what are your favorite infinitesimals?
i'd really like a simple system of infinitesimals that feels intuitive for derivatives AND integrals
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What do you mean a system? What’s systematic about them? Like axiomatization of epsilon-delta operations?
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i mean like a specific formalism (ideally one that can fade into the background once learned)
the dual numbers are a good example, you just make up a number e such that e * e = 0
then you can differentiate f(x) by just evaluating (f(x+e)-f(x))/e
(fun thing to try out)
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another "system" would be non-standard analysis
this is where you pretend there's a number H that's like infinity but still a natural number, and then you can use 1/H as your infinitesimal
this one makes integrals nice, since you're just adding up H slices of width 1/H
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but of course, it gets even more exotic
synthetic differential geometry requires you to give up the law of excluded middle ("anything is either true or false")
then your infinitesimals are numbers that are not *not* zero
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Damn this sounds shady af… you can just make up symbols and assign them properties? 🧐
I’d never heard of either
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hehe yes! 😈 in some sense that's all math ever is
the catch is that you have to stick religiously to the rules and properties you chose
otherwise they'll lose their magic power
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Sounds like you’d need at least 2 consistent properties to make a system work for both differentiation and integration… why would you expect this to exist? At least with e*e=1 or i^2=-1 you’re just making one thing up
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yeah, it might not exist, but on the other hand, the fundamental theorem of calculus is true!
maybe better to look for a way to move between two approaches in a simple way
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The Laplace transform seems rhyming to what you’re looking for. The frequency parameter s is differentiation and 1/s is integration. Maybe you could make up an infinitesimal Laplace transform.
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