Appendix 2: My Response to Some Critics, by Michael Grossman
This website concerns the systems of non-Newtonian calculus, multiplicative calculus, and nonclassical arithmetic created by Michael Grossman and Robert Katz between 1967 and 1970. The website was last edited on 24 June 2021.
It's an interesting and useful fact that the derivative and integral in any given non-Newtonian calculus can be expressed in terms of the classical derivative and classical integral, respectively. (Non-Newtonian Calculus [15], page 31) Unfortunately, from that fact some critics of non-Newtonian calculus have wrongly argued that the non-Newtonian calculi are useless.
In fact, if their argument is valid, then each of the absurd arguments indicated below would also be valid:
The operation of multiplication of two positive integers is useless because of the fact that it can be expressed in terms of (repeated) addition.
The operation of multiplication of two positive numbers is useless because of the fact that it can be expressed in terms of addition (by using logarithms).
The operation of division of two positive numbers is useless because of the fact that it can be expressed in terms of subtraction (by using logarithms).
The geometric average (of two positive numbers) is useless because of the fact that it can be expressed in terms of the arithmetic average (by using logarithms).
The well-known logarithmic derivative is useless because of the fact that it can be expressed in terms of the classical derivative.
The elasticity concept (used in economics, etc.) is useless because of the fact that it can be expressed in terms of the classical derivative.
The classical calculus is useless because of the fact that the classical derivative and classical integral can each be expressed in the context of the real number system (e.g., by using 'epsilon-delta' formulations).
Furthermore, if the non-Newtonian calculi are useless, then there would be no items listed in the Applications & Reception section of this website.
The great mathematician, astronomer, and physicist Carl Friedrich Gauss (1777-1855) discussed the usefulness of new calculi a long time ago:
"In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, indeed to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra [and] with the differential calculus ... . Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."
- Carl Friedrich Gauss, as quoted in Carl Friedrich Gauss: Werke, Volume 8, page 298; and as quoted in Robert Edouard Moritz's book Memorabilia Mathematica or The Philomath's Quotation Book, quotation #1215 (1914).
Finally, it's worth noting that the classical derivative and classical integral can each be expressed in terms of the corresponding operator of any given non-Newtonian calculus. (Please see page 28 of The First Nonlinear System of Differential and Integral Calculus [11], and page 34 of Bigeometric Calculus: A System With a Scale-free Derivative [10].)
- Michael Grossman
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