The Logic of the Enneads and its mathematical reflections

Plotinus was also a consummate logician in the sense of universal logic, ontological logic or logical ontology.  Buried deep and intricately interwoven within the Enneads we find a sophisticated Science of Logic in the sense of Hegel but expressing a radically different spiritual and philosophical framework and outlook. We will not discuss here the interesting question of a direct influence on Hegel or of Hegel having directly and extensively appropriated neoplatonic doctrines in his own system (we have already given some references concerning this topic). Nor will we discuss the radical rupture in the philosophy and spiritual practices of the neoplatonic school  that occurred after Iamblichus.  We can only stress again the importance of expounding systematically a non-Proclean and indeed anti-Proclean neoplatonism.  

 The project we sketched in our paper "Hegel and Modern Topology" was to given a Hegelian interpretation of key aspects of modern mathematics, logic and computer science and to in turn use these to furnish an interpretation of Hegel (the hermeneutic virtuous circle). We find that most of the insights contained in what we sketched of this project can be transposed from Hegel's Logic to Plotinus's Logic where they seem to find a even more adequate and natural setting. But it must be understood that, as we explained before, in this framework mathematics and formal structures function essentially as an elaborate system of symbolism and hierarchical reflection rather than as candidates for a direct and complete formalization.  The other aspect of mathematics and logic, culminating in dialects to be used as a tool for direct spiritual ascent, has to be considered separately.  

In order to embark on this project we should list the fundamental categories, concepts and processes of Plotinus' Logic which we can organized somewhat along the lines of Hegel's Science of Logic but with fundamental differences which are often the complete inversion of Hegelian conceptions or which collapse what Hegel separated or separate what Hegel joined. Bearing in mind Hegel's Logic of Being we can remark that Plotinus has an elaborate and deeply interesting theory of quantity, extension, number, quality, essence, determination and measure which permeates in the Enneads (and justifies our classification of Plotinus as a finitist) - together with a remarkable theory of relation which when deployed at the intelligible realm fits exactly our considerations of groupoids as models for essential relationality and interconnectedness.  The theory of a self-moving number and relationality of the nous fits well with our previous considerations on Galois groups and Galois extensions. We note that Proclus's theory of Henads is found in VI.6 but with the difference that for Plotinus they are a proto-determination in Being (as an infinite ouflow) which preceed the intelligibles rather than the One.  And also a highly developed and articulated theory of infinity (outflow, power) and limit (reversion, theoria)  and a theory of "matter" (both physical and intelligible) which amounts in the physical level to the poorest and emptiest determinations of being (strongly echoing the initial stages in the Logic of Being) - which also provides insofar as Plotinus states explicitly that indeterminate matter will display contradictory dualities - a striking possible interpretation of quantum theory.  A fundamental mathematical symbolism or reflection will be that of the limit (let us say, in the category theoretic sense) of a sequence or diagrams expressing an indeterminate outflow or process. This limit is a reversion, completion, perfecction and theoria in the Plotinean sense and an aufhebung and being-in-and-for-itself in the Hegelian sense.  The reversion to the level of Soul and Nous parallels to a certain extent to Hegelian levels of Essence of Concept but with many radical differences and fundamental corrections from a neoplatonic point of view.  

When using mathematical structures as neoplatonic symbols and reflections we find that it is best for formulate them and think of them in a strictly constructivist way.  Thus the number of objects in a category must be conceived as being indefinite and defined as one would do in a dependent-type based proof assistant - considering it to be the object object in the definition of an internal category.