Toward a History of Mathematics Focused on Procedures

Piotr Błaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, David M Sherry

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework.

Original languageEnglish (US)
Pages (from-to)1-21
Number of pages21
JournalFoundations of Science
DOIs
StateAccepted/In press - Sep 21 2016

Fingerprint

History of Mathematics
Elaboration
Gottfried Wilhelm Leibniz
Problem Solving
Pioneers
Accomplishment
Evaluation
Formalization
Fermat
Leonhard Euler
Calculi

ASJC Scopus subject areas

  • General
  • History and Philosophy of Science

Cite this

Błaszczyk, P., Kanovei, V., Katz, K. U., Katz, M. G., Kutateladze, S. S., & Sherry, D. M. (Accepted/In press). Toward a History of Mathematics Focused on Procedures. Foundations of Science, 1-21. https://doi.org/10.1007/s10699-016-9498-3

Toward a History of Mathematics Focused on Procedures. / Błaszczyk, Piotr; Kanovei, Vladimir; Katz, Karin U.; Katz, Mikhail G.; Kutateladze, Semen S.; Sherry, David M.

In: Foundations of Science, 21.09.2016, p. 1-21.

Research output: Contribution to journalArticle

Błaszczyk, Piotr ; Kanovei, Vladimir ; Katz, Karin U. ; Katz, Mikhail G. ; Kutateladze, Semen S. ; Sherry, David M. / Toward a History of Mathematics Focused on Procedures. In: Foundations of Science. 2016 ; pp. 1-21.
@article{84a82dff52ed4cb7b17a7fe210790122,
title = "Toward a History of Mathematics Focused on Procedures",
abstract = "Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework.",
author = "Piotr Błaszczyk and Vladimir Kanovei and Katz, {Karin U.} and Katz, {Mikhail G.} and Kutateladze, {Semen S.} and Sherry, {David M}",
year = "2016",
month = "9",
day = "21",
doi = "10.1007/s10699-016-9498-3",
language = "English (US)",
pages = "1--21",
journal = "Foundations of Science",
issn = "1233-1821",
publisher = "Springer Netherlands",

}

TY - JOUR

T1 - Toward a History of Mathematics Focused on Procedures

AU - Błaszczyk, Piotr

AU - Kanovei, Vladimir

AU - Katz, Karin U.

AU - Katz, Mikhail G.

AU - Kutateladze, Semen S.

AU - Sherry, David M

PY - 2016/9/21

Y1 - 2016/9/21

N2 - Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework.

AB - Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework.

UR - http://www.scopus.com/inward/record.url?scp=84988662718&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988662718&partnerID=8YFLogxK

U2 - 10.1007/s10699-016-9498-3

DO - 10.1007/s10699-016-9498-3

M3 - Article

SP - 1

EP - 21

JO - Foundations of Science

JF - Foundations of Science

SN - 1233-1821

ER -