你可能会对 John L. Bell 写的《A Primer of Infinitesimal Analysis》感兴趣，这是另一种利用直觉主义逻辑（intuitionistic logic）来研究无穷小量的方法。https://www.amazon.com/Primer-Infinitesimal-Analysis-John-Be... 这种逻辑的任何有趣性质和…
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…
…它们本身并不能约化为 0。这一切都可以用构造性的方式来完成，避免使用任何无穷大，或者依赖间接证明（如排中律）的经典逻辑。John Bell 的《A Primer of Infinitesimal Analysis》里有很详尽的介绍。自动微分（Automatic Differentiation）中的计算技术就是利用这类实体来精确计算导数，而不需要对无穷（极限）过程进行近似。微积分…
…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…
…在你的数学论证中（我承认这是个门槛，但它也能带来不少好处）。它所带来的优势是证明过程要简单得多，而且非常符合直觉推理。还有这本书，《A Primer of Infinitesimal Analysis》[0]，它仅仅使用无穷小量就构建了微积分和经典力学的大部分内容，而且在逻辑上完全严格。[0] https://www.cambridge.org/br/universitypress/subjects/mathem...
…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...
好，我想分享一下《A Primer of Infinitesimal Analysis》这个链接 https://www.cambridge.org/core/books/primer-of-infinitesimal... 这本书太太太赞了。它从零开始，最终构建出足够支撑牛顿式经典力学的体系，完全由几何…
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…
> Bell 在《A Primer of Infinitesimal Analysis》一书中那些看似非常符合直觉的论证，其实只是一种启发式（heuristics）。这差不多把整个想法都给毁了。整本书写得就好像这些直觉论证其实非常严谨一样；它们到底哪里启发式了？我的意思是，平滑无穷小分析（SIA）的核心就在于…
> 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…
真有意思——我在 Bell 的《A Primer of Infinitesimal Analysis》里读到过类似的说法，他说获得“平滑世界”无穷小量的代价是放弃排中律（LEM）。我不太明白为什么（他提到无限制地使用排中律会导致不连续…
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…
你可能会对 John L. Bell 写的《A Primer of Infinitesimal Analysis》感兴趣，这是另一种利用直觉主义逻辑（intuitionistic logic）来研究无穷小量的方法。https://www.amazon.com/Primer-Infinitesimal-Analysis-John-Be... 这种逻辑的任何有趣性质和…
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…
…它们本身并不能约化为 0。这一切都可以用构造性的方式来完成，避免使用任何无穷大，或者依赖间接证明（如排中律）的经典逻辑。John Bell 的《A Primer of Infinitesimal Analysis》里有很详尽的介绍。自动微分（Automatic Differentiation）中的计算技术就是利用这类实体来精确计算导数，而不需要对无穷（极限）过程进行近似。微积分…
…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…
…在你的数学论证中（我承认这是个门槛，但它也能带来不少好处）。它所带来的优势是证明过程要简单得多，而且非常符合直觉推理。还有这本书，《A Primer of Infinitesimal Analysis》[0]，它仅仅使用无穷小量就构建了微积分和经典力学的大部分内容，而且在逻辑上完全严格。[0] https://www.cambridge.org/br/universitypress/subjects/mathem...
…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...
好，我想分享一下《A Primer of Infinitesimal Analysis》这个链接 https://www.cambridge.org/core/books/primer-of-infinitesimal... 这本书太太太赞了。它从零开始，最终构建出足够支撑牛顿式经典力学的体系，完全由几何…
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…
> Bell 在《A Primer of Infinitesimal Analysis》一书中那些看似非常符合直觉的论证，其实只是一种启发式（heuristics）。这差不多把整个想法都给毁了。整本书写得就好像这些直觉论证其实非常严谨一样；它们到底哪里启发式了？我的意思是，平滑无穷小分析（SIA）的核心就在于…
> 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…
真有意思——我在 Bell 的《A Primer of Infinitesimal Analysis》里读到过类似的说法，他说获得“平滑世界”无穷小量的代价是放弃排中律（LEM）。我不太明白为什么（他提到无限制地使用排中律会导致不连续…
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…