Nonclassical Arithmetics


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. 



In 1967, Grossman and Katz created the  nonclassical arithmetics, i.e., the complete ordered fields distinct from the real number system, and used them to construct the non-Newtonian calculi. (They sometimes refer to the real number system as "classical arithmetic".) . Here are some important examples:

The nonclassical arithmetics are alternatives to the real number system. Like the non-Newtonian calculi, the nonclassical arithmetics provide a wide variety of mathematical tools for use in science, engineering, and mathematics. The nonclassical arithmetics are all isomorphic (structurally equivalent) to the real number system. Unfortunately, it had long been conventional wisdom that for all practical purposes, any two isomorphic systems are identical. But that is not the case! (For discussions about complete ordered fields, arithmetics, and the real number system, please see Non-Newtonian Calculus.)


"Conventional wisdom has it that, since all complete ordered fields are isomorphic (structurally equivalent), we might as well assume that there is only one complete ordered field, which is called the real number system. This view was at least partially responsible for the relatively late appearance (in 1967) of the non-Newtonian systems of differential and integral calculus, which were developed by recognizing that the various complete ordered fields are significantly different."

  - Michael Grossman and Robert Katz; from their 1984 article "Isomorphic calculi” [13].


After the publication of Non-Newtonian Calculus in 1972, various authors  have discussed useful applications of nonclassical arithmetics as well as non-Newtonian calculi.  (Please see the Applications & Reception page.)  For example, in their 2012 article “Some new results on sequence spaces with respect to non-Newtonian calculus” [122], Ahmet Faruk Çakmak and Feyzi Basar asserted: “In his Foundations of Science (formerly titled Physics: The Elements), Norman Robert Campbell, a pioneer in the theory of measurement, clearly recognized that nonclassical arithmetics might be useful in science for he wrote, ‘We must recognize the possibility that a system of measurement may be arbitrary otherwise than in the choice of unit; there may be arbitrariness in the choice of process of addition’”. 

Also, in his 1980 article "Non-Newtonian calculus applied to probability, utility, and Bayesian analysis" [16], James R. Meginniss (Claremont Graduate School and Harvey Mudd College) devised a nonclassical arithmetic for his "new theory of probability that is adapted to human behavior and decision making". His work is based on the idea that “probabilities do not obey the laws of ordinary arithmetic and calculus, but instead are governed by the laws of one of the non-Newtonian calculi and its corresponding arithmetic.”


"A substantial body of evidence indicates that people really don’t process probability information in the way that the classical or Bayesian models suggest. ... The purpose of this paper is to present a new theory of probability that is adapted to human behavior and decision making. Two basic ideas will be used: that probabilities do not obey the laws of ordinary arithmetic and calculus, but instead are governed by the laws of one of the non-Newtonian calculi and its corresponding arithmetic; and that ... . With these two ideas it is possible to reform the calculus of probabilities without abandoning the basic formalism of probability.”

  - James R. Meginniss, Claremont Graduate School and Harvey Mudd College, USA; from his 1980 article "Non-Newtonian calculus applied to probability, utility, and Bayesian analysis" [16].


As another example, in his 2019 article “Waves along fractal coastlines: from fractal arithmetic to wave equations” [487], Marek Czachor’s (Gdańsk University of Technology in Poland) devised a nonclassical arithmetic naturally suited to a Koch-type fractal curve. He then used that nonclassical arithmetic to construct a non-Newtonian calculus, yielding what “seems to be the first example of a truly intrinsic description of wave propagation along a fractal curve”. 


“The arithmetic perspective is simultaneously applicable to all the other aspects of mathematical modeling, including algebraic or probabilistic methods. The freedom of choice of arithmetic plays a role of a universal symmetry of any mathematical model.”-

 - Marek Czachor; from his 2019 article “Waves along fractal coastlines: from fractal arithmetic to wave equations” [487].


Furthermore,  in Non-Newtonian Calculus (page 76), Grossman and Katz used nonclassical arithmetics to define the concept of “*-metric”, later called “non-Newtonian metric” by some authors. An important special case of *-metric, based on geometric arithmetic and called “multiplicative metric” by some authors, is discussed in The First Nonlinear System of Differential and Integral Calculus [11] (page 62), and in Bigeometric Calculus: A System with a Scale-Free Derivative [10] (page 73). Subsequently, beginning in 2012, the concepts non-Newtonian metric-space and multiplicative metric-space were both used by various researchers to devise new theorems in fixed-point theory, an important tool in mathematics, biology, chemistry, economics, engineering, game theory, and physics. (For example: [117], [128], [182], [206], [213], [214], [234], [242], [243], [244], [249], [252], [254], [255], [257], [260], [261], [262], [263], [304], [305], [310], [311], [312], [313], [314], [317], [318], [320], [321], [325], [327], [335], [336], [337], [338], [339], [340], [341], [342], [343], [344], [346].  [353], [354], [355], [356], [357], [360], [362],[363], [364], [365], [366], [368], [371], [372], [375], [379], [386], [398], [402], [408], [409], [410], [411], [419], [424], [425], [426], [438], [439], [442], [448], [461], [462], [466], [473], [483], [484], and [486].) A comprehensive  discussion of multiplicative metric spaces is included in the chapter “Survey on Metric Fixed Point Theory and Applications” in the 2017 book Advances in Real and Complex Analysis with Applications. [372]



NOTE. In Grossman and Katz’s publications on non-Newtonian calculus, “arithmetic” means any complete ordered field whose elements are real numbers. (They defined an “arithmetic” to be a complete ordered field whose realm is a subset of R.) However, there was no need to restrict attention to real numbers. If they had simply defined an “arithmetic” to be any complete ordered field, all their results would still be valid.


NOTE. The six books on non-Newtonian calculus and related matters by Jane Grossman, Michael Grossman, and Robert Katz are indicated below, and are available at some academic libraries, public libraries, and booksellers such as Amazon.com. On the World Wide Web, each of the books can be read and downloaded, free of charge, at HathiTrust, Google Books, and the Digital Public Library of America.