![]() And that’s the telltale mark of an incomplete system. The problem isn’t necessarily that infinitesimals can’t seem to describe Weierstrass’s function, but rather that they can’t seem to describe why they can’t describe it. The idea of infinitesimals is to use local linearity: that, over a small enough interval, a function behaves like a straight line and can be modeled as such, but if a function just sprouts more and more jagged edges the further in you zoom then how can it be defined that way? And what does it even mean for a function to “act like a line?” And who’s to say it doesn’t behave that way at a certain level of infinitesimalness: after all, we just dropped certain infinities when it feels right anyway! it’s behavior is unusual, sure, but surely we can accurately describe its local behavior, right? Not exactly. Unlike the floor function, this function’s complications seem much less obvious. It was one of the first rigorously studied fractal curves: at any given point you can zoom further in and the curve will maintain its structure rather than smoothing out into a line, just like the red bubble shows when zooming in on a point.The use of infinite series as functions is completely legitimate: Leibniz and especially Newton used them frequently in their respective developments of Calculus. In fact, it’s his exact notation we generally use today for the derivative \( \frac\pi \). ![]() Leibniz in particular blurred the distinction between infinitesimals and real numbers, treating them as one in the same. The question Newton and Leibniz were asking throughout their development of “infinitesimal calculus” was how a tiny nudge of dx affects the value of various functions \( y=f(x), \) in other words, a tiny nudge dy! An infinitesimal describes a tiny nudge to the value of x far smaller than the degree of precision being used in conventional calculations… infinitesimally small, so to speak. Rather than simply writing it for the sake of convention within a limit-based derivative or integral, though, this symbol came to represent a quantity itself. ![]() Rather than relying on the rigor of axioms and limit postulates to develop a fully consistent system from the ground up, Newton and Leibniz- the pioneers of modern Calculus’s main ideas- built their new language around a murky idea that came to be known as “the infinitesimal.” That may sound mysterious, but I’ll bet that the symbol below is familiar to even the most limit-rooted Calculus students. What do I mean by this? Like all concepts we treat as mathematical law today, Calculus was as fluid with its rules as the continuous change it described. Plus, I could rest assured that my methods were backed by the fathers of Calculus themselves. Sounds stupid, I know, but at the time I was engrossed in an alternate system that explained the central ideas of derivatives and integrals seamlessly without the minefield of epsilons and deltas that scare away half of the first-time students opening a Calc textbook for the first time. When I was learning Calculus on my own, I decided to skip limits. ![]()
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