Suppose that we wanted to construct a mathematical universe where all objects were computable in some sense. How would we do it? Well, we could certainly allow the set into our universe: natural numbers are the most basic computational objects there are. (Notation: I’ll use to refer to when we’re considering it as part of […]
At Mathematics and Computation, there’s a really good accessible introduction to intuitionistic logic called Intuitionistic Logic for Physics. It also includes some nice accessible remarks on smooth infinitesimal analysis.
This is a continuation of my earlier post on smooth infinitesimal analysis. In this installment, I’ll show how the definition of a “stationary point” in Smooth Infinitesimal Analysis leads directly to a nice substitute for the Lagrange multipliers method. Then I’ll show how you can define differential forms as objects which assign a “signed volume” […]
Many mathematicians, from Archimedes to Leibniz to Euler and beyond, made use of infinitesimals in their arguments. These were later replaced rigorously with limits, but many people still find it useful to think and derive with infinitesimals. Unfortunately, in most informal setups the existence of infinitesimals is technically contradictory, so it can be difficult to […]