s called digital dyslexia.
Béna, F66760 -Enveitg
GENETICS.- Codons can be ciphered with binary sextuplets. The numerical value of these sextuplets should not be interpreted in conventional univocal arithmetics specific of homo sapiens only, an animal-meter able to make exact countings. The Equivocal Arithmétics' Theory provides the key to codon digitization and the corresponding data processing performed by NRA.
Thirty years ago, the Epistemological Béna Research Laboratory was engaged in a study on the natural foundations of digital data processing when the publication by Jacques Monod of his work " Chance and Necessity " lead us to believe that the results of this research could provide a strictly arithmetical explanation to genetic coding. On the occasion of a private interview with the author in 1972, we were able to tell him about this result which went against his theory : Monod considered that the coding was entirely unpredictable. Rather than revise his theory, Monod, who was interested but confused, quite rightly decided to remain cautious since the proposed explanation, although already solid, remained obscure on one essential point. We agreed therefore to meet again if and when we managed to clear up this point. This goal has now been achieved but it has taken no less than 28 years to come up with the result we were looking for. On the occasion of our meeting with Monod, we had no idea that this clarification would gradually lead to a fundamental surpassing of both conventional arithmetic and the logic which has governed western thought since Aristotle1 . If this finding applied to biology as described in Part Two is validated, it would not only call for a change of paradigm, it would result in there being a radically new abstract tool which, due to its efficiency in all scientific subjects, could establish itself in the future as the norm.
It is a well known fact that the genetic message is written with words of three letters, codons, taken from an alphabet of four letters, puric bases (in short U, C, A, G). The starting hypothesis retained is that this message is a coded message; in other words these four bases are the natural writing ciphers, namely the four representational figures of the numbers 0, 1, 2, 3 of the quaternary numeration system. In fact, if one writes these four figures by means of the four doublets of binary numeration :
Zéro=00, One=01,Two=10, Three=11,
then the codons become sextuplets consisting of the only two figures or digits, 0 and 1, used by the binary notation system. The 64 codons are therefore the first 64 integers, from 0 to 63.
Nowadays, this hypothesis is considered as untenable by most biologists because it would mean postulating that Nature invented digital data processing well before man and that it digitizes for its own account. There could be a natural correlation between the chemical identity of codons and their numerical identification, rather like in the case of individuals identified by a roll number allotted by an institution. Such a registration, furthermore, comes across a major objection: the numbering of the codons would appear to be partially muddled as if the allotted numbers were sometimes drawn randomly, sometimes consecutively like the tickets people draw according to their order of arrival in a queue. More precisely, the 64 codons could be compared with 64 telephone numbers. assigned by a phone agency to 22 callers having together at their dispiosal 64 lines. In this particular case, we are talking about 20 amino acids and two punctuation marks2 that are activated if and when called for. Telephone numbers are allocated as follows : three callers have only one single line, ten have two lines, one has three lines, five have four lines and three have six lines. The chemical identity of these callers and their respective numerical identification are reported further down this paper. Some geneticists calculated3 that the allocating of these numbers looked as if it were designed to reduce the risk of error. Indeed, by considering that some of the callers were more expose than the others, it seemed logic that the former should have a right to more lines. However, it could be pointed out that these differences of vulnerability attributed to the callers are inherent to the status of arithmetic as implemented by the calculations of probability of error. In other words, the logic used for allocating the numbers does not meet the needs of callers demanding extra lines but ensues from the logic of arithmetic used by the agency for the numbering of lines.
According to this analogy, NRA is the automated switchboard programmed to connect callers in compliance with the phone book. It is the interface between a number and a line. Even if, as biologists believe, codon identity is actually only chemical, the existence of such a switchboard holder of the interconnecting code between the codons' molecules and those of amino acids remains the main enigma raised by the appearance of life. It could also be noted that the author of the code who, by way of authority, assigns such and such a number to such and such a caller, assumes the role of any author who, by choosing a particular word to translate a particular idea, is creative of the relation between a signified and a signifier. In the case of the author of the phone book, the signified is a number, the signifier is the subscriber's name. Given that RNA could be considered as a switchboard, it is this author's coding function that needs to be clarified. The analysis and formalization of this function drags genetics into the field of semantics as performed by Nature.
One measures here the enormity of the questions entailed by the hypothesis of an assimilation of the genetic message to a coded message. One knows the difference between the uncoding of a cyphered message of which one possesses the key and the deciphering which aims at discovering the key of a cyphered message, as is the case here. More particularly in this case, one needs to know which agency possesses the key and which agency assigns such and such a number to such and such a caller. This natural, sovereign bureau, author of the phone book of numerical coding which owes nothing to the genius of man but to which the latter owes his own existence, may be an anthropomorphic fiction. One could apply to this office the objections based on the anthropological principle which states that, from Big Bang, a switchboard is in function. Its callers are physical realities such as gravitation or speed of light and the numbers are the numerical values that characterize these physical realities. Universal constants give evidence of an original correlation between the physical and the arithmetical. The correlation is a mystery to science, and all the more so since it has been established that life would not have appeared if these values were not very precisely the ones they actually are.
The trend is to ward off the mystery of this initial tuning by imagining quantities of other Universes. Each of these Universes being tuned differently, it was rather like a lottery in which our Universe must have drawn what we consider as the lucky number. But as the existence of these multiple Universes is by definition unverifiable, this non-refutable hypothesis is not an act of science but an act of faith. Thus, one can exorcise the hypothesis of a transcending anti-chance that would decide on the allocation to physical variables of numerical values calculated in such a way that matter appeared, followed by life and lastly human thoughts and man's attempt to clarify this genesis.
It is clear that the hypothesis of a natural digitization of codons calls determinism into question and as a result, one is inclined to immediately rule out intentionnality of this type in the name of scientific objectivity. Nevertheless, this same objectivity principle requires that one should not be halted by a metaphysical preliminary objection since a coherent, strictly arithmetical theory can account for the oddity of the allocation of the numbers. Besides, because it is likely to be submitted to the probation of facts, this explanatory theory is likely to be confirmed or invalidated. Furthermore, as we shall see, this theory, far from being anthropomorphous, denounces the univocal arithmetic anthropomorphism of human countings. Indeed, it proposes a generalization of the conventional arithmetic freed from the rigorous counting requirements specific of the sapiens, in the same way as geometry was generalized by freeing it from Euclid's postulate required for the measurement of the human properties.
What are the conditions of a univocal counting ? The works of Béna's laboratory on the natural foundations of digital data processing4 have shown that the manufacturer of an elementary meter of unitarian impulses should endow it, by construction and in the following order, with three settings that cancel the following three indeterminacies:
1 - Not discriminating between the numbers 0 and 1,
2- Not discriminating between the addition +1 and the substraction -1,
3 - Not discriminating between the multiplication by 2 and the division by 2.
If these settings are canceled in reverse order, deconstruction of the meter makes it the seat of an increasingly ambiguous arithmetic. In particular, the first canceling, that of the third setting, introduces indeterminacy into the discriminating of the multiple numbers. Only prime numbers remain univocal. Thus, a relationship appeared between the structure of this arithmetic deprived of the third setting and structure of the genetic coding as discovered by biologists. There are indeed 20 prime numbers between 0 and 63. These numbers are separated by a variable number of multiples represented here below by dashes.
0 1 2 3 -5 -7 - - -11 -13 - - -17 -19 - - -23 - - - - -29 -31 - - - - -37 - - - -41 -43 - - -47 - - - - -
53 - - - - -59 -61 - -0
It is like the results to a competitive exam in which the 64 candidates are ranked in order of merit, the dashes indicating the number of candidates placed equal. One notices now that distribution of the candidates is very similar to that of the numbers allotted to the 22 NRA callers. To make it identical, one would only have to include the number 9 or 32 and 27 or 33 among prime numbers. One would then obtain the sequence:
0 1 2 3 -5 -7 -9 -11 -13 - - -17 -19 - - -23 - - - 27 -29 -31 - - - - -37 - - - -41 -43 - - -47 - - - - -
53 - - - - -59 -61 - -0
in which the distribution of dashes is isomorphous of that of the calling codons among the 22 callers.
- 3 callers have only one single line (numbers 0 , 1, 2 followed by no dash),
- 10 callers each have 2 lines (numbers 3-, 5-, 7-, 9-, 11-, 17-, 27-, 29-, 41-, 59 - followed by a single dash)
- 1 caller has 3 lines (number 61 -- followed by two dashes)
- 5 callers each have 4 lines (numbers 13 - --, 19 - --, 23 - --, 37 - --, 43 - --, followed by 3 dashes)
- 3 callers each have 6 lines (numbers 31 - - - --, 47 - - - --, 53 - - - --, followed by 5 dashes)
This statistical isomorphism is certainly disturbing but in no way convincing unless one is able to justify the intrusion of the numbers 9 and 27 among the prime numbers. As we are about to see, a justification was obtained a posteriori from 1972. Indeed, owing to this intrusion, the numerical identity of the lines assigned to a caller, defined by a binary sextuplet, precisely reproduces the chemical identity of the relevant codons if one puts U=00 , C=01 , G=10, A=11. This isomorphism, of a total unlikelihood since it is no longer statistical but individual, encouraged us to continue researching a logical explanation to the intrusion of the powers of 3. Little by little it became quite obvious that this dark point was by no means a secondary point but a completely essential point that could only be cleared by the epistemology of logic and metaphysics.
Research at Béna on the natural foundations of digital data processing aims at the exegesis of machine languages. It postulates and exploits an ontological connection between the elementary logical functions and the physical functionings. In theory these functions are defined by conjugating three arithmetical idealities : the ideas of zero, unity and duality. The phenomenal expression of this conjugation is allowed by the fundamental variables Time, Force and Space, as well as being conjugated within any action. These first correlations among arithmetical signified and physical signifiers enable one to write the formula of three basic semantemes, or so called metasemes, the constituents of the meta-language of digital data processing. These three original metasemes are natural radicals of meaning, necessary and sufficient for the elaboration of all machine languages implemented by computers.
The purpose of this paper is not to present the theory of univocal arithmetic but its application to genetics1. Because of this, the study limits itself to the least one needs to know about these three original metasemes in order to give evidence that a fourth arithmetical ideality, the idea of three, is in fact implied by these three correlations.
As a broad outline, a digit is an elementary semanteme - let us agree temporarily to call it digital metaseme - the signifier of which is physically expressed by the appearance or the disappearance of the track of an action on a support recorder. Those unit events are usually coded by the graphics of figures such as 1 or 0. The signified of the digital metaseme is the counting 1 or 0 executed as a result of this event's appearance or disappearance, counting expressed by the numerical value of the numbers 1 and 0. The printing or deleting of the track of the unit event is therefore an action, the intensity of which is, in quantum scale, that of Planck's quantum. In the same way, it functions as an action unit at the resolution scale of any recorder, whatever it may be.
However, the relation is ambiguous between the physical expression of this event and its account for 1 or for 0. Indeed, depending whether the photographic counting is recorded in positive or in negative, the respective signifier of the numbers 1 and 0 could be :
- either the event constituted by the printing on a recorder of a dot implementing a quantum of action within the resolution scale of the recorder
- or the opposite event constituted by the deleting on a recorder of a dot first printed, that means the canceling of a quantum of action.
The temporality of any recording, or in other words, of any memorization, is characterized by the photographic notions of positive or negative. In positive, the film is blank or not excited before the appearance of the event which will be counted for One. It remains printed or excited more or less permanently later. In negative, it is the opposite; before and after exchange posts, the appearance becomes disappearance. Thus, it is more appropriate to call up henceforth "metaseme of the digital contingency" the metaseme referring to the contingency of an action likely to take place or not to take place.
When it is not known if the counting is done in positive or in negative, the numeric indecision of the event likely to be counted for 1 or for 0 is therefore in connection with the physical indecision of time likely to run backwards or forwards. Such would be the case of a movie reel if the difference between the rotational directions of reeling and unreeling were undecidable. Such is the case at the scale of quantum mechanics where the equations are temporarily reversible. But let us underline that, due to the fact that an event is likely or not to occur, the conventional definition of contingency implies that one does not mistake the past for the future. When the difference between past and future is not performed, one is in the presence of a quantum contingency that is not relative to the direction of the arrow of Time. It is an absolute contingency in which the indeterminacy is higher than that of the standard relative contingency. This absolute principle of contingency is not related to the direction of Time. Its meaning is that of the metaseme of a quantum contingency, a meaning that is not dependent on the human arbitrariness but ontologically registered in the initial tuning of the Universe.
In spite of its ambiguity, this meaning of the metaseme of quantum contingency proceeds from a natural correlation between :
- a physical signifier, namely the arrow of reversible Time aiming towards the future or the past,
- and an arithmetic signified, namely the exchangeable numbers 1 and 0.
Concerning this signified, let us call "quantum ratio" this numeric ambivalence which, it so happens, is the number 00 likely to take both values 1 or 0 indiscriminately5 . Concerning its signifier, temporal reversibility, characteristic of quantum micro-physical equations, is notably expressed during a quantum transition, a jump of an electron changing orbits and energy levels.
We can now underline a denial brought to the fundamental principle of the arbitrary status of meanings in human linguistics. As a matter of fact, we have found a meaning which is not arbitrary: namely the meaning of the principle of absolute contingency expressing this natural correlation between the reversibility of time and the ambivalence of the quantum ratio. Although it is true that in the human scale, the link between the signifier and the signified of a word is dependent on the arbitrary power of the author using this word to express himself. But this is no longer true in digital language which uses Nature in the quantum scale and in this case this singular correlation depends on the arbitrary power of an Antichance entity that one has no right to gloss over. Indeed, this correlation operation implicitly postulates a normalizer, initiator or implicit operator of a normative correlation. It is important to clarify this operator whose function is to establish within the collective of elementary particles a particular agreement on a standard. This norm is defined by the correlation which he imposes authoritatively between the reversibility of Time and the ambivalent quantum ratio. We know that this correlation is the source of the meaning of the metaseme of contingency. In so far as this singular standard is universal and invariant in our Universe, the author of this singular standard should not be considered as Chance but as Antichance. Axiomatization of the digital notation system means taking into account the function of this author who is the creator of the meaning of a natural metaseme to which man yields since he is not able to make any changes. The same thing applies for that matter to the other universal constants that give evidence of the existence of an ontological link between a phenomenal expression, such as the speed of light, and the numerical value of every constant. Man endeavors to measure those values with more and more accurately without being able to modify them.
To better clarify this natural tuning function between signifier and signified, it is a good idea to consider how the meaning of this quantum contingency metaseme becomes more accurate, when, passing from microphysics to macrophysics, the ambiguity of the quantum contingency metaseme stops because Time's course is no longer reversible. It is quite clear that an account in binary notation would be fundamentally ambiguous if the meter, be it a machine or a man, was incapable to discern if the digit was worth 1 or 0. Such is nevertheless the case of the arithmetic carried out by Nature on the scale of quantum micro-physics, the counting of which is affected by a bug that one could refer to as a dual bug. At the scale of molecular macro-physics, it turns out that this dual bug is henceforth corrected. Indeed, the one-way Time arrow, so called thermodynamics Time, supplies the natural criterion of discrimination between the photographic positive and negative that are missing at quantum scale. In reference to this singular criterion, appearance and disappearance become decidable. A normalization of the counting can occur in the same way as the conventional defining of the standard unrolling way of a movie reel would occur as a result of a consensus among duly informed projectionists. For example, this normalization could stipulate that, within matter, the appearance of the unitarian event counts as 1, its disappearance counts as 0. The opposite normalization is effective within antimatter. In both cases, the debugged contingency metaseme still means contingency ; but this contingency is no longer absolute like at the quantum scale, it is relative to the one-way direction of thermodynamics Time .
However, the projectionists' agreement on a common standard implies on their part that they submit to a standardization authority, rather like diapason manufacturers whose diapasons have to be tuned to the frequency of note A. The explicit taking into consideration of this author of a debugging, operator of the tuning of a material signifier on a formal signified, leads to the announced overtaking of Aristotelican logic. This agreement is similar to the one realized among accountants on the discrimination of figures 1 and 0. The principle of the excluded third term postulates indeed that there is no such term included between both opposite terms, 1 and 0, of the quantum ratio. As implied by the quantum theory, within natural reality, if one takes Planck's constant (by posing h=1) as a unit of action, no action would have an intensity included between the values 0 and 1. This principle of the excluded third term is not set back when a third debugger is taken into consideration because its intervention is not located between 0 and 1, like that of a mediator or an intermediary between opposite parties. The tuning operation of two contraries on a common discrimination standard is not included between 0 and 1; it occurs on another level where the normalization overrules the opposition between 0 and 1, like in the case of a referee who overrules without abolishing the antagonism between two opponents. Aristotelican logic with two opposite terms excluding any medium term is therefore saved; but it is then necessary for it to be inscribed within a more powerful logic with three terms. This logic could be called a trialectic logic6 . The understanding of the structure of genetic coding implies moving on from dialectic logic to trialectic logic, the third term of which is none other than the dark point we came up against in 1972. One can now understand why clarification of the structure of genetic coding was so laborious and took so long.
The metaseme of quantum contingency has been studied in detail so as to reveal the intervention of a tuning third party, author of the meaning of this metaseme as a principle of absolute contingency. At this point we will not develop the similar analyses that clarify at the quantum scale two other ontological metasemes:
- The metaseme of an interactive symmetry defined by the singular tuning between the ambivalence of the ratio ± 1 of an arithmetical progression and the reversibility of Force.
- The metaseme of genetic hierarchy defined by the singular tuning between the ambivalence of the ratio 2±1 of a geometrical progression and the reversibility of Space.
In order to allow a formalization of these metasemes, it is necessary to define the following terminology:
on the neurological register,
- The indeterminacy of the direction of the Time vector is called digital dyslexia.
- The indeterminacy of the direction of the Force vector 
is called chiral dyslexia
(confusion between both reading directions of a sequence of digits,
either leftwards or rightwards) -
- The indeterminacy of the direction of the Space vector 
is called fractal dyslexia
(confusion between the increasing and decreasing scales of Space
dimensions).
on the computing register,
- The ambivalence of the quantum ratio 00 is called dual bug,
- The ambivalence of the arithmetical ratio ±1 is called ordinal bug,
- The ambivalence of the geometrical ratio 2±1 is called cardinal bug,
By using the symbol Y of the diapason to represent the third term author, one can write therefore:
- contingency metaseme : dual bug Y digital bug, or in summary:
00 Y (1)
- symmetry metaseme : ordinal bug Y chiral bug, or in summary: ±1
Y (2)
- hierarchy metaseme : cardinal bug Y fractal bug, or in summary:
2±1 Y (3)
Furthermore, the function of this third term Y, third author of a normative tuning between a numeric signified and a physical signifier as defined by the genetic code, can and should be formalized. As it is entirely new, this formalization should be very rigorous, particularly concerning the unfamiliar phenomenological notions it uses. This demonstration is reported hereafter. For all but the most eager biologists, we suggest that this chapter be eluded. Readers should go directly to the following presentation of the key of genetic coding discovered thanks to this trialectic grid.
The function of the tuning operator symbolized by a diapason is to tune a collective on a standard norm. This tuning function is like any algebraic function : y = f (x). Each of these tunings is similar to that of the diapason. The letter "y" is the symbol of the materiality of the diapason, a physical signifier. The letter "x" is the symbol of the numerical value of its frequency, in other words a mathematical signified. Furthermore, the function of this third term Y is that of an operator, the author of a normative tuning between a numeric signified and a physical signifier. It can and should itself be formalized. Let us consider the physical expression of this function shown more particularly by the phenomena of resonance or syntonism. This function is that of devices called syntonizers which perform the tuning or the syntonism between a transmitter and a receiver by adjusting them to one single frequency. By construction, the syntonizer was been endowed with this syntonizing function by the manufacturer. This function is however too restrictive because the tuning between two oscillators can concern determinations other than the frequency. The physical expression of the function Y, specific of a third party tuner, should not only be defined regardless of the object being tuned, but also in the absence of every manufacturer endowing this original tuner with its syntonizing function. The syntonizer should be able to endow itself with the syntonizing function. As if it were self-building, it is originally self-syntonized, being capable of updating a syntonization power for which it is by definition a holder. Let us see how this potential and present self-syntonization function can be formalized.
The three formalisms (1), (2), (3), above define the Y function by the triple actualization of a tuning physically performed according to the three determinations T, F and L of any action (in short A). It remains therefore to be said that at the onset the third party author is endowed with the self-syntonization function both in power and in act. Thus, in principle, the actual realization of a physically shown or expressed tuning postulates a first third party author having by definition the power to tune, this power being used first to tune to the tuning function. This first third party author is the subject of a potential tuning or unexpressed tuning likely to be expressed7 . If it is accepted that the actual tuning is a degree 1 tuning : A1, the potential tuning is a degree 0 tuning : A0.
Let us now stop considering the physical signifier of this third party author's Y function and consider its numeric signified. It is numerically defined by the fact that it is as a third party that the tuning operator intervenes to fix the relation between two different entities expressed respectively by "y" physical reality, and by "x", mathematical ideality. It acts as a third regulating term of the relation between material sign and immaterial number, such a referee regulating the relation between two sides to which he does not belong. He is the holder of the rule governing their opposition. It is not for him to decide upon on the result of their competition but to proclaim it. He does not compromise with these sides and it is not his duty to seek a compromise like a mediator endeavoring to find the grounds of a conciliation between opponents. The referee is not a mixed, adjoining and mingled intermediary. His function is located on a different level where he overrules and oversees the interaction between the opposing sides; he governs the duel by demanding and making sure that it conforms to the norm for which he may not be the author but he is at least the guarantor and interpreter. He is not included between the opponents of this duel or match but a third party intruder who imposes the rules for this competition. He is not therefore a puppet but, acting as a normative authority, he is a full-time player in the competition. This norm stipulates the consensus of the opponents on the role and authority of this intruding third party. In other words this norm stipulates that any conclusion of an agreement is a triangular relation between three persons: two opponents initially in discord and a judge embodying the norm which, unanimously, presides over the settlement of their dispute.
Let us now consider what occurs when two different terms, the one a numeric signified and the other a physical signifier, are tuned by the operation of a third tuner. This tuner is the author of the meaning of this singular correlation. The numeric signified of this triangular relation is the idea of Three that one will call Ternarity or better Triality, because the idea of 1 is that of Unity, the idea of 2 that of Duality. The Y function, which means the operation of correlation between the tuning in act A1 and the idea of 31=3 or Triality, should be spelt Y1. Let us express now this function when it means not the operation of actual self-syntonization but this potential operation. This tuning or syntonization potency is potential tuning A0. It is not correlated to the idea of 31=3 but to that of the potential 3 or 30=1. Also, it is advisable to distinguish the function of third party author of degree 1 or Y1 when it is setting out to perform a tuning, from the function of third party author of degree 0, or Y0, when it refers only to the power of setting out to perform a tuning. This performing power, which is that of a first third party author, is not dependent on any overruling authority that may have conferred it on him. Also, it is advisable to distinguish the function of third author of degree 1 or Y1 when it is acting out a tuning, from the function of third author of degree 0, or Y0, when it refers only to the power to act out a tuning. This acting out power which is that of a primordial third author is not dependent on any superior authority which would have conferred it on him. It is characteristic of his double function to be therefore expressed on two different levels: that of potency and that of act.
In the same way it is essential to clarify this essential tuning function completed when axiomatizing arithmetic. A first axiom should then stipulate the agreement of axiomaticians on the meaning of the wording of axioms. This semantic agreement implies that the arithmetic is not only based only on the metanumbers 0, 1 and 2, arithmetical abstract signifieds, but on the graphic or phonetic figures which express them physically, concrete signifiers. It postulates the consensus of arithmeticians on the correlation between the ideality of the number and the reality of the figure that expresses it. This consensus results from the triple agreement operated by the third party author Y between three numeric ratios and three physical variables defined by the three metasemes We have already seen that these three ratios, expressed by means of the three metanumbers 0, 1, 2 are tuned by this third normative author with the three fundamental physic variables. Let us now see that the metanumber Three allows the expression of two new metasemes, the formula of which is given below. The first one expresses the ontological correlation operated by the potential tuning function Y0 between the number Three to the power of Zero i. e. 30 and the tuning A to the power of Zero i. e: A0; the second expresses the ontological correlation operated by the tuning function Y1 in act between the number Three to the power of One, i.e. 31 and the tuning A to the power of One , i.e. A1.
This last correlation is beginning to be recognized by physics through the notion of Information according to Shannon and its equivalence with the neguentropy 8. Indeed, an information supposes that two equiprobable terms of an alternative are decidable. This means that it implies the A1 tuning of the informed people on a common criterion of discrimination of these two terms. The existence of this criterion postulates a rupture of symmetry. However, so long as physics are not equipped with the appropriate trialectical logic tool, the controversy about the equivalence between information and entropy is not likely to be settled.
Because for the three first metasemes the numerical signified was qualified respectively as quantum ratio 00, as arithmetical ratio ±1, as geometrical ratio 2±1, one can suggest qualifying as triadic ratio the signified 31 of the metaseme ( 4 ). The latter can be qualified as metaseme of the triadic tuning, and as trinitarian ratio the signified 30 of the metaseme ( 5 ) which can be qualified as metaseme of the trinitarian tuning. It would however be a pity for this naming, because it evokes theological terminology, to arouse a blocking that could prevent one from verifying experimentally all the efficiency of this trialectic logic such as it is going to be sketched in the following paragraph.
The tuning function of the third party author Y is therefore liable to several degrees defining so many levels of application of this function. Following the recent experimental confirmations of non-separability of twin particles, some physicists are beginning to take note of the existence of such levels sometimes named levels of reality. But those levels are not only about levels of reality9 of tunings according to their degree, but also levels of ideality or of virtuality of the number 3 according to its degree. They are levels of a more and more strict natural normalization which can be understood by the analogy of the syntonization between a broadcasting station and a receiver. This latter is indeed liable to different degrees. For example, two identical pendulums are in echo when they oscillate with at same frequency. But their oscillation amplitudes can be different. Furthermore, they can turn in opposite directions, either rightwards or leftwards. Finally, they are not inevitably in phase.
Let us imagine a syntonizer which, having tuned the periods of two pendulums, is successively going to tune their amplitudes, their directions of rotation and their phases. The syntonization operation is gradually improved each time; the syntonizer perfects its tuning function A1 in the same way as it passed from the tuning in potency of degree 0 to the tuning in actuality of degree 1 without the intervention of any other another tuner. This gradual self-perfecting of the syntonization process can only proceed by self-syntonization. Since it renews the application on itself of this function Y1 of triadic ratio, its numerical signified will take successively the values 31, 32, 33, that is 3, 9, 27. On the condition that, in arithmetic, the indeterminacies defined by the three metasemes are erased, the implemented axiomatic implies these successive self-syntonizations, which, in other words, means that these powers of 3 are metanumbers expressing degrees of self-syntonization; they are constituent of the metalanguage of these arithmetics gradually being settled ; they are the bricks of the metalanguage in the same way as the metanumbers 0, 1, 2 and 3.
This result is essential to the intelligence of the less and less equivocal digitization that took place in Nature when particles, molecules, alive cells and thinking brains appeared successively. One could not detail here the economy of these qualitative jumps on digital data processing computers. It is analyzed in the report quoted in notes 1 and 4. The result of this analysis is given below. According to the normalization level which characterizes respectively the micro-physics, the macro-physics, the biology and the neurology of the sapiens, the function of the third normalizer is defined by the following metasemes:
Micro-physics: level of normalization of the first degree by initial agreement of the particles of the Universe on the standard values of the universal constants. Metaseme of the triadic tuning : 31 Y1A1
Macro-physics : normalization level of the second degree by tuning of the matter molecules on the one-way direction of thermodynamics time. Metaseme of the tuning in the square: 32 Y 2 A2
Biology: normalization level of the third degree by tuning of the living cells on the one-way direction of Earth's rotation. Metaseme of the tuning to cube: 33 Y 3A3
Neurosciences: normalization level of the fourth degree by tuning of the neurons of the neo-cerebral cortex on the one-way direction of gravitation. Metaseme of the tuning to the power of 4: 34 Y 4 A4
At the principle of this gradually increasing normalization is the degree Zero ontological level of normalization where the function of the normalizing Third is defined by the metaseme of the trinitarian tuning: 30 Y 0A0
It is enough here to retain from what precedes that, by the formalization of the tuning metaseme, the complement required by the axiomatic of the arithmetic is ensured. The analogy of radio communication can be enlightening in this respect : this axiomatic, reduced to the metasemes which expresses it, can be compared with the carrier wave, support of a modulation which contains a message. To collect this message it is necessary, by means of a detection operation, to separate the modulation from the carrier wave. Furthermore, we are going to check that to understand natural digitization, it is essential to dissociate the system of numbers from the metasystem of the metanumbers which, like a carrier wave, is the genetic matrix of the numbers. Let us also consider a stacking of sieves that are more and more fine as in the former sifting machines. Whatever the number and calibre of the sieves, the structure of this machine defines the sieving function, in the same way as the structure of a loom defines the weaving function. The metanumbers define the structure of the loom that weaves numbers. The powers of 3 characterize the degrees of tuning of this loom on settings made by the weaver. Those powers of 3, which interact with the structure of the loom, could not be confused with the multiple numbers because, like the other metanumbers, they are essential of the seed of the tree of numbers either multiple or prime.
Let us consider very concretely the construction of the system of binary numbers, limiting ourselves, however, to the first 64 integers. The construction is carried first out by univocal conventional arithmetic. Following this, we shall imagine a meter affected by the cardinal bug defined by the indeterminacy of the geometrical ratio 2±1.This construction is that of a family tree, the growth of which is liable to lead to a double reading, either external by scanning the display of the branches, or internal by scanning a section of the trunk, the sapwood including an extra ring in each generation. It is for that matter rather surprising that the construction of the system of binary notation is conventionally only represented by a dichotomous arborescence while Nature also records the annual growth of the tree thanks to the crowns of the sapwood. Moreover, it is interesting to note that this double expression of a growth was no mystery to the Chinese and Arabic arithmeticians. Let us show that the external reading of the branches is that of the value of cardinal numbers, while the internal reading of the sapwood is that of the rank of ordinal numbers. Let us represent indeed in a vertical plan the forks of the branches which spread upward by successive dichotomies and in a horizontal plan at ground level the annulate sapwood. Let us code in binary notations the embranchments by figures 0 and 1.
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As indicated in picture 1, that is limited to the first three generations of branches, one is indeed confronted with two readings of this coding ; in ascending reading of the successive generations, the numbers of the branches are not consecutive. Every number is a cardinal number that is characteristic of a numerical value obtained by reconstituting the various stages of numbering since the origin. This evaluation is made in accordance with the logic of comprehension : every cardinal number is a set defined by the elements it includes. In a genealogy, this number is characteristic of the identity of an individual defined by his heredity. In downward reading, the numbers are consecutive ; every number obtained by projection on the sapwood plane is a serial number, the position of which is defined with regard to its neighbors within the same generation. It is an ordinal number significant of a rank in a sequence as are in a genealogy the children of the same couple ranked according to age. In order to locate this rank, it is not necessary to reconstitute the history of the filiation. The rank of an ordinal number is defined in logic of the extension by its membership to a set.
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However, the representation of the sapwood is incomplete if we limit ourselves exclusively to the cross section of the trunk at ground level, that is to say the interface between the branches and the roots corresponding to the generation n°0. To encompass the entire spatial display of the tree as it grows, it is advisable also to consider the ordinal sequence of the generations by making successive cross sections of all the boughs at the level of every generation. Thus, one obtains a stratification of horizontal plans (picture 2), representing a projection of the vertical spreading of the tree. These plans are characterized by different scales that are representative of the extension in space, like superposed sieves of increasing calibre. The plane of the sapwood, at the level of generation n°0 in scale 1; the following plane at the level of generation n°1 is an enlargement of the sapwood in scale 2 ; that of above, an enlargement in scale 3, and so on... In other words we make a three-dimensional Eratosthene's sieve ; the plane of the sapwood contains the multiple of 1, and the following planes contain respectively the multiple numbers of 2, 3, 4, 5, 6, 7
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Let us now impair this construction with the cardinal bug, bearing in mind that multiplication and division then become undecidable for lack of distinguishing between the sign + from the sign - of the exponent ±1 of the geometrical ratio. The indecision penetrates the evaluation of the cardinal numbers carried by the various cross sections, the successive scales of which are then respectively in an undecidable way 2±1 (that is 2 or 1/2), 3±1 (that is 3 or 1/3), 4±1 (that is 4 or 1/4), and so on... In scale 1±1 of the sapwood's rings, the sequential ranking of the numbers is not affected because a number does not change if it is multiplied or divided by 1. Each maintains its localization within the sequence. However, if the rank of a number defined horizontally in extension does not change, its numerical value defined vertically in comprehension becomes indecisive if it is a multiple of a number other than 1. Only prime numbers escape from this indecision. Now, let us disregard the usual technique of spotting these prime numbers by means of the bidimensional Eratosthène's sieve in which the multiple is crossed out step by step. Instead, let us follow the process of spotting the prime numbers by means of a three-dimensional sieve made with stacked grids that retain multiple numbers in their meshes. On the calibre 1 grid, between two consecutive prime numbers, meshes remain bearing numbers for which the rank, but not the value, have been established.
Man has known how to make clocks with cogwheels for over a thousand years. These clocks are effectively automatic meters and the ancestors of all modern meters of unitarian impulses. But the counting performed by these machines is exclusively cardinal; the mechanisms of these clocks do not indeed reproduce the interaction between the cardinal branches and the ordinal sapwood of the arithmetical tree described above. Their working does not simulate that of the counting loom used by Nature. It would be enough to compose the vertical system of these cardinal gearings with the horizontal system of an ordinal sieving, such as Eratosthène's sieve, to reproduce the natural more or less bugged counting. The breakthrough of cyberscience relies on the manufacturing of such natural meters with, naturally, the replacement of ratchet wheels by electromagnetic mountings.
Eratosthène's sieve obtained by superimposing of various sapwoods is shown in picture 3. In order to physically materialize this sieve, it is a good idea to print its grids on a transparency, darkening on each grid the meshes corresponding to the multiple numbers they retain like fishes in a net. In Part One we described the isomorphism between the distribution of 64 codons among their 22 correspondents and the distribution of the first 64 numbers among the 22 metanumbers and prime numbers, subject to justifying the inclusion of the powers of 3 among metanumbers. This justification has now been achieved. In Figure 3, we have made a first attempt at appointing a series of numbers included between two prime numbers to 20 amino acids and 2 punctuation marks (Stop 1 and Stop 2). This attempt is temporary.
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Indeed, we tried to remain as close as possible to the conversion table which makes the NRA, experimentally established by biologists and reproduced in Figure 4. Indeed, if one adopts the following numeric coding using four binary doublets: U = 00 (that is 0 decimal) C = 01 (that is 1 decimal). A = 10 (that is 2 decimal) G=11 (that is 3 decimal), then, this table is a grid of quaternary digitization. It does contain, however, a big abnormality: one observes indeed that two Serine codons have fitted themselves amid 6 Arginine codons. This presentation has been taken up by biologists as it appears to be the most coherent and, moreover, it is imposed by various constraints: |
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.2 - Amino acid Gln, Lys, Arg, require that the couple A and G should not be separated while authorizing a permutation in their sequence; however, this permutation is forbidden by the amino acid Ile.
Eventually, since U and C could be permuted, this arrangement is an optimization in which there is a share of arbitrary power. One could well prefer other arrangements, which, by correcting the indicated abnormality, would recreate others somewhere else. It is precisely a matter of looking for an arrangement in which there would be no abnormality. Let us show that, by operating in ambiguous arithmetic, such an arrangement effectively exists.
As all human beings since the dawn of the thinking of sapiens, geneticists have not yet become aware of the fact that univocal classic arithmetic is anthropomorphic. This is required of accounting reports to ensure the harmony of social relationships. In order to avoid conflicts, it is absolutely essential to agree on the amount of debts and the claims. This agreement, however, on the exactness of the accounts, is only possible for the human brain. Now, the exegesis of the metalanguage of the digital data processing, which is here reviewed in substance, reveals that only the sapiens is capable of learning to count in an univocal way from childhood because his brain is not affected by the cardinal bug as in the case of the animal's brain. More precisely, unlike the animal, the sapiens is capable of performing mathematics and poetry because he can find a way inside the fractal piling of abstraction and symbolization levels. He has a compass allowing him to distinguish between the inductive ascent and the deductive descent. Man is a reasonable animal because, to him, the ambivalence of the geometrical ratio is not undecidable. It was, of course, admitted for a long time that man differed from the animal because he knows that he knows, and what is more, he knows that he knows that he knows, and so on..., according to the training acquired individually by intellectual exercising ; sapiens has crossed, it is said, the step of reflection, enabling thus his consciousness to be reflected. However, while this is universally acknowledged, one did not think that this power of reflection resulted from a difference of neuronal programming between the monkey and man. The difference was thought to be cultural and not natural. Now, this physiological discontinuity has been evidenced by the specifically human asymmetry of the functions of the left and right cerebral hemispheres. Whereas the left-hemisphere is specialized in the reductive descent towards increasing abstraction, the left-hemisphere is specialized in the creative ascent towards increasing symbolization ; in the same way as one is born left-handed or right-handed, some are more endowed for objective reasoning and others for subjective imagination. Other concomitant works of cerebral neurology demonstrate today this human privilege of an innate programming that allows only the human newborn child to learn the univocal numbering10 .
Biologists are anthropomorphic when they widen to all living beings this conceptual set of tools, the monopoly of which they alone possess. Indeed, the biologist's table of conversion (picture 4) is typically Cartesian. The coding of 64 codons is referred to a Cartesian system of coordinates of three trirectangular axes, Oxyz, as shown in picture 5. Each codon is spotted by its three coordinates: x first base, y second base, z third base. The three axes are each divided by four graduations U, C, A, G, which can equally well be coded by four doublets 00, 01, 10, 11, so that every codon is clearly identified by a sextuplet in its reference system.
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This is shown in picture 5 in which 4 codons are featured with their double chemical and numeric identities. But the fact that digitization is related to a reference system implies differentiating between the level of the referred object and the the level of the referring subject. Fractal dyslexia does not allow this differentiation for lack of an agreement of neurons on a common standard. Only our faculty of reflection, escaping this dyslexia, enables us to avoid confusing the inside representation from the outside reality. T his faculty of reflection enables us to grasp what has been lived from the outside so that it can be objectivized. In brief, we can step out of a reality in order to seize its form which mathematics will then translate by a formalism. |
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Let us imagine that this cube of 64 compartments is an automatic parking lot of 64 car-boxes, they are existing, where drivers leave their car at the entrance. A device consisting of elevators and conveyer belts carries the car to a free box. In his office outside the parking lot, the manager has at his disposal a luminous panel on which every car-box is spotted in Cartesian coordinates as shown in Figure 5. Let us suppose that the conveying device of the cars is a monorail that traverses the 64 car-boxes without actually going through same one twice. This monorail is inevitably therefore a double helix consisting of an external helix that passes through the 48 boxes situated in the periphery of the cube, prolonged by an internal helix that runs through the 16 axial boxes. One edge of the latter is situated on the central axis of the cube. To simplify the drawing, let us admit that the helix only contains straight sections; the turns are square; all the bends are at right angles, as on Figure 7. The internal helix passes through the 16 car-boxes on each of the four levels of the parking lot, passing four straight sections of each of the 4 car-boxes at every level. The internal helix passes through only four car-boxes at every level in four sections, each one including a single car-box.
This monorail is also used for the maintenance of the car-boxes thanks to a cradle into which the parking keeper climbs to go about his rounds. Let us consider the case whereby this keeper could have been born in the cradle and that he was held in this parking lot without ever getting out of it. If this were the case, the keeper could not even conceive that there was an outside to this this parking lot since he would completely incapable of objectivizing it. The Cartesian numbering system the manager uses is totally impenetrable to him, as it would be to a monkey that had been trained to do these rounds. The keeper is in fact stricken by fractal dyslexia. However, he needs to spot each of the car-boxes to perform his maintenance job. The double helix is his covered itinerary. On the way, he has to number the car-boxes in his dyslexic manner ; the manager, who has been informed of the keeper's fractal dyslexia, has a table to convert the keeper' numbering into Cartesian numbering. This numbering that the keeper uses is a sequential numbering as and when the cradle proceeds from car-box to car-box along the double helix monorail. The reference system of this numbering is the moving cradle. The keeper strapped to his seat is firmly attached to that system. Like Ampere's mate, he locates each car-box with regard to the reference defined by his body according to the following three axes:
- avertical axis OX oriented from his feet towards his head,
- a transverse axis OY, oriented towards his right-hand or left-hand side as the thread of the helix is oriented to the right or to the left,
- a longitudinal axis OZ oriented in the direction of the progress of the nacelle, from his back towards his front.
To perform this sequential numbering, the keeper has two different marks, for example white or black, allowing him to code four doublets 00 , 01 , 10, 11, which he always applies in this sequential order on the 6 faces of each of the cubic boxes. Penetrating one of these boxes, he places a mark characteristic of the floor where he stands according to the axis OX, for example a white mark on the ground and the other one on the ceiling if it is the level n°00=0. Theoretically, his marking should be the same for the 16 boxes on each floor. Then, according to the axis OY on each of both lateral faces of the box where he stands, he affixes a mark that is characteristic of the series of four boxes to which he belongs. For example he will affix on the four boxes of the same series a white mark on the face located on his left-hand side and another white mark on the face located on his right-hand side, so that on this level n°00, the first series of four boxes is also numbered 00=0. Eventually, he spots the four boxes of every series individually by writing successively the numbers 00=0 , 01=1 , 10=2, 11=3 on their faces situated in front of him and behind him. In this moving system of reference OXYZ, each box is therefore spotted by a sextuplet which, as a rule, is not the same as the sextuplet defined in the Cartesian Oxyz reference system. However let us not forget that this keeper stricken by fractal dyslexia is indeed capable of numbering the boxes step by step according to the ordinal sequence of the metanumbers 0, 1, 2, 3. Concerning these numbers' cardinal identity, however, i.e., their numerical value, he is confused. He discerns in an univocal way only the numbers which are prime numbers. For him, the numerical values of the numbers included between two prime numbers are numbers placed equal with one of these prime numbers; which one? It depends which way he is going.
To establish the table of correspondence between both numberings, it would be necessary to know which way the cradle was going, in particular to know if on occasions, the keeper proceeded backwards on his rounds. It would also be necessary to know which box is at the start of the keeper's round. For him, fastened in his cradle, the round in double helix is indeed a genuine toboggan. Depending whether his head is up or down, the orientation of the vertical axis OX is reversed with regard to that of the axis Ox, as well as that of the transverse axis OY with regard to that of axis Oy. Furthermore, when moving in reverse, the progress movement according to the axis OZ is not defined by the orientation of the axis Oz. How can one know the choices of the keeper in the presence of different options concerning the round and markings? Because biologists know the result of this marking, one is in the situation of the manager who, thanks to internal television, could observe the keeper going about his round from his cabin outside the parking lot. .Here is the result of this observation.
It is advisable at first to forget the trajectory of a monorail which, fixed at the time of the construction of the parking lot, would force the keeper to follow that particular course. The idea was only brought up for educational purposes since it is actually the keeper himself who draws up his round from box to box and indeed, as he proceeds, he has the power to move step after step and from floor to floor. It looks, therefore, as if the helical itinerary chosen by the keeper is very coherent. In particular, it respects the parity that ensues from the universal principle of symmetry. One knows that the proteins of live cells are characterized by homochirality whereas the cells of a lifeless body are heterochiral since they are affected by chirale dyslexia. His round is divided into two symmetric circuits with regard to the center of the parking lot (Figure 7). The helix threads being in opposite directions, these circuits are enantiomorphic like our two hands. The Northern circuit goes through both upper floors of the parking lot in double helix ; the Southern circuit goes through both lower floors of the parking lot in double helix ; there is an inversion of the helix thread of the Southern circuit when it changes floor, the same observation applies to the Northern circuit.
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Finally these changes of floor are not reached through bends as in the case of a continuous monorail. It is necessary to substitute a ribbon for this monorai ; the turns, like the rings of sapwood, are fragments of ribbon ; the numbers of the boxes are to be read like the figures on a tape measure or like the squares on a Snakes and Ladders board game (the French version of Snakes and Ladders has 63 squares !). But unlike in the case of Snakes and Ladders, the bends of the ribbon are at an angle of 90° and in each of these bends, the ribbon is folded as indicated in Figure 6. Furthermore, when it is necessary to change floors, the ribbon lying flat on the ground during the three straight sections of a floor becomes vertical, like a slip-road up to the next floor. This rotation of the horizontal ribbon becoming vertical entails a permutation of Axes OX and OY. |
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This route required by the keeper's enantiomorphism entails a problem for every circuit at the connection of these two floors. Indeed the terminal compartment of the turn of the first floor is already occupied as an initial compartment of the turn of the upper floor (Figure 7). Boxes 18 and 42 are already numbered as box n°30, as if they were shunted. This shunting, however, of the numbers 18 and 42 is compensated with the redoubling of the numbers 30 and 62 because a similar problem arises at the start of each round. As far as the keeper can see, no two boxes carry the same number 30. According to him there are two distinctive boxes : a box 30, which he considers as placed equal with the prime number 29 which precedes it, and a box number 30 which he considers as placed equal with the prime number 31 which follows it. It is exactly the same thing in the case of the two starting boxes to which we Cartesians assign the same number 62. But the keeper considers the one as placed equal with the prime number 61 which precedes it, the other placed equal with the metanumber 0 which follows it (n°63 also being placed equal with the number 0). It would be wrong therefore to believe that it was curious for the keeper choose these boxes 62 as starting squares of his numbering. To him, these squares are respectively the terminal compartment N° 61 in two copies (61 and 62) and the initial square n°0 in three copies (62, 63, 0).
This natural coding is illustrated in sketches 7, 8 , 9 and 10. The ribbon laying flat like a closed buckle is shown in Figure 9. The double coding in both Oxyz and OXYZ reference systems is reviewed in the table of sketch 10. It should also be noted that the doublets 00 and 11 coding metanumbers 0 and 1 are palindromic doublets, i.e., they do not change their value if one reads them from right to left or from left to right. They are therefore indifferent to the direction of the helix' thread. It is not the same case with asymmetrical doublets 01 and 10 of the metanumbers 1 and 2 which exchange if one changes the direction of reading. That is why, as indicated at the bottom of table 10, the sequence of graduations 0, 1, 2, 3, on the axes OX and OY becomes 0, 2, 1, 3, when the orientation of these axes with the inversion of the helix' thread is reversed. On the other hand, the orientation of the axis OZ is not bound to the chirality of the helix; it indicates if the keeper is going forward or backward, because his axis is directed from his back towards his front in the direction of the arrow of Time or in reverse. The sequence of graduations of the axis OZ read 00, 01, 10, 11, when going forward or in photographic positive becomes 11, 10, 01, 00, when going backward or in photographic negative.
This natural coding, although completely coherent in arithmetic affected by the cardinal bug, can seem very complicated to a Cartesian spirit. It is difficult for the Cartesian to put himself into the skin of a monkey or another living being and follow him in the elaboration of this route corresponding to his logic. The round that is actually presented here with the help of sketches is likely to be programmed with a software for processing images similar to those that architects use when their computer enables them to embark on a virtual visit of the various rooms of a house, the plans of which they have drawn. Programming of such a software for a cubic parking lot of 64 alveoli is currently underway. It will feign the behavior of living people suffering from fractal dyslexia and it will save effort here and relieve the individual from the anthropomorphism of univocal arithmetic. In fact, this virtual visit of the parking lot was preceded by a real visit of a rather big constructed model so that one can wind one's way around inside and become identified with the keeper. It is from this hand made coding that the report presented to Jacques Monod was drafted in 1972. In spite of the obtained promising results, our mastery of the trialectic logic was greatly insufficient to attain a fully intelligent behavior on behalf of the keeper suffering from fractal dyslexia. It would seem possible that, once this behavior has been formalized, the virtual itinerary programmed will contain light variations with regard to the one described here, so laboriously hand-elaborated, that it is bound to harbor some minor errors. These corrections of detail should contribute towards a better understanding of the theory as described in this paper and which has been developing for the past 28 years, always in the direction of an improved confirmation of its accuracy. NRA is indeed on the one hand the corresponding interface between 64 codons, and on the other, 20 amino acids and two punctuation marks.
1-That problematics has been widely developed and displayed in the essay " Le métalangage naturel de l'informatique digitale ". It may be found on Bena website : http://www.bena.org
2- It appears as if Nature was using two punctuation marks like comma and full stop. Full stop is supposed to be activateds only by codon called opal. Comma is supposed to be activade by both codons called ochre and amber
3-In particular, French " Gamow Institut " has performed such works. Several papers have been comunicated on that subject to the " Académie des sciences de Paris " (T. 296- Série III- 767 -1983 - t. 401 Série III n°5, 157 -1985)
4-All informations on the genesis, the stages and the last state of that long search may be found on Bena website.
5-It may be demonstrated that number Nn, when both N and n tend towards 0, tends towards 1 if n tends more quickly towards 0 than N. It tends towards 0 if N tends more quickly towards 0 than n. So, the numerical indeterminacy of numbers 0 and 1, if interchangeable, is related to the temporal indeterminacy of a reversible Speed vector expressing the relative decreasing of the numbers N and n
6-Stéphane Lupasco has been a precursor of three terms logic called logic of contradiction. Since then other approaches are nowadays been searched, particularly by Basarab Nicolescu (Included third term logic), Thierry Magnin (logic of complementarity), Élie Bernard-Weil (ago-agonistic logic)
7- It is obvious that we stand here in the heart of auto-manifestation phenomenology according to Husserl that , nowadays, Michel Henry is relieving. For better information, reader would report to recent books of that philosopher such as " C'est moi la Vérité " Seuil 1997 and " Incarnation " Seuil 2000
8- A fourth fundamental constant is here implied. It is the Boltzmann constant that is characteristic of the Universe's initial tuning together with Planck constant, Gravitational constant, and Light's velocity.
9- In Particular , Basarab Nicolescu, the author of several books, notably the transdisciplinaity's manifesto.
10- This counting programming of a newborn child has been brought recently to the fore by several researchers, in particular by Stanislas Dehaene &endash; French INSERM ; cf : "Comment notre cerveau calcule-t-il ?" in "Pour la Science" n°236 - Juin 1997 &endash; and ; "Sources of mathematical thinking" in "Science magazine" - 7 Mai 1999
