Independence of the Grossone-Based Infinity Methodology from Non-standard Analysis and Comments upon Logical Fallacies in Some Texts Asserting the Opposite

Abstract

This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ① (this symbol is called grossone). The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts (Gutman et al. in Found Sci 22(3):539–555, 2017; Gutman and Kutateladze in Sib Math J 49(5):835–841, 2008; Kutateladze in J Appl Ind Math 5(1):73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics.

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Notes

  1. 1.

    Khwarismi International Award, assigned by The Ministry of Science and Technology of Iran, 2017; Honorary Fellowship, the highest distinction of the European Society of Computational Methods in Sciences, Engineering and Technology, 2015; Outstanding Achievement Award from the 2015 World Congress in Computer Science, Computer Engineering, and Applied Computing, USA; Degree of Honorary Doctor from Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine, 2013; Pythagoras International Prize in Mathematics, Italy, assigned by the city of Crotone (where Pythagoras lived and founded his famous scientific school) and the Calabria Region under the high patronage of the President of the Italian Republic, Ministry of Cultural Assets and Activities and the Ministry of Education, University and Research, 2010; Lagrange Lecture, Turin University, Italy, 2010; etc.

  2. 2.

    In section 8, the authors of Gutman et al. (2017) inform the reader that before appearing in Foundations of Science their paper has been 5 times rejected by The Mathematical Intelligencer.

  3. 3.

    A numeral is a symbol (or a group of symbols) that represents a number that is a concept. The same number can be represented by different numerals. For example, symbols ‘4’, ‘four’, ‘IIII’, and ‘IV’ are different numerals, but they all represent the same number.

  4. 4.

    Recall that numerical computations are performed with floating-point numbers that can be stored in a computer memory. Since the memory is limited, mantissa and exponent of these numbers can assume only certain values and, therefore, the quantity and the form of numerals that can be used to express floating-point numbers are fixed. Due to this fact, approximations are required during computations with them because an arithmetic operation with two floating-point numbers usually produces a result that is not a floating-point number and, as a consequence, this result should be approximated by a floating-point number. In their turn, symbolic computations are the exact algebraic manipulations with mathematical expressions containing variables that have not any given value. These manipulations are more computationally expensive than numerical computations and only relatively simple codes can be elaborated in this way.

  5. 5.

    A logical fallacy (see https://www.yourlogicalfallacyis.com) is a flaw in reasoning. Logical fallacies are like tricks or illusions of thought, and they are often very sneakily used by politicians and the media to manipulate people. This and the following footnotes explaining the meaning of fallacies were taken from https://www.yourlogicalfallacyis.com.

  6. 6.

    You could say that this is the mother of all biases, as it affects so much of our thinking through motivated reasoning.

  7. 7.

    Complex subjects require some amount of understanding before one is able to make an informed judgement about the subject at hand; this fallacy is usually used in place of that understanding.

  8. 8.

    It is entirely possible to make a claim that is false yet argue with logical coherency for that claim, just as it is possible to make a claim that is true and justify it with various fallacies and poor arguments.

  9. 9.

    Many ‘natural’ things are also considered good, and this can bias our thinking; but naturalness itself does not make something good or bad.

  10. 10.

    By exaggerating, misrepresenting, or just completely fabricating someone’s argument, it is much easier to present your own position as being reasonable, but this kind of deceitfulness serves to undermine rational debate.

  11. 11.

    Notice that the finiteness of the number of symbols in the numeral is necessary for executing practical computations since we should be able to write down and store values we execute operations with.

  12. 12.

    It should be noticed that the astonishing numeral system of Pirahã is not an isolated example of this way of counting. In fact, the same counting system, one, two, many, is used by the Warlpiri people, aborigines living in the Northern Territory of Australia (see Butterworth et al. 2008).

  13. 13.

    For instance, ①-based numerals can be used for working with functions and their derivatives that can assume different infinite, finite, and infinitesimal values and can be defined over infinite and infinitesimal domains. The notions of continuity and derivability can be introduced not only for functions assuming finite values but for functions assuming infinite and infinitesimal values, as well. Limits \(\lim _{x \rightarrow a} f(x)\) are substituted by expressions and f(x) can be evaluated at concrete infinite or infinitesimal x in the same way as it is done with finite x. Series are substituted by sums having a concrete infinite number of addends and for different number of addends results (that can assume different infinite, finite or infinitesimal values) are different as it happens for sums with a finite number of summands. There are no divergent integrals, limits of integration can be concrete different infinite, finite or infinitesimal numbers and results can assume different infinite, finite or infinitesimal values. A number of set theoretical paradoxes can be avoided, etc.

  14. 14.

    Notice that the paper (Sergeyev 2015a) does not say that the rank (9) is ‘the best one’. It is just one more way to rank countries that can be useful in certain situations.

  15. 15.

    Notice that this point of view implies that the first competition is more important than the second one, etc. violating so the principle of equality of all sportive disciplines.

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Acknowledgement

The author thanks four unknown reviewers for their valuable comments. The author thanks Prof. Daniel Moskovich, Ben-Gurion University of the Negev, Beer-Sheva, Israel for providing a preliminary list of logical fallacies present in Gutman et al. (2017).

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Correspondence to Yaroslav D. Sergeyev.

Appendix

Appendix

The ①-based methodology is one of the possible views on infinite and infinitesimal quantities and Mathematics, in general. It is added to other existing philosophies of Mathematics such as logicism, formalism, intuitionism, structuralism, etc. (their comprehensive analysis can be found, e.g., in Linnebo 2017; Lolli 2002; Shapiro 2001). Three postulates and an axiom that is added to axioms for real numbers form the methodological platform of the proposal. They are given below to make this paper self-contained. A special attention in the ①-based methodology is paid to the fact than numeral systems that are among our tools used to observe mathematical objects limit our capabilities of the observation. A detailed discussion on this methodological platform can be found in Sergeyev (2017).

Methodological Postulate 1

We postulate existence of infinite and infinitesimal objects but accept that human beings and machines are able to execute only a finite number of operations.

Methodological Postulate 2

We shall not tell what are the mathematical objects we deal with; we just shall construct more powerful tools that will allow us to improve our capacities to observe and to describe properties of mathematical objects.

Methodological Postulate 3

We adopt the principle ‘The part is less than the whole’ to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite).

The Infinite Unit Axiom The infinite unit of measure is introduced as the number of elements of the set, \({\mathbb {N}}\), of natural numbers. It is expressed by the numeral ① called grossone and has the following properties:

Infinity Any finite natural number n is less than grossone, i.e., \(n <~\textcircled {1}\).

Identity The following relations link ① to identity elements 0 and 1

$$\begin{aligned} 0 \cdot \textcircled {1} = \textcircled {1} \cdot 0 = 0, \quad \textcircled {1}-\textcircled {1}= 0,\quad \frac{\textcircled {1}}{\textcircled {1}}=1, \quad \textcircled {1}^0=1, \quad 1^{\textcircled {1}}=1, \quad 0^{\textcircled {1}}=0. \end{aligned}$$
(12)

Divisibility For any finite natural number n sets \({\mathbb {N}}_{k,n}, 1 \le k \le n,\) being the \(n\hbox {th}\) parts of the set, \({\mathbb {N}}\), of natural numbers have the same number of elements indicated by the numeral \(\frac{\textcircled {1}}{n}\) where

$$\begin{aligned} {\mathbb {N}}_{k,n} = \{k, k+n, k+2n, k+3n, \ldots \}, \quad 1 \le k \le n, \quad \bigcup _{k=1}^{n}{\mathbb {N}}_{k,n}={\mathbb {N}}. \end{aligned}$$
(13)

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Sergeyev, Y.D. Independence of the Grossone-Based Infinity Methodology from Non-standard Analysis and Comments upon Logical Fallacies in Some Texts Asserting the Opposite. Found Sci 24, 153–170 (2019). https://doi.org/10.1007/s10699-018-9566-y

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Keywords

  • Numerical infinities and infinitesimals
  • Grossone
  • Infinity Computer
  • Non-standard anallysis
  • Logical fallacies