Zero Divided by Zero − and the Concept of Differentiation

Tür:

Kitap Bölümü

Yayın yılı:

2016

Yayıncı:

BENTHAM SCIENCE PUBLISHERS

This chapter begins with a brief historical introduction, wherein the seeming paradox of Achilles and the turtle is examined. Although this treatise does follow part of the tradition and progresses to the concepts of limits and continuity, including the more formal perspective of using epsilon and delta type proofs that have been the touchstone of calculus’s foundation for over three centuries, no further development of such a protocol is undertaken. In its place a very different “Weltanschauung” (world philosophy) that focuses on what this author asserts is the appropriate underlying foundation of calculus is promulgated; namely, the relation of the concepts of none, some and all (algebraically expressed as 0, 1 and ∞) to the six fundamental operations of numbers (addition, subtraction, multiplication, division, raising to a power and extracting a root). From the 54 potential binary combinations of these sets, the seven traditional indeterminate l’Hôpital forms, as well as three additional related forms that mathematicians have missed for over three centuries are distilled. In the process, attention is focused on combinations deliberately disallowed in previous mathematics courses; especially those that arise with respect to infinity and division by zero. One particular combination, which has as its objective the determination of those extreme values that the given function can reach both globally (over all of space), and locally (in a given interval), is postulated to be…

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