Mythopoetics

As expressed in another entry: "The world of the life of a historical community, the Lebenswelt (L) (in Habermas’ [2010] sense of the concept) is in a close connection with the realm of experience that today is under the scrutiny of life sciences. In the philosophical milieu, one is spontaneously drawn to consider such a realm exclusively under the scope of contemporary science, and therefore, systematically , but if we want to elucidate the concept of system , we should proceed more carefully, for the semantical actions which lead us to distinguish something as a system are conditioned by some automatic psycho-biological protocols which belong to a different realm, let us call it Unterlebenswelt (U) . The acritical knowledge which constitute the communicative actions of L is the result of an evolutive process of communication and complexification which started beyond human grounds, in the communicative actions of U. Since communication is a social homeostatic tool, the basic...

 Let us construct a symbolic formal system with the following elements.     1. An arbitrary axiomatic system which contains Gödel’s axiomatic system together with its rules of inference (Ga)     2. The functions and relations of the system are recursively defined and free from contradiction.     3. We construct an isomorphic representation of the subsystem of non-numerical symbols by a system of positive integers, ascribing natural numbers to the symbols. Therefore, we can express any formula in numerical terms (particularly as a sequence of primes), and proofs as sequences of positive integers.     4. We construct a set of formulas F which are directly deducible within the system and which represent common expressions of our calculus. For every formula fi ∈ F, there is a numerical formula pi ∈ P, for P⊂ F, such that Ga ⊢ pi.  Construct a fj which expresses “this formula is not deducible”, a valid and meaningful expression of our calculu...

  Brouwer’s understanding of language uniquely on human terms justifies his separation between language and mathematics but does not make any psycho-biological sense. The act of distinction which characterizes the intuition of time in Brouwer’s sense precedes the use of our human languages but not of languages in general, i.e., of communication among living creatures. A twoity, therefore, precedes also the linguistic construction that we call number -even the concept of numerosity- and denotes an action of separation. For this reason I will call merisma (part), to such an action of distinction. A merisma  is a precondition for any action, no matter whether such distinction is accompanied by second-order distinctions in relation to the cognitive actor. A merisma has the irreversible character of the intuition of time, of life experience, and therefore, implies a fundamental or primitive determination for later distinctions. We can notate this primitive merisma by ‘≠’. The commu...

 If the Kantian criterion that distinguishes synthetic from analytic propositions, whether they are axioms or postulates, is the non-inclusion of the synthetic ones versus the inclusion in the subject of the analytic propositions, an analytic proposition could not be reduced or transformed to a synthetic one, nor vice versa, since something is or is not included in something else. Well, the very concept of inclusion is not exactly clear, modern set theory has opted for an extensional definition, that is, giving a list of the things that are included in another given. Inclusion thus conceived is less problematic, although it is not without paradoxes. Let's skip this for a moment and focus on the Kantian distinction.    Kant gives us his well-known example of a priori synthetic proposition, capable of giving us knowledge independently of experience: 7+5=12 But we can transform this equality into this other: 7=12-5 And then into: 7=7 That is an analytical proposition. And co...

Hypothesis 1: False formulas are unprovable. Hypothesis 2: Formula A is unprovable. 3. If A is false then A is not-unprovable. 4. Then, by modus tollens H2 and 3.: A is true. 5. Ergo, when A is true it is unprovable. We cannot define provability in semantical terms. The concept of provability has only syntactical meaning, therefore is useless for life.

  Si el criterio kantiano que distingue las proposiciones sintéticas de las analíticas, ya sean axiomas o postulados, es su no inclusión o su inclusión en el sujeto, una proposición analítica no podría ser reducida o transformada a una sintética, ni a la inversa, pues algo está o no está incluido en otra cosa. El propio concepto de inclusión no es precisamente algo claro, la teoría de conjuntos moderna ha optado por una definición extensional, es decir, dando una lista de las cosas que están incluidas en otra dada. La inclusión así pensada es menos problemática aunque no exenta de paradojas. Obviemos esto por un momento y centrémonos en la distinción kantiana.   Kant nos da su conocido ejemplo de proposición sintética a priori, capaz de darnos conocimiento de manera independiente a la experiencia: 7+5=12 Pero podemos transformar esta igualdad en esta otra: 7= 12-5, y después en 7=7, que es una proposición analítica. Y a la inversa...

Given a set S, a cardinal number M is the invariance of S after changes in the properties and relations of its objects. Two sets have the same cardinality if we can construct a one-to-one correspondence between its elements. These definitions are highly psychological. It implies a deeper identity of a set than the identity given by the extension of a property. Such identity would have to be a property since it can be predicated of more than one set. But how are we to know such substance in relation to which there occurs an invariance after changing properties and relations of objects? From a constructive point of view it would only make sense that such unmovable and unchangeable “Terminus” was to be introduced by definition (recursive). Then cardinal number would mean simply that we can construct a countable algorithm (in ℤ +) for a given set, and two sets have the same cardinality if for a given machine their countable algorithms stop at the same time.

There is nothing new in the mathematical approach to music composition which characterized many of the main stream pieces after the 60’s of last century. Recall Leibniz’s words: music is the hidden exercise of arithmetic performed by the soul though unaware of its process of counting. To a Platonic and Pythagorean oriented mind that would give an explanation for the universal power of music, its qualification as a mathematical discipline and thus as a subjacent structure of reality. Music would represent a kind of algebra for the intuition of time just as geometry deals with our intuition of space. Obviously, from a Kantian point of view, where space and time are pure forms of intuition or the conditions for our intuition of the world, such idea is meaningless: space and time are not objects, nor relations among objects, but simply the way that occurs our perception of the world. Other minds (non-human) would perceive it differently. In any case, mathematics prestige and strength, h...

    Let us construct a symbolic formal system with the following elements: 1. An arbitrary axiomatic system which contains Gödel’s axiomatic system together with its rules of inference, say G a . 2. The functions and relations of the system are recursively defined and free from contradiction. 3. We construct an isomorphic representation of the subsystem of non-numerical symbols by a system of positive integers, ascribing natural numbers to the symbols. Therefore, we can express any formula in numerical terms (particularly as a sequence of primes, as Gödel did, but the fundamental theorem of arithmetic is irrelevant for our argument), and also express proofs as sequences of positive integers. 4. We construct a set of formulas F which are directly provable within the system and which represent common expressions of our calculus.  Then, for every formula f i ∈ F, there is a numerical formula p i ∈ P, for P ⊆ F, such that G a ⊢ p i . Construct now a f...

It took two hundred years for the concept of system to make its way into a science of its own. Curiously enough, if we compare Condillac’s [1798] definition [1] to that of Ackoff [1973], [2] we observe that the general understanding of the concept as a set of interrelated parts is basically the same, furthermore, if we filter out the different precisions made from a wide scope of ontoepistemological stand points, we might conclude that most definitions of system since Condillac share a common conceptual core that since the 1960’s has been formalized in set theoretic terms as: S ⊂ x {V i | i ∈ I}       [Mesarovic, 1972] Where “x” is a Cartesian product, “I” an index set, and V is a set of V i relations . The concept of system grew before Condillac linked to epistemological intuitions in natural philosophy, as we can see en the works of Galileo [3] and Newton [4] in relation to astronomical order and the unity of that order. In this sense, we...