https://bmcr.brynmawr.edu/2010/2010.02.17/
Ennead VI,6, which deals with Plotinus' philosophy of number, is a very difficult text to understand. However the text offers prima facie strong evidence for enlisting Plotinus squarely in the ultrafinistic and constructivist camp (not only for the essential qualitative numbers but for their monadic quantitative emanations as well). We need to investigate how the second kind of number may be traced back to the first kind. We suggest this be effected by viewing the sequence of natural numbers as embodying increasingly rich qualitative and relational properties and use this as a basis for a spiritual exercise. That is, we view the sequence of positive integers as embodied in the corresponding sequence of finite groups and their group rings over finite fields, finite fields (for powers of primes), their irreducible polynomial and primitive elements and their matrix rings and finite projective planes (for numbers of the form $n^2 + n + 1$) which has a connection to the game Spot It!, general finite rings and other finite algebraic and combinatorial structures, different kinds of automata, algorithms, games, graphs, formal grammars of classical languages etc., and visualize the unfolding of their structure and relations. Note that the finite necessarily algebraic extensions of finite fields have cyclic Galois groups.The metareflection principle (which includes the principle of induction) expresses an ascent and return and conversion. Of great interest are the finite-field techniques used by the Singular algebra software, including Rational Reconstruction via the Extended Euclidean Algorithm (also known as the Farey Map). So, contrary to certain opinions, the finite fields $\mathbb{F}_q$ can indeed be considered as good approximations of $\mathbb{Q}$. Note that Book X of Euclid is a structural study of towers of quadratic field extensions over the rational numbers $\mathbb{Q}$.
Let the following postulates be assumed in common for the totality of the objects of mathematics: that their mode of existence is incorporeal and selfsubsistent, holding an intermediate rank between indivisible beings and those that are divided about bodies, both as forms and as reason- principles, being assigned the middle status between the simple and the divisible, being purer than the latter, more variegated than the former; that they make use of composition and division, but oversee the synthesized and the divided without coming to be and eternally; they are inferior to intelligible beings but prior to natural ones; they are in beauty, in order, and in exactitude superior to things visible but inferior to the intelligible, and similarly they have an intermediate symmetry and compatibility; they have the power to transport and lead over to the indivisible forms, since they are akin to them, and they lead away from corporeal concerns those who have become accustomed to them, converting them to the divine beings, as if on a ladder leading up to the heights. - Iamblichus, De Com. Math. Sci.
But we must emphasize just how profoundly obscure, opaque, incomplete, vague and metaphorical all the above neoplatonic and neopythagorean accounts of mathematics are, cosmo-ontologically and with regards to spiritual realization - and even to basic epistemology and psychology.
While mathematics is placed at an intermediate rank, we can certainly understand their theory of how mathematical objects operate in nature and at the same tie subsist in the human mind (but then they must be finite in number). We can also to some extent understand how, in this framework, the study of mathematics is spiritually beneficial, helping to energize the soul and turn her away from lower passions, sense-impressions and opinions. However it remains totally unknown and incomprehensible how mathematical objects or theories could be considered lower-level emanations of the intelligible realm and what their higher principles and prototypes could be. And beyond the basic understanding of the cathartic and anagogic function of studying mathematics outlined above it remains totally incomprehensible just how mathematics (or what branch or version of mathematics, ancient or modern) is to be studied as part of a spiritual path leading to the intelligible realm and exactly how mathematics could effect such an ascent and reversion. Our authors speak in hints and riddles and vague but attractive metaphoric suggestion.
Here is our proposal for a solution. The neoplatonists and neopythagoreans did not grasp Plato's genuine thought concerning the cathartic and anagogic function of mathematics (such as expounded in Book VII of the Republic). Plato is quite clear that the key role here is not to be played by mathematics but by dialectic, by pure logic. The key spiritual role of mathematics is as a starting point from the exercise of pure logic or dialectics. Just as the visible triangle is a tool for grasping the idea of triangle so too are mathematical objects and theories the tools for the self-revelation of pure logic. The fatal flaw of neoplatonism, as mentioned before, is not taking up the most sophisticated developments of formal logic and dialectics inherited from the Stoics and Megarians (and attested by Galen and Boethius). That is, they did not comprehend or develop or deploy Plato's hymn of dialectic. This only is the anthos tês psychês which can lead the soul back to the unity of the intelligible realm. Dialectics is both the most sophisticated formal logic (which realizes logicism by extracting theories from pure thought and analyses hypothesis while being beyond them or extracting them directly from itself) and pure power and energy (pure general fluid intelligence and concentration) which creates and dissolves, separates and combines - and we may interpret the cryptic account given by Plotinus himself in Ennead I,3. This is why it is important to understand the deep spiritual meaning and function of the dialectics of the later Academy, the Megarians, the Stoics, Pyrrho and Sextus - in light of the correct spiritual understanding of Nagârjuna, Yogâcãra and other dialectical Buddhist traditions as well as their Vedantic counterparts. The finitism and constructivism above must be understood as furnishing the clearest, cleanest and richest ground for dialectics to reveal itself and blossom. But note that the range of dialectic transcends mathematics and encompasses all concepts and theories.
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| Finite projective plane with $13$ points and $13$ lines, $13 = 3^2 + 3 +1$ |
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