"The limits of my language mean the limits of my world" (Wittgenstein, Tractatus 6.421).
"That which is expressed in language we cannot express by language" (Wittgenstein, Tractatus 4.121).
"Thus the limit can only be drawn in language, and what lies beyond the limit is simply absurd" (Wittgenstein, Prologue to Tractatus).
The Church-Turing Thesis
According to the so-called "Church-Turing thesis," the limits of computability, that is, of what is computable, are reduced to what is computable by a Turing machine or by recursive functions.
According to the Church-Turing thesis, the intuitive concept of computation can be implemented by a Turing machine or by recursive functions. Both approaches (functional and operational, respectively) bound the maximum known possible computational power, and are equivalent:
Every recursive function is computable by a Turing machine.
Every computational process performed by a Turing machine can be expressed by a recursive function.
The MENTAL Thesis
In MENTAL the Church-Turing thesis is generalized, in the sense that we speak of the limits of the expressible in general, and not only of the computable. It is the "thesis of MENTAL":
Only that which is expressible by MENTAL language is expressible
The MENTAL thesis generalizes the Church-Turing thesis, for language is operative and descriptive:
With the operational aspect, we can express what is computable, that is, what is constructible by means of a finite expression, which produces as a result a finite expression in a finite time.
With the descriptive aspect, we can express the finite and the infinite by a finite expression representing several expressions.
MENTAL implies a higher level of abstraction than the Turing machine and the computable (recursive) functions, since it is possible to specify both a Turing machine and the recursive functions of Church's lambda calculus, which means a unification of computer science and mathematics in the computational aspect, that is to say, it unifies the concept of algorithm and function.
MENTAL clarifies the deep nature of computation, of the operational. And, at the same time, it clarifies the nature of the descriptive. Both are the two aspects of universal semantic primitives.
MENTAL's thesis is based on a set of principles, which are the primitives of language themselves, that establish the degrees of freedom. Expressions of language are concrete manifestations of those degrees of freedom.
The expressible and the modes of consciousness
If an entity can be constructed by an expression, then the entity is associated with the mode of consciousness of the left hemisphere of the brain. For example, natural numbers, integers and rationals, finite sets, and so on.
If the entity cannot be constructed, only described (because, for example, it implies infinity in some way), then the entity belongs to the right brain hemisphere mode of consciousness. For example, irrational numbers, infinite sets, etc.
In short, the expressible is that which conforms to the semantics-syntax of MENTAL. The universal semantic primitives are inexpressible, since they belong to the internal and deep mental level. Only their external or superficial manifestations are expressible in the form of concrete (operative or descriptive) expressions.
The limits of MENTAL
What lies beyond language, beyond the expressible? The mystical, as Wittgenstein asserted, that is, the ineffable, the indistinguishable, that which cannot be expressed, the absolute:
"The feeling of the world as a limited whole is the mystical" (Tractatus 6.45).
"The inexpressible certainly exists. It shows itself, it is the mystical" (Tractatus 6.522).
"Of what one cannot speak one must be silent" (Tractatus 7).
But MENTAL allows us to intuit the inexpressible, since the perception of the archetypes of the consciousness places us on the frontier between the expressible and the inexpressible, between mysticism and scientism, in a transcendent consciousness, which brings us closer to the frontier of the absolute.
Therefore, MENTAL has intrinsic limits, a lower limit, but no extrinsic limits. There is no upper limit, because the fractal nature of language (its combinatorial capacity) allows the existence of an unlimited number of levels.
Consistency and completeness
Any formal axiomatic system must have two characteristics: consistency and completeness.
A consistent system is a system in which it is not possible to arrive at a contradiction, that is, that it is possible to arrive at an expression that is both true and false.
A system is complete if every mathematical expression that is related to the axioms can be asserted to be true or false.
These two properties are meaningless in MENTAL for two reasons:
Because MENTAL is not a formal axiomatic system. It is a semantic axiomatic system. Axioms are primary concepts that are inexpressible. Only their particular manifestations are expressible: the expressions.
Because the concepts of truth and falsity are meaningless. Expressions are neither true nor false. They are simply constructed, and by constructing them they have existence in abstract space.
What does make sense to ask is whether the proposed set of semantic primitives are sufficient for expressivity purposes. It is not possible to prove this formally, as it is not possible in the case of the Church-Turing thesis, but the numerous examples of application of language to very different domains seem to confirm it, hence the proposed thesis. In any case, it is perfectly possible to establish other alternative sets of semantic primitives, although we believe that the selected ones constitute a very adequate proposal.
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