#Positive and negative infinitesimals how to

The subject had its origins in Alexander Grothendieck’s work on Algebraic Geometry, as interpreted by F. Smooth Infinitesimal Analysis (abbreviated SIA) is a variant of Non-classical Real Analysis which uses nilpotent infinitesimal quantities to deal with concepts such as continuity and differentiability. Is Smooth Infinitesimal Analysis constructive? 1. Where can I find all the axioms of Smooth Infinitesimal Analysis?Ħ.2. Can I prove the non-infinitesimal form of Taylor’s theorem in SIA?Ħ.1. Can I prove the Intermediate Value Theorem in SIA?ĥ.3. Can I prove the Extreme Value Theorem in SIA?ĥ.2. Infinitesimals don’t have reciprocals, so how does the quotient rule work?ĥ.1. What is the deal with the square root function?Ĥ.4. How do I prove that the exponential function is its own derivative?Ĥ.3.

#Positive and negative infinitesimals how to

I don’t know how to prove the chain rule. I believe I’ve found a contradiction using microcancellation. Can you show me a non-zero nilsquare infinitesimal?ģ.2. Yet my SIA book calls the real line a field. I’ve heard in Algebra that fields don’t have nilpotent elements. What kind of reasoning am I allowed to use?Ģ.2. I find Intuitionistic Logic… unintuitive. Making infinitesimals nilsquare seems arbitrary. What’s the intuition behind Smooth Infinitesimal Analysis?ġ.3. This is Revision (30 July 2016) of the Smooth Infinitesimal Analysis FAQ.ġ.2.