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What is hidden under the cover of apparentness in mathematics and in the usual world around us

Arkady G. Aseev

With heartfelt thanks to Alexander Sazonov for deep comments to this article during its creation; without his input this work would not reach such integrity.

In the everyday life we are surrounded by many apparent things. We are used to them and do not ask: why? Moreover it is not in our nature to argue about things, which we learnt long ago, in our childhood. We take it for granted.

Though indeed it could be far otherwise. Keys to the secrets of our world are often hidden under the cover of apparentness. And I need to note that this is one of the most reliable places to guard the secret doors to the other worlds.

In this article I just want to take the cover off one of the apparent things: junior school, where we learnt arithmetic — addition, diminution, multiplication and division. There we were urged to remember that it is impossible to divide by zero. And if we multiply by zero, we get zero. We remembered this quite all right, but why it is just so — we do not know. Let’s hunt down this question.

At school we were taught that multiplication is the shorter record of addition process of equal numbers several times. For example, record 2x5 (two is taken 5 times) is equivalent to the record 2+2+2+2+2. Similarly division process is the shorter record of multiple diminution of one and the same number. For example, 10:2 means how many times there is 2 in 10, and is equivalent to diminution record 10-2-2-2-2-2. So according to the modern mathematical theory the process of multiplication if equivalent to the process of addition, and division — to diminution.

And further it becomes more interesting. If we multiply any number by zero, we shall get zero. But indeed taking any number zero times we either receive zero — as we did not take anything, or the original number itself — as we did not add anything to it, or exactly, we added it to itself but zero times. (Original number does exist, but we can add anything to it, or add nothing — that is zero.) At this stage we have already double uncertainty with the multiplication, in comparison with how it was given at school. At school there is full uniqueness of multiplication by zero, as a result each number will turn to zero. But let us look back at the process of addition, which is equivalent to multiplication. 2*0 is equivalent to nothing being added to zero, or why then should we write that we multiply some number by zero?

And here we get an interesting paradox. For all numbers we observe one-one transformation of addition to multiplication, and vice versa, with the only exclusion — number zero. Zero can give us initial number if we come from addition; or it gives us zero if we follow multiplication logic. It means that for operations with zero, addition is not equivalent to multiplication, as it is for any other numbers.

Let us observe division by zero. By its meaning — it is how many times zero is found in some number. And now it becomes clear, that we can get various answers, depending on how we understand what is zero. It could be one, several, infinity or nothing. Indeterminacy becomes really unlimited here.

What if we ask how much it would be if we divide 12 by zero? How many times zero is contained in 12? None at all. There is no zero. There is no zero in number 12. So the result of division will be zero. Again we see that addition is not equivalent to multiplication in case of zero.

Let us review simple cases.

For example, we have an object, it could be a book. We can a book e.g. 5 times, and get 5 books. This operation could be written also as multiplication 1*5. Then we have the same book, it does exist, and we want to take it let’s say — zero times. What can we say about the result? The book was at the shelf, and is there now. We did not take it. So the answer could be zero, and could be number one. Let’s look more closely into this. The book was located in space at a certain place, and it is there. At this there was no process of movement in time. In our world time is connected with the processes of condition change, with the processes on-going. Directly we do not perceive pure time. It means that if zero for us is connected to the dimension of space, and we multiply an object, let it be a certain number, by zero, also connected to the dimension of space, we do not provide a new space to that object. The object stays where it was. While if we get zero related to dimension of time, it means — absence of time. Then multiplying by zero, we as if do not allow the object to live further on, it disappears from time. In that case we get zero as a result.

Let us consider division process. E.g. we have 10 books. Let us divide them by zero. In its meaning, it is how many zero books are in 10 books. If we mean absence of anything by zero and not infinitesimal small number, then correct number will be none, a zero. At this division by zero the books do not disappear from the pile of books, as it could be if we divide by 5 and could distribute books in 5 piles, 2 books in each pile. So the answer to our question could be number 10 as well. Here talking about books we mean the space volume. We are used to it and take it for granted. But we also have time. And if we mean absence of time by zero, then 10 books divided by zero will give us zero as well — complete absence of anything. No infinity at all, as we were taught in the childhood.

As you see, nature of zero is ambiguous. Let us try to clear up a little bit more.

If we refer to the history of mathematics, we shall see that modern mathematical theory was created as multi-layer superstructure upon miscellaneous theories and practical methods, which appeared in various civilizations for different purposes. E.g. from the beginning the people worked with the numbers in a very practical way — counting (objects). Later squares and volumes were added, which led to direct necessity to invent broken numbers, But even among the nations who used broken numbers effectively, they were not considered true numbers. One of the main purposes of inventing zero was necessity to use positional notation of numbers (it means that the value of a number is defined not by itself, but by its place — empty places were filled by zero as we do it now). Zero was not used as a self-sustaining number, and did not participate in calculations. Opposite, negative numbers appeared due to the need to keep intermediate results, while solving systems of linear equations, and they had other name (equivalent to «debt», «accrued expenses»), also they were not considered «true» numbers and so were not used in the final result. Similar examples are related to the surd and complex numbers, which appeared from the requirement to solve new more difficult tasks.

Modern theory looks more homogenous than previous ones. At secondary schools we learn rather early to switch from real numbers to abstract «x», «y», and so on. It is good as a whole, as ability to work with various levels of abstraction actively develops intellect. Here we have but one of the main «hidden rocks» of modern mathematic — we do not return to the basic things (axioms) after we have learnt them. We are more disposed to move up and forward (imagine modern theory as a «tree»). When we need to broaden theory for the new class of problems, we move further to the «branches» and «leaves», grow the new «branches», but we do not return back to the «trunk» and «roots». So we lose the opportunity to climb the other «trees», moreover we cannot even imagine that other «trees» exist.

Let’s also consider the meaning of the number zero; and how it is represented and taken in physics and mathematics.

In mathematics use of zero mainly depends on its branch, to which refers the considered problem. E.g. zero is «pure» zero in arithmetic, infinitesimal small number in the theory of limits; and both in integral calculus.

In physics zero could appear infinitesimal small number and could be «pure» zero (in elementary particle physics: neutron charge is zero, and not infinitesimal small; also spin, magnetic moment and other quant numbers could be zero.) In physics we always meet dimensions of quantities. In physics zero is always zero of something (meters, seconds, coulombs, grams and derived quantities). So physical zero is loaded with dimension. It is not allowed in physics to use zeros of different dimensions in operations of addition and equalization, while in mathematics it does not create a problem. It is unacceptable to equalize even formal dimensionless quantities of different origins (e.g. solutions concentration and wave phase). Division is an example of another ambiguousness in physics as well: if relation result of two (indeterminate) quantities is zero, we apply mathematical approach — dividend is not a zero, but divider is zero. This approach does not take in consideration neither dimension, or physical attributes, or history of quantities in relation (while for physics all this is important).

Invention of equivalents «addition — multiplication» and «diminution — division» refers more to the modern stage of theory evolution, but does not give an answer to the following question: «how to apply equivalence principle when among quantities participating in operation, there is zero or surd numbers». Surd number is infinite nonrepeating fraction (in any calculation system), so we even in theory are not aware of its «complete» record; and direct application of equivalence principle gives us algorithm with infinite number of steps. In the meanwhile there is no problem with solving such equations even at school: just multiply something by «pi» or «root of 2» and use symbolic form till the end of calculations, not bringing it to uniformity (e.g. decimal notation).

And now it appears that zero was created not as a full valid number, but as a substitute to fill gap digits. A zero was recognized equal to other numbers, when modern classifications came to life (natural numbers, whole numbers, real numbers, etc.); and by this time theory of arithmetic operations with numbers was quite worked in details.

So modern mathematical theory is in a sense a «patchwork quilt», collected from various perceptions of different nations and epochs, where calculations were used for different aims and purposes. So often meaning of «zero» was different from the current one.

In modern understanding, zero is a transition point, above all. It could be transition between world spaces, or between sets of numbers, just a lot of between this and that. And there is complete ambiguousness in transition point.

What is special with these transition points: full compliance disappears for addition and multiplication, and as well for diminution and division. It means that modern mathematics is not completely valid. And all earlier mathematically proven theories are not always valid for the processes describing transition via zero. It is important to note that all theories, related to microworld and cosmology, are bound with zero point transitions. So we can state that modern concepts of reality at the current stage of humankind development do not always correspond with reality, and require global revision.

Noocosmology as a science is operating both with visible and invisible world. We can perceive physical world by our physical sense organs and by devices, which are kind of extensions of physical organs. We cannot perceive invisible world directly, but we know about it by spiritual experience and/or indirectly.

E.g. in Christianity it is said that souls of those who die, go to after world. So that after world is the one, which is beyond our reach for usual human perception. At the same time after world is an integral world with its own life and special lifestyle.

Very interesting example is the time. We measure time by the speed of the processes, indirectly. We cannot measure time directly. We cannot store the time in order to use it as resource. At this we feel the time, understand that it is reality, which impacts and influences us. But in most cases there is nothing we can do with it.

Curiously, in the book «Collected works in physics and philosophy» by R. Bartini, our world is observed as consisting of 3D space and extended 3D time, orthogonal to it. Bartini describes existence of an object in our world as (3+3) D complex formation, which consists of 3D space multiplied by 3D time-like extension, orthogonal to it; both have orientation. And Bartini proves this statement.

Also Bartini describes in his book that various objects can pass from our 3D space dimension to 3D time dimension, thus becoming invisible in space. It is obvious that zero is transition point between the worlds in the described picture; or at least transition from space to time coordinate.

Building mathematical graph of transition from one to another world, we can see that the graph in transition point will look the same as before it or after, while we observe an object disappearing from our world. Probably the object will suddenly re-appear later. While the graph will be smooth, with no special features.

This example shows discrepancy between theory and reality. At least we can say that our perception of the world is very limited in these critical transition points. By this objects do not stop to exist when they pass through zero, they continue their existence. But we cannot register them by routine methods. While mathematics does not describe in its theories our process of reality perception, our limits. And mathematics cannot adequately describe these worlds, which are yet unavailable for our perception.

Don’t think that other worlds are something unreal. Ordinary man does not feel these worlds. E.g. healers who work with energy, know that if you remove pathogenic energy from the field (and better take away its cause at the thin layers), then pathogenic energy related to the physical appearance of the diseases, e.g. tumor, then the disease will disappear, and the man will recover. The same happens if we use the opposite approach, apply traditional medicine, e.g. provide surgical intervention and remove cancer — pathogenic energy disappears from the human’s field. One is connected to another, provides impact and flows from one to another. And this is just one example that perception of the world for the ordinary man is significantly limited. At the same time, another world could be very near, not far away.

One more thing: in all theories about space humankind was for a long time using the idea of flat or non-flat isotropic spaces with whole-number dimensionality (Euclid and non-Euclid geometry). Geometry of curved spaces (non-Euclid) though has differences in axiomatic (e.g. axiom about parallels) mainly gets ideas from the «flat» geometry. And all modern physics is based on this. There are questions related to various anisotropic curved spaces, where its curvature depends on scale, meaning that space curvature has different values for various scales (micro-, «human» and astronomic) or even different characteristics, or there are spaces with non-whole-number dimensionality — these questions though have been raised by the scientists, are not practically used. And here very significant role is played by the patterns, which we learnt in our childhood, first of all — at school. Researcher wishing to plumb the depth of a mystery of anisotropic spaces or spaces with non-whole-number dimensionality, does not want to leave school axioms and rules, but prefers to build the new theory on the old background, rather than go to the new field of knowledge.


Main conclusion of this article is the following: under the cover of routine apparent things we often do not see real living world in all its variability. It is very difficult to see that not all is that simple and definite as we are used to from the childhood, because you think it is solid and apparent. But if not remove constantly the cover of routine, which catches our consciousness, one can live close-minded all life through and not discover that the real world is quite different, and is not limited to the one which description we learnt at school. If not remove covers of apparent things, you will never come to the broad full perception of the world. But this way is very difficult and requires constant work, not to come back to routine world perception, which is setting a limit to our knowledge.

Modern mathematical theory is fragmentary and contains many discrepancies and mistakes. Thus our knowledge of the world is very limited, and as for specially micro- and macro-world — it is not quite correct. It is because there is no full correspondence between the processes of multiplication and addition, and division and diminution when there is transition via zero point. There are also other inaccurate, non-unique or erroneous things in mathematics, not covered in this article.

If humankind does not want to be isolated in its perception of this known world, but wishes to be acquainted with other existing worlds, than collaborative work of all scientific society is required to re-consider existing scientific paradigm.

It is not possible to perceive the whole, if you analyze it, thus disjoint it in order to understand who it works; and then collecting these parts to get the whole working system again. We have to elaborate and develop other scientific methods of perception.

November, 15th, 2013

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