Awakening

 We mentioned before that the is a difference between a true spiritual path (leading beyond anguish and suffering) and a false spiritual path. A true spiritual path involves essentially a habit and state of mind which  we call philosophical awareness or consciousness. The false spiritual path completely lacks it. This state of mind we could attempt to describe as transcendental philosophical consciousness (i.e. awareness, analysis, inquiry)  of the totality of consciousness itself (the immanent world). We use the abbreviation TPC.  One of its oldest and most magnificent attestations is found in the Pali nikayas (with the Socratic dialogues and Pyrrho) and more recently to a certain degree in Hume and Kant. This is what Schopenhauer said concerning Kant's Critique of Pure Reason (the analogies of experience): never has the world been looked at from a greater distance.  Our project is not only to say something about what TPC involves but also to discuss its epistemic and logical aspects, a subject of uttermost difficulty. And of central concern is of course its close relationship to introspective psychology and so-called psychologism (as well as the ambiguous hydra of the term 'idealism'). There is always a danger that TPC may lapse back into the modes of non-TPC consciousness. That is, a false objectification of TPC insight and knowledge. Such an objectification involves forgetfulness (of constitution, condition and presupposition) and loss of insight.   TPC crucially involves a way of seeing and questioning (bracketing, suspending) the totality of (conceptual, intuitive, affective, volitional, egological, spatial-temporal, somatic, etc) experience. The total awareness of the total sphere and process of thought and what is beyond all thought and what thought is actually behind and the fact that it is behind it. TPC is a gradual affair,  an ideal which is worked towards by practice and training. TPC traces back and unveils the unified common source of all branches of philosophy. Poetical language achieves its highest function in the expression of TPC. 

All this we have written about in our compiled Philosophical Monologues collection.

Notes

The embedding matrices used for instance in ChatGPT-3 are a vector space representation of co-occurrence frequency matrices for a given context-window size. This matrix can also be seen as a complete graph with edges labelled by probability values (we can assign values also to polyhedra).  It is important to study the properties of these matrices.

Locally integrable functions (fundamental to distributions and weak solutions to PDEs) are a kind of sheaf-theoretic completion of the $L^p$ spaces. On the other hand from a finitist point of view simple functions are the basic kind of function and these are 'dense' in many fundamental spaces in an appropriate sense. What is the meaning of a PDE in a scientific context ? A positing of certain recursive numerical algorithms and approximation ideals.  How curious that numerical conditions for stability involve the pairing of space and time.  And fascinating is that smooth initial conditions for PDEs as simple as $u_t - g(u)u_x = 0$ may generate shock waves. And what is, from a computational finitist point of view, a weak solution ? A good philosophical goal: to gain a deeper understanding of distributions (cf. Sato's hyperfunctions).

How could we prove that a physical system is computing beyond the Turing limit ? We could of course produce experimental evidence to the contrary, producing a certain algorithm that agrees with know observations. There are irrational numbers whose sequence of digits are not computable. How are we to view scientific theories about such numbers (which could represent measurements of some fundamental physical constant), calculations and approximations of such numbers, and confrontation with experimental evidence ? Obviously such a question is only interesting from a non-finitist perspective.

The interest in solving the P=NP problem hinges on the complexity of the algorithm for transforming a NP-machine into a P-machine. 

Completed papers

Quantifier Reasoning and Multiple Generality in Aristotle and Ancient Logic

Aristotle's Topics and Extended Second-Order Logic

On Analyticity and the A Priori 

Natural Term Logic (in preparation)