Authors: Mark C Marson
To gain true understanding of a subject it can help to study its origins and how its theory and practice changed over the years – and the mathematical field of calculus is no exception. But calculus students who do read accounts of its history encounter something strange – the claim that the theory which underpinned the subject for long after its creation was wrong and that it was corrected several hundred years later, in spite of the fact that the original theory never produced erroneous results. I argue here that both this characterization of the original theory and this interpretation of the paradigm shift to its successor are false. Infinitesimals, used properly, were never unrigorous and the supposed rigor of limit theory does not imply greater correctness, but rather the (usually unnecessary) exposition of hidden deductive steps. Furthermore those steps can, if set out, constitute a proof that original infinitesimals work in accordance with limit theory – contrary to the common opinion that the two approaches represent irreconcilable philosophical positions. This proof, demonstrating that we can adopt a unified paradigm for calculus, is to my knowledge novel although its logic may have been employed in another context. I also claim that non-standard analysis (the most famous previous attempt at unification) only partially clarified the situation because the type of infinitesimals it uses are critically different from original infinitesimals.
Comments: 18 Pages.
[v1] 2019-01-10 21:01:16
Unique-IP document downloads: 58 times
Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.
Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.