True Infinitesimal Differential Geometry Mikhail G. Katz∗ Vladimir
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An Introduction to Nonstandard Analysis 11
AN INTRODUCTION TO NONSTANDARD ANALYSIS ISAAC DAVIS Abstract. In this paper we give an introduction to nonstandard analysis, starting with an ultrapower construction of the hyperreals. We then demon- strate how theorems in standard analysis \transfer over" to nonstandard anal- ysis, and how theorems in standard analysis can be proven using theorems in nonstandard analysis. 1. Introduction For many centuries, early mathematicians and physicists would solve problems by considering infinitesimally small pieces of a shape, or movement along a path by an infinitesimal amount. Archimedes derived the formula for the area of a circle by thinking of a circle as a polygon with infinitely many infinitesimal sides [1]. In particular, the construction of calculus was first motivated by this intuitive notion of infinitesimal change. G.W. Leibniz's derivation of calculus made extensive use of “infinitesimal” numbers, which were both nonzero but small enough to add to any real number without changing it noticeably. Although intuitively clear, infinitesi- mals were ultimately rejected as mathematically unsound, and were replaced with the common -δ method of computing limits and derivatives. However, in 1960 Abraham Robinson developed nonstandard analysis, in which the reals are rigor- ously extended to include infinitesimal numbers and infinite numbers; this new extended field is called the field of hyperreal numbers. The goal was to create a system of analysis that was more intuitively appealing than standard analysis but without losing any of the rigor of standard analysis. In this paper, we will explore the construction and various uses of nonstandard analysis. In section 2 we will introduce the notion of an ultrafilter, which will allow us to do a typical ultrapower construction of the hyperreal numbers. -
Actual Infinitesimals in Leibniz's Early Thought
Actual Infinitesimals in Leibniz’s Early Thought By RICHARD T. W. ARTHUR (HAMILTON, ONTARIO) Abstract Before establishing his mature interpretation of infinitesimals as fictions, Gottfried Leibniz had advocated their existence as actually existing entities in the continuum. In this paper I trace the development of these early attempts, distinguishing three distinct phases in his interpretation of infinitesimals prior to his adopting a fictionalist interpretation: (i) (1669) the continuum consists of assignable points separated by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, with no gaps between them; (iii) (1672- 75) a continuous line is composed not of points but of infinitely many infinitesimal lines, each of which is divisible and proportional to a generating motion at an instant (conatus). In 1676, finally, Leibniz ceased to regard infinitesimals as actual, opting instead for an interpretation of them as fictitious entities which may be used as compendia loquendi to abbreviate mathematical reasonings. Introduction Gottfried Leibniz’s views on the status of infinitesimals are very subtle, and have led commentators to a variety of different interpretations. There is no proper common consensus, although the following may serve as a summary of received opinion: Leibniz developed the infinitesimal calculus in 1675-76, but during the ensuing twenty years was content to refine its techniques and explore the richness of its applications in co-operation with Johann and Jakob Bernoulli, Pierre Varignon, de l’Hospital and others, without worrying about the ontic status of infinitesimals. Only after the criticisms of Bernard Nieuwentijt and Michel Rolle did he turn himself to the question of the foundations of the calculus and 2 Richard T. -
Cauchy, Infinitesimals and Ghosts of Departed Quantifiers 3
CAUCHY, INFINITESIMALS AND GHOSTS OF DEPARTED QUANTIFIERS JACQUES BAIR, PIOTR BLASZCZYK, ROBERT ELY, VALERIE´ HENRY, VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G. KATZ, TARAS KUDRYK, SEMEN S. KUTATELADZE, THOMAS MCGAFFEY, THOMAS MORMANN, DAVID M. SCHAPS, AND DAVID SHERRY Abstract. Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson’s frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz’s distinction be- tween assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robin- son’s framework, while Leibniz’s law of homogeneity with the im- plied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide paral- lel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz’s infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Eu- ler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infi- nite exponent. Such procedures have immediate hyperfinite ana- logues in Robinson’s framework, while in a Weierstrassian frame- work they can only be reinterpreted by means of paraphrases de- parting significantly from Euler’s own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson’s framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of de- arXiv:1712.00226v1 [math.HO] 1 Dec 2017 parted quantifiers in his work. -
Do We Need Number Theory? II
Do we need Number Theory? II Václav Snášel What is a number? MCDXIX |||||||||||||||||||||| 664554 0xABCD 01717 010101010111011100001 푖푖 1 1 1 1 1 + + + + + ⋯ 1! 2! 3! 4! VŠB-TUO, Ostrava 2014 2 References • Z. I. Borevich, I. R. Shafarevich, Number theory, Academic Press, 1986 • John Vince, Quaternions for Computer Graphics, Springer 2011 • John C. Baez, The Octonions, Bulletin of the American Mathematical Society 2002, 39 (2): 145–205 • C.C. Chang and H.J. Keisler, Model Theory, North-Holland, 1990 • Mathématiques & Arts, Catalogue 2013, Exposants ESMA • А.Т.Фоменко, Математика и Миф Сквозь Призму Геометрии, http://dfgm.math.msu.su/files/fomenko/myth-sec6.php VŠB-TUO, Ostrava 2014 3 Images VŠB-TUO, Ostrava 2014 4 Number construction VŠB-TUO, Ostrava 2014 5 Algebraic number An algebraic number field is a finite extension of ℚ; an algebraic number is an element of an algebraic number field. Ring of integers. Let 퐾 be an algebraic number field. Because 퐾 is of finite degree over ℚ, every element α of 퐾 is a root of monic polynomial 푛 푛−1 푓 푥 = 푥 + 푎1푥 + … + 푎1 푎푖 ∈ ℚ If α is root of polynomial with integer coefficient, then α is called an algebraic integer of 퐾. VŠB-TUO, Ostrava 2014 6 Algebraic number Consider more generally an integral domain 퐴. An element a ∈ 퐴 is said to be a unit if it has an inverse in 퐴; we write 퐴∗ for the multiplicative group of units in 퐴. An element 푝 of an integral domain 퐴 is said to be irreducible if it is neither zero nor a unit, and can’t be written as a product of two nonunits. -
Fermat, Leibniz, Euler, and the Gang: the True History of the Concepts Of
FERMAT, LEIBNIZ, EULER, AND THE GANG: THE TRUE HISTORY OF THE CONCEPTS OF LIMIT AND SHADOW TIZIANA BASCELLI, EMANUELE BOTTAZZI, FREDERIK HERZBERG, VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G. KATZ, TAHL NOWIK, DAVID SHERRY, AND STEVEN SHNIDER Abstract. Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding “negligible” terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern the- ories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions. Contents 1. Introduction 2 2. Methodological remarks 4 2.1. A-track and B-track 5 2.2. Formal epistemology: Easwaran on hyperreals 6 2.3. Zermelo–Fraenkel axioms and the Feferman–Levy model 8 2.4. Skolem integers and Robinson integers 9 2.5. Williamson, complexity, and other arguments 10 2.6. Infinity and infinitesimal: let both pretty severely alone 13 3. Fermat’s adequality 13 3.1. Summary of Fermat’s algorithm 14 arXiv:1407.0233v1 [math.HO] 1 Jul 2014 3.2. Tangent line and convexity of parabola 15 3.3. Fermat, Galileo, and Wallis 17 4. Leibniz’s Transcendental law of homogeneity 18 4.1. When are quantities equal? 19 4.2. Product rule 20 5. Euler’s Principle of Cancellation 20 6. What did Cauchy mean by “limit”? 22 6.1. Cauchy on Leibniz 23 6.2. Cauchy on continuity 23 7. Modern formalisations: a case study 25 8. A combinatorial approach to decreasing rearrangements 26 9. -
0.999… = 1 an Infinitesimal Explanation Bryan Dawson
0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder . -
Understanding Student Use of Differentials in Physics Integration Problems
PHYSICAL REVIEW SPECIAL TOPICS - PHYSICS EDUCATION RESEARCH 9, 020108 (2013) Understanding student use of differentials in physics integration problems Dehui Hu and N. Sanjay Rebello* Department of Physics, 116 Cardwell Hall, Kansas State University, Manhattan, Kansas 66506-2601, USA (Received 22 January 2013; published 26 July 2013) This study focuses on students’ use of the mathematical concept of differentials in physics problem solving. For instance, in electrostatics, students need to set up an integral to find the electric field due to a charged bar, an activity that involves the application of mathematical differentials (e.g., dr, dq). In this paper we aim to explore students’ reasoning about the differential concept in physics problems. We conducted group teaching or learning interviews with 13 engineering students enrolled in a second- semester calculus-based physics course. We amalgamated two frameworks—the resources framework and the conceptual metaphor framework—to analyze students’ reasoning about differential concept. Categorizing the mathematical resources involved in students’ mathematical thinking in physics provides us deeper insights into how students use mathematics in physics. Identifying the conceptual metaphors in students’ discourse illustrates the role of concrete experiential notions in students’ construction of mathematical reasoning. These two frameworks serve different purposes, and we illustrate how they can be pieced together to provide a better understanding of students’ mathematical thinking in physics. DOI: 10.1103/PhysRevSTPER.9.020108 PACS numbers: 01.40.Àd mathematical symbols [3]. For instance, mathematicians I. INTRODUCTION often use x; y asR variables and the integrals are often written Mathematical integration is widely used in many intro- in the form fðxÞdx, whereas in physics the variables ductory and upper-division physics courses. -
Leonhard Euler: His Life, the Man, and His Works∗
SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St. -
Do Simple Infinitesimal Parts Solve Zeno's Paradox of Measure?
Do Simple Infinitesimal Parts Solve Zeno's Paradox of Measure?∗ Lu Chen (Forthcoming in Synthese) Abstract In this paper, I develop an original view of the structure of space|called infinitesimal atomism|as a reply to Zeno's paradox of measure. According to this view, space is composed of ultimate parts with infinitesimal size, where infinitesimals are understood within the framework of Robinson's (1966) nonstandard analysis. Notably, this view satisfies a version of additivity: for every region that has a size, its size is the sum of the sizes of its disjoint parts. In particular, the size of a finite region is the sum of the sizes of its infinitesimal parts. Although this view is a coherent approach to Zeno's paradox and is preferable to Skyrms's (1983) infinitesimal approach, it faces both the main problem for the standard view (the problem of unmeasurable regions) and the main problem for finite atomism (Weyl's tile argument), leaving it with no clear advantage over these familiar alternatives. Keywords: continuum; Zeno's paradox of measure; infinitesimals; unmeasurable ∗Special thanks to Jeffrey Russell and Phillip Bricker for their extensive feedback on early drafts of the paper. Many thanks to the referees of Synthese for their very helpful comments. I'd also like to thank the participants of Umass dissertation seminar for their useful feedback on my first draft. 1 regions; Weyl's tile argument 1 Zeno's Paradox of Measure A continuum, such as the region of space you occupy, is commonly taken to be indefinitely divisible. But this view runs into Zeno's famous paradox of measure. -
Connes on the Role of Hyperreals in Mathematics
Found Sci DOI 10.1007/s10699-012-9316-5 Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics Vladimir Kanovei · Mikhail G. Katz · Thomas Mormann © Springer Science+Business Media Dordrecht 2012 Abstract We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in func- tional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being “vir- tual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnera- ble to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace − (featured on the front cover of Connes’ magnum opus) V. -
Arxiv:1405.0984V4 [Math.DG] 23 Dec 2015 Ai N Vryte Pwt Lsia Nltcntos Ntepresen D the finite in by Defined Notions
DIFFERENTIAL GEOMETRY VIA INFINITESIMAL DISPLACEMENTS TAHL NOWIK AND MIKHAIL G. KATZ Abstract. We present a new formulation of some basic differential geometric notions on a smooth manifold M, in the setting of nonstandard analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M, we define a prevector field, which is an internal map from ∗M to itself, implementing the intuitive notion of vectors as infinitesimal displacements. We introduce regularity conditions for prevector fields, defined by finite dif- ferences, thus purely combinatorial conditions involving no analysis. These conditions replace the more elaborate analytic regularity conditions appearing in previous similar approaches, e.g. by Stroyan and Luxemburg or Lutz and Goze. We define the flow of a prevector field by hyperfinite iteration of the given prevector field, in the spirit of Euler’s method. We define the Lie bracket of two prevector fields by appropriate iteration of their commutator. We study the properties of flows and Lie brackets, particularly in relation with our proposed regularity conditions. We present several simple applications to the classical setting, such as bounds re- lated to the flow of vector fields, analysis of small oscillations of a pendulum, and an instance of Frobenius’ Theorem regarding the complete integrability of independent vector fields. 1. Introduction We develop foundations for differential geometry on smooth manifolds, based on infinites- imals, where vectors and vector fields are represented by infinitesimal displacements in the arXiv:1405.0984v4 [math.DG] 23 Dec 2015 manifold itself, as they were thought of historically. Such an approach was previously intro- duced e.g. -
Nonstandard Analysis (Math 649K) Spring 2008
Nonstandard Analysis (Math 649k) Spring 2008 David Ross, Department of Mathematics February 29, 2008 1 1 Introduction 1.1 Outline of course: 1. Introduction: motivation, history, and propoganda 2. Nonstandard models: definition, properties, some unavoidable logic 3. Very basic Calculus/Analysis 4. Applications of saturation 5. General Topology 6. Measure Theory 7. Functional Analysis 8. Probability 1.2 Some References: 1. Abraham Robinson (1966) Nonstandard Analysis North Holland, Amster- dam 2. Martin Davis and Reuben Hersh Nonstandard Analysis Scientific Ameri- can, June 1972 3. Davis, M. (1977) Applied Nonstandard Analysis Wiley, New York. 4. Sergio Albeverio, Jens Erik Fenstad, Raphael Høegh-Krohn, and Tom Lindstrøm (1986) Nonstandard Methods in Stochastic Analysis and Math- ematical Physics. Academic Press, New York. 5. Al Hurd and Peter Loeb (1985) An introduction to Nonstandard Real Anal- ysis Academic Press, New York. 6. Keith Stroyan and Wilhelminus Luxemburg (1976) Introduction to the Theory of Infinitesimals Academic Press, New York. 7. Keith Stroyan and Jose Bayod (1986) Foundations of Infinitesimal Stochas- tic Analysis North Holland, Amsterdam 8. Leif Arkeryd, Nigel Cutland, C. Ward Henson (eds) (1997) Nonstandard Analysis: Theory and Applications, Kluwer 9. Rob Goldblatt (1998) Lectures on the Hyperreals, Springer 2 1.3 Some history: • (xxxx) Archimedes • (1615) Kepler, Nova stereometria dolorium vinariorium • (1635) Cavalieri, Geometria indivisibilus • (1635) Excercitationes geometricae (”Rigor is the affair of philosophy rather