Numerical Computations with Infinite and Infinitesimal Numbers: Theory and Applications

Even the brilliant efforts of the creator of the nonstandard analysis Robinson that were made in the middle of the twentieth century have been also directed to a reformulation of the classical analysis (i.e., analysis created 200 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. In fact, he wrote in Sect. 1.1 of his famous book [28]: “It is shown in this book that Leibniz’s ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical analysis and to many other branches of mathematics” (the words classical analysis have been emphasized by the author of this chapter).

We remind that numeral is a symbol or group of symbols that represents a number. The difference between numerals and numbers is the same as the difference between words and the things they refer to. A number is a concept that a numeral expresses. The same number can be represented by different numerals. For example, the symbols “3,” “three,” and “III” are different numerals, but they all represent the same number.

In connection with Cantor’s \(\aleph _{0}\) and \(\aleph _{1}\) it makes sense to remind another Amazonian tribe—Mundurukú (see [27]) who fail in exact arithmetic with numbers larger than 5 but are able to compare and add large approximate numbers that are far beyond their naming range. Particularly, they use the words “some, not many” and “many, really many” to distinguish two types of large numbers. Their arithmetic with “some, not many” and “many, really many” reminds strongly the rules Cantor uses to work with \(\aleph _{0}\) and \(\aleph _{1}\), respectively. For instance, compare “some, not many”  + “many, really many” = “many, really many” with \(\aleph _{0} + \aleph _{1} = \aleph _{1}\).

As it was already mentioned, in 1900, at the second Mathematical Congress in Paris, David Hilbert has presented his 23 problems for the twentieth century promoting the abstract philosophy in Mathematics that was close to Kant. However, before this event, at the first Congress 3 years earlier Henri Poincaré has given a general talk emphasizing the connection of Mathematics with Physics sharing this point of view with Fourier, Laplace, and many others. Clearly, in this dispute between Poincaré and Hilbert the present chapter is closer to the position of Poincaré.

It is important to emphasize that we speak about axioms of real numbers in sense of Postulate 2, i.e., axioms define formal rules of operations with numerals in a given numeral system. Therefore, if we want to have a numeral system including grossone, we should fix also a numeral system to express finite numbers. In order to concentrate our attention on properties of grossone, this point will be investigated later.

First, this quantity is inexpressible by numerals used to count the number of elements of finite sets because \(\mathbb{N}\) is infinite. Second, traditional numerals existing to express infinite numbers do not have the required high accuracy (remind that we would like to be able to register the alteration of the number of elements of infinite sets even when one element has been excluded). For example, by using Cantor’s alephs we say that cardinality of the sets \(\mathbb{N}\) and \(\mathbb{N} \setminus \{ 1\}\) is the same—\(\aleph _{0}\). This answer is correct but its accuracy is low—we are not able to register the fact that one element was excluded from the set \(\mathbb{N}\). Analogously, we can say that both of the sets have “many” elements. Again, this answer is correct but its accuracy is low.

At the first glance the record (14) [and, therefore, the numerals (15)] can remind numbers from the Levi–Civita field (see [20]) that is a very interesting and important precedent of algebraic manipulations with infinities and infinitesimals. However, the two mathematical objects have several crucial differences. They have been introduced for different purposes by using two mathematical languages having different accuracies and on the basis of different methodological foundations. In fact, Levi–Civita does not discuss the distinction between numbers and numerals and works with generic numbers while each numeral (15) represents a concrete number. His numbers have neither cardinal nor ordinal properties; they are built using a generic infinitesimal and only its rational powers are allowed; he uses symbol ∞ in his construction; there is no numeral system that would allow one to assign numerical values to his numbers; it is not explained how it would be possible to pass from d a generic infinitesimal h to a concrete one (see also the discussion above on the distinction between numbers and numerals).

In no way the said above should be considered as a criticism with respect to results of Levi–Civita. The above discussion has been introduced in this text just to underline that we are in front of two different mathematical tools that should be used in different mathematical contexts.

Naturally, if we speak about limits of sequences, lim _{n → ∞} a(n), then \(n \in \mathbb{N}\) and, as a consequence, it follows that n should be less than or equal to grossone.

Note that even if \(a ={ \textcircled{1}}^{2} + 2+\varepsilon\), where \(\varepsilon\) is an infinitesimal number (remind that all infinitesimals are not equal to zero), we are able to establish that the function has discontinuous formulae.