On language yet again

 What is language, after all ? Consider a system.  Let this system have subsystems capable of epistemic, volitional and doxastic states. The epistemic and doxastic states can refer to (partial) aspects of the system, its subsystems and their relations, both present, past, future, possible or alternative, counterfactual. Between two subsystems there are channels. The structure of a channel at a given time or interval is called communication data. The meaning of the data is the communion data plus the context which roughly speaking is nothing but the global present and past history of the system, but more specifically the present and past history of the system as it relates to the sender and receiver. This history will of course in turn involve communication.  But subsystems' doxastic, epistemic and volition states also employ data in the configuration of their internal states - thus the symbolic mirroring of the macrocosm in the microcosm - which must bear at least analogy to the communication data of channels. These states evolve, are learned or discovered in the context of communication.  I am happy must include both a state for happiness and a state for awareness with a parameter which represents the happiness state and the time in which the actual happiness state obtains. Of course internal representation can depend on context too. The ideal of mathematical and scientific language and knowledge is that it is maximally context-independent. A major feature that cannot be overlooked involves the growth, disappearance and reproduction of subsystems. UDIL is a good tool for formalizing the objects of knowledge, the states of affairs of subsystems and the total system. Another problem is the existence of a plurality of languages, including modes of expression and interpretation. One can even wonder if there are as many languages as subsystems.

Thus the meaning of communication data or representation data is in general  a function of particular subsystems (for instance the sender and receiver) as well as their state and state-history which will implicate large parts of the whole system and its history.

A very good formalism for these kind of systems is found in the Object Specification Logic of Sernadas et al. Takahara's general systems theory, which is clearly inspired by classical control theory, needs to be improved and reworked in the following form. We are given an input space $X$ with special do-nothing element $\bullet$, a state space $Q$ and an output space $Y$ also with a show-nothing element $\bullet$. Let $T$ represent linearly-ordered time.

Then a (in general non-deterministic) system is a correspondence $\mathcal{S}:  X^T \rightarrow \mathcal{P}((Q\times Y)^T)$. We can easily define all the core notions as well as feedback and interconnection of systems.  But of course each $Q_i$ will be very complex to be able to represent states of the entire system, etc. Also $Q$ will in general have a part $Q'$ which will have spatial determination $Q' \rightarrow \mathbb{R}^3$. If the spatial extent of the system varies then in a way $Q$ also varies. Thus there are reasons to extend the definition of system to sheaves.

Hegel is not an easy or even very clear writer. But therein are many important concepts and observations couched in an unfamiliar language that can be studied in a perhaps more through and lucid way both in previous (there is a huge amount of implicit reference and borrowing in Hegel)  and posterior philosophy. Aristotle, Sextus Empiricus, Augustine, Proclus,  Cusanus, Spinoza, Malebranche, the 'esoteric' Leibniz of Russell and Gödel,  Lotze, Bolzano, Brentano, McTaggart's commentary of Hegel's Logic and  Hofstadter's Gödel, Escher, Bach are to be noted. Hegel's core notions and valid observations are found in a more thorough and lucid form both before and afterwards, even rediscovered in mathematical logic,  category theory, computer science and general systems theory.

Grothendieck topologies and mereology

Suppose we considered arrows $f :B \rightarrow A$ in a category as expressing generalized parthood and read them as $A$ is an $f$-part of $B$. But what if we also consider a $g : C \rightarrow B$ ? Then by composition $C$ is a $ f \circ g$-part of $A$. What is the concept of sieve, mereologically speaking ?A sieve on $x$ is a set (class) of parts $y$ of $x$, $Pyz$, such that if $z$ is a part of $y$ then $z$ also belongs to this set.\[Siev(A,x) \equiv \forall y. ((y \in A \rightarrow Pyx) \& \forall z. Pzy \rightarrow Pzx) \]A sieve on $x$ is a set of parts of $x$ plus all the parts of those parts. It is a $P$-filter on $z$.A covering sieve is a generalization of fusion: $x$ is to be seen as a generalized fusion of any one of its covering sieves. The axioms for a Grothendieck topology make sense for fusions. If $z$ is a fusion of the $\phi(x)$ and $Pyz$ then we can consider $\phi_y(x)$ (the $y$-restriction of $\phi(x)$) expressing elements which are overlaps of elements which satisfy $\phi(x)$ with $y$. Then we should have that $y$ is the fusion of the $\phi_y(x)$.Suppose $z$ is a fusion of the $\phi(x)$. And consider a $P$-filter on $z$ given by $\psi(y)$. If for every element $x$ which satisfies $\phi(x)$ we have that $x$ is a fusion of the $\psi_x(y)$ then $z$ is also a fusion of the $\psi(y)$.Finally each $z$ is the fusion of all its parts.\\

What about the topological concept of boundary ? The Aristotelic concept of the topos of $x$ involves considering a $z$ with $Pxz$ such that for all $y$ such that $Pxy$ we have $Pzy$ (a minimal cover). A modal version would be more appropriate. Then the topos itself should be a kind of completement $t$ such that $t$ and $x$ do not overlap and $t + x = z$.

Talking about what will never be specified

There real line has an uncountable number of elements. Natural language and all our formal languages have at most a countable number of possible expressions. Hence there are real numbers r which we will never be able to define in the sense that r = the x. P(x) for some property P(x). But we are referring to these numbers hence defining them. Is there a contradiction here ? No, because we are defining them as an indefinite plurality. We still cannot separate them or distinguish one of them. No matter what specification we propose to single out an indefinable r it will always escape through our fingers. There will always be more than one indefinable satisfying the specification.
Perhaps this might offer a clue for a new interpretation of quantum theory ? Note that if particles behaved classically and had smooth trajectories specified by a differential equation, then the position and momentum at any definable time would be itself definable (even if not computable). Hence we could postulate that in quantum theory the particle evolves through indefinable states and hence cannot be described by a differential equation - only in the sense of indefinite pluralities as above - that is, a stochastic differential equation or wave-function differential equation. In this way there would be no need of a 'collapse of the wave function'. What are some objections to this interpretation ?