(Was chapter 21 in version 1)
What I think is the most exact to say, about i, basis of the imaginary numbers, is that it results of a skilful use of a paradox: although it does not exist, it happened that it was very useful to do as if it existed, with a special axiom, passing over the very visible contradiction.
If the trick is blatant, in the case of the imaginary numbers, it is not unique. I would even say that it is rather general.
As a matter of facts, all the mathematic constructions contain such tricks, without which they would simply not work.
To start with the famous Sets Theory that we saw in chapter I-2. We remember that this theory, basis of today mathematics and its more complex buildings, contains two contradictions, two paradoxes, two unsettlable statements, which were however fixed with the help of the second and third axiom. Only the first of the three axioms of the Sets Theory is really useful and founder of something, the two others were created only to hide the appearance of paradoxes. I clearly say to hide: if in the second axiom one says «a set cannot be an element of itself» we only refuse to consider sets which would contain themselves. This is only a convention, because we can find such objects (note 36). Moreover some mathematicians study them, and this leads to the Supersets Theory. As to the third axiom, it is about, as we noted, just a single paradox, that of «the set of all the sets». This object exists, it matches a definition, but, cumbersome and useless, it was decided to leave it on the touchline, with the other hooligans which are contained into themselves. What is important to retain about all this, is that things like imaginary calculation or the Sets Theory contains contradictions in their base. And if we excluded all what is self-contradictory, then mathematics would be impossible, they would not exist.
Of course, mathematicians do not like to see things presented in this way, since they are engaged in their quest for a perfect system explaining everything in an univocal way. However, demonstrations of the impossibility of such a system were brought several times in the early 20th Century. The most well known is the theorem of incompleteness of Gödel, which says that any set of axioms, complex enough to describe the set of natural numbers, leads inevitably to indemonstrable statements (paradoxes). Several similar theorems, more complex, were also demonstrated in the same epoch. Gödel's work remained for long in an obscure cupboard (with its buddies the merry gang of the odd consequences of Quantum Mechanics, you guess the party they did in there) because it was somewhat annoying for the mathematicians of the time. It even inspired masochist or down in the mouth philosophies about reality being definitively unknowable for us. The theorem of Gödel however does not prevent nature from being beautiful under the sun, and I even find it rather funny, standing there in the middle of the field. Because let us remind: reality exists, and it never lets itself stop by a paradox. When it meets one of them, it just uses this freedom to do what it wants.
This joke is not gratuitous: it means that when a natural law is no longer logically determined (for example when it leads to a logical paradox, or to infinite values) then other usually hidden laws, inhibited, will take the initiative and will determine what will occur. Reality never stops to exist.
Let us take an example: the logical statement «A equals non-A» is always false and it does not allow to calculate A. In the material world, were created electronic circuits called logical circuits, which voltages on their pins can take only two states: for example all the voltage corresponds to a logical yes, and no voltage corresponds to a logical no. The output pins achieve logical combinations of the states present on the input pins: OR, AND, NOT... These circuits are designed to function under all conditions, and even if the voltages at the inputs are weakened or has parasites, it is always yes or not. Thus we obtain that this inevitably imperfect material object behaves nevertheless like a perfect logical machine, free from any material defect. But we can now have fun to materialise the preceding paradox, by connecting an input A to an output non-A (the wire achieves the equality of the voltage, and thus the logical equality). Whatever it will happen, it will nevertheless happen something. Let us connect... Piiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii! It is not a logical state (impossible to determine) the circuit started to oscillate in high frequency, in an entirely different operation mode, determined by its material defects such as its inductance or parasitic resistors which do not play any role in its normal logical operation. This operating mode appears only when a logical paradox makes impossible a normal operation. This effect of masked laws which appear when the foreground laws are logically undetermined is very common in nature, and we could quote many other cases in physics, electronics and elsewhere. We shall even see in chapter IV-9 how it could even make appear new laws of physics!
We can always try to reassure ourselves with thinking that the theorem of incompleteness is a concern for only some stale mathematicians occupied in hair-splitting devoid of any practical interest. False hope: in addition to mathematics, the physics theories all are axiomatic systems, together with all the systems in religion, philosophy, metaphysics, ethics, society, politics, economy, industry, games, agriculture... In other words we have to expect to find paradoxes, statements impossible to settle, contradictions in every aspects of our lives.
Admittedly, most philosophical, political, ethical or religious disputes are caused by reasoning mistakes (dogmatism, ideologies, and all the troubles seen in chapter I-9) or prejudices on life (which are necessarily incompatibles). But we can find cases, in physics, in ethics, in spirituality, etc. where a paradox needs to be solved of one of the two possible way, so that the theory can work.
What I try to introduce here is that there is nothing intrinsically bad to this situation: just like it was done with the imaginary numbers or with the sets, we can take profit of a paradox to modify any theory in a way which fits our purposes. This is what I call the creator paradox, or the founder absurdity, a key concept for the continuation.
As a matter of facts, in more of having very useful practical uses, this method also works naturally, into the very way reality creates itself, including the physical reality. It is this way because physics obeys to logical laws, which can be described with an axiomatic system. If this axiomatic system has logical indeterminism, see paradoxes, this however not forbids the physical reality to continue to exist and to function: it reificates one of the two terms of the paradox, or it finds other laws allowing to skirt the problem. A paradox never stopped the world of running. (Note that at this point, we still consider the physical reality as something which behaves according to logical laws, while being distinct of them).
We can see this statement at work on a classical example of a paradox: the paradox of the barber, which is described in this way: «In a town, a barber shaves all the men who do not shave themselves. Does he shave himself?» Presented in this way, it is impossible to reply yes or no to the question, as both replies are in logical contradiction with the statement. However, if ever this paradox is physically implemented, nothing prevents the barber of following the behaviour of his choice, or even a third like to let his beard grow. If we want to represent this full system with an axiomatic logical system, we are constrained to add a second axiom describing the behaviour of the barber. But we have several choices to do so: a hairless theory and a bearded one!
And without sad philosophy about unknowability...
We can find many examples in physics, where a logical indeterminism sees one of the possibilities reified, and only one. The clearest and most fundamental example is the quantum random: a particle will choose a state, and only one, among several logically allowed, without any logical determinism for one choice or another.
For readers who are sensitive to the poetry of mathematics, there is a very beautiful image which visually illustrates this idea of creative absurdity, of contradiction necessary for the foundation of any logical system. They are the Penrose networks (named after their inventor Roger Penrose). They are tilling made from two basic tiles with a rhombus shape, and ornamented with patterns:
The rule for building the tiling is that the coloured curves must match from a tile to the other. This tiling makes possible to completely cover the plane (to pave the kitchen without leaving holes between the tiles) just as we do with square, hexagonal or triangular tiles. However they do this in a curious way: the pattern of tiles and the drawings formed by curved lines are both at random and never reproduce identically. It is a Penrose network.
A Penrose network, once started, can continue ad infinitum, by adding tiles according to given rules. But the most curious is that it seems that the matching rule of the tiles must be violated at the beginning, and then strictly respected (if not the network gets blocked and we cannot continue it). Here is an example of Penrose network, able to grow perfectly and ad infinitum, but which shows two initial defects (close to the centre):
In conclusion of this chapter, we shall give a definition of the existence of an axiomatic system which will be one of the most significant bases for the continuation:
1) An axiomatic system, whatever it is, exists when this existence does not generate inner logical contradiction (paradox, indecidability...)
2) Any axiomatic system contains internal contradictions (Theorem of Gödel)
3) Rules 1 and 2 being contradictory, so that a axiomatic system can exist, it is always necessary (rule 2) to accept some founding contradictions (exceptions to rule 1) and to arbitrarily set a value for them. But once this condition fulfilled, the logical rules must then be strictly respected.
It is to be noted that these three statements themselves comply to the conditions they express!
The Sets Theory itself complies to these rules: two internal contradictions needed the second and third axiom, to ensure its coherence. A different second axiom leads to a different theory, the Supersets Theory.
In the history of my though, as young as my teens, I envisioned a «mathematics of objects which does not exist», especially like i or the set of all the sets. It is these speculations which led me to the above definition of existence.
This definition is equivalent to the first principle of metaphysics of the chapter III-2, in more detailed, for axiomatic systems.
We conclude from this, about the individual logical objects that this axiomatic system contains:
4) In a axiomatic system, an individual object, whatever it is, exists only if this existence does not generate any logical contradiction in its system (paradox, indecidability...) (with of course the exception of the objects involved into the founding paradox of the system).
The imaginary number i is an example of object involved in a founding paradox.
If there is a quasi absolute choice of the starting axioms, if we have some choice of values for some statements in order to settle the founding contradictions, we on the other hand do not have any more choice on their logical implications:
5) From a set of axioms and values to founding contradictions, our axiomatic system is completely and rigorously determined, up to an infinite number of logical inferences. We can only explore it, but no more modify it.
So it is now easy to understand how choosing arbitrarily one of the two incalculable values in a contradiction can found something, in «illogical» examples like the Set Theory, the calculation with the i number, or the Penrose network, and especially in the following examples in this part of the book. This strangely looks like an act of creation of a reality, comparable with the Divine Creation. But for now we can create only axiomatic systems!
A more subtle case is when logical indeterminisms (chapter I-10) appear, not at the moment of the foundation, but later in the development. This can happen in many ways: paradoxes, infinite values, and especially actualisation indeterminism (chapter I-10). So what could happen?
6) If a lone indeterminism is isolated in a «region» of our axiomatic system, we are free to actualise one of the contradictory values, but the effect only takes place in this region, and not in the remaining part of the system. It then can exist several variants of this region, but we can consider only one at a time. We shall see the incredible consequences of this in chapter IV-9 in the fourth part on physics.
7) If there are actualisation indeterminisms at each logical implication in our axiomatic system (at each step of its building) so an amount of freedom of evolution appears. This seems to be the case in Quantum Mechanics, basis of our physics, and this arises very important questions which will be discussed in the fourth part on physics.
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