A proposal for a new transcendental idealism in which pure mathematics plays a central role - much as it did for Plato and Proclus. Here are some of the tasks:
1. Study the continuity between ancient Greek mathematics and modern mathematics. The Greeks had an arithmetic and a geometry, but also the idea of a common mathematics which, it seems, attempted to study common structures of both arithmetic and geometry. Also there was applied mathematics like optics, mechanics, astronomy, etc. Greek arithmetic corresponds to modern algebraic number theory. Greek geometry to modern algebraic geometry. Common mathematics to commutative algebra. Thus the incredible and unique development of pure mathematics in France by the Bourbaki group and Grothendieck's school of Algebraic Geometry represents a genuine continuation of the spirit of Greek mathematics. We must study the philosophical significance of the structures of (finite) group, commutative ring and field (and also linear algebra, group representations and non-commutative algebras and rings).
2. Setting aside category theory we must abstract and study universal forms and processes manifest in modern 'common mathematics', i.e. modern algebra, and how these express the fundamental architectonics of Proclus' metaphysics, specially as outlined in the 'Principles of Theology'. For example, how the complex can be built up from the simple (decomposition theorems). Or how the complex can be restricted to express simplicity is different ways (different localizations of a ring). How simple objects function as generators and measures or numbers (cf. discrete valuation rings). The omnipresence of hierarchies (in particular their completions such as in the construction of the algebraic closure), finitude conditions, dualities, symmetries (in particular of figures in $\mathbb{R}^n$ or the famous Coxeter groups) and different ways of reading finite spectra of structures (Noetherian and Artinian conditions, compositions series, derived series, (co)homology sequences, etc.) . Read Albert Lautman. The 'simplicity' of a structure is sometimes related to the action it has upon itself, in particular in the sense of being able to 'return to itself', that is, the action being transitive - (for instance the case of a group being simple or the fact that all Sylow $p$-subgroups are conjugate). Note the curious duality of $S_n$ with regards to $A_n$ which are simple for $n\geq 5$.
3. Another approach to logic. Abstract logic vs. the concrete living enriched logic - mediated by appropriate special domains - which reveal the essence of logic in away which necessarily requires the energeia of logic itself. This is to be investigated in conjunction with Proclus' theory of logoi (see Gregory MacIsaac, The Soul and Discursive Reason in the Philosophy of Proclus, 2001).
4. Consider the sheafication operation and its application to the construction of the inverse image functor for sheaves $f^*$. There is a moment of atomization by taking stalks and the there is a re-integration by re-instating local coherence.
This is a very interesting post. You write, among other things:
ReplyDelete"For example, how the complex can be built up from the simple (decomposition theorems).".
And contemporarily, such reasoning is used both in theology (God as a simple entity or the One as a simple entity and the Nous, Psyche and other entities that come from it) or the example of the density matrix of states in quantum mechanics and the illustration of pure and mixed states on the Bloch sphere. Very many aspects of Proklos' mathematics seem to have contemporary applications.
On the other hand, what do you actually want to formalise and why?