Zooming in on infinitesimal 1–.9.. in a post-triumvirate era

@article{Katz2010ZoomingIO,
  title={Zooming in on infinitesimal 1–.9.. in a post-triumvirate era},
  author={Karin U. Katz and Mikhail G. Katz},
  journal={Educational Studies in Mathematics},
  year={2010},
  volume={74},
  pages={259-273}
}
The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the key ideas about infinitesimals via a proceptual analysis of the meaning of the ellipsis “...” in the real formula $\hbox{.999\ldots = 1}$. Infinitesimal-enriched number systems accommodate quantities in the half-open interval [0,1) whose extended decimal… 

THE TENSION BETWEEN INTUITIVE INFINITESIMALS AND FORMAL MATHEMATICAL ANALYSIS

Abstract: we discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating

Meaning in Classical Mathematics: Is it at Odds with Intuitionism?

We examine the classical/intuitionist divide, and how it reflects on modern theories of infinitesimals. When leading intuitionist Heyting announced that "the creation of non-standard analysis is a

Namur Is mathematical history written by the victors ?

We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leibniz, Euler, Cauchy

Ju n 20 13 IS MATHEMATICAL HISTORY WRITTEN BY THE VICTORS ?

We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leibniz, Euler, Cauchy

Ten Misconceptions from the History of Analysis and Their Debunking

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number

Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s

Hyperreals and Their Applications

NSA can be introduced in multiple ways. Instead of choosing one option, these notes include three introductions. Section 1 is best-suited for those who are familiar with logic, or who want to get a

Almost Equal: the Method of Adequality from Diophantus to Fermat and Beyond

We analyze some of the main approaches in the literature to the method of ‘adequality’ with which Fermat approached the problems of the calculus, as well as its source in the παρισότης of Diophantus,

When is .999... less than 1?

We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is "an infinite number of 9s" merely a figure of speech? How are such alternative interpretations

Proofs and Retributions, Or: Why Sarah Can’t Take Limits

The small, the tiny, and the infinitesimal (to quote Paramedic) have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that

References

SHOWING 1-10 OF 95 REFERENCES

Elementary Axioms and Pictures for Infinitesimal Calculus

is to conceive it as a system of real numbers given by decimal expansions or, more formally, as a complete ordered field. But it was not always so; prior to the formalisation of the real number

INFINITESIMALS.

. Leibniz entertained various conceptions of infinitesimals, considering them sometimes as ideal things and other times as fictions. But in both cases, he compares infinitesimals favorably to

Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes

Among his achievements in all areas of learning, Leibniz's contributions to the development of European mathematics stand out as especially influential. His idiosyncratic metaphysics may have won few

The Concepts of the Calculus

IN the historical survey of mathematics, as of any branch of science, it is customary to ascribe a great discovery to one or more individuals. Without, however, seeming in any way to detract from the

Nonstandard analysis and the history of classical analysis

The history of classical analysis seems to be the most thoroughly investigated part of the history of mathematics. This is not surprising. Mathematical analysis was considered to be "the simplest and

Nonstandard Student Conceptions About Infinitesimals

This is a case study of an undergraduate calculus student’s nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include

Non-Standard Analysis

1. As early as 1934 it was pointed out by Thoralf Skolem (see [17]) that there exist proper extensions of the natural number system which have, in some sense, ‘the same properties’ as the natural

When is .999... less than 1?

We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is "an infinite number of 9s" merely a figure of speech? How are such alternative interpretations

Concerning Progress In The Philosophy Of Mathematics

Lectures on the hyperreals : an introduction to nonstandard analysis

I Foundations.- 1 What Are the Hyperreals?.- 1.1 Infinitely Small and Large.- 1.2 Historical Background.- 1.3 What Is a Real Number?.- 1.4 Historical References.- 2 Large Sets.- 2.1 Infinitesimals as
...