![]() This is the real numbers plus the infinitesimal. Robinson defines a new set of numbers, the hyperreals. There is another solution, found in the 20th century by Abraham Robinson, which is to formalize the concept of the infinitesimal. The solution Bolzano found was to eliminate the use of that tool. The one fundamental problem was that Newton and Leibniz depended on an informal tool to build their formal system. infinitesimal calculus with infinitesimals And the book must also include tons of “real world examples” in order to get into print and into schools. The end result is that students are expected to perform derivatives and integrals using a formalization that bears no resemblence to the intuition they’re being taught-which they need to understand what they’re doing. ![]() ![]() This should be obvious, since Newton and Leibniz started with derivatives and integrals respectively and not limits, which are completely tangental to all of the things they actually wanted to calculate. Furthermore, the restatements of the derivative and the integral with limits are less intuitive. This leads to absurdities like learning the formal definition of the limit after two weeks of computing them with various formulas and heuristics, which my wife is enduring now. It would probably be impossible to understand or use calculus without an intuitive grasp of infinitesimals, yet all the textbooks for calculus instead use the epsilon-sigma definition of the limit as their basis. Thanks to Karl Weierstrass, this is the formulation used by textbooks today: a formulation without actual infinitesimals, but retaining Leibniz’s notation that uses them. From there, all the rest of calculus can be rebuilt. Other mathematicians (Bolzano, Cauchy) found a way to skirt the issue by defining the limit formally without using infinitesimals. This kind of thing leads to great discomfort in mathematicians. Now, Newton and Leibniz managed to make great use of calculus, but they never improved on their own informal definition of what an infinitesimal was. The second problem with calculus instruction is that we no longer use infinitesimals! No wonder the word has fallen out of use how hard would it be to understand infinitesimal calculus without infinitesimals in it! Instead of teaching math, metaphor gets taught, and we pretend that distilling the commonality between six metaphors is easier than just understanding the math. So then you get unlimited examples, which wind up creating more confusion. So simple, the applications are unlimited. They’re not a difficult concept, they’re actually quite simple. Frankly, this is a lot like the confusion in Haskell with monads. This leads to the first problem of calculus instruction: lack of clarity about what calculus is. The tools provided, the derivative and the integral, happen to be incredibly general tools. The total distance, likewise, is the sum of all of the individual distances moved, instant to instant, along this function. Velocity is defined by direction plus change in position over the length of time, but there could be such a thing as an instantaneous velocity everybody with a car is familiar with this notion, it’s your speedometer. Newton conceived this infinitesimal calculus as merely this tool he needed to solve certain general physics problems like the above. ![]() Alternately, if I have a function of distance, what is the total distance? These are things you need in Newtonian physics, because you’re interested in saying, if I have this function of distance, what is the rate of change, in other words, the velocity. So why do we call Newton’s calculus “calculus” and not something more specific? Looking at history, you find that “calculus” used to be referred to as “infinitesimal calculus.” This is an accurate and appropriate name, because calculus is the foundation of studying rates of change and summations over time. There are lots of other calculii computer science has lambda, pi and join calculii. The word “calculus” is actually a generic word for a mathematical system. The blame for this is usually left squarely on the academic establishment, who are certainly at fault, but there is more to it than that. If you took calculus in school, you probably remember it as a blurry mismash of baffling equations, difficult algebra, many seemingly irrelevant “real-world” examples and overall confusion. ![]()
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