II         -           HIERARCHY OF SCIENCES.

 

            The diverse types of judgment determine diverse levels of sciences. Here, let us understand “science” in the wider sense: “the group of knowledge acquired and organized methodically”. This knowledge constitutes a technical field (the “applied” sciences: “the discipline which sees the adaptation of theoretical knowledge with a productive end”), a practical field (knowledge that considers the knowledge of the norms of human action”), and the theoretical field ("science” in the strict sense, common today: “knowledge which sees the reasonable knowing of the phenomena of nature”). All these knowledge are scientific if they conclude progressively. But, only the theoretical sciences know their methodological principles of research. They include not only awareness and knowledge, but also the reflection; in it the comprehension touches its vertex. We will stop, then, on theoretical sciences.

 

A.        Physics.

 

            1.         The Qualitative Abstraction.

 

            Any science has a corresponding level of abstraction. In the first sense, to abstract means: “to draw outside”, this mental operation is common and inevitable. It consists in paying attention to this or that aspect of the real, without accounting the totality of its presence. Sciences are abstract in this first sense, because they consider only a face of the real. Abstraction reduces our approach towards beings from a particular point of view of the object.

 

            Qualitative abstraction is first. In fact, intelligence exercises its power, in the first instant, on the sensible objects, in as much as it sees one being in itself, in its alterity guaranteed by its sensibility. The sensible experience attains being as one and total. Qualitative abstraction puts outside of its consideration the individual character of the sensible, and therefrom, it remains only the abstracted quality, this color, this sound, or this body chemically pure, this typical behavior.  Abstraction dematerializes the quality of the sensible, and universalizes it, thus making the object worthy of knowledge.

 

            This abstraction universalizes. The abstract universal is a particular form, born from the sensible and from the foundation of the intelligence, which relates itself universally to the perceived world. So, the sensible, sensed in a particular way, is perceived in the universal way. The universal form is not only in the object; it is the condition of possibility because the sensible object can appear to the spiritual subject intelligibly, with all its proper characteristics. But, the qualitative abstraction regards of fact the universalizing relation of the spirit to the perceived things in the sensible mode. However, the sensible object must present itself to such treatment through becoming intelligible; the perceived is not other than the sensed.

 

            The sciences through which the spirit (mind) refers itself to the sensible world are multiple. They diversify themselves from their material object and from their formal object. The material object is considered according to its material type, determined naturally: mineral, vegetable, animal, human. According to this point of view, the law of universality is similar to that of logic. Extension is inversely proportioned to the comprehension. The human sciences have minor extensions with respect to those mineralogical, but more comprehension. For the formal object it understands the particular aspect of the object determined by the subjective power, which receives it. To any subjective potency corresponds a formal object. That which is colored, sonorous, etc., is received by the sensible potencies which the formal objects are respectively the colored, the sonorous, etc., that is by the seen, by the heard, etc. Therefore, the formal object mediates the material object and the subject; this mediation limits the approach of the material object to this or that point of subjective unconquerable view.

 

            The theoretical sciences are determined by the formal object of the intelligence in as much as this perceives being. Physics is determined particularly by the universality of the perception, because we cannot know the sensible and observable being in other mode than universally. Physics studies therefore being according to a specific objectivity and its state in the ensemble of the world.

 

            2.         The Physical Sciences.

 

            The quality defines the field of physics, the typical science of nature. This study the diverse concrete beings from the point of view of their appearances related formally before the spirit (mind) in abstract community. The ancient physics, in as much as the quality is conceived by the perception, roots itself in the perception universally and abstracted of the diverse beings, which accept however this universalization. For this reason, ancient physics treats particularly the problem of movement where it studies in the privileged mode the relation of the diverse to the common. The movement is in fact constituted by the diversity of the successive positions of a mobile. The identity of the mobile, continues between the diverse positions, is the principle of objective unification in the diverse moments of the succession, united in a space and a continuous time. The ancient physics comprehend sensible and mobile being departing from irreversible regularity of the successive apparition. The regularity of the modification of the sensible gives place to affirmation of the cause of these modifications; the regular modification of the appearance must be, for the spirit which dedicates itself to the universal and to the necessary, a reasonable passage and provoked by a state to the other. Then, physics investigates analytically the cause of change of the sensible appearances.

 

            The physical universality serves the model to experimental sciences or of observations. The experimental method discovers the common constants to several individual phenomena; referring to the abstract quality, the hypothesis to the observable, exercises in the empirical sciences the same structure of the qualitative abstraction; investigating the universal formally present in the diverse particular. In this, the physical explanation rather approximates the real than exhausting it; physics tends to become provisionary; the universal is overcome by the singular, and never totally reducible to that which it observes.

 

            The human sciences follow the same inspiration. The historical explanation completes itself when the particular phenomenon enters in a general picture; in sociology, the regularity of the phenomena grounds the statistical investigations which allow anticipating the arrival by a projection of the possible on the real future. However, the same modern sciences reject the exigency of a total explanation of man. They content of a universality of what it cannot know more than necessary.

 

B.        Mathematics.

 

            1.         From Quality to Quantity.

 

            The qualitative abstraction must be overcome, the universal, put in this level of abstraction, in fact, is not derived simply from the sensible. The sensible appears, modifies itself, and disappears. Now, the regularity of these movements is a law, in such a way that a common pattern can serve as measure and giving its norm as its cause. To measure a phenomenon implies a knowledge, which is not based on the sensible, but that which the observer uses to its benefit. This knowledge belongs to a quantitative abstraction. The foundation of the qualitative universality is the quantitative unity.

 

            The quantitative abstraction surpasses the individuality of the sensible. It only remains in a formal and universal point of view, lacking of the sensible material. The qualitative abstraction pursues the movement it so began. The qualitative abstraction has put in light forms which the universality is unity. Such forms represent common matters, considered as ideal individuals and intelligible in their proper unity. These individuals are considered intelligible matter and among them is affirmed the law, a relation that is of necessity. So the universality of the quality is together confirmed and surpassed.

 

            In comparison with the qualitative abstraction, quantitative abstraction is more universal. Contemporary sciences are constructed all on the model of the universal mathematics. All scientific research and propositions can use mathematics because its formality renders it available to such a use; but there is no a posteriori in it, but here all is contained a priori.

 

            However, the word “abstraction” means in some way a particularity treated outside of a totality. But, in fact, to abstract the quantity consists in the recognition of the unity of the ideal being (a triangle defined by its form) or sensible (in fact attended by diverse sense, but which is one itself). The quantitative abstract is equal to itself, it is which exists empirically, it is which form the content of an act only of thought. Quantitative abstraction, considers itself the unity of being, precedes logically the qualitative abstraction.

 

            2.         Unity and Mathematical Laws.

 

            Intelligibility implies the necessity of their elements and of their relations; mathematical laws are the guarantee. The logical anteriority of mathematics establishes the intelligible form to the conditions of being considered as a unitary being. The unity of the form is the necessary means of establishing physics. Mathematics considers the form itself, according to its formal unity, as purely the laws that unite the forms independently from matter coming from the senses.

 

            The unity of the perceived being or thought is a priori. This unity enters to be part of a formal law, that is, of an ensemble. The succession of numbers, of the ensemble that is of the unity in ordered relationship strictly, is also a priori. For example, the law of recurrence is true through the infinite series of the entire numbers; now this series is not covered never entirely, through definition. The law is strictly a priori. The point of departure and the development of mathematics are entirely a priori.

 

            The contemporary formal system radicalizes the sense of quantitative abstraction. The elements that constitute them are determined by a rule, which defines the formal and legal unity in which they receive their sense. The sense of the element is constructed in function of the rule from which it derives, and not of its proper unity. Mathematics, so conceived, is an axiomatic formal system, in which the laws of deduction is controlled formally, being included state in the system of the first axioms.

 

            Therefore, mathematical universality contains specific treatment, the transcendence of the formal laws in confrontation of the unity of the existent, and the reflectiveness. The transcendence of the formal systems is very evident; it comes from their origin a priori. These systems try to re-ascend as far as a minimum number of fundamental axioms, which define the original system, from which, it could deduce all the subsystems which give intelligence of diverse aspects of our world. This original system would be much real by being capable of reflection. Reflectiveness grounds the transcendence of laws. The reflectiveness is a mode of presence of itself, in the adequation of the act to its operation. The operative activity would affirm then its operation effecting it according to the mathematical necessity. The reality of the formal systems comes in such mode consolidated by their reflectiveness in which it affirms itself their original necessity.

 

            3.         The Unity of Physics and of Mathematics.

 

            The independence of mathematical systems does not give, however, their total sense, because these systems have pressed on the real and then be applied. Without laws of concordance between the formal system and the real, without rules of interpretation, the system turns to a vacuum. The system cannot be interpreted if it is not intrinsically interpretable. Therefore, the formal system, by touching effectively the real, must be capable. The application to the facts is essential. Certainly, mathematics is sufficient itself in its formal constructions entirely a priori; it needs to ground it and use it as efficacious instrument of manipulation of the real.

 

            Empirical sciences consider their object, which is one concrete unity, according to the diverse levels of its abstraction. The unity of the existent objects or ideal manifests their irreducibility, as the formal unity of the totality in which they take place. The formula of a chemical body isolates it in its nominal specificity, and relegates it among other chemical bodies similarly formulated. The natural sciences are in the level of overcoming the evidence of the senses through constituting a new synthesis, grace to the mediation of the analytic intelligence. Therefore, they resolve a concrete synthesis through concluding in an abstract synthesis.

 

            The way of mathematics is opposite. Mathematics supposes at the start a synthesis, which isolates a priori the analytic elements; it proceeds from the abstract synthesis to the concrete synthesis where the analytical elements receive a sense of recomposition.

 

            The foundation of apriority of the sensible and intelligible unity is spiritual. The unity of sensible being is confirmed and rendered intelligible by mathematics. The unity of the formal system is to image the unity of the reflective movement of the spirit. Leaving partly any acquired quality from the sensible experience, the mathematical unity becomes constructed by the relations of being one in the formal ensemble where the spirit articulates the intelligible structure of the real; this system is based, ultimately, in the essential operations of the spirit which, negating whatever validity to its conditioned axioms, it negates at the same time the validity of the ensemble of the relations according to which they defined the mathematical unity.  The empirical unity of being and formal unity of the system thus become the base of the dynamic unity of the spirit. The mathematical unity gives then to natural science its foundation. This assumes its intelligibility by the activity of the spirit, which guarantees ultimately its true reality.

 

            4.         The Overcoming of Mathematics.

 

            The formal universality is based on the exigency of the rigor and of coherence of which the formal systems themselves do not have the reason; they receive it of fact as a destiny to which they cannot but submit themselves. The formal system depends on an ensemble of the irreducible axioms one to the other, which cannot be deduced one from the other, and all necessary in rendering account the totality of logic effectively practiced. This assemblage is constructed making of diverse propositions and of diverse definitions the elements of a proposition more simple as possible and always more extensive. To the foundation of the ultimate propositions, synthesis connected by a synthetic activity. The spirit assumes itself the responsibility of this activity. The complex proposition is then the same exposition of the spirit, its symbol.

 

            Now, the formal system is reflective because it integrates its proper laws of constitution. To affirm that the formal system constitutes propositions with the help of a reflection on the constitution of these propositions, and that the rule of deduction is integrated as axiomatic proposition, it makes of this an element of itself sensed, a form departing from other synthetic propositions would be deduced. At this moment, the spirit would be ideally integrated in the axiomatic system.

 

            However, the law obliges it, without which the formal system can justify that in a diverse way than in the arbitrary mode. The system knows only that every logical fact must be overcome a priori by a proposition more elevated. Logic would be so overcome and grounded by meta-logic, then by metameta-logic, indefinitely, if it wishes to bring back to the internal of the formal system the integrity of its foundation. Such foundation can not, but be sending back infinitely. The foundation of mathematics is not internal to the same mathematics. No one system can develop itself without using a proposition which cannot be neither demonstrated nor refuted, which then is not resolved. The explanation grounds itself, higher than mathematics.