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Is a new arithmetic growing?

CONFERENCE PAPER · JANUARY 2007

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2 AUTHORS:

Raúl M. Falcón

Universidad de Sevilla

101 PUBLICATIONS  112 CITATIONS 

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Juan Núñez Valdés

Universidad de Sevilla

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Is a new arithmetic growing?

R. M. FALCON and J. NUNEZ

Department of Geometry and Topology.Faculty of Mathematics. University of Seville.

Aptdo 1160. 41080-Seville (Spain).rafalgan@us.es jnvaldes@us.es

Abstract

It is not exaggerated to say that all of the people, children included, know theimportant role played by the elements 0 and 1 in Mathematics. One of the mainreasons for these elements to be fundamental is their roles of uniqueness withinany mathematical structure. However, the main goal of this paper is to showthat these roles last fact could be discussed by redefining the usual mathematicallaws, quite more in agreement with some aspects of the reality.

Introduction

Since our more youthful years, we have been aware of the existence of unit elements. In thefirst place, we are been taught to count one-to-one: 1, 2, 3, 4,...  Secondly, we are told thatthese elements, named  numbers   are related between themselves by different laws. So, welearn to add, firstly from one to one: 1 + 1 = 2, 2 + 1 = 3, ... Teachers tell us that thesymbol + represents the  addition   and that such a law has an unit element: the zero (0),which means that:   a + 0 = 0 + a =  a, whichever the number  a is.

Later, we are taught to multiply and, in a similar way as before, we are told that thesymbol  ×   represents the  multiplication or product  and that the unit element of this law is1, that is:   a  ×   1 = 1  ×   a  =  a, and we are also taught that + and  ×   are related betweenthemselves by the   distributive property .

These two unit elements and such a property allow to form in our mind the idea of a  mathematical structure . At the same time, we easily understand that several types of numbers have to be consecutively appearing:   Z-numbers, which get round the problem of computing the  opposite  of a  N-number;  Q-numbers, which do the same with respect to theinverse  of a  Z-number; later  R  and  C, and so on, in the way that each type of them extendsthe previous one. All these sets have something in common: elements 0 and 1 are the unitelements in all of them, with respect to the two laws previously indicated.

Here, we do not pretend to discuss the importance of such unit elements, that is to say,we do not try to put in doubt the transcendence or their uniqueness, but, why have these

1 International Conference on Dynamical systems ICDS 2007, 4

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elements to be precisely the 0 and the 1 and not others different? For example, why notconsider   5

6√ 7

, instead of 1 ? or why not to define:   a  ×   5

6√ 7

  =   5

6√ 7  ×   a  =   a? Note that

5

6√ 7

 could be in fact any real number, although we have chosen it to do more amazing this

subject.

1 Do others possibilities exist?

It is reasonable to think that 0 and 1 are the unit elements for different reasons: by agreement,by notation or, even, by logic (please, if    5

6√ 7

 is even an irrational number!). However, a mostserious response would be to say that, in any case, the laws + and  ×  should be redefined toaccept other units. Well, let us redefine them. How long the equality 2  ×  2 = 4? is goingto be imposed to us? Naturally, someone told us that 2 × 2 = 1 (mod 3), but this response

would not satisfy to us either. In fact, we do not want to reduce our sets of numbers. Whydo we have to restrict the imagination of our students by obliging them to accept the uniquepossibility 2  × 2 = 4? Can it be thought that in our world this is truly so? Note that, if wethink in Mathematics as a tool to understand the Universe, some theories, like Relativity,for instance, have already become phased out.

Indeed, it is sufficiently proved that (R3, +,×) is not a real model for understanding theUniverse. In fact, what we call  unit :

I 3 =

1 0 0

0 1 00 0 1

is not such an  unit   in the physical consideration of Universe.

To get a better understanding of these subjects, let us say that Mathematics generallyused in quantitative sciences of the 20-th century were based on ordinary fields with charac-teristic zero, a trivial (left, right) unit  I  = +1 and on an ordinary associative product betweengeneric quantities  a, b of a given set, such as matrices, vector fields, etc. Such a Mathemat-ics is known to be linear, local differential (beginning from its topology), and Hamiltonian,thus solely representing a finite number of isolated point-particles with action-at-a-distanceforces derivable from a potential. Such a Mathematics was proved to provide an exact andinvariant representation of planetary and atomic systems as well as, more generally, of allthe so-called exterior dynamical systems   in which all constituents can be well approximatedas being point-like.

By contrast, the great majority of systems in the physical reality are nonlinear, nonlocal(of integral and other type) and not completely representable with a Hamiltonian in thecoordinates of the experimenter. This is the case for all systems historically called interior dynamical systems,   such as the structure of: planets, strongly interacting particles (suchas protons and neutrons), nuclei, molecules, stars, and other systems. The latter systemscannot be consistently reduced to a finite number of isolated point-particles. Therefore, themathematics so effective for exterior systems is only approximate at best for interior systems.

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For these reasons, even with no necessity of referring to hole black dynamic or to Einstein-Podolski-Rosen’s paradox, it is necessary to keep in mind that the unit has not got theexclusive character from which it is always presupposed, but it depends on several external

factors, like coordinates, speed, aceleration, temperature, density, ...That is, the unit matrix above indicated is not constant, but we have to consider it

variable:   I 3   =   I 3(x,  x, x,t,m,...). The question is how can we vary a mathemathical unit,which, by agreement, is unique. A possible response to this question appeared in 1980, whena group of theoretical physicists began to construct a new mathematical foundation, which is,at present, known as Isomathematics  [4].  This new theory, which consists on a generalizationof the current mathematics, begins with a step by step construction of new mathematicalstructures (named   isostructures ) verifying the same axioms and properties as usual ones,although the dependence of external factors for the unit elements (named  isounits)   is nowallowed. Such a construction makes possible that these new mathematics put all together the

usual ones, since they admit, as a particular case, that the unit elements remain invariable,as it actually happens.

Then, as a part of this new construction, which is not curiously the biggest possible, sincethe named genomathematics and hypermathematics are being also dealt), we will show nextthe construction in a schematic way of the   isonumbers , which will allow us to answer thequestion above formulated. To distinguish between isonumber and conventional number, thefirst ones will be written by using bold characters.

Let us fixe, for instance, the field of the real numbers  R and let us suppose that we wishto construct the set of isonumbers associated with  R, which have an arbitrary real numberI  as isounit. This new set will be denoted by 

R and it is defined: R = {

 a =  a  ×   I   : a  ∈ R}.

Then, it is immediate to note that 1 = 1 ×  I   =   I   ∈ R. Apart from that, it will be necessaryto redefine the usual product, which can be made in the following way: a × b = (a  ×  b)  ×  I ,where × is now named  isoproduct .

Obviously, all these changes must be accompanied by the use of new laws [1]. However,it is sufficient for our aim to realize that the answer to the question:   “can   I   =   5

6√ 7

  be the unit element of the real numbers with respect to the product ?” is affirmative. In any case,note that the choice of such an element  I  is merely anecdotal, since, as we have previouslymentioned, any real number could have been chosen as unit. Particularly, under such achoice, the following computations are obtained:

 2 = 2  ×   56√ 7  =   5

3√ 7  = (2  ×  1)  ×   5

6√ 7  = 2 × 1.

 3 = 3  ×   5

6√ 7

 =   5

2√ 7

 = (3  ×  1)  ×   5

6√ 7

 = 3 × 1. 6 = 6  ×   5

6√ 7

 =   5√ 7

 = (2  ×  3)  ×   5

6√ 7

 = 2 × 3.In this way, a new mathematical structure has been reached. It is formed by the same

set of elements as the initial one, which constitutes a difference with the case of  Z3. Manyother examples can be shown. If we think in an equality already considered in this paper,

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2  ×  2 = 4, we would have now, under the new conditions, that 2 × 2 =   10

3√ 7. Moreover, it is

not difficult to prove that 2 × 2 = 1 when  I  =   1

4 is the isounit, without working in  Z3.

2 Final conclusions

We would like to conclude the paper by commenting that the changes previously described,which can seem very simple and non transcendent, have involved, among many other sub-

 jects, the appearance of a new cleaning energy, named  hadronic energy , a possible gener-alization of Einstein’s Relativity Theory, the prediction of the existence of the antimatterand, as a consequence, of the antigravity, the modification of the Einstein-Podolski-Rosen’sTheory and the generation of new codification theories, due to the use of prime numbers.For a more complete information about all of the subjects here comented the reader can

consult both the text and the references of [3, 5], for instance.

Nowadays, mathematicians and physicist throughout the world are trying to endow thisnew theory with a right mathematical foundation which makes it serious and consistent. Itrequires a first step by step construction of each usual mathematical structures (see [2], forinstance). The main driving of this project is R.M. Santilli, at present President of the I. B.R. in Florida.

So, according with the previous exposition, it is not hazardous to affirm that we aredived in the beginnings of a new arithmetic which could supply non imaginable facts in anon distant future. We think that this impression should be transmitted to young students

to extend their minds and to allow them to elaborate their own conjectures about this newsituation.

References

[1] R. M. Falcon and J. Nunez, La isoteorıa de Santilli. International Academic Press,America-Europe-Asia, ISBN 1-57485-055-5 (2001).

[2] R. M. Falcon and J. Nunez, Isorings and related isostructures,  Bollettino dell’ Unione Matematica Italiana  8-B (2005), 437 - 452.

[3] C.X. Jiang, Foundations of Santillis Isonumber Theory With Applications to New Cry-tograms, Fermats Theorem and Goldbachs Conjecture. International Academic Press,AmericaEurope-Asia, ISBN 1-58485-056-3 (2002).

[4] R. M. Santilli, On a posible Lieadmisible covering of the Galilei Relativity in NewtonianMechanics for nonconservative and Galilei noninvariant systems,   Hadronic Journal   1

(1978), 223 - 423.

[5] R. M. Santilli, Isominkowskian geometry for the gravitational treatment of matter andits isodual for antimatter,  Intern. J. Modern Phys.  D:17 (1998), 351 - 407.

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