You must have remarked, that what I have hitherto produced in support of the system of monads is destitute of solidity. In denying monads, it is impossible to talk any longer of ultimate particles, and still less of the number of them which enters into the composition of each body.
he is under the necessity of maintaining, that it is always possible to carry this divisibility further, without ever being obliged to stop and this is the case of divisibility in infinitum, on which system we absolutely deny the existence of ultimate particles: consequently the difficulties resulting from their infinite number fall to the ground themselves. Whoever denies the system of monads, must likewise, then, deny that the divisibility of bodies is limited. The system of monads is established in the former case, since after having divided a body into indivisible particles, these very particles are monads, and there would be reason for saying that all bodies are composed of them, and each of a certain determinate number. It is admitted on both sides, that bodies are divisible: the only question is, “Whether this divisibility is limited?”, or “Whether it may always be carried further, without the possibility of ever arriving at indivisible particles?” You must be perfectly sensible that one of the two systems, which have undergone such ample discussion, is necessarily true, and the other false, seeing they are contradictory. Principle of the satisfying Reason, the strongest Support of the Monadists. Our educational experience and the student reactions to our approach are detailed in this recent publication.This is the seventh of eleven Letters of Euler I will rewrite and post on the subject of infinitesimals (the infinitely small), an idea that is fundamental to a good understanding of calculus. Thus the students receive a significant exposure to both approaches.
To respond to the recent comment, a difference between our approach and Keisler's is that we spend at least two weeks detailing the epsilon-delta approach (once the students already understand the basic concepts via their infinitesimal definitions). In fact, I did a quick straw poll in my calculus class yesterday, by presenting (A) an epsilon, delta definition and (B) an infinitesimal definition at least two-thirds of the students found definition (B) more understandable. Infinitesimals provide an alternative approach that is more accessible to the students and does not require excursions into logical complications necessitated by the epsilon, delta approach. The epsilon, delta techniques involve logical complications related to alternation of quantifiers numerous education studies suggest that they are often a formidable obstacle to learning calculus. To answer your question about the applications of infinitesimals: they are numerous (see Keisler's text) but as far as pedagogy is concerned, they are a helful alternative to the complications of the epsilon, delta techniques often used in introducing calculus concepts such as continuity. The real numbers $\mathbb$ is algebraically simplified to $2x+\Delta x$ and one is puzzled by the disappearance of the infinitesimal $\Delta x$ term that produces the final answer $2x$ this is formalized mathematically in terms of the standard part function.
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