Categorial Method

Conjugation via Adjunction

We consider a problem in the context of a categorial domain whose objects and relations are directly inaccessible. We may think of this domain as a particular level in a universe accommodating other possible levels as well. First, we move out of the context of the problem, formulated at the level of the inaccessible domain, by adjoining to it architectonically another accessible domain, through which a partition spectrum can be built regarding the former under some viable hypotheses.

In order to accomplish this, we have to set-up an encoding bridge from the level of the inaccessible domain to the level of the accessible domain, such that a partial or local congruence can be established between these two domains entering into communication. The partition spectrum is the outcome of this congruence and amounts to the evaluation of the hypotheses.

The process is completed by setting up an inverse decoding bridge from the level of the accessible domain back to the level of the directly inaccessible one.

In this way, the available means and knowledge pertaining to the accessible domain can be lifted at the initial context of the problem. This process accomplishes a metaphora, which is usually pertaining to structure. Therefore, the problem can be effectively resolved in the context of its initial formulation by the embracing of the obstacle it engulfs via the adjunction channels opened up through the functorial encoding/decoding bridges with the other categorial domain.

Metaphora can be iterated through the adjunction of more than one controllable domains adjoined in succession to the inaccessible domain. Therefore, the circulation achieved through metaphora is capable of resolving the problem spectrally under the adjunction of one or more levels in communication via the bridges. Finally, the resolution of the initial problem translates to the qualification and quantification of the invariants associated with the obstacle imposing direct inaccessibility, expressed in terms of the congruence relations of the spectra.

Since the central issue is the notion of an obstacle, and the metaphora devised to circulate around it via different domains capable of establishing congruence relations with the domain where this obstacle is located, the notion of a rigid and absolute foundation for all mathematical objects is not applicable.

Instead, what is crucial always is the conception and effectuation of a functorial architectonic scaffolding that potentially is able to bridge bidirectionally, bound, and bond together these domains via natural transformations, so that the metaphora can be performed successfully unveiling the invariants that underpin the congruences.

The synthesis of the invariants is not dependent on the sequential ordering of events, that is, it is not dependent on their chronological ordering. Rather, it requires tempering, balancing through the means, and a critical temporal state of attunement, what was called Kairos in the ancient literature.

From this viewpoint, the metaphora enacting the circulation through another spectral domain may be though of as a motif that can be iterated, and as such, it guides the evolution of mathematical thinking by opening up functorially more and more elaborate channels, depending on the ingenuity of the architectonics. Due to tempering and attunement, the motif energetically transforms to a motivic key that bears the capacity to unravel the invariants.

Modes of Unfolding

Congruence and Invariance

The diachronic character, intricacy, and value of mathematical thinking, especially in correlation with its communicative articulation, requires a substantially more rich thinking about the notion of time, not only pertaining to the linear ordering of events, but bearing the capacity to unravel the fibres of the weave making up a good mathematical theory, by which we mean what gives the character of abstraction, as well as the character of the historically diachronic and persistent to such a theory.

A characteristic instance is Goedel's first incompleteness theorem, which can be summarized in the assertion that if a formal system containing arithmetic is consistent, then it contains undecidable propositions, namely statements whose truth or falsity cannot be expressed within the language of this formal system.

Although Goedel's theorem bears the tag of a "metamathematical proposition'', the inertia against incorporating any viable temporal notion in the so called foundations by certain schools of purists has been so prevailing that the "meta'' characterization is devoid of any temporal connotation despite the meaning of this term, which, as a consequence, is interpreted at a purely formal skeletal logical level. From our perspective, this is a case par excellence of the conflation and confusion between Logos and Formal Logic in the continuity of Mathematical Thinking.

Instead of abolishing time completely from Mathematical Thinking, it is worth exploring the implications of a richer conception of time in relation to the genesis and growth of mathematical concepts. This is a fundamental aspect of the body of Mathematical Thinking that constitutes its integrity, its communicative capability, and its overall functionality as a living temporal entity, thus, certainly not as a formal skeleton.

Mathematical Forms-Abstraction-Axiomatics

According to Bourbaki, a collective name for a group of renowned French mathematicians, who struggled with the elucidation of the notion of structure in the continuity of mathematical thinking:

From the axiomatic point of view, mathematics appears thus as a storehouse of abstract forms-the mathematical structures; and it so happens-without our knowing why-that certain aspects of empirical reality fit themselves into these forms, as if through a kind of preadaptation. Of course, it cannot be denied that most of these forms had originally a very definite intuitive content; but, it is exactly by deliberately throwing out this content, that it has been possible to give these forms all the power which they were capable of displaying and to prepare them for new interpretations and for the development of their full power.

It is only in this sense of the word "form" that one can call the axiomatic method a "formalism". The unity which it gives to mathematics is not the armour of formal logic, the unity of a lifeless skeleton; it is the nutritive fluid of an organism at the height of its development, the supple and fertile research instrument to which all the great mathematical thinkers since Gauss have contributed, all those who, in the words of Lejeune-Dirichlet, have always laboured to substitute ideas for calculations.

The objective is to comprehend the conditions of partial congruence among abstractions and modular substitutions due to different types of obstacles in the course of unfolding of historical time, irrespectively of the temporal distance among events, which is actually the decisive factor for both, the qualification of diachronic validity, and the success of abstract thinking.

This richer conception of time is always implicit in the unfolding of structure from some domain to another one via the architectonics of the process of circulation around a pertinent obstacle, and in essence pertains to the distinguishable harmonics of the "Mathematical Logos'', the characteristic invariants of the appropriate topos of a theory, where the theorems assume their validity and truth.