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). I mention the ancient antecedents of calculus (the Methods of Exhaustion and of Mechanical Theorems) in context. 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 argue that these schools are equivalent in effect but that the former is more convenient. These are also the infinitesimals of smooth infinitesimal analysis (SIA) which I contrast with the more widely known discipline of non-standard analysis (NSA). The infinitesimals considered are nilpotent – a property uncontroversially possessed by infinitesimals before the 20th century. This contrasts with the prevalent opinion that the two methodologies are incompatible.
I give a novel proof (the nilsquare-limit theorem) that these concepts are two aspects of the same thing – indefinite precision. This has been taken to be either infinitesimals or limits at different times in history. Submission statement: Next to Nothing - a Single Paradigm, Abstract I here tackle the most enduring controversy in mathematics, namely the question of what is the correct foundation for calculus.
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