The following note sketches some ideas that attempt to make sense of Proclus' theory of mathematics and dialectic in the Commentary of the First Book of Euclid and Commentary on the Parmenides. How does the study or doing of mathematics lead to the unveiling of the system of the essential logoi in the soul and consequently the souls reversion (according to its mode) to the nous ? What mathematics should be studied or done and how should it be approached ? Is there an essential philosophical difference between ancient and modern mathematics ?
To attempt to answer some of these questions we propose the following theory of mathematics. The structure of mathematics (be it ancient or modern) resembles the structure of living tissues, it is composed of a grid, a tiling, of 'cells' which are also evidently (logically and conceptually) interconnected. But each cell (even if incomplete and fragmentary from a purely formal mathematical point of view, from the point of view of concepts employed and results derives) exhibits a certain essential unity and sufficiency from a higher perspective.
In the figure above the lowermost layer of cube represents mathematics with its natural division into cells (small cubes), each representing an autonomous intelligible unit of mathematical theory. It is important to be abe to carve out mathematics according to its natural cells or units. Now mathematics is constantly growing (both in scope and in detail) and self-revising. But this growth should be represented as a horizontal growth represented by the expansion of the lower layer of the cube (adding new cubes). Over each cube in the upper layer is a column of cubes progressing in the upward direction. These represent the progressive unveiling of the logoic and noetic content of that particular mathematical cell: for each mathematical cell is like a microcosm of self-sufficient intellectual and noetical content and potential. It would be more accurate to represent the cube as converging like a cone in the upward direction, for the ultimate goal of the vertical process of every cell is the same. It is this upwards interpretation which is also a source of synthesis and progress in mathematics. It is clear that Proclus' anagogic process cannot depend in any way on the further horizontal progress of mathematical theory (or on the difference between ancient and modern mathematics). Rather it must be sufficient to consider one (or a few) genuine mathematical cells and use it a starting point for the anagogic process.
Common mathematical practice is concerned almost exclusively with horizontal expansion and the birth of more cells, a frenzy for finding proofs, defining concepts and producing new results - which justifies in a certain sense some of the censure addressed at mathematics in Hegel's Science of Logic (the proofs are left behind like a ladder). There is not so much of a return-to-self via dwelling on a given cell, or a gradual development and deepening philosophical and spiritual intuition of a given organic unit of mathematical theory. All genuine units of mathematical theory have at first sight something 'difficult', 'mysterious' , 'non-evident' or 'surprising' about them (and this is the source of the addictive nature of mathematics), even if this be regarded as proceeding from a mere fortuitous combinations of clever tricks.
Thus for the Proclean anagogic and reversion process based on mathematics our first, vitally important, task is to identify and natural intelligible cells, noetically self-sufficient units, in the great body of mathematical literature and knowledge.
And yet there are so many factors and qualities involved in a portion of mathematical theory that it seems difficult to assign perfection, completeness and sufficiency to any given theoretical portion (either ancient or modern). So the corresponding anagogic process will, it seems, always be approximative only, if we consider merely its dependency with regards to its purely mathematical basis. Something else will be required to supplement the defect.
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