A primer of infinitesimal analysis

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1. nextaccounticAug 2026
   > 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 you really can have a triangle where one side is infinitesimal, because infinitesimals are part of the number system, it's not something extra.

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2. nextaccounticAug 2024
   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 arguments using infinitesimals, and being fully rigorous. Also has an appendix where it gives the construction using category theory

   Anyway so does this mean that Z3 can't be used to prove intuitonistic arguments? That's surprising to me. What exactly makes SMT tied to classical logic?

   edit: okay check out this https://dl.acm.org/doi/10.5555/1983702.1983729 the author used an embedding in order to represent intuitionistic formulas inside classical logic, so they could use Z3 anyway

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3. nextaccounticApr 2024
   > Those geometric arguments are essentially less rigorous versions of limits! And that lack of rigor (...)

   Yes it's equivalent to limits, but limits are a very cumbersome machinery, specially if you use the epsilon delta definition (there exists .. such that all ..).

   But note that I just linked you a PDF that does fully 100% rigorous calculus using only infinitesimals with no limits. Yhey aren't disregarding small terms willy nilly (like it was done in the early history of calculus)

   The only catch about SIA is that it requires you to use intuitionistic logic rather than classical logic 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...

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4. bmitcMay 2020
   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 model is that all functions are infinitely differentiable.

   Other introductions are An Invitation to Smooth Infinitesimal Analysis by John L. Bell and Synthetic Differential Geometry by Michael Shulman.

   http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf

   http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.p...

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5. hardboiledApr 2017
   Reals are not equivalently as unrealizable or dubious as infinitesimals. Infintesimals can be constructively specified as nilpotent/nilsquare entities, numerical entities which when squared equal 0 but where those entities 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 can be done constructively without infinite limiting processes purely algebraically using these nilpotents. And from a geometric interpretation there is nothing nonsensical about a tangent line to a curve.

   Also you don't differentiate numbers (real or rational), you can only differentiate functions.

   Also the idea of a actual infinity is a poetic mathematical one. It doesn't have to fit reality. The issue is whether it is useful and to what extent.

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6. ogogmadAug 2026
   Nonstandard analysis reduces to 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 by using NSA and combining it with little-o notation (on hyperreal numbers, and NOT on functions and their asymptotic growth rates). Then the geometric arguments in Bell's "a primer of infinitesimal analysis" can still be carried out, but are more easy to unravel back to core definitions.
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