A primer of infinitesimal analysis
by John L. Bell · first published 1998
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…themselves aren't reducible to 0. All of this can be done in a constructive manner avoiding any use of infinity or classical logic that depends upon indirect proofs (ie excluded middle). John Bell's 'Primer of Infinitesimal Analysis' has good details. The computational techniques from Automatic Differentiation use these types of entities to calculate derivatives exactly without approximating infinite (limiting) processes. Calculus…
on “How real are real numbers? (2004)” ↗You might be interested in the book A Primer of Infinitesimal Analysis by John L. Bell, which is another approach to infinitesimals that uses intuitionistic logic. https://www.amazon.com/Primer-Infinitesimal-Analysis-John-Be... Any interesting property of this logic and…
on “Calculus Made Easy (1914) [pdf]” ↗…in your mathematical arguments (which I admit is a barrier, but it also buys you some things). And what it offers is much simpler proofs that support intuitive reasoning. There is also this book, "A Primer of Infinitesimal Analysis" [0], which develops a big chunk of calculus and classical mechanics using only infinitesimals, and is fully rigorous. [0] https://www.cambridge.org/br/universitypress/subjects/mathem...
on “Calculus Made Easy” ↗Ok so I want to link A Primer of Infinitesimal Analysis https://www.cambridge.org/core/books/primer-of-infinitesimal... this book is so so so good. It begins with nothing and eventually builds enough for Newton-style classical mechanics, made entirely of geometric…
on “Knuckledragger, a Semi-Automated Python Proof Assistant” ↗> The seemingly very intuitive arguments in Bell's book "a primer of infinitesimal analysis" are only heuristics. This kinds of kill the whole idea. The book is written as if the intuitive arguments were actually rigorous; what's heuristic about them? I mean the whole point of SIA is that…
on “Simplifying and Refactoring Introductory Calculus” ↗That's interesting - I read something similar in Bell's 'A primer of infinitesimal analysis' where he said the price for 'Smooth World' infinitestimals is giving up the Law of Excluded Middle (LEM). Don't really understand why (he said something about unconstrained use of LEM allows discontinuous…
on “Multiplicative Infinitesimals” ↗…just the Compactness Theorem of First-Order Logic. One happy consequence of this is that it's compatible with classical logic, unlike SIA/SDG. The seemingly very intuitive arguments in Bell's book "a primer of infinitesimal analysis" are only heuristics. It's also disturbing how the nilsquare "infinitesimals" in SIA/SDG satisfy δ^2 = ε^2 = 0 but NOT necessarily εδ = 0! By contrast, you can more easily obtain nilpotent infinitesimals…
on “Simplifying and Refactoring Introductory Calculus” ↗