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An Very Brief Overview of Surreal Numbers for Gandalf MM 2014
An very brief overview of Surreal Numbers for Gandalf MM 2014 Steven Charlton 1 History and Introduction Surreal numbers were created by John Horton Conway (of Game of Life fame), as a greatly simplified construction of an earlier object (Alling’s ordered field associated to the class of all ordinals, as constructed via modified Hahn series). The name surreal numbers was coined by Donald Knuth (of TEX and the Art of Computer Programming fame) in his novel ‘Surreal Numbers’ [2], where the idea was first presented. Surreal numbers form an ordered Field (Field with a capital F since surreal numbers aren’t a set but a class), and are in some sense the largest possible ordered Field. All other ordered fields, rationals, reals, rational functions, Levi-Civita field, Laurent series, superreals, hyperreals, . , can be found as subfields of the surreals. The definition/construction of surreal numbers leads to a system where we can talk about and deal with infinite and infinitesimal numbers as naturally and consistently as any ‘ordinary’ number. In fact it let’s can deal with even more ‘wonderful’ expressions 1 √ 1 ∞ − 1, ∞, ∞, ,... 2 ∞ in exactly the same way1. One large area where surreal numbers (or a slight generalisation of them) finds application is in the study and analysis of combinatorial games, and game theory. Conway discusses this in detail in his book ‘On Numbers and Games’ [1]. 2 Basic Definitions All surreal numbers are constructed iteratively out of two basic definitions. This is an wonderful illustration on how a huge amount of structure can arise from very simple origins. -
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. -
Some Mathematical and Physical Remarks on Surreal Numbers
Journal of Modern Physics, 2016, 7, 2164-2176 http://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196 Some Mathematical and Physical Remarks on Surreal Numbers Juan Antonio Nieto Facultad de Ciencias Fsico-Matemáticas de la Universidad Autónoma de Sinaloa, Culiacán, México How to cite this paper: Nieto, J.A. (2016) Abstract Some Mathematical and Physical Remarks on Surreal Numbers. Journal of Modern We make a number of observations on Conway surreal number theory which may be Physics, 7, 2164-2176. useful, for further developments, in both mathematics and theoretical physics. In http://dx.doi.org/10.4236/jmp.2016.715188 particular, we argue that the concepts of surreal numbers and matroids can be linked. Received: September 23, 2016 Moreover, we established a relation between the Gonshor approach on surreal num- Accepted: November 21, 2016 bers and tensors. We also comment about the possibility to connect surreal numbers Published: November 24, 2016 with supersymmetry. In addition, we comment about possible relation between sur- real numbers and fractal theory. Finally, we argue that the surreal structure may pro- Copyright © 2016 by author and Scientific Research Publishing Inc. vide a different mathematical tool in the understanding of singularities in both high This work is licensed under the Creative energy physics and gravitation. Commons Attribution International License (CC BY 4.0). Keywords http://creativecommons.org/licenses/by/4.0/ Open Access Surreal Numbers, Supersymmetry, Cosmology 1. Introduction Surreal numbers are a fascinating subject in mathematics. Such numbers were invented, or discovered, by the mathematician John Horton Conway in the 70’s [1] [2]. -
Calculus for the Life Sciences I Lecture Notes – Limits, Continuity, and the Derivative
Limits Continuity Derivative Calculus for the Life Sciences I Lecture Notes – Limits, Continuity, and the Derivative Joseph M. Mahaffy, [email protected] Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA 92182-7720 http://www-rohan.sdsu.edu/∼jmahaffy Spring 2013 Lecture Notes – Limits, Continuity, and the Deriv Joseph M. Mahaffy, [email protected] — (1/24) Limits Continuity Derivative Outline 1 Limits Definition Examples of Limit 2 Continuity Examples of Continuity 3 Derivative Examples of a derivative Lecture Notes – Limits, Continuity, and the Deriv Joseph M. Mahaffy, [email protected] — (2/24) Limits Definition Continuity Examples of Limit Derivative Introduction Limits are central to Calculus Lecture Notes – Limits, Continuity, and the Deriv Joseph M. Mahaffy, [email protected] — (3/24) Limits Definition Continuity Examples of Limit Derivative Introduction Limits are central to Calculus Present definitions of limits, continuity, and derivative Lecture Notes – Limits, Continuity, and the Deriv Joseph M. Mahaffy, [email protected] — (3/24) Limits Definition Continuity Examples of Limit Derivative Introduction Limits are central to Calculus Present definitions of limits, continuity, and derivative Sketch the formal mathematics for these definitions Lecture Notes – Limits, Continuity, and the Deriv Joseph M. Mahaffy, [email protected] — (3/24) Limits Definition Continuity Examples of Limit Derivative Introduction Limits -
Hegel on Calculus
HISTORY OF PHILOSOPHY QUARTERLY Volume 34, Number 4, October 2017 HEGEL ON CALCULUS Ralph M. Kaufmann and Christopher Yeomans t is fair to say that Georg Wilhelm Friedrich Hegel’s philosophy of Imathematics and his interpretation of the calculus in particular have not been popular topics of conversation since the early part of the twenti- eth century. Changes in mathematics in the late nineteenth century, the new set-theoretical approach to understanding its foundations, and the rise of a sympathetic philosophical logic have all conspired to give prior philosophies of mathematics (including Hegel’s) the untimely appear- ance of naïveté. The common view was expressed by Bertrand Russell: The great [mathematicians] of the seventeenth and eighteenth cen- turies were so much impressed by the results of their new methods that they did not trouble to examine their foundations. Although their arguments were fallacious, a special Providence saw to it that their conclusions were more or less true. Hegel fastened upon the obscuri- ties in the foundations of mathematics, turned them into dialectical contradictions, and resolved them by nonsensical syntheses. .The resulting puzzles [of mathematics] were all cleared up during the nine- teenth century, not by heroic philosophical doctrines such as that of Kant or that of Hegel, but by patient attention to detail (1956, 368–69). Recently, however, interest in Hegel’s discussion of calculus has been awakened by an unlikely source: Gilles Deleuze. In particular, work by Simon Duffy and Henry Somers-Hall has demonstrated how close Deleuze and Hegel are in their treatment of the calculus as compared with most other philosophers of mathematics. -
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. -
Limits and Derivatives 2
57425_02_ch02_p089-099.qk 11/21/08 10:34 AM Page 89 FPO thomasmayerarchive.com Limits and Derivatives 2 In A Preview of Calculus (page 3) we saw how the idea of a limit underlies the various branches of calculus. Thus it is appropriate to begin our study of calculus by investigating limits and their properties. The special type of limit that is used to find tangents and velocities gives rise to the central idea in differential calcu- lus, the derivative. We see how derivatives can be interpreted as rates of change in various situations and learn how the derivative of a function gives information about the original function. 89 57425_02_ch02_p089-099.qk 11/21/08 10:35 AM Page 90 90 CHAPTER 2 LIMITS AND DERIVATIVES 2.1 The Tangent and Velocity Problems In this section we see how limits arise when we attempt to find the tangent to a curve or the velocity of an object. The Tangent Problem The word tangent is derived from the Latin word tangens, which means “touching.” Thus t a tangent to a curve is a line that touches the curve. In other words, a tangent line should have the same direction as the curve at the point of contact. How can this idea be made precise? For a circle we could simply follow Euclid and say that a tangent is a line that intersects the circle once and only once, as in Figure 1(a). For more complicated curves this defini- tion is inadequate. Figure l(b) shows two lines and tl passing through a point P on a curve (a) C. -
SURREAL TIME and ULTRATASKS HAIDAR AL-DHALIMY Department of Philosophy, University of Minnesota and CHARLES J
THE REVIEW OF SYMBOLIC LOGIC, Page 1 of 12 SURREAL TIME AND ULTRATASKS HAIDAR AL-DHALIMY Department of Philosophy, University of Minnesota and CHARLES J. GEYER School of Statistics, University of Minnesota Abstract. This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible. §1. Introduction. To perform a hypertask is to complete an uncountable sequence of tasks in a finite length of time. Each task is taken to require a nonzero interval of time, with the intervals being pairwise disjoint. Clark & Read (1984) claim to “show that no hypertask can be performed” (p. 387). What they in fact show is the impossibility of a situation in which both (i) a hypertask is performed and (ii) time has the structure of the real number system as conceived of in mainstream mathematics—or, as we will say, time is R-like. However, they make the argument overly complicated. -
Limits of Functions
Chapter 2 Limits of Functions In this chapter, we define limits of functions and describe some of their properties. 2.1. Limits We begin with the ϵ-δ definition of the limit of a function. Definition 2.1. Let f : A ! R, where A ⊂ R, and suppose that c 2 R is an accumulation point of A. Then lim f(x) = L x!c if for every ϵ > 0 there exists a δ > 0 such that 0 < jx − cj < δ and x 2 A implies that jf(x) − Lj < ϵ. We also denote limits by the `arrow' notation f(x) ! L as x ! c, and often leave it to be implicitly understood that x 2 A is restricted to the domain of f. Note that we exclude x = c, so the function need not be defined at c for the limit as x ! c to exist. Also note that it follows directly from the definition that lim f(x) = L if and only if lim jf(x) − Lj = 0: x!c x!c Example 2.2. Let A = [0; 1) n f9g and define f : A ! R by x − 9 f(x) = p : x − 3 We claim that lim f(x) = 6: x!9 To prove this, let ϵ >p 0 be given. For x 2 A, we have from the difference of two squares that f(x) = x + 3, and p x − 9 1 jf(x) − 6j = x − 3 = p ≤ jx − 9j: x + 3 3 Thus, if δ = 3ϵ, then jx − 9j < δ and x 2 A implies that jf(x) − 6j < ϵ. -
~Umbers the BOO K O F Umbers
TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex planation of the multitudinous ways in which the word "number" is used. -
The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 Email: [email protected]
The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 email: [email protected] In the last issue of SIGACT NEWS (Volume 45, No. 3) there was a review of Martin Davis's book The Universal Computer. The road from Leibniz to Turing that was badly typeset. This was my fault. We reprint the review, with correct typesetting, in this column. In this column we review the following books. 1. The Universal Computer. The road from Leibniz to Turing by Martin Davis. Reviewed by Haim Kilov. This book has stories of the personal, social, and professional lives of Leibniz, Boole, Frege, Cantor, G¨odel,and Turing, and explains some of the essentials of their thought. The mathematics is accessible to a non-expert, although some mathematical maturity, as opposed to specific knowledge, certainly helps. 2. From Zero to Infinity by Constance Reid. Review by John Tucker Bane. This is a classic book on fairly elementary math that was reprinted in 2006. The author tells us interesting math and history about the numbers 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; e; @0. 3. The LLL Algorithm Edited by Phong Q. Nguyen and Brigitte Vall´ee. Review by Krishnan Narayanan. The LLL algorithm has had applications to both pure math and very applied math. This collection of articles on it highlights both. 4. Classic Papers in Combinatorics Edited by Ira Gessel and Gian-Carlo Rota. Re- view by Arya Mazumdar. This book collects together many of the classic papers in combinatorics, including those of Ramsey and Hales-Jewitt. -
Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot Game
Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot Game Will Johnson June 9, 2011 Contents 1 To Knot or Not to Knot 4 1.1 Some facts from Knot Theory . 7 1.2 Sums of Knots . 18 I Combinatorial Game Theory 23 2 Introduction 24 2.1 Combinatorial Game Theory in general . 24 2.1.1 Bibliography . 27 2.2 Additive CGT specifically . 28 2.3 Counting moves in Hackenbush . 34 3 Games 40 3.1 Nonstandard Definitions . 40 3.2 The conventional formalism . 45 3.3 Relations on Games . 54 3.4 Simplifying Games . 62 3.5 Some Examples . 66 4 Surreal Numbers 68 4.1 Surreal Numbers . 68 4.2 Short Surreal Numbers . 70 4.3 Numbers and Hackenbush . 77 4.4 The interplay of numbers and non-numbers . 79 4.5 Mean Value . 83 1 5 Games near 0 85 5.1 Infinitesimal and all-small games . 85 5.2 Nimbers and Sprague-Grundy Theory . 90 6 Norton Multiplication and Overheating 97 6.1 Even, Odd, and Well-Tempered Games . 97 6.2 Norton Multiplication . 105 6.3 Even and Odd revisited . 114 7 Bending the Rules 119 7.1 Adapting the theory . 119 7.2 Dots-and-Boxes . 121 7.3 Go . 132 7.4 Changing the theory . 139 7.5 Highlights from Winning Ways Part 2 . 144 7.5.1 Unions of partizan games . 144 7.5.2 Loopy games . 145 7.5.3 Mis`eregames . 146 7.6 Mis`ereIndistinguishability Quotients . 147 7.7 Indistinguishability in General . 148 II Well-tempered Scoring Games 155 8 Introduction 156 8.1 Boolean games .