Finitism and resource consciousness

In formal logic and theoretical computer science not enough attention has been payed to the explicit study of resource limitations and their implications (cf. the study of the length of proofs).  We should reform these disciplines by explicitly postulating finite bounds in every aspect of the object of study and attach importance to the quantitative study of the interdependence of these bounds.  For example: study the length of proofs in function of the formula to be proved in system based on a finite language. Study resource limitation in the grammar of natural language. Study the computational capacities of automata seen as finite approximations of Turing machines. This may be a way a developing abstract theories with a higher degree of fundamenta in re.  Study the bounds and limitations of encodings of information and attempt to understand the question: can there be provable mathematical statements which have proofs which are too big to be written down or processed by a computer given our physical limitations ? Are there numbers to big to be represented by any means in the physical universe ? Finitism means not only finitely many basic or 'atomic' elements but finitely many higher-order relations between them.  This all is related to the mysterious problem of qualitative changes in scale (and also the relativity of scale as illustrated in Gulliver's travels).  Remaining in the finite domain by increase the quantity of a certain parameter (size, speed, etc.) suddenly beyond a certain limit the behavior of the system can change radically.  This is usually studied using the infinitary models of mathematical analysis and geometry, but surely this occurs in a finitary  context at a basic level.  The question of finitism is related to some of the most fundamental questions both of phenomenology and the very concept of 'analysis' and 'scientific consciousness'.  An authentic phenomenology must give a central place to the concepts of 'illusion' and 'delusion' as well as the two-pronged nature of the 'social' and the 'cultural'.  

To understand the synthesis of phenomenology, logic and analytic scientific consciousness (in a playful symbolic way we could say: the synthesis between Hume, Sextus and Democritus - and we would add Epictetus for moral theory)  it is good to look at the structure and dynamics of the theories of biology and biochemistry and their surprising correspondence with certain aspects of contemporary mathematics.  The analogy between a tissue and its cells and the definition of algebraic variety (or better sub-analytic objects or general stratified objects) which has both a 'smooth' and 'extensive' aspect and a local, discrete, algebraic 'intensive' aspect.  The algebraic locality is much like the discrete biochemistry of a cell and homology theory a kind of genome or centrifuge of the algebraic structure (so too are graded ring constructions).  What is lacking is of course the dynamic, transformative aspect.  The connection between biochemistry and logic (or between processes and inferences) is profound and Girard explicitly acknowledges this in his foundational papers on linear logic. We propose that the basic local-global (sheaf theoretic) concepts of geometry be applied to logic as well.  We have an organism made up from local logics considered as individual cells. For each cell hypothesis are what are given from without and what is deduced (asserted) is what exits the cell. The difference from standard sheaf (and topos) theory is that we have a dynamic global aspect related to (rapid) transport and distant interactions (cf. the $\pi$-calculus).  This TeX package is interesting because it bridges the gap between logical (and linguistic) syntax and chemical syntax by developing a linear system to represent two and three dimensional chemical diagrams (just as logical expressions codify syntactic trees).

If reality is represented by $U$ then a 'perspective' or 'approximation' or 'abstraction' or 'construction' is a system of relational logical atomism $A \rightarrow U$.  These perspectives can be organized by a partial orders $A < A'$ signifying that $A'$ extends $A$ or that $A$ abstracts from details of $A'$. A very interesting aspect of this abstraction are the convenient fictions of mathematical analysis, the 'passage to the infinite', seen also in statistical mechanics and the kinetic theory of gases. Some comparisons might be made with Jain logic. The theory of scale is fundamental here.

All formal systems involve the characteristic of proliferation, generativity (cf. our previous discussion on semi-Thue systems).  This is also omnipresent in algebra and geometry. The philosophical and scientific ideal of a 'system'.  This reproduces fundamental characteristics of mental processes (note how Chatterjee's book on the Yogacara makes a connection to the broadly Hegelian subjective idealism of Gentile).  But this generativity is limited by resources and is also error prone. Pyrrho and Sextus have a radically distinct approach and goal. 

Addendum: Boethius and the medieval tradition support out interpretation of a topic as a 'maximal proposition', i.e. as an axiom.  There is an interesting notion of definition as an 'unfolding'. We should mention Boethius' 'hypothetical syllogisms' in our note on Kant's logic.  Can we find where Boethius is more explicit about a universally quantified conditionals ? Note that his indefinite  propositions is our indefinite article selector and that singular propositions are introduced. Disjunction seems to have been studied from a profoundly mereological perspective (see also Kant and Hegel) which echoes our treatment. 

Another view of philosophy

Philosophy is divided into pure grammar, ethics and liberation practices (eleutherology).  Pure grammar holds that the main legitimate domain and goal of philosophy is the description and direct awareness of the deep structures and processes of natural language (and this includes exploration and classification of the total semantic universe).  It is called 'pure' because this is to be done without any extraneous reductionism and ideological, theoretical presuppositions or interpretation, anything beyond the domain of the analytic a priori computationalist  'core logic' discussed in our previous work. Pure grammar criticizes standard philosophy (from its historical roots onwards) for its shaky or questionably linguistic confusions and simplifications (rather than taking the formalist logical-dialectical approach we have discussed in the past). There is also the problem of ideological distortions and arbitrary dogmas both in their views and applications of language, specially the most recent physicalist and behaviorist ones (and this applies equally to many contemporary linguistic theories with physicalist, behaviorist and social-pragmatic premises, or the dubious theories which reject meaning altogether.). Though of course historically many philosophers in some of their works have approached to realm of pure grammar (Aristotle's Topics, the Stoics, Buridan, Leibniz, Peirce, Frege, Montague, etc.).  Some of the great linguists are revealed to have been great philosophers (Pânini,  the (nava)Nyâya school, Saussure, Anna Wierzbicka). It is also 'pure' because it claims to be (or shows itself to be) cognitively and epistemically a priori and more fundamental than all the sciences: pure grammar is 'first psychology' and 'first cognitive science' (of course we distinguish between surface and depth grammar and bring to the foreground the plurality of natural languages and the study of universals). Pure grammar shows the limits of the world (without making claims about the world) and its shows what must and can be transcended in order to go beyond the world. Pure grammar challenges not so much the idea of formal logic in itself but the faulty paradigms which dominated the extensionalist logic of the 20th century (in particular its inability to distinguish between distributive and merely intensional quantification). Pure grammar  challenges both dialectics (Madhyamaka, Hegel) and idealism (Yogâcâra, Plotinus) - while at the same time ascribing a certain value to both of these (and more can be said in relation to eleutherology).  Perhaps Ch'an (Zen) - which we might call linguistic buddhism (the practice of the Koan) -  can be seen as related closely to pure grammar in its eleutheric aspect. Pure grammar should not be confused with the ideas of Mauthner, Wittgenstein and 'ordinary language philosophy'  - certain superficial similarities mask a profound, radical and irreconcilable difference and opposition (there is no question of pure grammar attempting to reduce language and meaning to any alleged social act, game or toolset: also a language need not be in perfect working order): a task of pure grammar is critical analysis and refutation of such schools. The same goes for structuralism (including the Prague School),  'grammatology' and postmodernist linguistic philosophy.  Comparison with Husserl's Logical Investigations (and to Bolzano, Lotze, Brentano, Meinong,  Twardoswski, Mally and specially Marty)  is an interesting and involved subject that is well worth exploring.  An important work for pure grammar is Bernard Pottier's Linguistique Générale (1974). Also Jean-Louis Gardies. Of interest is also medieval scholastic logic and grammar.

Wilhelm Wundt - Psychologismus und Logizismus (1910)

Some of our preliminary studies for our project of a 'pure grammar': Bealer's Intensional Logic, Aristotle's Second Order Logic,  On Analyticity and the A Priori.  It is important to compare the theory of definition in the Topics with modern 'analytic semantics' and its sememes. Furthermore, our papers  Differential Models, Computability and Beyond and Hegel and Modern Topology can be considered essays on regional ontologies and the pure categories of the understanding.

Pure grammar: an extended universal philosophical semantics and grammar as transcendental knowledge in the Kantian sense. Pure grammar is about seeing, about coming to awareness. It has no method. And it has nothing to do with social acts, behaviors or games. Learning language is a linguistic activity and is not a game or an action (for games presuppose language).  We cannot act without language (or logic). Sneezing is not an act.  Philosophy does not result from linguistic confusions and misapplication: rather the vice of philosophy is simply the ignorance of the importance and complexity of the pure linguistic a priori.

Pure grammar puts forward linguistic (arti)facts which being surveyed automatically  lead to pure knowledge.  That is, the artifacts are presented which disclose immediately cognitive-linguistic-semantic structural symmetries and dualities and triads - as well as proto-spatial and proto-temporal a priori categories. Thus pure grammar contains a kind of pure  a priori immediately evident geometry.  Pure grammar has not as primary aim the foundation of science but rather eleutherology. Kant anticipated much of contemporary linguistics.

The recursion of subordinate clauses is limited in natural language (maybe maximum three levels before intelligibility is lost ?).   We should investigate how subordinate clauses can be interpreted or transformed away.  This is of course intimately connected to intensional logic.  John believes that Mary believes that John believes something about her.  Can we transform this multiply embedded subordinate clause sentence into a sentence without subordinate clauses ?  John believes something about Mary. That is something Mary could be considered to believe.  And John believes that Mary believes that. This is of course just a way of disguising nominalization of sentences through backwards pointing demonstrative pronouns.

So, there is the triad of psychological (phenomenological) a priorism, logical a priorism and grammatical-semantic a priorism.  A particular kind of psychological a priorism is psycho-somatic a priorism with profound eleutheric consequences. As Wundt discusses in the text linked above, there is a deeply confusing entanglement, dance  and even contest between all the members of this triad. Which one, we can wonder, holds the key to a synthetic illumination of human existence ? To pañña and the overcoming of dukkha ?

The correct 'phenomenism': we note  that the naive, default world-view is actually 'idealist' in the sense that it is the result of an unconscious projection and constitution by an ego, a mine-making force and determination. Correct phenomenism  looks at things as they really are, looks at what really is in fact there and discerns the fundamental properties shared by the  things* that are there, not what is made from, by, in, or with them by the ego and mine-making force and its constitutive conceptual proliferation. False idealism attempts to logically and causally derive the world from the ordinary ego or subject (or will).  True phenomenism lets what really is shine forth as it really is and liberates from the world and from the world-tending ego and its superimposition, distortion  and positing.  Seeing psycho-physicality from a neutral third-person perspective. It is the coming to intimate yet detached awareness of the process of one's own thoughts qua such and merely as such as well as the perception of the world (and the mundane ego) as immanent and constituted in this thought-current (into which is woven sense-imagination). yogascitta vritti nirodhah. Correct phenomenism unveils the right reference point which synthesizes and makes a compendium of the sphere of core semantic categories of human existence (and hence thought and language and life). This is the critical 'phenomenology' we have discussed extensively in this blog.

*we do not mean 'things' in the ontological sense, but in the sense of mind-stuff manifestations.