Ghosts of Departed Quantities
Towards the end of The Analyst, Berkeley addresses possible justifications for the foundations of calculus that mathematicians may put forward. In response to the idea fluxions could be defined using ultimate ratios of vanishing quantities (Boyer 1991), Berkeley wrote:
It must, indeed, be acknowledged, that used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?
Edwards describes this as the most memorable point of the book (Edwards 1994). Katz and Sherry argue that the expression was intended to address infinitesimals directly and not Newton's theory of fluxions. (Katz & Sherry 2012) Berkeley's logical criticism is answered in the framework of the Leibnizian calculus by pointing out that the term is not set equal to zero but rather merely rejected at the end of the calculation so as to arrive at the value for the differential quotient, as an application of Leibniz's transcendental law of homogeneity.
Today the phrase "ghosts of departed quantities" is also used when discussing Berkeley's attacks on other possible foundations of Calculus. In particular it is used when discussing infinitesimals (Arkeryd 2005), but it is also used when discussing differentials (Leader 1986), and adequality (Kleiner & Movshovitz-Hadar 1994).
Read more about this topic: The Analyst
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