Infinitesimal Time Scale Calculus
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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 . -
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. -
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. -
From Pythagoreans and Weierstrassians to True Infinitesimal Calculus
CORE Metadata, citation and similar papers at core.ac.uk Provided by Keck Graduate Institute Journal of Humanistic Mathematics Volume 7 | Issue 1 January 2017 From Pythagoreans and Weierstrassians to True Infinitesimal Calculus Mikhail Katz Bar-Ilan University Luie Polev Bar Ilan University Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Analysis Commons, and the Logic and Foundations Commons Recommended Citation Katz, M. and Polev, L. "From Pythagoreans and Weierstrassians to True Infinitesimal Calculus," Journal of Humanistic Mathematics, Volume 7 Issue 1 (January 2017), pages 87-104. DOI: 10.5642/ jhummath.201701.07 . Available at: https://scholarship.claremont.edu/jhm/vol7/iss1/7 ©2017 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. From Pythagoreans and Weierstrassians to True Infinitesimal Calculus Cover Page Footnote Israel Science Foundation grant 1517/12 (acknowledged also in the body of the paper). This article is available in Journal of Humanistic Mathematics: https://scholarship.claremont.edu/jhm/vol7/iss1/7 From Pythagoreans and Weierstrassians to True Infinitesimal Calculus Mikhail G. -
Arxiv:1404.5658V1 [Math.LO] 22 Apr 2014 03F55
TOWARD A CLARITY OF THE EXTREME VALUE THEOREM KARIN U. KATZ, MIKHAIL G. KATZ, AND TARAS KUDRYK Abstract. We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches to the extreme value theorem. We argue that a given pre-mathematical phenomenon may have several aspects that are not necessarily captured by a single formalisation, pointing to a complementar- ity rather than a rivalry of the approaches. Contents 1. Introduction 2 1.1. A historical re-appraisal 2 1.2. Practice and ontology 3 2. Grades of clarity according to Peirce 4 3. Perceptual continuity 5 4. Constructive clarity 7 5. Counterexample to the existence of a maximum 9 6. Reuniting the antipodes 10 7. Kronecker and constructivism 11 8. Infinitesimal clarity 13 8.1. Nominalistic reconstructions 13 8.2. Klein on rivalry of continua 14 arXiv:1404.5658v1 [math.LO] 22 Apr 2014 8.3. Formalizing Leibniz 15 8.4. Ultrapower 16 9. Hyperreal extreme value theorem 16 10. Approaches and invitations 18 11. Conclusion 19 References 19 2000 Mathematics Subject Classification. Primary 26E35; 00A30, 01A85, 03F55. Key words and phrases. Benacerraf, Bishop, Cauchy, constructive analysis, con- tinuity, extreme value theorem, grades of clarity, hyperreal, infinitesimal, Kaestner, Kronecker, law of excluded middle, ontology, Peirce, principle of unique choice, procedure, trichotomy, uniqueness paradigm. 1 2 KARINU. KATZ, MIKHAILG. KATZ, ANDTARASKUDRYK 1. Introduction German physicist G. Lichtenberg (1742 - 1799) was born less than a decade after the publication of the cleric George Berkeley’s tract The Analyst, and would have certainly been influenced by it or at least aware of it. -
The Hyperreals
THE HYPERREALS LARRY SUSANKA Abstract. In this article we define the hyperreal numbers, an ordered field containing the real numbers as well as infinitesimal numbers. These infinites- imals have magnitude smaller than that of any nonzero real number and have intuitively appealing properties, harkening back to the thoughts of the inven- tors of analysis. We use the ultrafilter construction of the hyperreal numbers which employs common properties of sets, rather than the original approach (see A. Robinson Non-Standard Analysis [5]) which used model theory. A few of the properties of the hyperreals are explored and proofs of some results from real topology and calculus are created using hyperreal arithmetic in place of the standard limit techniques. Contents The Hyperreal Numbers 1. Historical Remarks and Overview 2 2. The Construction 3 3. Vocabulary 6 4. A Collection of Exercises 7 5. Transfer 10 6. The Rearrangement and Hypertail Lemmas 14 Applications 7. Open, Closed and Boundary For Subsets of R 15 8. The Hyperreal Approach to Real Convergent Sequences 16 9. Series 18 10. More on Limits 20 11. Continuity and Uniform Continuity 22 12. Derivatives 26 13. Results Related to the Mean Value Theorem 28 14. Riemann Integral Preliminaries 32 15. The Infinitesimal Approach to Integration 36 16. An Example of Euler, Revisited 37 References 40 Index 41 Date: June 27, 2018. 1 2 LARRY SUSANKA 1. Historical Remarks and Overview The historical Euclid-derived conception of a line was as an object possessing \the quality of length without breadth" and which satisfies the various axioms of Euclid's geometric structure. -
Hyperreal Calculus MAT2000 ––Project in Mathematics
Hyperreal Calculus MAT2000 ––Project in Mathematics Arne Tobias Malkenes Ødegaard Supervisor: Nikolai Bjørnestøl Hansen Abstract This project deals with doing calculus not by using epsilons and deltas, but by using a number system called the hyperreal numbers. The hyperreal numbers is an extension of the normal real numbers with both infinitely small and infinitely large numbers added. We will first show how this system can be created, and then show some basic properties of the hyperreal numbers. Then we will show how one can treat the topics of convergence, continuity, limits and differentiation in this system and we will show that the two approaches give rise to the same definitions and results. Contents 1 Construction of the hyperreal numbers 3 1.1 Intuitive construction . .3 1.2 Ultrafilters . .3 1.3 Formal construction . .4 1.4 Infinitely small and large numbers . .5 1.5 Enlarging sets . .5 1.6 Extending functions . .6 2 The transfer principle 6 2.1 Stating the transfer principle . .6 2.2 Using the transfer principle . .7 3 Properties of the hyperreals 8 3.1 Terminology and notation . .8 3.2 Arithmetic of hyperreals . .9 3.3 Halos . .9 3.4 Shadows . 10 4 Convergence 11 4.1 Convergence in hyperreal calculus . 11 4.2 Monotone convergence . 12 5 Continuity 13 5.1 Continuity in hyperreal calculus . 13 5.2 Examples . 14 5.3 Theorems about continuity . 15 5.4 Uniform continuity . 16 6 Limits and derivatives 17 6.1 Limits in hyperreal calculus . 17 6.2 Differentiation in hyperreal calculus . 18 6.3 Examples . 18 6.4 Increments . -
Arxiv:1204.2193V2 [Math.GM] 13 Jun 2012
Alternative Mathematics without Actual Infinity ∗ Toru Tsujishita 2012.6.12 Abstract An alternative mathematics based on qualitative plurality of finite- ness is developed to make non-standard mathematics independent of infinite set theory. The vague concept \accessibility" is used coherently within finite set theory whose separation axiom is restricted to defi- nite objective conditions. The weak equivalence relations are defined as binary relations with sorites phenomena. Continua are collection with weak equivalence relations called indistinguishability. The points of continua are the proper classes of mutually indistinguishable ele- ments and have identities with sorites paradox. Four continua formed by huge binary words are examined as a new type of continua. Ascoli- Arzela type theorem is given as an example indicating the feasibility of treating function spaces. The real numbers are defined to be points on linear continuum and have indefiniteness. Exponentiation is introduced by the Euler style and basic properties are established. Basic calculus is developed and the differentiability is captured by the behavior on a point. Main tools of Lebesgue measure theory is obtained in a similar way as Loeb measure. Differences from the current mathematics are examined, such as the indefiniteness of natural numbers, qualitative plurality of finiteness, mathematical usage of vague concepts, the continuum as a primary inexhaustible entity and the hitherto disregarded aspect of \internal measurement" in mathematics. arXiv:1204.2193v2 [math.GM] 13 Jun 2012 ∗Thanks to Ritsumeikan University for the sabbathical leave which allowed the author to concentrate on doing research on this theme. 1 2 Contents Abstract 1 Contents 2 0 Introdution 6 0.1 Nonstandard Approach as a Genuine Alternative .