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.

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