Mathetic Laws

Speculative science – as the science of understanding – has as its object properly immaterial, necessary beings. It tends towards establishing truths or determining the falsehood of postulates. With it, we build schemes with features of necessity and immutability. Speculative science is therefore the science of the Intellect and of the necessary. It is important to clearly establish the objects of this science, which are proportional to the degree of the scibilis’ intelligibility, according to the level of abstraction.

Our knowledge – chronologically outlined – comes from empiria. It comes, therefore, from material objects, since they firstly stimulate our exterior senses and it is upon such materials that our sense provides the phantasma, which was spoken by Aristotle. The mind constructs such schemes, which are properly intellectuals. In the first level are those that depend on the matter, according to the being as well as the intellect. Namely, one can not conceive such schemes – or the thing – according to the being without the matter, and one can not also intellectually define it without the matter. For instance, the abstractions of man, plant, house, tree, etc.

Abstractions of second level are those that depend on the matter according to the being but do not depend according to the intellect, i.e., they need the matter to existentialize but can be defined without it, such as the geometrical figures and mathematical numbers.

Abstractions of third level depend not on the matter, either according to the being nor to the intellect. They are the metaphysical schemes. They can be defined without resorting to the matter. We can even subdivide it: those that are never within matter, such as God, and those that are not in the matter for absence of necessity, such as potentiality and actuality, one (ontological or transcendental, not the mathematical one), etc.

These are the three levels of human intelligibility, upon which we build the speculative science – in accordance with the Aristotelian view. In the first level, speculatively, we have Cosmology – the philosophy of Physics; in the second level, we have Mathematics; and in the third level, Metaphysics.

One cannot mistake abstraction for mere mental separation. The first one implies the formation of formal schemes; it is not the attention towards something by disregarding something else, as said by some modern philosophers. It is a formal construction, and if we abstract the white from this paper sheet, it is not the white “of” this paper that we highlight, but “the” white, upon which we can first-level-abstract. Herein we have the solution to a new classification of science, for with the construction of Mathesis we verified that the philosophy of any science is that that studies the principles of that science, its arkhai. Arkhe whilst entity is object of Ontology; arkhe whilst meon (non-being) is object of Meontology; arkhe whilst divine and uncreated entity is of Theology; arkhe whilst material and created belongs to a science the ancients called Pneumatology; that that studies principles whilst principles of natural, corporeal entity, is the Philosophy of Physics or Philosophy of Science. Therefore, a philosophy of a science is that that studies the principles of that science. Thus one can talk about a philosophy of anthropology, dedicated to the study of the principles of man; or a philosophy of mathematics, etc.

When studying dependency, we verify that there are several species of dependency, such as the ontical dependency (which is one), the real-real dependency (of this or that being), the ontological dependency (which is the reason of being), the logical dependency (that to which we reduce the concepts, such as specie to genre), mathetic dependency (which is that of the logoi to the logoi, of the eternal laws, one to another). It is said to be schemes of first intention those that correspond to the beings of our common experience (abstraction of first level). Beings of second intention is when the object is a rationate being, such as the logical and the mathetical beings – properly built through sapiential speculation. Thus the necessity to distinguish onticity from ontologicity, and that from logicity, and all of them from matheticity, when referring to terms that are objects of our speculations.

Mathetic laws preside the mind, but each sphere has its own features that only allows analogical reduction (never direct), in a way that if we want to reduce the facts of Physics to the eide of Metaphysics we have to forgo certain aspects of the physical sphere in order to achieve the Logical sphere, and then once again until the sphere of Ontology. That is important to avoid errors in the field of demonstrations, such as the ones committed by some philosophers who judged that that that is true in Logic is necessarily true in Ontology – which is wrong, although that that is true in Ontology is necessarily true in Logic.

A logical truth is not yet an ontological truth, otherwise we could reach ontological truths through Logic, as intended by the idealists and rationalists. Logical truth occurs through adequation that not always correspond to reality. For instance, the judgement “God exists” is a logically truth statement, since the predicate exists, attributed to Him, is a necessary predicate, for it is of its own essence and conditions to exist (an inexistent God is not God at all). So that to say “God exists” is logically true, but ontologically demands another proof.

Logic is founded on the coherence of eidetic-noetic concepts, due to the clarity given by Ontology and Mathesis, by the precision of the analogizing logoi. Thus we obtain more accurate and pure concepts – presided by logical laws, since they are the same laws of Mathesis. Ontological principles are also logical principles, examined by Mathesis. Thus we find concretion that gives the true meaning of concrete philosophy, for we can work from onticity to matheticity without violating neither sectors. Though we were doing logic alone, nothing was being done but continuing within what was already done and we could not isolate the problems.

Logics is an auxiliary science that cannot be separated from the whole philosophizing. Mathesis cannot forego of Logics, although one cannot forget that from the coherence of ideas one cannot conclude an evidence. Moreover, it is a mistake to consider the immediate subjective evidence as founded on logical coherence. We can find examples of perfect logical coherence without correspondence to reality. Logics by itself is not sufficient since we could create concepts that correspond not to reality and then deduce – from them – a series of very coherent judgements, with logical precision but without real validity. Mathesis intends to offer a content of real validity to Logic itself in a way that that is free from the danger of becoming a mere discipline of coherence – since that is not sufficient for it is not a guarantee of truth whatsoever. That is the case of non-euclidian mathematics: it is coherent, but that does not mean it is necessarily – and for that reason – truth, although they can correspond to the practical reality, that we shall investigate latter on.

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”.

Mathesis Megiste

Chapter One[1]

Mathesis Megiste is a Pythagorean expression that, translated into English, means supreme instruction. Pythagoras taught his disciples[2] about a knowledge of which man has a potential ownership, partially actualized only by a few sages, and transmissible to learners once duly prepared to assimilate it.

The fundamental first steps were Mathematics, Logic – the art of Reasoning – and Music. With those three preliminary disciplines, the disciple’s mind was equipped to penetrate the mathesis, synthesis and apex of all possible knowledge and means of leading mankind to the transparent comprehension of all fundaments of every science, although lost or veiled pathway.

Mathesis Megiste is the supreme instruction. The term mathesis origins from two roots: ma or man, which means thought, and thesis, meaning position. Properly, Mathesis means positive thought and megiste, superlative of mega, means maximum. The maximum positive thought.

There are four forms of human language: the pragmatic, common in daily conversations; the religious, which – used in religious texts – is a symbolic, metaphorical and allegoric language; the scientific, in which man confers clear concepts to the things he classifies, in order to transform those concepts in instruments of man’s work; and, finally, the divine or philosophical language, which is the Mathesis Megiste, apex of all languages, where the concepts reach its maximum purity and validity in all fields of human knowledge.

Mathesis Megiste constructs a universe of valid discourse for all spheres of human knowledge whilst each different discipline has its universe of discourse restricted to its field. Mathesis Megiste seeks a universal language.

One sector of Mathesis well known to mankind is Mathematics, which language is universal. Since Mathematics is a part of Mathesis, one can say that Mathesis – a generic – is a type of Meta-mathematics, its meta-language, to which one could similarly reduce the other languages related to the different disciplines.

One can easily observe that, throughout the work of all great philosophers, all of them have grasped – at least partially – the matter of the Mathesis. All through Philosophy there is the “discovery” – let’s for the interim use this term – or the revelation of a series of axioms, which are considered as truth per se notas, i.e., as self-evident truths. They do not require demonstration and provide the means to the discipline. Accentuated in the work of the truly great philosophers, these axioms – found either inductively or through eidetic reductions – will constitute a type of “cosmos” with such rigid cohesion that, far downward, they are all united by a single principle, their eidetic source and origin, from which they could have been deducted once there is a method able to extract the virtually enclosed judgments within these fundamental principles. Based in twenty-five centuries of philosophical thought, the purpose of this book is to demonstrate that mankind can achieve it at the present time.

The Pythagoreans affirmed that this science was known by Pythagoras, who could not transmit it to his disciples for not having found them able to receive it, for it is said that of such he complained to his wife Theano and his daughter Myia – his best disciples – as the reason he kept it as a secret, which was not because he wanted to, but for the absence of whoever had ears to hear or eyes to see. Pythagoras was only a practical esoteric, not a theoretical one. He had not intent to keep the knowledge to an elite, as stated by his enemies; he actually wanted that knowledge widespread. Nonetheless, the superior knowledge could not be accessible without previous training and initiation. Otherwise such knowledge would not be properly assimilated, and could be a factor of new confusions. Therefore, it was essential to form groups able to spread it, but not without preparing in advance those who should receive it. Pythagoras wanted to be the good sower: to sow not on dry land, but on properly prepared ground, able to receive the seed.

This is one of the most important and greatly ethical points of all esoteric movement, namely Platonic, Aristotelian and Pythagorean. The secrete thought was hidden not for the mere intention of to be given only to a select group or to remain hidden, but for the reason that it could not be just given to those with no prior preparation. The intention of Pythagoras with building his famous Institute was to create masters, able to disseminate the knowledge and previously prepare those who would receive it. Thus being good seeders, as in the parable of Jesus.

Mathesis Megiste, therefore, aims to be the apex of knowledge, the point of convergence of all disciplines when speculatively treated. Therefore, it has as its object – or its search – the principle (arkhé). The philosophical speculation ought to spin around the principle – or the arkhé, as called by the Greeks – as expressed in the Gospel of St. John, when he says: “In the beginning (arkhé) was the Word (Logos), and the Word (Logos) was with God”. Logos, here, is the ultimate sense of wisdom (sophia), as we shall see in due time.

For obvious reasons, the principle must naturally be confused with what in all religious ideas is called God, since He is always considered, in all of them, as the principle of all things. Mathesis does not dedicate to study God as God, but the principle as principle. It is a different speculation, without excluding the possibility of offering a mathetic foundation to the theological thought.

As our world is dominated by banausia, i.e., the spirit of the specialist, forcedly brought by the technical, scientific and economic development, inevitably these specialists find it difficult to have a common language so they can communicate amongst each other. However, over the past twenty-five centuries, all of the great philosophers intended to transform Philosophy into Mathesis Universalis, able to serve as a point of communication. They all intended to build a philosophical language capable of unifying mankind so that everyone could be understood regardless the variety of disciplines. That is a fair intention, and also, as we shall see, reasoned and justified, and man must necessarily seek it. Subsequently, the purpose of Mathesis is not only the knowledge of the truth, but also the search for an instrument, a suitable language to allow mankind to communicate, not descending to the inferior matters (such as sports, a common subject amongst men of culture) but ascending to a language higher than their own disciplines, upwards and converging to a common point.

It is convenient to distinguish between speculative philosophy and practical philosophy. The first seeks to establish the true and the false, whilst the practical philosophy, a philosophy built by man in his factual activities, naturally does not concern about the truth, but about what is right or wrong.

When analyzing the human ideogenesis and the volitional operations of the human beings, inasmuch as mankind develops a philosophical language, reaching the expressing intelligible species, it works with more pure concepts. Understanding usually tends to the speculative. The Speculative Philosophy is the result of higher human understanding operations. Now, one can observe that the will – orexis, a rational appetite for the goodness – tends to choose between things because those things that are considered desirable benefit the one who desires. The will rationally tends towards goodness, to the convenient things. This orexis naturally drives the human being to all practical achievements and it is an exclusive aspect of the human beings apart from other beings, deprived of will. One can thus consider Practical Philosophy as the higher product of will put into action, i.e., the intellectual achievement of man on his own practice, on his drama, on his action of dominating things and organizing it, on having a vision of his world, which is, still, a product of the will. Therefore, the understanding, taken to its ultimate consequences, reaches Speculative Philosophy, and the will, in its maximum achievement, reaches Practice Philosophy. It is important to properly distinguish these two worlds.

One of the major problems that arises in Psychology – and the subject here is not the trivialities of today but the speculative approach, the Philosophy of Psychology – is to know the nature of the understanding and the nature of the will. Are they both functions of a single principle or derived from two different principles? Later on, those discussions should be easily answered.

Nevertheless, the result – as Mathesis shall demonstrate – can only be one: they have the same origin. Human wills must have the same source. It cannot be explained by physical chemistry, as some modern psychologists do, because the choice of the good, the choice of values, cannot be solved by physical chemistry. Nor can it be explained mechanically. Therefore, another source to explain the will ought to be found. The will manifests in man, but not in animals, in which as estimative aspects is found, other than the human cognitive one. An animal can, by instinct, estimate one thing or another, but never execute a reduction to the universal, with which the will woks, such as humans do.

This theme of Psychology requires lengthy preparation and, since it is not fundamental at this point, shall be discussed later. As for now, it is only important to distinguish between speculative and practical philosophy and provisionally accept that the first would have to be predominantly a work of the understanding, and the practical is primarily a work of the will. Although, they cannot be separated or to be considered as having an abyss between them. Nor should we think, as many do, that man can just stick to the practical philosophy, moving away from the speculative. Discussions around this matter can become sterile. Inasmuch as the validity of the foundations of speculative philosophy are demonstrated, so practical philosophy stands confirmed and modern arguments that these speculatively established postulates does not apply to practical philosophy are results of poor analysis of unprepared people or merely bad faith trying to hide reality. One can find that there is no abyss between speculative and practical philosophy: they not only can work with each other but Dialectics, from this point of view, is the art of working with both simultaneously. True Dialectics must be able to apply the fundamentals of speculative philosophy into practical philosophy as well as, in turn, rise from the practical to the speculative. Action of back and forth or regressive operations, such as the one studied in Psychology, are characterized precisely in the understanding of these two directions: one parts from the universal ideas to the individuals in its general aspects, and the other one parts from the individual to achieve universals of first, second and third levels.

First levels abstractions are those in which one abstracts from own experience, such as the concepts of table, chair, tree. Second levels are those that leave out materiality. The first only elides the accidents, but considers materiality, and the second level elides materiality and only consider the quantitative aspect, such as the abstractions of mathematics in its vulgar sense, and finally the third levels abstractions, that elide materiality and accidents to only consider the formal aspects, independently from the presence of concrete things. The mind takes them only under maximum abstractions, as the metaphysical ones, such as cause, effect, anteriority, object, genus, species, subject and predicate. Those concepts belong to third-level abstractions, however man uses them alongside with the ones from first level, since man always “do” metaphysics whether he likes it or not.

Mathematics has become a language of reality since one can easily put aside the material and accidental aspects and consider all things only from a quantitative aspect. Man has the ability to “mathematize the world”. And it was that mathematization of the world that allowed the development of science, which, no doubt, began to develop when mathematics started to be used by the orientation of the Pythagoreans. Without Mathematics, no higher intellectual development would have been possible.

Science only truly develops insofar as mathematics is inserted into it. Only then, science increases its body of knowledge and achieves consistency in its operations. The same is true in Philosophy as to Logic, since it is a type of Mathematics, which is, in return, a type of Logic, as we shall see.

Logic plays the same role in its relation to Philosophy, since Philosophy becomes increasingly secure insofar as the philosopher becomes more logical, i.e., the more logically proficient the philosopher becomes, the better he philosophizes and draw more accurate conclusions.

The fundamental pillars of the development of human knowledge are in Logic and Mathematics, reason why they are the two key materials of all classical curricula of philosophy. Security in logical and mathematical thinking allows security throughout the rest of human knowledge. Mathematics and Logic have become a common language of our world and serves as meta-language of all disciplines, since mankind can reduce all of them, in their higher aspects, to Mathematics[3] and Logic.

Mathematics, in the Aristotelian sense, should be based on things that were abstracted from their materiality and considered in their accidentality. So, as one is able to build a mathematics of quantity, one could also build a mathematics of qualities and so on, since all other accidents studied by Aristotle are, in some way, linked to these two fundamentals of quality and quantity. And, subsequently, to relation. Meaning that, from these three one could build a mathematics of quantity, a mathematics of quality and a mathematics of relation. Now, as for the mathematics of relation and of quality one could ask whether there should be a withdrawal from the quantitative aspect. And the answer is “no”, since there is a quantitative aspect also in quality, as there is a qualitative aspect in quantity, as well as a quantitative and qualitative aspect in relation. Modern mathematics has pierced the field of relation, such as in the idea of functions, as relation of relations. Nevertheless, it has never developed in the field of quality. Hence it is inapplicable in the field of practical sciences, related to mankind’s ethical life. Thus, Mathematics has failed to provide great contributions to Sociology, Economics, Law, Religion, Art, etc., since it remained only founded in quantitative schemes. Therefore, it is imperative that Mathematics ought to be based on the qualitative aspect so can cover these fields of human knowledge.

To justify a Qualitative Mathematics, that must be built sooner or later, as on all unitive aspects according to their schemes of participation, there should be built its corresponding mathematical segments, since mathematics is an instrument of knowledge, as it is Logic. As Logic mathematizes concepts, Mathematics should logicize the schemes of participation.

The argument of Zeno of EIea offers an example, for in quantity and space – that can be potentially divisible since considering only the extension as such – one could get stuck in the concept of infinity. The mistake of Zeno was to disregard the qualitative aspect of the steps of the tortoise, when in fact, they were also there.

From the moment one works simultaneously with the concepts of quality and quantity, Zeno’s argument becomes puerile. Zeno’s mistake was to think that Achilles steps progressed by points when they were processed by totalities that qualitatively represent substantive determinations. Meaning that they are qualitative determinations, which, consequently, would give a finite number, and then Achilles would overcome the tortoise anyway. In addition, Achilles steps are greater than the tortoise steps, as well as performed in less time. The entire argument of Zeno was founded only quantitatively, therefore, not concretely founded. It was founded only abstractly, and even within that abstraction it was refutable.

A mere quantitative Mathematics cannot be applied to social issues. But the return to Set Theory, unfurled by the Pythagoreans (who studied the theory of Plethoi), can pave the way for qualitative mathematics since the sets, sooner or later, will have to undertake typical qualitative aspects, thus allowing the construction not properly of calculations in the quantitative sense of mathematics, but of other operations typical of the qualitative, such as the operation of analogy, and then, consequently, participation, and others, which may be tomorrow mathematized.


[1] Fragment of the book The Wisdom of the Principles, unpublished.

[2] When referring to the Pythagoreans, we denote those who actually followed the genuine teachings of the master of Samos, even if this is a debatable matter, as we shall see. To the Pythagoreans, Mathesis Megiste was a language used as meta-language to all mankind’s reachable science.

[3] Mathematics should not be considered only from the aspect given by Western culture, which is the sense of second level abstractions with mere quantitative aspect.