Functions and Analysis

   

Next to Nothing - a Single Paradigm

Authors: Mark C Marson

I here tackle the most enduring controversy in mathematics, namely the question of what is the correct foundation for calculus. This has been taken to be either infinitesimals or limits at different times in history. I here give a novel proof (the nilsquare-limit theorem) that these concepts are two aspects of the same thing – indefinite precision. This contrasts with the prevalent opinion that the two methodologies are incompatible. The infinitesimals considered are nilpotent – a property uncontroversially possessed by infinitesimals before the 20th century. These are also the infinitesimals of smooth infinitesimal analysis (SIA) which I contrast with the more widely known discipline of non-standard analysis (NSA). I argue that these schools are equivalent in effect but that the former is more convenient. I give a graphical demonstration of the proof and a corollary which explicates the old idea of ‘degrees of smallness’; and then use the new perspective offered by the proof to reinterpret the history of calculus, placing particular emphasis on Leibniz’s efforts to justify his notation for calculus and Lagrange’s later efforts to do the same. I mention the ancient antecedents of calculus (the Methods of Exhaustion and of Mechanical Theorems) in context. I then cover the crisis of foundations in mathematics in the late 19th and early 20th centuries with emphasis on the role (or lack thereof) of Cauchy, pathological functions, and the philosophies of Cantor and formalism (also mentioning their antitheses – namely intuitionism and constructivism). In conclusion I clarify the role of series in calculus, discuss how to reconcile the new paradigm with the law of excluded middle (LEM), and explain the close connection between this approach and finite difference calculus (FDC). In the Continuation I respond to a criticism of Version 1 by explaining how ‘microlinearity’ originally justified calculus, I elaborate on an ‘increment free’ approach to calculus pioneered by Carathéodory, and I address a related issue – namely how the absence of a properly algebraic approach to calculus for most of the 20th century led to widespread confusion about how calculus actually works i.e. I explain Leibniz’s higher derivative notation and discuss a mistaken attempt to reformulate it. I then give an example of the use of differentials in ratios with an illustration. I finally conclude by appealing that the philosophical rift between most mathematicians and the rest of science be remedied. Other topics covered include: the attempts of the formalists to free mathematics from contradiction while also admitting the Axiom of Choice (ref 23); the need for FDC together with a simple numerical example (refs 24 to 26); and, one of the consequences of the period of hegemony enjoyed by formalism – namely the independent rediscovery of various aspects of its antithetical philosophies by various researchers (ref 37).

Comments: 39 Pages. Original published - 11 January 2019

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Submission history

[v1] 2019-01-10 21:01:16
[v2] 2021-12-04 23:31:30

Unique-IP document downloads: 179 times

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