Approaches to analysis with infinitesimals following Robinson, Nelson, and others

This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis. We highlight some applications including (1) Loeb's approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson's and related frameworks to the multiverse view as developed by Hamkins. Keywords: axiomatisations, infinitesimal, nonstandard analysis, ultraproducts, superstructure, set-theoretic foundations, multiverse, naive integers, intuitionism, soritical properties, ideal elements, protozoa.

[1]  Benno van den Berg,et al.  A functional interpretation for nonstandard arithmetic , 2012, Ann. Pure Appl. Log..

[2]  Yves Péraire.  Infinitesimal Approach of Almost-Automorphic Functions , 1993, Ann. Pure Appl. Log..

[3]  E. I. Gordon.  Nonstandard Methods in Commutative Harmonic Analysis , 1997 .

[4]  Petr Vopenka,et al.  Alternative Set Theory , 2009, Encyclopedia of Optimization.

[5]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[6]  Mikhail G. Katz,et al.  Cauchy's Continuum , 2011, Perspectives on Science.

[7]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[8]  Sam Sanders,et al.  Reverse formalism 16 , 2017, Synthese.

[9]  Mikhail G. Katz,et al.  EDWARD NELSON (1932–2014) , 2015, The Review of Symbolic Logic.

[10]  Abraham Robinson.  Selected papers of Abraham Robinson , 1978 .

[11]  Mikhail G. Katz,et al.  Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond , 2012, 1205.0174.

[12]  Michel Fliess,et al.  Analyse non standard du bruit , 2006, ArXiv.

[13]  Vladimir Kanovei,et al.  Interpreting the Infinitesimal Mathematics of Leibniz and Euler , 2016, 1605.00455.

[14]  David Ballard.  Foundational Aspects of “Non”standard Mathematics , 1994 .

[15]  J. Harthong.  Etudes sur la mécanique quantique , 1984 .

[16]  Saharon Shelah,et al.  A definable nonstandard model of the reals , 2004, J. Symb. Log..

[17]  Heinz Weisshaupt,et al.  Diffusion processes via parabolic equations: an infinitesimal approach to Lindeberg's limit theorem , 2009, J. Log. Anal..

[18]  Karel Hrbacek,et al.  Axiomatic foundations for Nonstandard Analysis , 1978 .

[19]  W. Luxemburg.  Non-Standard Analysis , 1977 .

[20]  Philip Ehrlich,et al.  The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes , 2006 .

[21]  Karin U. Katz,et al.  Euler’s Lute and Edwards’s Oud , 2015, 1506.02586.

[22]  K. Zdanowski,et al.  The Intended Model of Arithmetic. An Argument from Tennenbaum's Theorem , 2006 .

[23]  M. Schützenberger,et al.  Triangle of Thoughts , 2001 .

[24]  Piotr Blaszczyk,et al.  Is mathematical history written by the victors , 2013, 1306.5973.

[25]  Alain Robert.  Analyse non standard , 1985 .

[26]  Ulrich Kohlenbach,et al.  Applied Proof Theory - Proof Interpretations and their Use in Mathematics , 2008, Springer Monographs in Mathematics.

[27]  Mariam Thalos,et al.  Why is there Philosophy of Mathematics at all , 2016 .

[28]  C. Gomes,et al.  Structure and Randomness , 2006 .

[29]  Hisahiro Tamano,et al.  On Rings of Real Valued Continuous Functions , 1958 .

[30]  Toru Kawai.  Axiom systems of nonstandard set theory , 1981 .

[31]  Yves Péraire,et al.  THEORIE RELATIVE DES ENSEMBLES INTERNES , 1992 .

[32]  Jeremy Avigad.  Weak Theories of Nonstandard Arithmetic and Analysis , 2000 .

[33]  Mikhail G. Katz,et al.  From discrete arithmetic to arithmetic of the continuum , 2013 .

[34]  Euler’s definition of the derivative , 2007 .

[35]  Karel Hrbacek,et al.  Analysis with Ultrasmall Numbers , 2010, Am. Math. Mon..

[36]  Alexandre Borovik,et al.  Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus , 2011, 1108.2885.

[37]  Karin U. Katz,et al.  Gregory’s Sixth Operation , 2016, The Best Writing on Mathematics 2019.

[38]  P. Vopenka,et al.  Mathematics in the alternative set theory , 1979 .

[39]  T. Tao.  Compactness and Contradiction , 2013 .

[40]  Judith V. Grabiner,et al.  The origins of Cauchy's rigorous calculus , 1981 .

[41]  Guy Wallet,et al.  Entrée-sortie dans un tourbillon , 1986 .

[42]  Gert Schubring,et al.  Conflicts between Generalization, Rigor and Intuition. Number Concepts Underlying the Development of Analysis in 17th-19th Century France and Germany , 2005 .

[43]  Piotr Blaszczyk,et al.  Leibniz versus Ishiguro: Closing a Quarter Century of Syncategoremania , 2016, HOPOS: The Journal of the International Society for the History of Philosophy of Science.

[44]  Hilary Putnam,et al.  Models and reality , 1980, Journal of Symbolic Logic.

[45]  Jeremy Gray.  The Real and the Complex: A History of Analysis in the 19th Century , 2015 .

[46]  Renling Jin.  The sumset phenomenon , 2001 .

[47]  Mikhail G. Katz,et al.  Commuting and Noncommuting Infinitesimals , 2013, Am. Math. Mon..

[48]  Bar-Ilan University,et al.  From Pythagoreans and Weierstrassians to True Infinitesimal Calculus , 2017 .

[49]  Peter A. Loeb,et al.  Conversion from nonstandard to standard measure spaces and applications in probability theory , 1975 .

[50]  Joel David Hamkins,et al.  Is the Dream Solution of the Continuum Hypothesis Attainable? , 2012, Notre Dame J. Formal Log..

[51]  Joel David Hamkins,et al.  THE SET-THEORETIC MULTIVERSE , 2011, The Review of Symbolic Logic.

[52]  Tools , 2019, Oncology Issues.

[53]  H. Keisler.  Elementary Calculus: An Infinitesimal Approach , 1976 .

[54]  H. Gaifman.  Non-Standard Models in a Broader Perspective , 2005 .

[55]  La mathematique non standard vieille de soixante ans , 1981 .

[56]  Vladimir Kanovei,et al.  Internal approach to external sets and universes , 1996, Stud Logica.

[57]  Robert Lutz.  Nonstandard Analysis.: A Practical Guide with Applications. , 1981 .

[58]  Vieri Benci,et al.  Alpha-theory: An elementary axiomatics for nonstandard analysis , 2003 .

[59]  Isaac Goldbring,et al.  Hilbert's Fifth Problem for Local Groups , 2007, 0708.3871.

[60]  Mikhail G. Katz,et al.  Almost Equal: the Method of Adequality from Diophantus to Fermat and Beyond , 2012, Perspectives on Science.

[61]  Leif Arkeryd.  Nonstandard Analysis , 2005, Am. Math. Mon..

[62]  Terence Tao,et al.  Sum-avoiding sets in groups , 2016, 1603.03068.

[63]  Vieri Benci,et al.  The eightfold path to Nonstandard Analysis , 2004 .

[64]  H. Jerome Keisler,et al.  On the strength of nonstandard analysis , 1986, Journal of Symbolic Logic.

[65]  Mikhail G. Katz,et al.  Meaning in Classical Mathematics: Is it at Odds with Intuitionism? , 2011, 1110.5456.

[66]  Dalibor Pražák,et al.  Nonstandard analysis of global attractors , 2015, Math. Log. Q..

[67]  M. D. Nasso,et al.  Iterated hyper-extensions and an idempotent ultrafilter proof of Rado’s Theorem , 2013, 1304.3009.

[68]  T. Tao.  Hilbert's Fifth Problem and Related Topics , 2014 .

[69]  H. Jerome Keisler,et al.  An Infinitesimal Approach to Stochastic Analysis , 1984 .

[70]  Heinz Weisshaupt,et al.  Radically elementary analysis of an interacting particle system at an unstable equilibrium , 2011, J. Log. Anal..

[71]  Mikhail G. Katz,et al.  The Mathematical Intelligencer Flunks the Olympics , 2016, ArXiv.

[72]  H. Keisler.  Foundations of infinitesimal calculus , 1976 .

[73]  Errett Bishop,et al.  Review: H. Jerome Keisler, Elementary calculus , 1977 .

[74]  Karel Hrbacek,et al.  Standard Foundations for Nonstandard Analysis , 1992, J. Symb. Log..

[75]  K. Easwaran.  Regularity and Hyperreal Credences , 2014 .

[76]  Paul R. Halmos,et al.  I Want to be a Mathematician , 1985 .

[77]  Vladimir Kanovei,et al.  Proofs and Retributions, Or: Why Sarah Can’t Take Limits , 2015 .

[78]  P. V. Andreev,et al.  An Axiomatics for Nonstandard Set Theory, Based on von Neumann-Bernays-Gödel Theory , 2001, J. Symb. Log..

[79]  Vladimir Kanovei,et al.  Nonstandard Analysis, Axiomatically , 2004 .

[80]  Peter Fletcher,et al.  Nonstandard set theory , 1989, Journal of Symbolic Logic.

[81]  Nonstandard Asymptotic Analysis , 1987 .

[82]  K. D. Stroyan,et al.  Introduction to the theory of infinitesimals , 1976 .

[83]  W. A. J. Luxemburg,et al.  Applications of model theory to algebra, analysis, and probability , 1971 .

[84]  Joel David Hamkins,et al.  A Natural Model of the Multiverse Axioms , 2010, Notre Dame J. Formal Log..

[85]  G. Ferraro.  Differentials and differential coefficients in the Eulerian foundations of the calculus , 2004 .

[86]  Mikolás Janota,et al.  Digital Object Identifier (DOI): , 2000 .

[87]  Vladimir Kanovei,et al.  Controversies in the Foundations of Analysis: Comments on Schubring’s Conflicts , 2016, 1601.00059.

[88]  Vladimir Kanovei,et al.  Toward a History of Mathematics Focused on Procedures , 2016, 1609.04531.

[89]  Karel Hrbacek,et al.  Relative set theory: Internal view , 2009, J. Log. Anal..

[90]  Karel Hrbacek.  Axiom of Choice in nonstandard set theory , 2012, J. Log. Anal..

[91]  L. Dries,et al.  Hilbert's 5th problem , 2015 .

[92]  Mikhail G. Katz,et al.  Leibniz's laws of continuity and homogeneity , 2012, 1211.7188.

[93]  David A. Ross.  Loeb Measure and Probability , 1997 .

[94]  C. Henson,et al.  FOUNDATIONS OF NONSTANDARD ANALYSIS A Gentle Introduction to Nonstandard Extensions , 1996 .

[95]  TO BE OR NOT TO BE CONSTRUCTIVE , 2017 .

[96]  Vladimir Kanovei,et al.  Is Leibnizian Calculus Embeddable in First Order Logic? , 2016, 1605.03501.

[97]  Mikhail G. Katz,et al.  Ten Misconceptions from the History of Analysis and Their Debunking , 2012, 1202.4153.

[98]  Stephan Korner.  Realism in mathematics , 1991 .

[99]  R. Goldblatt.  Lectures on the hyperreals : an introduction to nonstandard analysis , 1998 .

[100]  Frank Quinn.  A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today , 2012 .

[101]  Alexandre V. Borovik,et al.  A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos , 2016, Logica Universalis.

[102]  D. Hilbert.  Über das Unendliche , 1926 .

[103]  Edwin Hewitt,et al.  Rings of real-valued continuous functions. I , 1948 .

[104]  Markus Schweizer.  Nonstandard Asymptotic Analysis , 2016 .

[105]  Mikhail G. Katz,et al.  Differential geometry via infinitesimal displacements , 2014, J. Log. Anal..

[106]  Leif Arkeryd,et al.  Intermolecular forces of infinite range and the Boltzmann equation , 1981 .

[107]  Vladimir Kanovei,et al.  Small oscillations of the pendulum, Euler’s method, and adequality , 2016, 1604.06663.

[108]  Per Martin-Löf,et al.  Mathematics of infinity , 1988, Conference on Computer Logic.

[109]  A Nonstandard Proof of the Jordan Curve Theorem , 1996, math/9608204.

[110]  Frederik Herzberg,et al.  Radically Elementary Probability Theory , 2013 .

[111]  Joel David Hamkins,et al.  Some Second Order Set Theory , 2009, ICLA.

[112]  Edward Nelson.  Internal set theory: A new approach to nonstandard analysis , 1977 .

[113]  Paul R. Halmos,et al.  I Want to Be A Mathematician: An Automathography , 1986 .

[114]  A. Pillay.  Models of Peano Arithmetic , 1981 .

[115]  Robert Goldblatt,et al.  Lectures on the hyperreals , 1998 .

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

[117]  Joel David Hamkins,et al.  The Set-theoretic Multiverse : A Natural Context for Set Theory( Mathematical Logic and Its Applications) , 2011 .

[118]  Detlef Laugwitz.  Infinitely small quantities in Cauchy's textbooks , 1987 .

[119]  V. A. Molchanov.  The use of double nonstandard enlargements in topology , 1989 .

[120]  Robert M. Anderson,et al.  A non-standard representation for Brownian Motion and Itô integration , 1976 .

[121]  S. Albeverio.  Nonstandard Methods in Stochastic Analysis and Mathematical Physics , 1986 .

[122]  Vladimir Kanovei,et al.  Problems of set-theoretic non-standard analysis , 2007 .

[123]  Analyse Non Standard et Représentation du Réel: Deux exemples en Automatique , 2007 .

[124]  Patrick Riley,et al.  Leibniz's Philosophy of Logic and Language , 1973 .

[125]  J. E. Rubio.  Optimization and Nonstandard Analysis , 1994 .

[126]  K. Gödel,et al.  Review of Skolem's Über die Unmöglichkeit Einer Vollständigen Charakterisierung der Zahlenreihe Mittels Eines Endlichen Axiomensystems , 1990 .

[127]  Paolo Mancosu,et al.  MEASURING THE SIZE OF INFINITE COLLECTIONS OF NATURAL NUMBERS: WAS CANTOR’S THEORY OF INFINITE NUMBER INEVITABLE? , 2009, The Review of Symbolic Logic.

[128]  Abraham Robinson.  Concerning Progress In The Philosophy Of Mathematics , 1975 .

[129]  George Wilmers,et al.  Models OF Peano Arithmetic (Oxford Logic Guides 15) , 1993 .

[130]  Yevgeniy Gordon.  AN AXIOMATICS FOR NONSTANDARD SET THEORY , .

[131]  Vladimir Kanovei,et al.  Undecidable hypotheses in Edward Nelson's internal set theory , 1991 .

[132]  R. Werner,et al.  Classical mechanics as quantum mechanics with infinitesimal ħ , 1995 .