I have been teaching Calculus for over 20 years now, in both Spanish and American institutions. Over time, I have encountered the same struggles with the subject among my students. In my opinion, those difficulties are related, at least in part, to the form Calculus is presented to students nowadays and the extent to which it has departed from its roots.

I believe that the best way to present a new idea is by mimicking the historical development. Calculus was born as a tool to describe motion (for Newton) or as an algebra of infinitesimal quantities (for Leibniz). Leibniz introduced notations for derivatives and integrals that reflected his understanding of the underlying concepts, connecting the latter to their “finite” counterparts. His notation was so well thought that it stuck among his contemporaries and was adopted by the following generations of mathematicians and, importantly, engineers and applied scientists in general. The main reason it was adopted by applied scientists is rooted in its suggestive character, in its connection to our natural intuition. Even today, it is customary to read expressions like “consider an infinitesimal amount of work ” or “the force exerted on an infinitesimal element of current in Physics and Engineering books.

But..wait a minute! We do not teach Calculus using that language anymore! Why so? Why is there a disconnection between the way Calculus is taught and the way it is used by Engineers? Well, one of the main reasons is that Math is supposed to be rigorous, and Leibniz’s infinitesimals lack rigor. What that means is that they cannot be properly defined, at least if we stick to the standard conception of the continuum (real numbers). Calculus was born at the end of the XVII century, and it was not until around 1850 that the mathematical community (mostly thanks to Cauchy’s and Weierstrass’s work) was able to define the basic concepts of Calculus in a satisfactory manner. The new key concept that allowed to put previous intuitions and vague ideas on a firm basis was that of “limit”. Undoubtedly, the concept of limit helped dispel the obscurity surrounding infinitesimals. Moreover, it allowed to define more abstract concepts of Topology, Functional Analysis and other modern fields of Mathematics.

So, all modern Calculus textbooks start with a chapter on the concept of limit. That can’t be wrong, right? First things should go first. But there is a problem: it took Mathematics over 150 years to come up with the concept of limit, formalized for the first time by Weierstrass in his lectures (1861) and, in a more explicit form, by Dini in 1878. Exposing students to such sophisticated concept is doomed to be unsuccessful. They can hardly understand the concept itself, let alone its significance for a proper construction of Calculus. Here is what happens in practice: We briskly go over the definition, knowing from experience that the vast majority of students will just rot-memorize the procedure to prove that a limit has a certain value, subsequently explaining some practical tools to compute limits. Limits appear again briefly in the chapters on derivatives and integrals, but this is perceived rather as an annoyance which is not needed anymore once the rules to calculate and operate with derivatives and integrals are presented. For most students, Calculus thus becomes a bunch of recipes to compute things like velocities, work, areas and other extensive quantities, etc. The supposedly unifying principle is missed, and the intuition was not properly developed because now we have a rigorous, unifying principle!

I would like to add here that, over the 150 years period between Leibniz and Weierstrass, major advances in Calculus were made by the Bernoullis, Euler, Lagrange, Laplace and many others. And all that without the rigorous concept of limit.

One of my goals in this blog is to present examples of the use of infinitesimals in Calculus, stressing its dynamic side as opposed to the somehow sclerotizing effect of favoring derivatives over differentials as the protagonists of Differential Calculus. Among other things, I would like to restore as the expression of a quotient, for all practical purposes. It is my hope that these examples help students understand “the true metaphysics of Calculus”.