A New Computational Methodology Using Infinite and Infinitesimal Numbers Yaroslav D. Sergeyev Universit`a della Calabria, 87030 Rende (CS), Italy and N.I. Lobatchevsky State University, Nizhni Novgorod, Russia
[email protected] http://wwwinfo.deis.unical.it/~yaro
Abstract. Traditional computers work numerically only with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this lecture, a new computational methodology (that is not related to non-standard analysis approaches) is described. It is based on the principle ‘The part is less than the whole’ applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The new methodology has allowed the author to introduce the Infinity Computer working numerically with infinite and infinitesimal numbers. The new computational paradigm both gives possibilities to execute computations of a new type and simplifies fields of Mathematics and Computer Science where infinity and/or infinitesimals are required. Examples of the usage of the introduced computational tools are given during the lecture. Keywords: Numeral systems, infinite and infinitesimal numbers, Infinity Computer, numerical computations, Turing machine.
There exist different ways to generalize traditional arithmetic for finite numbers to the case of infinite and infinitesimal quantities (see, e.g., [1, 2, 4, 5] and references given therein). However, arithmetics that have been developed so far to deal with infinite quantities are quite different with respect to the finite arithmetic we are used to work with. In fact, arithmetics working with infinity can ∞ have undetermined operations (for example, ∞ − ∞, ∞ , etc.) or they use representation of infinite numbers based on infinite sequences of finite numbers. These difficulties did not allow people to create computers working with infinite and infinitesimal quantities numerically. In this lecture, we describe a new methodology (see survey [8] and applications in [6, 7, 9, 11, 14, 15]) for treating infinite and infinitesimal quantities expressed in a new numeral1 system. It has a strong numerical character and is based on the principle ‘The part is less than the whole’ applied to all numbers 1
We remind that numeral is a symbol or group of symbols that represents a number. A number is a concept that a numeral expresses. The same number can be represented by different numerals. For example, the symbols ‘8’, ‘eight’, and ‘VIII’ are different numerals, but they all represent the same number.
2
Yaroslav D. Sergeyev
(finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The new methodology has allowed the author to introduce the Infinity Computer (see European patent [12]) working numerically with infinite and infinitesimal numbers. The new computational paradigm both gives possibilities to execute computations of a new type and simplifies fields of Mathematics and Computer Science where infinity and/or infinitesimals are required. In order to understand how it is possible to look at the problem of infinity in a new way, let us consider a study published in Science by Peter Gordon (see [3]) where he describes a primitive tribe living in Amazonia - Pirah˜a - that uses a very simple numeral system for counting: one, two, many. For Pirah˜a, all quantities bigger than two are just ‘many’ and such operations as 2+2 and 2+1 give the same result, i.e., ‘many’. Using their weak numeral system Pirah˜a are not able to see, for instance, numbers 3, 4, 5, and 6, to execute arithmetical operations with them, and, in general, to say anything about these numbers because in their language there are neither words nor concepts for that. Moreover, the weakness of their numeral system leads to such results as ‘many’ + 1 = ‘many’,
‘many’ + 2 = ‘many’,
which are very familiar to us in the context of views on infinity used in the traditional calculus ∞ + 1 = ∞, ∞ + 2 = ∞. This observation leads us to the following idea: Probably our difficulty in working with infinity is not connected to the nature of infinity but is a result of inadequate numeral systems used to express infinite numbers. Thus, it is proposed to introduce a new numeral system having a possibility to express finite, infinite, and infinitesimal numbers in a unique framework and to execute arithmetical operations with all of them. An infinite unit for measuring infinite sets is used as the radix of the new positional numeral system. It is necessary to emphasize that the new approach is not a contraposition to the ideas of Cantor and Robinson. In contrast, it is introduced as an applied evolution of their ideas. The problem of infinity is considered from the point of view of applied Mathematics and theory and practice of computation. The following methodological consideration should be also mentioned. Note that foundations of the Set Theory dealing with infinity have been developed starting from the end of the XIX-th century until more or less the first decades of the XX-th century. Foundations of the classical Analysis dealing both with infinity and infinitesimal quantities have been developed even earlier, more than 200 years ago, with the goal to develop mathematical tools allowing one to solve problems arising in the real world in that time. As a result, they reflect ideas that people had about Physics more than 200 years ago. Thus, these parts of Mathematics do not include numerous achievements of Physics of the XX-th century. Even the brilliant results of Robinson were made in the middle of the XX-th century and have been also directed to a reformulation of the classical Analysis
A Computational Methodology Using Infinite and Infinitesimal Numbers
3
(i.e., Analysis created two hundred years before Robinson) in terms of infinitesimals and not to the creation of a new kind of Analysis that would incorporate new achievements of Physics. The point of view on infinite and infinitesimal quantities presented in this lecture uses strongly two methodological ideas borrowed from the modern Physics: relativity and interrelations holding between the object of an observation and the tool used for this observation. The latter is directly related to connections between numeral systems used to describe mathematical objects and the objects themselves. Numerals that we use to write down numbers, functions, etc. are among our tools of investigation and, as a result, they strongly influence our capabilities to study mathematical objects. Moreover, we are able to write down and to study only those numbers that are expressible by numeral systems we know. The new methodology and the corresponding language allow one to look at problems related to infinity with a higher precision with respect to traditional numeral systems. In [14], infinite processes and Turing machines have been studied using the new approach. In that paper, a deep investigation is performed on the interrelations between mechanical computations and their mathematical descriptions emerging when a human (the researcher) starts to describe a Turing machine (the object of the study) by different mathematical languages (the instruments of investigation). Together with traditional mathematical languages using such concepts as ‘enumerable sets’ and ‘continuum’ the new computational methodology allowing one to measure the number of elements of different infinite sets is used. It is shown how mathematical languages used to describe the machines limit our possibilities to observe them. This analysis is done with respect to deterministic and non-deterministic Turing machines. Acknowledgments. This research was partially supported by the Russian Federal Program “Scientists and Educators in Russia of Innovations”, contract number 02.740.11.5018.
References 1. Cantor, G.: Contributions to the Founding of the Theory of Transfinite Numbers. Dover Publications, New York (1955) 2. Conway, J.H., Guy, R.K.: The Book of Numbers. Springer, New York (1996) 3. Gordon, P.: Numerical Cognition without Words: Evidence from Amazonia. Science. 306(15 October), 496–499 (2004) 4. Mayberry, J.P.: The Foundations of Mathematics in the Theory of Sets. Cambridge Univ. Press, Cambridge (2001) 5. Robinson, A.: Non-Standard Analysis. Princeton Univ. Press, Princeton (1996) 6. Sergeyev, Ya.D.: Arithmetic of Infinity. Edizioni Orizzonti Meridionali, CS (2003) 7. Sergeyev, Ya.D.: Blinking Fractals and Their Quantitative Analysis Using Infinite and Infinitesimal Numbers. Chaos, Solitons & Fractals. 33(1), 50–75 (2007) 8. Sergeyev, Ya.D.: A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities. Informatica. 19(4), 567–596 (2008)
4
Yaroslav D. Sergeyev
9. Sergeyev, Ya.D.: Evaluating the Exact Infinitesimal Values of Area of Sierpinski’s Carpet and Volume of Menger’s Sponge. Chaos, Solitons & Fractals. 42(5), 3042– 3046, (2009) 10. Sergeyev, Ya.D.: Numerical Point of View on Calculus for Functions Assuming Finite, Infinite, and Infinitesimal Values Over Finite, Infinite, and Infinitesimal Domains. Nonlinear Analysis Series A: Theory, Methods & Applications. 71(12), e1688–e1707 (2009) 11. Sergeyev, Ya.D.: Numerical Computations and Mathematical Modelling with Infinite and Infinitesimal Numbers. J. Applied Mathematics & Computing. 29, 177–195 (2009) 12. Sergeyev, Ya.D.: Computer System for Storing Infinite, Infinitesimal, and Finite Quantities and Executing Arithmetical Operations with Them. EU patent 1728149 (2009) 13. Sergeyev, Ya.D.: Counting Systems and the First Hilbert Problem. Nonlinear Analysis Series A: Theory, Methods & Applications. 72(3-4), 1701–1708 (2010) 14. Sergeyev, Ya.D., Garro, A.: Observability of Turing Machines: A Refinement of the Theory of Computation. Informatica. (in press) (2010) 15. The Infinity Computer web page, http://www.theinfinitycomputer.com
