Infinitesimals, used properly, were never unrigorous and the supposed rigor of limit theory does not imply greater correctness, but rather the (usually unnecessary) exposition of hidden deductive steps. I argue here that both this characterization of the original theory and this interpretation of the paradigm shift to its successor are false.
But calculus students who do read accounts of its history encounter something strange – the claim that the theory which underpinned the subject for long after its creation was wrong and that it was corrected several hundred years later, in spite of the fact that the original theory never produced erroneous results. After all, as Erickson so well puts it: “What is humanly unreachable is not the same as that which cannot be.To gain true understanding of a subject it can help to study its origins and how its theory and practice changed over the years – and the mathematical field of calculus is no exception. But this is only when considered within the context of the infinitesimal, a perspective that is not assumed in pre-calculus classes and may never even be discussed in classes of calculus and beyond.Īnyone who enjoys deeply pondering the idea that a point must occupy an absolute minimum space (albeit below the level of finitude), and therefore that two points can exist ‘side by side’ will find much to consider here. He does offer a different way of viewing, among other things, the number line as finite rather than infinite, and he even proposes that division by zero is possible. While it may at first seem that Erickson is about to challenge everything most mathematicians hold dear, this isn’t necessarily so. What appears as quite daunting initially turns out to be a real learning opportunity for serious students or teachers of mathematics. Erickson’s reader-friendly style brings these (almost wildly) theoretical concepts into focus and manages to describe metaphysical mathematical entities so that they can be “seen” in the mind’s eye. And although this is heady material, not intended for novices, those who approach it with effort and patience will find that it’s actually a smooth, albeit slow, read. Laid out in as orderly a fashion as that first chapter, The Nature of Infinitesimals is well edited, with only minor punctuation flaws.
In Chapter 1: The Plan of the Book,” Erickson lists the mathematical constructs he will support or refute - and he delivers on those as the book progresses. Erickson states that the purpose of his book is “to prove the existence of the infinitesimal and reveal some important facts regarding the nature of numbers and the outer infinity…” While it is difficult to “prove” the existence of things that cannot be seen, Erickson makes a reasonable and valuable argument. Welcome to the world of the finite, sub-finite, and infinite actual infinity versus potential infinity the measurable and the immeasurable.
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