A PENN'ORTH OF GROSSONE

The Interfax Agency and other mass media announced on November 3, 2010 that Yaroslav Sergeyev, a professor at Lobachevsky State University of Nizhny Novgorod, has received the Pythagoras Award. It was mentioned that “the professor constructed and patented a new ‘Infinity Computer’” and “he suggested a new mathematical language that enables one to record various infinitely large and infinitely small numbers.” This information requires commenting.

Sergeyev’s idea is to introduce into arithmetic some infinitely large number— grossone, consider only the numbers that are less than the grossone, and operate exclusively on these numbers. Sergeyev embellishes his idea with shallow arguments, emphasizing that he does not use Cantor’s approach and returns to Ancient Greeks. He formulates the three postulates of his own:

Postulate 1. Existence of infinite and infinitesimal objects is postulated but it is also accepted that human beings and machines are able to execute only a finite number of operations.
Postulate 2. It is not discussed what are the mathematical objects we deal with; we just construct more powerful tools that allow us to improve our capacities to observe and to describe properties of mathematical objects.
Postulate 3. The principle formulated by Ancient Greeks ‘The part is less than the whole’ is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite).

The scientific depth of Sergeyev’s postulates transpires.

Elliot Mendelson, an outstanding American logician whose textbook was reprinted many times and is very popular in teaching throughout the world, published the following review on Sergeyev’s book Arithmetic of Infinity which was printed in Italy in 2003:

The author attempts to introduce new kinds of number systems that he claims have important applications. First, he reviews some facts about the ordinary number systems and set theory. (Here, there is some confusion about alephs and the continuum hypothesis. For example, he defines aleph-one to be the size of the power set of the set of natural numbers.) The systems he deals with consist of objects which are called extended real numbers, but the descriptions of these objects and their properties are not clear enough to permit any warranted judgments about the assertions made by the author about these systems.

Sergeyev confronts his ideas with the famous nonstandard analysis of Abraham Robinson. Robinsonian infinitesimal analysis is rightfully considered as one of the most brilliant achievements of mathematics in the twentieth century. Using the sophisticated methods of the recently-invented model theory in the beginning of the 1960s, Robinson synthesized the approaches by Newton and Leibniz in a new mathematical language and technique. Nonstandard analysis not only inherited all technical tools of Newton’s method of prime and ultimate ratios and Leibniz’s monads but also explained and justified the ingenious tricks of Euler who freely dealt with actual infinites and infinitesimals.

Sergeyev has poor knowledge of these classical scientific achievements, counterposing his bizarre surmisals to the modern analysis. But all linguistic and mathematical tools that are needed to Sergeyev are readily available within nonstandard analysis.

Sergeyev defines his grossone as the “number of elements of the set of natural numbers.” In fact, the role of this would-be mysterious entity can happily be performed by the factorial of an arbitrary infinite number which are galore in nonstandard analysis. This circumstance is absolutely evident to specialists but nonetheless it was meticulously elaborated in Siberian Mathematical Journal, 49:5, 835–841 (2008) in order to demonstrate the humble place of Sergeyev’s speculations. This article also revealed the principal shortcomings of Sergeyev’s attempts at implementing calculations with a grossone on a computer. Unfortunately, it turned out impossible to stop the flood of Sergeyev’s publications in the variety of the international journals having little if any in common with foundations of analysis. Miraculously, there are no Sergeyev’s publications on his grossone in Russian in the Russian mathematical database Math-Net.Ru.

Sergeyev’s writings about invention of new numbers and the “Infinity Computer” are speculations of negligible interest to mathematics but exuberant with pretensions, which might be perilous to science. The fact that these speculations underlie the Pytharogas Award by the City of Crotone on behalf of the University of Calabria having Sergeyev on staff cannot improve neither the content of the writings of Sergeyev nor his attitude to the precious knowledge of the treasure-trove of mathematics.

Ancient Italian grossones are linguistically close to Sergeyev’s grossone but differ in value.

Appendices:

Information from Italy:

Published as
Newsletter of the European Mathematical Society, No. 79, March, 2011, p. 60;
J. Appl. Indust. Math., 2011, V. 5, No. 1, 73–75.