alt Hacker NewsIt is an interesting piece but to claim that no heavy machinery is used is a bit disingenuous at best. You have defined some purely algebraic operation “differentiation”. This operation involves a choice of infinitesimal. Is it trivial to show that the definition is independent of infinitesimal? especially if we are deriving at a hyperreal point? I doubt it and likely you would need to do more complicated set theoretic limits rather analytic limits. How do you calculate the integral of this function? Or even define it? Or rather functions, since it’s an infinite family of logistic functions? To even properly define this space you need to go quite heavily into set theory and i doubt many would find it simpler, even than working with distributions
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Even just defining the hyperreals and showing why statements about them are also valid for the reals needs to go through either ultrafilters (which are some rather abstract objects) or model theory. Of course you can just handwave all of that away but then I guess you can also do that with standard analysis.
The machinery of mathematics goes arbitrarily deep. I think the interesting thing here is that with relatively little training you can start to compute with these numbers, which is definitely not the case with analysis on distributions.
Or put differently - here you can kinda ignore the deeper formalities and still be productive, whereas with distributions you actually need to sit down and pore over them before you can do anything.
That said, I'm curious why infinitesmals never took off in physics. This kind of quick, shut-up-and-calculate approach seems right up their alley.