We met in this first part various situations where our logical reasoning led to indeterminations, to indemonstrable statements.
-In an Aristotelian axiomatic system, we discovered the paradoxes. But other more subtle cases can be encountered.
-In more complex mathematical systems, where are used numbers, functions, equations... various cases can lead to indeterminations: infinite values, several solutions to an equation... It is not a matter of paradoxes, but of impossible calculations. Physics theories are peppered with this.
-Useless Axiom (its presence or its absence does not change anything) It is the case of the «fourth axiom» of the Sets Theory, known as axiom of the choice (in connection with the fact of ordering the elements of a set) This problem was discussed for a long time by mathematicians, who settled the dispute with agreeing that this axiom does not change anything. It is thus useless, although it exists.
-In an axiomatic system, new logical statements result from the axioms, through the logical play of implications. It may happen that an opposite reasoning carries out to demonstrate the axioms from some of these logical statements, which then become the axioms of this new demonstration. There are thus several ways to describe such a system, according to the statements we will choose as axioms. But in this case there is no logical criterion to say that one in these ways is better or more exact than the other... Especially this is not a demonstration of any of these axioms! Nor the firsts, neither the seconds! Only the observation of the reality of theses axions will dictate which to use. If both sets are true, then we have the freedom to choose one of these ways, according to practical reasons, and even to cultural reasons.
In non-Aristotelian logics, a new form of indeterminism appears, specific to these logics. Progressive logics can exhibit, under certain conditions, what is called sensitive dependency to initial conditions (butterfly effect) which make a later result unpredictable. In fuzzy logics or probabilistic logics the use of probabilities makes impossible to predict the state of a statement or a variable, at a given moment or under given conditions. In the case of non-duality, Yin-Yang dialectics or quadripolar logics, the two (or four) terms are always simultaneously present, passing from a potential form to an actuated form according to the situation, even from a moment to another in a given situation. We shall call that the actualisation indeterminism© (note 93 on ¬©), which does that some statements cannot be predicted without referring to a particular situation of the objects to which they apply. However in the actual situation, the statement will do take a value. That we cannot predict. This situation is rare in Aristotelian logic (only in the case of paradoxes) but it becomes a common case in non-Aristotelian logics.
In the case of industrial microprocessors performing functions in fuzzy logic, the final stage of calculation precisely consists in assigning an actualised value, possibly Aristotelian, to a command, starting from fuzzy conditions or probabilities. Example: In a factory, a tank fills in a random and unforeseeable way. It is necessary to start a pump to empty it, sufficiently in advance not to be surprised by a too abrupt filling, but not too often, not to waste energy. The microprocessor will have to ensure an Aristotelian run/not run command of the pump, starting from assumptions and probabilities on the future filling. It is thus impossible to predict if the pump will run at a given moment. More generally, even in the case of Aristotelian softwares as those I designed in my professional activities, it may happen that such a software is regarded as «indeterminist» because we cannot predict how it will behave at a given moment. (Of course in these examples, the indeterminism appears only from the point of view of a user who does not know the internal state of the software, because in the absolute an Aristotelian software is always deterministic).
In the case of probabilistic (stochastic) logic, the state of a signal at a given moment is unspecified, as only the probability of this state is useful to know. This time there is a true indeterminism.
In the case of non-dualities, for example authority and freedom in education, then we shall see one aspect appearing sometimes, and sometimes the other. This actualisation will at need change from a second to another, or according to very different proportions from one child to another. Only the situation dictates these changes, and even if one of the aspects appears alone at a given moment, the other is always alert and ready to emerge. This other aspect is even indispensable to the first, it is its basement. It is indeed easy to understand that if authority does actuate in education, it is fair only if it is based on a respectful attitude toward the child's profound wishes. The aspect of freedom is thus also present, as a basement for the authority, even if it does not actuates at this moment. In reverse way, we soon become a lump in allowing a child to do whatever he wants without any restrain from respect of others and of ourselves. Only some discipline over himself will really allow the child to really fulfil his profound aspirations in his future adult life. One of the two aspects actuates, the other gives it its correct meaning or power to achieve its purpose. So both contrary aspects are always really present together, inseparable and non-dual, although we cannot determine the one which will express at a given moment.
At last, quantum indeterminism, as seen by the Copenhagen school, seems to be a perfect case of actualisation indeterminism, due to the probabilistic style of mathematical laws which govern the propagation of particles. We shall speak of this again in the part four on physics.
A general approach of a non-Aristotelian statement will only be able to define probabilities or thresholds (Measurable in the case of fuzzy or probabilistic logics, and non-measurable in the case of higher non-Aristotelian logic)
The logical reasonings and axiomatic constructions whatever they are, are in facts only human constructions, which exist only in our intellect, as means to try to understand reality (We cannot however conclude that they do not exist: they simply have this mode of existence. Various persons reasoning separately on the same axiomatic system will come to the same conclusions, and this shows that such systems exist objectively). However this means is effective only when reality agrees to behave according to such a logical system. Fortunately, real objects are likely to do this, since logic is one of their properties. For the majority, they behave in a manner which matches with the operation of our reasonings and axiomatic systems. However when these systems lead to paradoxes or indeterminisms, then our reasoning is wedged. But reality shall not stop existing with only such a pretext; it simply adopts other paths. Generally it then behaves according to other laws usually hidden (We shall study an example about logical integrated circuits into chapter III-3). Sometimes it seems to express a true indeterminism, when no law nor subjacent causality intervenes (this seems to be the case with the quantum random, according to the interpretation known as of Copenhagen, that we shall see in the fourth part on physics). In certain precise cases, not any logical reasoning of any kind does seem effective; it is the case of questions like the origin of the existence, the Big Bang or God. It seems that the more we are non-Aristotelian, the more things become complicated; but sometimes it is the reverse: curiously a non-Aristotelian reality can manage to solve an Aristotelian paradox, by more or less arbitrarily actualising one of the two contradictory terms against the other.
We shall see further in chapter III-3 the strange case of a paradox which seems to be a founder of something. It seems that the very operation of reality systematically uses such founding absurdities© (note 93 on ¬©), without which nothing would exist, neither universe, nor matter... Who knows, we could even use such founding absurdities to create something!
I point out that this notion of actualisation indeterminism can be, rather strangely, used as a mean of mind control. We noted, for instance in a social situation where the motives of partners are fuzzy or not yet determined, our attitude may be a «quantum blend», for instance of confidence and credulity. If our partner makes the decision to cheat us, the credulity only «becomes true»! An example is about a love affair: if any of the two partners attempts to cheat the other, so this other will feel raped, even if before the relation was pleasant. (This differs from the classical deliberated crockery, in the meaning that the decision which made the situation what it is, this decision was taken after, as in crockery it was taken before). So it is easy to induce somebody in a given action or behaviour, and then to «change» the significance of this person's behaviour, in order to «show» that this behaviour was incorrect. This happened once to me, and I was very bad for weeks, feeling guilty of something I did not. It is only years after I understood how I was made by this guy. This kind of manipulation is very common in politics, social struggles and family affairs, were one blames us for something we were nevertheless asked to do before. So I feel I must warn everybody. This process subtly differs of the one described in Orwell's book «1984», where there is an Aristotelian lie on the motive, prior to the action. In the process I describe here the meaning of the action is changed after the action, with playing on a logical indeterminism. Please note that the similitude with quantum physic is only formal (same logical law applying in both domain, note 91 on the use of the word «quantum») and we must not extrapolate a science-fiction story from this! But it is rather exciting to find the so strange laws of Quantum Mechanics in a so daily field...Maybe they are not so strange after all.
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