Mathesis Universalis

Chapter Three

Mathesis, as we intend to reconstruct, is the tenth science, the supreme instruction of the Pythagoreans, the sapiential contemplation of Saint Bonaventure. We have presented a series of aspects that motivates us to pursue the reconstruction of such wisdom, indispensable for an accurate philosophical reasoning, in a way to disencumber philosophy from the current state of confusion and provide mankind with an able instrument of solving any defiant aporia.

Our starting point is a fundamental postulate of our philosophy that we shall, in an opportune time, demonstrate: “in Philosophy, there are no unsolvable questions, but merely improperly placed ones”. Philosophers who placed such problems in a wrong manner caused any apparent insolvability and allowed the emergence of such seemingly invincible aporias.

Now, the progress of modern science is mainly due to an instrumental discipline of great value, i.e., Mathematics. The simplification of calculations and the solution of a series of problems that seemed impossible but became simple were acquired not through complexity, but through simplification. Newton amazed his contemporaries after achieving astonishing mathematical results, which later on was found to actually be able means of efficient and rapid calculations. Nowadays, we can observe the clarity achieved by Mathematics through the study of the sets, and even those who manifested aversion towards it can find, now, an unpredictable pleasure in learning it, due to its clarity and didactical aspects.

Mathematics is a language, applicable to the corporeal world, founded on second level abstractions[1], what allows it to divest of material conditions and, therefore, serves as an instrument of connection to the world of third level abstractions, the world of Metaphysics. For that reason, Mathematics is part of the field of Speculative Philosophy. At the same time, it has a function of metalanguage[2], since it can partially substitute the language that correspond to the sciences formed by species specialissima, i.e, that have immediate correspondence to the chronotopically existent individuals – which are first level abstractions – mainly those of Physics. It dispenses the formal differences that distinguish the objects of the various sciences, so to consider them under a common aspect. Such is the reason for the extraordinary development of Science, only happening where the mathematical methods are applied, according to what was intended by the Pythagoreans (and, nowadays, factually demonstrated).

Sciences with greater progress are the ones that allowed the mathematization of their objects. When Pythagoras realized such metalinguistic function of Mathematics, at the same time that it was merely applied on its quantitative aspect (to which the Greek called logistike, the mathematics of calculation), he proposed the surpassing of the quantitative towards the qualitative, the relational, and beyond, so to achieve a metalanguage for all sciences. Within his so-called secret thought, there was four types of language: the pragmatic, of the common people; the scientific, of the diverse disciplines with their particular objects; the religious, of symbols and analogies; and, finally, the divine, which precisely reaches the higher level, the metamathematical language. The latter would be the language of the Supreme Instruction, of supreme knowledge, in which the Mathesis Megiste is founded – the superior search of a philosopher. The philosopher is the lover of such supreme wisdom, such sophia, which he tries to reach through different pathways (methods). Such was the answer of Pythagoras to a certain tyrant who asked him who he was: “a lover of wisdom, a wanderer who searches such wisdom”.

He built his Institute to help others in their search for such supreme instruction. It cannot be confused – as done by certain esoteric and occultist ideas – with a merely mystical thought, but with a genuinely scientific one, that can serve of metalanguage, of metamathematics and of sapiential legality. Therefore, instead of working with aspects of material conditions, it would work only with general laws (logoi, arkhai, arkhetypoi, paradeigmata, exemplary ideas), not of human thought (which belongs to the field of Psychology and – partially – of Logic), but of supreme ideas that rule all things, independent of man. That is to say that Philosophy would start with man yet not submitted to him, but to a superior knowledge. Those supreme laws would be the true knowledge of divinity, belonging to what he called the gods in his exoteric language, such as in the Golden Verses, so to be understood by his listeners, although his true conception was monotheistic, as one can realizes by the study of the Pythagorean fragments.

Human language emerges as a human necessity to communicate with one another. Words have intentionality – conventions of common language – since we employ terms so others can understand what we intend to express. In a way that the common language we develop from childhood is part of our pragmatics, i.e., belongs to the human pragma[3]. Now, a merely pragmatic word could not serve as an instrument of science, because of its vulgar acceptions. Due to the works of linguists from the XIII, XIV and XV centuries and their famous speculative grammars, which studied the intentionalities within language pragmatics, it was demonstrated an intentional universality.

Semantics is the section of speculative pragmatics that studies the meanings or, better yet, the acceptions of verbal terms. A single word cannot grant a meaning: the word book, for instance, does not offer a guarantee of reference to the object book or the verb to book. The voice is the same, but the acception can be different. Moreover, words placed in face of other words assume diverse acceptions. There is, therefore, a certain law of correlation of words.

In Philosophy, as well as in Mathematics, one can construct syntax without semantics, i.e., one can work with syntactical signs with no determinate meaning. By examining the correlatives – the relations between beings – we observe that there is that of which actuality necessarily implies the actuality of another. One is the term of the other and vice versa, such as in the father and son correlation. Once discovered the law of correlation, we can reduce it to a syntactical form, such as the following inevitable rules of correlation: 1) Actuality of terms of correlation; 2) The definition of a term necessarily implies the definition of the other; 3) The terms are always simultaneous, insofar as considered within the relation (the father is only so when there is a son; the son is only so when there is a father: the simultaneity of terms within the correlation is inevitable). In the concrete section of this work we shall extract a series of laws of correlation and mathematise them. Mathesis is precisely the construction of such metamathematics that studies such laws in every aspect of reality, distinct from all particular sciences, each one dedicated to a single, particular aspect of existence.

Superordinate sciences consider the common aspect of various particular sciences. Above them, we have the superior superordinates that take that common aspect from the inferior superordinates until reaching a science wearer of a language valid for all the others, with the same rigor as mathematics, founded on apodictic demonstrations and able to offer us the desiderated certainty of Mathesis. Wherever there are correlatives, correlation is valid and predominant, since wherever it cannot rule, there would be no correlation. Such law is universal, eternal, with no temporal beginning and it is not a creation of ours. We have merely found it, discovered it, through the analysis of the logoi[4].

When such set of laws are revealed to human intellect, we can finally have an universal view: we reach what is called Mathesis Universalis, as referred by Leibnitz and many other philosophers, the ability to guide Philosophy and practical sciences. We can reach a set of perfect laws that are valid regardless of our mind.

One of the terrible errors of modern philosophy consists in the gnoseological defect of disbelieving human knowledge based on the fact that humans can err. Human errors are a fact, but to induct a general law from such fact is to induct a consequence that was not in the premises (in the antecedences). Such consequent does not have, within its antecedents, its reason for being; it affronts the law of antecedence and consequence that rules syllogism. We can verify that man reaches such laws that could not be otherwise. Even if there were no mankind, the laws of antecedence and consequence and of correlation would be valid by an objective value, independent from our subjectivity.

Mathetical laws could not have a temporal beginning: could not have been started nor preceded by an anterior absence of such laws. Law of, for instance, correlation surpasses time; it is eternal. And even if there were no correlatives, such law would be capable of being intellected by a mind and would be independent from such mind. An intelligent mind is able to apprehend it. It cannot occur as something that happens here and there, in time and space, but as something beyond materiality. It is a stable, immutable and eternal thing. And that is what we shall observe as we progress in the study of Mathesis, as we get used to auscultate the eternal: at some point, it is no longer hard to discover such laws, since they present themselves as immediate and rigorous result of others.

When we descend, it is back to reality and we are able to scrutinize problems that seemed so complex and difficult, such as the concept of term, which, once duly understood as beginning, middle and end, become clear and simple. We can signify as term that of what marks a determination, offers a vestige much as in the beginning, in the middle and in the end. We can conceive a potentially infinite series of such terms, as done in Mathematics, and study the theory of terms and its laws, so to realize that it is a possible or actual determination – and not an essential constituent – of something.

Terms are not necessarily causes of things, since, for instance, a point, being the term of a line, is not its cause. It can be the beginning of a line or the end of it; it can also be a medium term, a final term, and in all intermediate terms we can visualize a point. Nonetheless, it cannot be the cause of the line, nor, consequently, the matter, the form, the final cause, and the efficient cause of it. And that being so, it should be enough to dissolve all the arguments of Zeno of Elea, since he considers the point as cause. All argumentation that considers the line as composed by points is unfounded, since it considers line in a terminative aspect – what in it we can determine, mark, signalize – that is not the material constituent of the line.

Absolute nothingness is refuted in all forms by our very presence and its postulation as a philosophical problem is a false postulation and a false problem. Nevertheless, nothingness, relatively considered, is a mathetical theme and, as such, object of Mathesis in a field called Meontology, the study of nothingness, of non-being (while Ontology studies the being). Although Ontology cannot be separated from Meontology and Mathesis, such triangle constitutes a unity and demands different methods of investigation, each one considering one formal aspect. Ontology considers the being whilst being and Meontology considers nothingness whilst nothingness, i.e., no longer work with a positive term, but a negative one. Consequently, Meontology cannot scrutinize the negative ideas without the presence of Ontology, the positive term. Meontology proposes to solve the problems of modern speculation, which is extremely concerned with the problem of nothingness, relegate to a second plane in classical philosophy – although still object of scrutiny back then. The regression of modern philosophy, here, has a positive aspect, since we can look back to what have been studied in classical philosophy on the subject. Due to the great development of Mathematics, modern mankind is familiar with the abstractive work.

Mathesis is not a simplification of calculation. When a mathematician simplifies a calculation with elements of Algebra – what signifies a great progress and an ingenious work of mankind – he is not properly creating, but merely discovering what has been revealed to his intellect. He is not a creator, but a discoverer. The laws of Algebra were already there and man finally found them. And, as Mathematics developed, such laws became a simplification, a superior language within Mathematics itself, since also within it there are a series of strata of languages, from Arithmetic to the more complex calculus. Such was a great achievement, which also occurs in Philosophy. Certainly, the great philosophers, such as Pythagoras, did not exhaust all the laws of Mathesis and they surely knew less then we can know today. Although, Pythagoras knew about the existence of a metalanguage – the language of languages – that he searched and investigated in order to reach it. Such language (the tenth promised science) was not transmitted to us. We shall, therefore, seek to discover it.

[1] “Second level abstractions are those in which the mind, aside from abstracting from matter (such as in first level abstractions), also abstracts from sensible properties, and considers only the extension of things under their continuous or discrete aspects – its quantitative aspects. They are precisely the abstraction of Mathematics: the continuous provides the geometric figures and the discrete, the numbers.”

[2] “A specialized form of language or set of symbols used when discussing or describing the structure of a language”, cf. Cambridge Advanced Learner‘s Dictionary, 2010.

[3] Pragma is the term that indicates the content of praxis, of human actions, of human accomplishments.

[4] Opportunely, we shall demonstrate that can only be a tripartite division of the being: a) A semetipso et ab aeterno; b) Ne cab aeterno nec asemetipso; c) ab aeterno sed non semetipso. The mathetical laws belong to the third type, “eternal but not from itself”.

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