Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems

There has been controversy about Sergeyev’s work and its applications. It is the contention of the reviewer that the Sergeyev grossone, , can simply be interpreted as a generic large natural number and that there is no new theory of infinity involved. Part of the controversy stems from the fact that Sergeyev does not make an attempt to give a logical foundation for his new idealized infinite and infinitesimal elements. For example, we can write

[1]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[2]  J. Brian Conrey,et al.  The Riemann Hypothesis, Volume 50, Number 3 , 2003 .

[3]  R. Sarpong,et al.  Bio-inspired synthesis of xishacorenes A, B, and C, and a new congener from fuscol† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sc02572c , 2019, Chemical science.

[4]  K. Gödel.  Consistency of the Continuum Hypothesis. (AM-3) , 1940 .

[5]  Harold M. Edwards,et al.  Riemann's Zeta Function , 1974 .

[6]  A. Zhigljavsky,et al.  Classical areas of mathematics where the concept of grossone could be useful , 2016 .

[7]  Vieri Benci,et al.  Numerosities of labelled sets: a new way of counting , 2003 .

[8]  Louis D'Alotto,et al.  Cellular automata using infinite computations , 2011, Appl. Math. Comput..

[9]  Gabriele Lolli,et al.  Infinitesimals and infinites in the history of mathematics: A brief survey , 2012, Appl. Math. Comput..

[10]  Yaroslav D. Sergeyev,et al.  Computations with Grossone-Based Infinities , 2015, UCNC.

[11]  Renato De Leone,et al.  The use of grossone in Mathematical Programming and Operations Research , 2011, Appl. Math. Comput..

[12]  I. Stanimirović,et al.  COMPENDIOUS LEXICOGRAPHIC METHOD FOR MULTI-OBJECTIVE OPTIMIZATION , 2012 .

[13]  Alfredo Garro,et al.  Single-tape and multi-tape Turing machines through the lens of the Grossone methodology , 2013, The Journal of Supercomputing.

[14]  Davide Rizza.  Supertasks and numeral systems , 2016 .

[15]  P. Gordon.  Numerical Cognition Without Words: Evidence from Amazonia , 2004, Science.

[16]  H. Isermann.  Linear lexicographic optimization , 1982 .

[17]  J. M. Child.  The Early Mathematical Manuscripts Of Leibniz , 1921, The Mathematical Gazette.

[18]  Franco Montagna,et al.  Taking the Pirahã seriously , 2015, Commun. Nonlinear Sci. Numer. Simul..

[19]  K. Knopp.  Theory and Application of Infinite Series , 1990 .

[20]  Hartmut Jürgens,et al.  Chaos and Fractals: New Frontiers of Science , 1992 .

[21]  A. Kanamori.  The higher infinite : large cardinals in set theory from their beginnings , 2005 .

[22]  Yaroslav D. Sergeyev,et al.  Counting systems and the First Hilbert problem , 2010, 1203.4141.

[23]  Alfredo Garro,et al.  Observability of Turing Machines: A Refinement of the Theory of Computation , 2010, Informatica.

[24]  L. Corry.  A Brief History of Numbers , 2015 .

[25]  Yaroslav D. Sergeyev,et al.  Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge ☆ , 2009, 1203.3150.

[26]  L. Brugnano,et al.  Solving differential problems by multistep initial and boundary value methods , 1998 .

[27]  Robert Spira,et al.  Zeros of sections of the zeta function. II , 1966 .

[28]  Maurice Margenstern.  Infinigons of the hyperbolic plane and grossone , 2016, Appl. Math. Comput..

[29]  Donald A. Martin,et al.  Mathematical Problems. Lecture Delivered Before the International Congress of Mathematicians at Paris in 1900 , 1979 .

[30]  Brian Butterworth,et al.  Numerical thought with and without words: Evidence from indigenous Australian children , 2008, Proceedings of the National Academy of Sciences.

[31]  C. J. Keyser.  Contributions to the Founding of the Theory of Transfinite Numbers , 1916 .

[32]  Yaroslav D. Sergeyev,et al.  The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area , 2015, Commun. Nonlinear Sci. Numer. Simul..

[33]  Francesca Mazzia,et al.  A new mesh selection strategy with stiffness detection for explicit Runge-Kutta methods , 2015, Appl. Math. Comput..

[34]  Louis D'Alotto.  Finite and Infinite Computations and a Classification of Two-Dimensional Cellular Automata Using Infinite Computations , 2017, PaCT.

[35]  Fabio Caldarola.  The Sierpinski curve viewed by numerical computations with infinities and infinitesimals , 2018, Appl. Math. Comput..

[36]  Yaroslav D. Sergeyev,et al.  Solving ordinary differential equations by working with infinitesimals numerically on the Infinity Computer , 2013 .

[37]  Massimo Veltri,et al.  Usage of infinitesimals in the Menger's Sponge model of porosity , 2011, Appl. Math. Comput..

[38]  Yaroslav D. Sergeyev,et al.  Numerical Methods for Solving Initial Value Problems on the Infinity Computer , 2016, Int. J. Unconv. Comput..

[39]  Akihiro Kanamori,et al.  The Mathematical Development of Set Theory from Cantor to Cohen , 1996, Bulletin of Symbolic Logic.

[40]  M. Balazard,et al.  Sur l'infimum des parties réelles des zéros des sommes partielles de la fonction zêta de Riemann , 2009, 0902.0923.

[41]  R. Guy,et al.  The Book of Numbers , 2019, The Crimean Karaim Bible.

[42]  Manlio Gaudioso,et al.  Numerical infinitesimals in a variable metric method for convex nonsmooth optimization , 2018, Appl. Math. Comput..

[43]  E. Bombieri.  Infinity: The Mathematical Infinity , 2011 .

[44]  Yaroslav D. Sergeyev,et al.  Higher order numerical differentiation on the Infinity Computer , 2011, Optim. Lett..

[45]  W. Hugh Woodin,et al.  The Continuum Hypothesis, Part I , 2001 .

[46]  Jean-Michel Muller,et al.  Elementary Functions: Algorithms and Implementation , 1997 .

[47]  Gebräuchliche Fertigarzneimittel,et al.  V , 1893, Therapielexikon Neurologie.

[48]  M. Potter.  Orders of infinity , 2004 .

[49]  John P. Mayberry.  The Foundations of Mathematics in the Theory of Sets , 2001 .

[50]  Petr Hájek,et al.  The theory of semisets , 1972 .

[51]  Yaroslav D. Sergeyev,et al.  A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities , 2008, Informatica.

[52]  M. Colyvan.  An Introduction to the Philosophy of Mathematics , 2012 .

[53]  Y. Zou.  Single Variable Calculus: A First Step , 2018 .

[54]  E. Hairer,et al.  Introduction to Analysis of the Infinite , 2008 .

[55]  R. Devaney.  An Introduction to Chaotic Dynamical Systems , 1990 .

[56]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[57]  Renato De Leone,et al.  Nonlinear programming and Grossone: Quadratic Programing and the role of Constraint Qualifications , 2018, Appl. Math. Comput..

[58]  Gabriele Lolli,et al.  Metamathematical investigations on the theory of Grossone , 2015, Appl. Math. Comput..

[59]  Edward Sapir,et al.  Selected Writings of Edward Sapir in Language, Culture and Personality , 1950 .

[60]  Martin Berz,et al.  Automatic differentiation as nonarchimedean analysis , 2003 .

[61]  M. Zarepisheh,et al.  A dual-based algorithm for solving lexicographic multiple objective programs , 2007, Eur. J. Oper. Res..

[62]  Yaroslav D. Sergeyev,et al.  Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers , 2007 .

[63]  One , 2004 .

[64]  P. J. Cohen.  Set Theory and the Continuum Hypothesis , 1966 .

[65]  Roman G. Strongin,et al.  Introduction to Global Optimization Exploiting Space-Filling Curves , 2013 .

[66]  George Sugihara,et al.  Fractals: A User's Guide for the Natural Sciences , 1993 .

[67]  H. Sagan.  Space-filling curves , 1994 .

[68]  Joel S. Cohen,et al.  Computer Algebra and Symbolic Computation: Mathematical Methods , 2003 .

[69]  Jeff R. Cash,et al.  An MEBDF code for stiff initial value problems , 1992, TOMS.

[70]  Yaroslav D. Sergeyev,et al.  Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains , 2009, 1203.4140.

[71]  Yaroslav D. Sergeyev,et al.  Numerical computations and mathematical modelling with infinite and infinitesimal numbers , 2012, ArXiv.

[72]  Yaroslav D. Sergeyev,et al.  On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function , 2011, 1203.4142.

[73]  Christian Bischof,et al.  Computing derivatives of computer programs , 2000 .

[74]  Yaroslav D. Sergeyev,et al.  The Olympic Medals Ranks, Lexicographic Ordering, and Numerical Infinities , 2015, 1509.04313.

[75]  Roberto Barrio,et al.  Performance of the Taylor series method for ODEs/DAEs , 2005, Appl. Math. Comput..

[76]  Group radicals and strongly compact cardinals , 2013 .

[77]  Yaroslav D. Sergeyev.  Using Blinking Fractals for Mathematical Modeling of Processes of Growth in Biological Systems , 2011, Informatica.

[78]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

[79]  A. Soyster,et al.  Preemptive and nonpreemptive multi-objective programming: Relationship and counterexamples , 1983 .

[80]  Maurice Margenstern.  Using grossone to count the number of elements of infinite sets and the connection with bijections , 2011, ArXiv.

[81]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[82]  Anatoly A. Zhigljavsky,et al.  Computing sums of conditionally convergent and divergent series using the concept of grossone , 2012, Appl. Math. Comput..

[83]  Francesca Mazzia,et al.  Solving ordinary differential equations by generalized Adams methods: properties and implementation techniques , 1998 .