Yet, the biggest effect I think this will have is pedadogical. I’ve always found the definition of a limit kind of unintuitive, and it was specifically invented to add post hoc coherence to calculus after it had been invented and used widely. I suspect that formulating calculus via infinitesimals in introductory calculus classes would go a long way to making it more intuitive.
Different people will have different intuitions. I’ve always found the epsilon-delta method clear and simple, and infinitesimals made of shadows and fog when used as a basis for calculus. Every infinitesimals-first approach I have seen involves unexplained magic or papered-over cracks at some point, unexplained and papered-over because at the stage of first learning calculus the student usually doesn’t know any formal logic. There’s a reason that infinitesimals were only put on a sound footing a century after epsilon-delta. Mathematical logic had to be invented first.
Here the magic lies in depending on the axiom of choice to get a non-principal ultrafilter. And I believe I see a crack in the above definition of the derivative.f is a function on the non-standard reals, but its derivative is defined to only take standard values, so it will be constant in the infinitesimal range around any standard real. If f(x)=x2, then its derivative should surely be 2x everywhere. The above definition only gives you that for standard values of x.
I also think that making it more intuitive is missing the point of learning—really learning—mathematics. The idea of the slope of a curve is already intuitive. What is needed is to show the student a way of thinking about these things that does not depend on the breath of intuition to keep it aloft.
Here the magic lies in depending on the axiom of choice to get a non-principal ultrafilter. And I believe I see a crack in the above definition of the derivative. f is a function on the non-standard reals, but its derivative is defined to only take standard values, so it will be constant in the infinitesimal range around any standard real. If f(x)=x2, then its derivative should surely be 2x everywhere. The above definition only gives you that for standard values of x.
Yep, the definition is wrong. If f:R→R then let ∗f denote the natural extension of this function to the hyperreals (considering ∗R behaves like R this should work in most cases). Then, I think the derivative should be
f′(x)=st(∗f(x+Δx)−∗f(x)Δx)
W.r.t. what the derivative of ∗f should be, I imagine you can describe it similarly in terms of ∗∗R, which by the transfer principle should exist (which applies because of Łoś′s theorem, which I don’t claim to fully understand).
Different people will have different intuitions. I’ve always found the epsilon-delta method clear and simple, and infinitesimals made of shadows and fog when used as a basis for calculus. Every infinitesimals-first approach I have seen involves unexplained magic or papered-over cracks at some point, unexplained and papered-over because at the stage of first learning calculus the student usually doesn’t know any formal logic. There’s a reason that infinitesimals were only put on a sound footing a century after epsilon-delta. Mathematical logic had to be invented first.
Here the magic lies in depending on the axiom of choice to get a non-principal ultrafilter. And I believe I see a crack in the above definition of the derivative.f is a function on the non-standard reals, but its derivative is defined to only take standard values, so it will be constant in the infinitesimal range around any standard real. If f(x)=x2, then its derivative should surely be 2x everywhere. The above definition only gives you that for standard values of x.
I also think that making it more intuitive is missing the point of learning—really learning—mathematics. The idea of the slope of a curve is already intuitive. What is needed is to show the student a way of thinking about these things that does not depend on the breath of intuition to keep it aloft.
Yep, the definition is wrong. If f:R→R then let ∗f denote the natural extension of this function to the hyperreals (considering ∗R behaves like R this should work in most cases). Then, I think the derivative should be
f′(x)=st(∗f(x+Δx)−∗f(x)Δx)
W.r.t. what the derivative of ∗f should be, I imagine you can describe it similarly in terms of ∗∗R, which by the transfer principle should exist (which applies because of Łoś′s theorem, which I don’t claim to fully understand).
For f(x)=x2, the derivative then is:
f′(x)=st((x+Δx)2−x2Δx),=st(2x+Δx),=2x.