such as the Spanish language that uses an infinitive like regresar meaning to return. A word like regresar is governed by conjugation rules. So we can say that in the Spanish language, as opposed to the English language, the rules of the language apply at a morphological level. However, unlike the Spanish language, whereby the conjugation rules are limited to the infinitive or irregular verbs, the Sesotho language conjugates infinitives, verbs and nouns. An expression like “Retswa moletolong” meaning we come from the vigil uses Moletelong as a conjugated form of the Moletolo noun. In my scholarly paper The Golden Rays of the African Sun I go into details about this syntactic feature of the Sesotho language.

The Spanish language does not have the ability to conjugate a noun because it was never designed that way. The Sesotho language was designed to conjugate nouns because of its strong morphological syntax. Moreover, the Sesotho language has different forms of verbs including the phrase form of a verb. The suffix ng describes the ontological state of the object. Ontology is built into the language meaning that philosophizing is not a separate thing but is also an integrated part of the language as is the case with numerical reasoning.

As previously mentioned, the Sesotho language appies its rules at level one and level two abstraction using sentence and word formation. To show an example of a morphology that is governed by the syntactic rules of the Sesotho language, let’s look at a word like Sekhiye that is coined from the English word Key. Not only did the Basotho take the word from the English to make it their own but they subjected the word to the syntactic rules of their language so that they developed the syntactically correct infinitive Hokhiya meaning to utilize the key. In this way, if you are not privy to the word formation, you are going to think that the word is originally Sesotho which is a great credit to the language.

What is also interesting is that the Basotho boasted many words for Key before the landing of the English here in South Africa such as Senotlolo; Sengomela; Seqhoboshela. All these words refer to a key object in one form or another depending on the ontological state of the key object. With the word Sekhiye, the Basotho have enriched their vocabulary without really having to. Instead of the Basotho being driven by lexical necessity, the Basotho were motivated by intellectual curiosity to learn about how the English conceive of a key. The English language uses one word Key whereas the Basotho have many words for key. Perhaps next time I can write a separate article about why the Basotho have many words for a key.

The Sesotho word for key along with infitives and conjugated forms are as follows:

Key Concepts Sesotho {Sekhiye, hokhiya, khiya, mokhiyo, hotlakhiya, bakhiile, batlakhiya, hokhiyela}

An English translated version of the list is:

Key Concepts {Key, to apply the key, apply the key, state of the key, to apply the key in future, they have applied the key, they shall apply the key, to apply the key on behalf of}

In the above list, the Sesotho language has encoded concepts like object, infinitive, state, future infinitive, conjugation, singular future infinitive, plural future infinitive, plural infinitive and cooperative infinitive. There are more philosophical concepts of the language but for now this list should suffice.

This specific example of Sekhiye confirms that a new word formation in the Sesotho language is governed by the syntax of the language even if it is an etymological derivative from another language. The syntax facilitates new word formation through the mechanism that expands the language and adds some sophistication with respect to its lexical, numerical, morphological and semantic extent and richness.

The Numerical Logic

It’s noteworthy that numerical reasoning is a vital part of the Basotho people and their Sesotho language. This language boasts expressive power and knowledge translation as part of the facility of the language. The language allows for an abstract form of logic like numerical reasoning as a native part of the language. This allows the native speaker to traverse the abstract, the pragmatic, the numerical and the symbolic domains and back again with ease.

Symbolic Domain

Language is essentially a symbolic system with many levels of abstraction. The Sesotho language is a highly coded system. Let’s take an example such as a cow. The Basotho are a cattle herding society. In a Western society, a cow is a cow and there is nothing mysterious about that. This is because the Western society is highly influenced by a philosophical system that says that the only thing that exists is a material and without a material basis of existence there is no existence. in order for an object to exist, it must have a material component. Thus, material is a fundamental basis of existence in a Western society and the scientific method is built on that ontology. Philosophies like Materialism, Dialectical Materialism, Ontological Materialism, Physicalism, Ontological Physicalism, and Reductionism are the theoretical guides of this materialist thought.

For the Basotho, a cow is not just a physical object. A cow is also a symbolic and spiritual object. A cow can atone for your wrongs and reconnect you with the cosmos and your ancestors. A cow is also an investment bank system. When a Mosotho needs to marry he withdraws the capital from Lesaka (the pen where the cows are kept). This means that the cow is also a symbol of an economic system in which every member of society owns a piece of that economy as opposed to the English system of capitalism whereby only a small fraction of society (a fifth of the North American population) own the economy (more than three quarters of the economy). A shepherd among the Basotho owns a portion of the cows he herds. This ensures a selfless committment to the preservation, defense and enrichment of that economy by the members of that society.

The Basotho pay tribute to a cow and even produce poetic chronicles that are meant to venerate the attributes of the cow. Even when a Mosotho departs from this planet, she or he is accompanied by a cow and is clad in a sacred cow skin. The Basotho do this because they believe that the cow has a sacred spirit. In this case, the Basotho are like the Hindus of India who also venerate the spirit of the cow. Thus the Indians and the Basotho are like brothes and sisters.

The Basotho’s philosophy of science, like other Africans, declares that in the universe there is Mokedi (wave-based material) and Moya (spirit) that coexist. An interesting note here is that Moya in Sesotho means something that transforms between the material and the spirit. An axiomatic statement like “Eya moyafala” means the material is transforming into spirit. This gives the Africans a metascientific knowledge domain with respect to the ontology of the universe. This coexistentialist ontology requires a language with a built-in facility for symbolic logic and numerical logic in order to transcend a materialist description of existence and thus give us an epistemic access to a higher form of science. The Sesotho language provides that facility and describes this kind of metascientific phenomenon. Using this Africanist worldview, a material becomes a subset of the metascientific superset.

Other African languages possess this kind of metascientific language such as was the case with Dr. Cheikh Anta Diop when he wrote a book on the physics of Theory of Relativity using his home language of Woolof in the venerated country of Senegal. Another country that achieved this higher science of symbolic logic is Cameroon that gives us the sophisticated writing system of Bamoun under their great King Njoya.

Numerical Logic

It is the numerical logic that provides another structure of the language in the form of cardinality. Cardinality is a numerical form of structure that sets the African language of Sesotho apart from the English. In the above example of Sekhiye, we generated a set of infinitive and conjugated forms of the word formation and demonstrated how cardinality is part of the structure of the language with respect to its numerical logic. In the above case, we generated a set of seven elements relating to the concept of Key.

Now let’s take another concept of Circumference which is a mathematical concept meaning distance around a geometric shape. When you generate a list of words with that concept in the English language you don’t go that far. The language soon hits a lexical roadblock. That lexical roadblock also means loss of epistemic access to the knowledge power encapsulated in the language.

You get Circumference and probably perimeter even though these concepts don’t belong in the same category since circumference deals with curvilinear objects whereas perimeter deals with rectilinear objects. Check the table 1 and table 2 with respect to Sesotho and English concepts of circumference as shown below.

Table 1 – Sesotho set of circumference concepts.

Table 1 - Sesotho set of circumference concepts.

Table 2 – English set of circumference concepts.


The English concept gives us a maximum of two words with respect to distance around. If we insist on the taxonomy of the concepts then there would only be one word.

The above tables show that if we confined ourselves to the English language as a language of academic discourse and prestige we would be shutting ourselves out of very rich knowledge associated with an African language. Have you explored your mother tongue to find out how much knowledge is locked inside the language? Some of the concepts cannot even be translated because those concepts do not exist in the English language as we have shown above.

To be continued…


Zulu, Z. (2014). The Sacred Knowledge of the Desert: African Philosophical Transcendence. Madisebo University College Press: Johannesburg.

Zulu, Z. (2012). Sesotho Dictionary of Mathematics. Madisebo University College Press: Johannesburg.

Zulu, Z. (2008). ‘Ontological States of the Object’, Unpublished, Ottawa,

Zulu, Z. (2014). The Basotho Origin of Mathematics – A Public Lecture, Central University of Technology, Bloemfontein, Free State.

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