Calculus without infinitesimals5/31/2023 ![]() ![]() Part of Hobbes' strategy included a campaign against the infinitesimal, championed in England by Hobbes' greatest rival, a mathematician named John Wallis. ![]() Establish a state that is absolutely logical, where the laws of the sovereign have the force of a geometrical proof," Alexander says. "He thought the only way to re-establish order was much like the Jesuits: Just wipe off any possibility of dissent. Thomas Hobbes, remembered today for his works of political philosophy like Leviathan, was also acknowledged at the time as a mathematician. The aristocracy and propertied classes were desperate to hold onto their traditional power while lower class dissent fermented underneath. Meanwhile a similar situation was playing out in England, where civil war was also threatening upheaval. That was a sharp contrast with the dependable outcomes of geometry. But in the 17th century, those questions didn't yet have satisfying answers - and worse, the results of early calculus were sometimes wrong, Alexander tells NPR's Arun Rath. Today, mathematicians have found ways to answer that question so that modern calculus is rigorous and reliable. The fight over how to resolve it had a surprisingly large role in the wars and disputes that produced modern Europe, according to a new book called Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by UCLA historian Amir Alexander. That's the paradox lurking behind calculus. And if they're zero, well, then no matter how many parts there are, the length of the line would still be zero. But how long are those parts? If they're anything greater than zero, then the line would seem to be infinitely long. You can keep on dividing forever, so every line has an infinite amount of parts. Just how many parts can you make? A hundred? A billion? Why not more? Then you can cut those lines in half, then cut those lines in half again. Here's a stumper: How many parts can you divide a line into? Your purchase helps support NPR programming. Pretty cool.Close overlay Buy Featured Book Title Infinitesimal Subtitle How a Dangerous Mathematical Theory Shaped the Modern World Author Amir Alexander I've never heard of this "method of Lagrange," so maybe someone on r/math can shed light on this? Are there less magical explanations of how this method was derived? Can you get by in differential geometry using only polynomials? Are his concerns about infinitesimals legitimate?ĮDIT: I just watched the next lecture, and Lagrange's technique generalizes to partial derivatives of polynomials of any number of variables. I don't see how this is anything more than a fancy way of calculating the Taylor series for a polynomial.Īdmittedly, the large advantage of this method is that it holds for all fields, as it only deals with algebraic numbers. He just shows that you get the same result as you would get using limits. The problem is that he never gives any kind of proof that his way of finding derivatives actually resembles tangent approximations. The resulting expression is what he calls the "k-th tangent to p(x) at r." Finally, he substitutes every x with x-r. The p_n(r) polynomials turn out to be terms in the Taylor series of the original polynomial. Then he factors out all the x powers so the expression looks like this: Then, using the binomial theorem, he expands the expression. The way it works is he substitutes x with x+r in his polynomial of choice, p(x). In any case, he continues in his lecture to define derivatives algebraically using something like the Taylor series, and he points to Lagrange as the method's inventor. In the comments of the above video, he seems to oppose the existence of infinite sets, though the set of rationals is infinite (maybe he means only countably infinite sets exist?). In the previous lectures, he made it clear that he is not a fan of transcendentals (sin, cos, etc.) and prefers parametrizations of curves that are polynomials with rational coefficients. Wildberger begins by challenging the traditional approach to calculus education which uses limits and infinitesimals. A glance through the comments will give you a rough impression of what's going on. I'm not saying it was bad, but it was definitely controversial. I've recently begun watching a lecture series on differential geometry by Norman Wildberger, and I thoroughly enjoyed the first two lectures. ![]()
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