ABOUT INVERTERLOGIC

I introduce here an "inverterlogic", which allows to base mathematic as well as geometry on a single axiom.
This axiom is an abstracted transistor and therefore there is not to dispute, that it is a matter of fact. It can be imagined as an abstracted seasaw too and is therefore much more simple than a pair of compasses and a ruler or as an abacus. My idea might be more acceptable, if I start stating, that I did not intend to make a new base for mathematics, but wanted to solve the problem to solder standardized electronical "logic devices" in such a way, that I could reach my aims. Normally Boole's algebra is used for this purpose. But as that algebra seemed to me to be too complicated and in some respect even not useful, I aimed to a less complicated consideration enabling me to consider the needed timing too. As in fact those electronical devices are made only out of transistors in source circuits, it was next to me to consider those transistor circuits, which are familiar to me and could be easily soldered by me for experimental purpose too. This was upon a time of about 15 years ago the first step to the idea, which I discuss below.

For the first I was happy to be able to solve my problems really fast and perfect.
More than one year passed by till I realized, that Boole's operators (sometimes called "junctors" too) AND and NOT, which are demonstrated by deMorgan's theorem to be fundamental, are not as fundamental as that peculiar NOT, which is an inverter logically considered. But new expressions then were needed, which I evaluated in the meantime. These expressions are in general the same as used to design electronical circuits.
The NOT, given by an inverter, is not only more fundamental but a matter of fact opposite to Boole's operators, which are in principle not more than the syllogisms of Aristoteles, created more than 2000 years ago. Till Boole the relation to the usage of language did not disappear, a relation, which of course instantly provokes the idea, that logic is not a matter of fact, but exists only in our minds. Spinoza as well as Kant tried to dispute these ideas needing a lot of words, which critically considered did not proove evidence but look more like an incantation. Also in modern "mathematical logic" after Boole the relation to usage of language is only disguised and not antiquated, not leading to evidence but only new signs and formulas, which can not be used without taking a strenuous run. That is why matters can still be seen opposite to spirit, world spirit or any other holy spirit.
But in respect to an inverter there is only to state, that it is oscillating, if the exit is connected to the entry, and the evidence is prooved, that logic is a part of and in this world. If connected inverters are real, i.e. as a transistor circuit, the represented logic is independant of a human observer always the same and real as well. Thus logic is not an idea or reasoned by any usage of language.
But you will have to consider truth in a very different way, because the commom way in fact prevented of relating logic to reality. Although the intention to use logic to distinguish true from false is formalized now to "values" "true" and "false", logic was not made real by this and besides did not keep truth in certain expressions out of doubt.
Truth in inverterlogic is a physical effect instead, which can be considered anyway as other matters of fact, which can not be touched as electrons or black holes...
Nevertheless the dream of logical verification depending on sentences is not antiquated, but on the contrary is related to reality and is physical existant. I demonstrate under 2.1.3.1. how inverterlogic enables automatically done verification of algebraic equations using dull computers.
Of course there is to say some more to demonstrate, that the whole logic can be made out of inverters.
The reader of this can trust in the evidence of this idea, because there were nothing to read, if not at least the solderable inverters were logical enough. But that an inverter, abstracted as inverter-axiom, can oscillate too, you can read for the first time here...

I demonstrate with an abstracted monoflop under 2.2.1., that the logic and therefore the whole mathematic includes at least a dimension of time and is therefore a part of reality. The dimension of energy has to be there too, because nothing in this world can effect anything, if it is not at least a Planck's quantum (existing in dimensions of time and energy).
This is ignored by abstraction in common mathematic, which else is scarcely changed.
In a very different way, but so to speak by short-cut too, the dimension of time appears too considering a register under 2.2.2. Then not times between argument and result as short as may please are to notice, but times as long as may please, which appear evidently as a state of storage in computers too.

These statements considering a logic, extended with a dimension of time, are not a joke, but allow far reaching conclusions especially in physic, which seems to make facts not intelligible without mathematic, but presents only phenomenas and effects.
Some readers, familiar with the formula from Einstein "E is equal to M multiplied by the power of 2 of c", basing the idea of an universe, which appeared out of nothing with a big bang, may be not familiar with the fact, that he thought very little such formulas to be useful to express physical facts perfectly enough. Other immortal physicians had a lot of trouble too to use mathematical methods making clear and calculatable, what they could find in experiments. Especially Bohr, Heisenberg, Schrödinger, Dirac and others needed to think of a particle-wave duality and probability, not only irritating Einstein, but also elsewhere provocing exasperated philosophical discussions about to be or not to be. But there was no chance to prevent of this because of experimental stated facts.
Considering this in respect to inverterlogic makes you see, that not a lack of judgement or loyalty to philosophical maximes caused the irritating duality, but a lack of fitting mathematical expressions. Inverterlogic annihilates that to and from, because the inverter contains such a "duality" in one singular thing, caused by taking time between argument and result. I base geometry under 2.2.1.1. with the side-effect too, which shurely will enjoy some physicians: Not only time appears quantized, but space too, which becomes a dimension related to time, while Einstein was only able to combine time insoluble with space, stating a space-time.
Thus the inverterlogic is the mathematical way to prevent of the contradiction between theories of relativity and quantum, which is in principle caused by a not quantized geometry, which Einstein had to use to express the curvature of space due to matter. There is to say, that mathematicians can imagine one more point between every two points...
String-theory, based on the inverter, which is capable of oscillating, will become new and real, while dimensions need not be arbitrarily supposed. The possible dimensions are considered under 2.2.1.2.1. All dimensions are derived from one, which is to divide to get more than one. This text is visible in two dimensions for a reader, because the picture is just made by this way...

The logic, based on the inverter-axiom, is not suited for haunting between points and especially between numbers. In the inverterlogic both elements in mathematical theory are based on the inverter, but not only one. Then one thing becomes elementary, which is familiar neither to mathematicians nor to physicians - the adress. By the way there is demonstrated, what about truth in the logic.
The adress, coherent to inverterlogic, is not seen as an adress, familiar to programmers, but as a circuit, familiar to engineers. They know that circuit as a "wired OR". In the inverterlogic this circuit is changed to connected exits causing the unsoluble statement, that there exist two kinds of inverters, which nevertheless are effecting the same values and thus are given by the same inverter-axiom. This is also much more of interest in physical consideration than logical relevant, because the matter-antimatter duality thus can be reasoned logical, while till now it is only stated. This analogy supports the idea too, that there are in fact inverters and their connexions in the nature. But there is to say, that there seems to be no chance to identify the inverter as any known particle. I speculate, that it is equal to Planck's time ( or length) - extremely tiny.

Besides that, engineers are used to solder outputs of transistors to make them acting like Boole's OR. But they can make adresses too by this way, which are a thing, Boole did not consider about hundred years ago. The adress does appear neither in his logic nor in modern mathematical logic, where only a caricature is to find as order of quantities and a way to count.
In the inverterlogic you can connect as much exits as you want, but this will be an "edge", which becomes an adress only under certain circumstances to consider next. I make that notion "edge", because already facing this banal connexion you are able to see more than means for adressing or ORing. Especially the knowledge of the state at the inputs disappears in case of a certain result, which then can be interpreted as ORing. Thus OR is not an operator as seen by Boole or Aritoteles, but a state, caused by certain values and connexions in the eyes of an observer, who can not see behind the connexions, if he does not see more than values of results. An "edge" is in fact a horizone of event, hiding every lots of places and values there and by that every sense. But the value at the edge exists and can get any meaning a posteriori(!), if the value is not the result of ORing. If you prevent of assigning a meaning to ORed values, you prevent of the known fuzzyness caused by algebraic negation. Then making truth evident is possible.

This allows more far reaching conclusions too, because an existing physical logic only allows many meanings but excludes every sense in this world, while the meaning is not caused by any creator or most important programmer, but made by that thing, which gets the result at an edge as an input. This supports the evidence of the theory of evolution, which does not plaid for creatures made out of loam and breath of an creator, but a fitting meaning due to changed circumstances. Making the meaning is done in a region, which nearly everyone accepts as code, the DNA.
Much more imposing is to think of ORing existing everywhere in the nature. This ORing as derived in the inverterlogic is even not familiar to mathematicians, but well suited to handle probability in such a mathematical way, which does not look like incantation of ghosts.

But mathematicians will mainly wonder about operators, which are not boxed elsewhere than numbers. They are familiar instead with making heaps out of numbers or elements and operators, but need great efforts making this evident in one case or to distinguish it in other cases.
Mathematicians know the "incompleteness sentences" of Gödel too, which prevent them to dream of a theory, which bases the whole mathematic on some or even one Axiom. As about hundred years ago Hilbert had such a dream, his student Gödel made the proof, that this is not possible using the natural numbers given by the axioms of Peano and then valid operators (Peano-arithmetic"="PA"). Because of this the incompleteness sentences depend on everything based on PA too. In every case there are to find antinomy and not-calculability.
This depends not on the inverter-logic, because it is not based on natural numbers. Also the means, used by Gödel in his proof, especially the operators, are in the inverterlogic derived from the inverter-axiom and nothing else and because of that are quite different, if there is not common calculating of interest, but i.e. "calculability". These operators can not be represented by numbers or used as numbers in any case - which is the starting point of Gödels proof and named "Gödelizing". In short: The very elegant proof, made by Gödel, says, that PA and on this based other things are incomplete, while the inverterlogic says, where it is incomplete (i.e. the dimension of time and the consequences of oscillation and and storage) - but of course, there is to say some more depending on this...
Besides logical connexions, which touch known things, I introduce under 3. a new kind of logical connexions, which enable you to cause those few possible connexions of single inverters using a code and to get by this way every logical machine you could like as a set of values.
Nevertheless this representation is not done by Gödelizing and is obviously not made out of numbers, but a code, which is to use due to peculiar rules and is insignificant without an 'efficacy-giver'. Only by this way, you can finally get 'artificial intelligence' (="AI").
My idea of an 'efficacy-giving' demontrates at the one hand side, that every tried way towards artificial intelligence is completely missing the mark. It will never be available neither using known computer structures nor "higher" languages. It will also not be available using any circuits, which are suggested to imitate human information computing (i.e.'neuronal networks'). By this way only a (much too complicated) programed adaption is available, but not learning and thinking. At the other hand side, my idea demonstrates, that artificial intelligence is not as desireable, as you like to think. Some people achieve to get pleasant tin-niggers without a will. But in fact thinking is not possible without a free will (some people know this already), and chiefly not without a feeling. This feeling in machines can never be aequivalent to the known common one. And if it could be aequivalent to known feeling, it would be not less to avoid. Such machines indeed can be much more incalculateble dull, bad, maniacal and old. They must never leave the laboratory!
I intentionally evaluated the here presented inverterlogic complete deductively and interpreting. A discussion of such problems is therefore not included. But I added some discussion at places, where the roots of such problems are. An example are sets of elements, which include themselves as an element - discussed under 2.1.3.
In inverterlogic numbers are available only as the significance of the values of inverters. Operators are made as well as numbers by connecting inverters and are available in the same variety at pleasure as number systems. Numbers are derived under 1.2.2. considering elementary connexions, while operators are derived not before making clear, what numbers are. Thus I prevent of something, which irritates at least me facing the known axioms.
The axioms of Peano define indeed the quantity of natural numbers, but stating, that they are already out there and thus there are to define only rules to find the "next" one in that heap. This can only be done, if a addition of a 1 is already available too.
The definition starting at prime numbers additionally needs the division available, which can only be done, if subtraction and therefore negative numbers are already given. The natural numbers need to be given before they are to define, else there could not exist any prime number.
In inverterlogic instead the numbers are derived from the possibilities to make the values of inverters appear by connecting them in a certain way, while a single inverter can only result values H or L and thus can only define one place amongst lots of needed places for numbers.

At the one hand side inverterlogic makes immense amounts of number systems and operators manifest, but on the other hand side antiquates some common ideas, which are in fact superfluous or even troublesome - especially, if calculations have to be done by computers. While till now computers are supposed to exist in quite an other world than numbers and operators, inverterlogic enables you to derive the computer only from the inverter-axiom, making a formula out of the computer as well as out of that, which is known as program, but not meant in the way, inverterlogic makes it appear. There it appears namely as a manner of adressing and not as means to make money.
You might not be pleased by the effect, that digital circuits as well as programcode can be no longer granted for patent ( at least in respect to german patent law). But besides more insight you will get instead rules for shortening without fee, which make devices and code much simpler and better. Finally the construction of digital circuits is simplified (which was my starting point).

Now I gave a sketch of the significance of the following text. The details you will have to explore there.
At an additional page, I publish consequences of the inverterlogic, which here are only sketched, but thoroughly discussed there: