Thus the correct prototype of the concept of category is a collection of objects each pair of which may be subject to a plurality of different relations. This prototype concept is superior because the relations are implicitly classified - and this is what is often done in practice in category theory. Morphisms (as usually considered) are rather redundant as relations and only special classes of morphisms become in fact relevant (and relations worthy of that name). Another problem is that 'naive' category theory assumes we have a notion of equality between 'morphisms' (for instance in universal constructions). But when are two relations 'equal' ? Clearly mere extensional equivalence is not sufficient or at least problematic.
But is not a category a collection of objects that are like different species of the same genus ? And according to our discussion above, species which can have a multiplicity of different relations between them ? But this is precisely the theory in Aristotle's Topics. The immediate species of a given genus are subject to many possible relations (perhaps involving a third factor): opposition, more-or-less, more desirable, better known, etc.
What are some of the most significant results in philosophy ? The clarification of the pure concept of computability (both in terms of machines and term rewriting: Turing, Church) and the pure general concept of axiomatic-deductive system and in particular the clarification of axiomatic-deductive systems which to some extent mirror actual human reasoning processes (natural deduction, dependent type theory). The complex interaction between psychological experience and objectivism (if not already thoroughly explored by Hegel) made definite progress with Brouwer's intuitionism and subsequent advances in constructivism (or intuitionism) - specially dependent type theory (Martin-Löf type theory). A major philosophical error: Zermelo-Frankel foundations and the standard proliferation of the concepts of 'topological space' and abstract theory of rings and fields. Superior to the concept of topological space is J.R. Isbell's concept of 'locale' (1972) which recaptures aspects of Aristotle's, Leibniz's and Kant's (and Hegel's) concept of space - or better still, the notion of Heyting algebra.
Originally there was a profound unity between algebraic geometry, analytic geometry, mathematical analysis and differential equations - between algebra, geometry and the method of infinitesimals and indivisible points. This 'good' mathematics was neglected or became marginal during the disaster of 20th century mathematics. It is represented by lesser known disciplines of 'real algebraic geometry' (which we can also say is the 'real' algebraic geometry having roots in the work of the Italian algebraic geometers), the study of analytic, semi-analytic and sub-analytic sets, singularity theory and Thom's theory of stratified morphisms: some of this mathematics seems to have been adumbrated by Hegel's long notes on the section on Quantum in the Science of Logic. The bad mathematics consisted in wrongly abstracted algebraic geometry based on Noetherian rings and fields (A. Weyl, Grothendieck) and the wrong abstraction of analysis based on general topology and 'infinite-dimensional' vector spaces (all ultimately based on the Bourbaki framework and Zermelo-Frankel set theory). The theory of finitely determined germs and universal unfolding is 'true algebraic geometry' which is also the approach to the calculus based on the basis of powers found in Hegel (also we should prefer étale spaces to the abstract definition of sheaf and covering spaces to locally constant sheaves). The opposition between Thom and Grothendieck (we mean here scheme theory not his later work on topoi, homotopy theory, dessins d'enfants, etc. ) might be seen broadly as the opposition between authentic and false mathematics. A nice project would be to continue Lawvere's work on synthetic differential geometry and Grassmann and also go back to the great algebraic geometers of the past (such as Bonaventura Cavalieri) and give a rigorous foundation to their infinitesimal and 'indivisible' techniques
It is interesting to read Husserl's Philosophy of Arithmetic in light of Hegel's treatment of Quantity in his Science of Logic. Indeed Hegel's treatment of number, magnitude, infinite progression, ratio, measure, etc. can be given interesting interpretations in terms of category theory (specially the treatment of the natural number objet and computability in a topos) and modern singularity and bifurcation theory.
The theory of knowledge involves the analysis of the essence of reason. But it assumed that human consciousness cannot completely abrogate and go beyond reason (and thus ultimately see reason), not in the direction of something 'inferior' to reason (presumably the realm of dangerous lower instincts or to a kind of 'irrationalism' which is a real problem of our times), but something superior to reason, super-rational. The same goes for the 'self' and 'volition'. Indeed Hume's theory of the self does not deny that there is systematic 'energy' at work causing the impression of 'self' (even if allegedly this 'idea' is unfounded). Modern western theory of knowledge has the fatal error of ignoring the fundamental principles of ancient philosophy (both western and eastern) relating to the necessity of engaging in systematic practices relating to the purification of consciousness in order to be able to gain access to knowledge (and in particular self-clarity relating to consciousness itself). But note that these practices themselves already require (partial) philosophical insight. See MacIsaac's thesis on 'The Soul and Discursive Reasoning in the philosophy of Proclus' for some interesting and overlooked ideas regarding the theory of knowledge. Sausurre's theory of syntagmas and association is an example of truly insightful phenomenology (or cognitive depth-psychology). Interesting also is Peirce's phenomenology.
If to be at home in the world of objectivity much practice and preliminary methodology is required, a cyclic return and refinement, why should not the same hold for what Peirce called 'phaneroscopy', direct internal spiritual cognition ? And the 'subjective' and 'objective' being subtly connected, when easy with the objective world is attained some subjective intuitive clarity is automatically attained. And indeed introspection involves an object, consciousness itself or aspects thereof become objects themselves.
We do not understand what consciousness is nor how consciousness can directly perceive itself nor what in that case would be perceived and how it would perceive and how this perception relates to objectivity. What is clear is that we never perceive atomic sensations. The concepts of 'subject' and 'object' are liable to criticism. Maybe consciousness can only at first look at itself indirectly when engaged in some cognitive activity.
Speak not of subject or object but of ceasing to look without and starting to look within (Plotinus). And look within in a way in which you are looking at what is really there without distortions and projections. But what is this 'looking without' and 'looking within' and first of all 'looking' ? (Husserl himself quotes Augustine's noli foras ire, in te ipsum redi, in interiore homini habitat veritas.).
If we need logic to grasp and operate an axiomatic-deductive system, what sense is there is trying to capture logic itself as an axiomatic-deductive system ? (it is a kind of reflection-into-self). Cf. also our previous criticism of Sextus. The net of logic is very vast yet there is no easy escape from it.
The situation of ordinary consciousness: it cannot just immediately gaze inward on itself, see itself through inner perception and perceive the truth (it will just loose itself in a pool of shadows and mirages and fleeting dreams). It can only manifest itself to itself gradually in and through its activity and specially clear logical cognition. However there is also the possibility of making a constant progressive effort at 'conversion' or 'turning inward' its aim and target (its intention).
Stoic logic was not a kind of 'propositional logic', rather it was closer to quantifier free many-sorted first order logic.
Just as different physical phenomena can be described and studied by the same mathematical structure, so too different mathematical theories can contain the same unity and energy of reversion arising from the same source.
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