Brief History

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. 



The non-Newtonian calculi were created in the period from 1967 to 1970 by  Michael Grossman and Robert Katz. In July of 1967, they created an infinite family of calculi that includes the classical calculus, the geometric  calculus, the harmonic calculus, and the quadratic calculus. In August  of 1970, they created infinitely-many other calculi,  including the bigeometric calculus, the anageometric calculus, the biharmonic calculus, the anaharmonic calculus, the biquadratic calculus, and the anaquadratic calculus. All of the calculi can be described simultaneously within the framework  of a general theory. They decided to use the adjective "non-Newtonian"  to indicate any of the calculi other than the classical calculus. Infinitely many non-Newtonian calculi are multiplicative calculi, including the geometric and bigeometric calculi. 

In 1972, Grossman and Katz completed their book Non-Newtonian Calculus. [15] It contains discussions of the nine aforementioned non-Newtonian calculi, the  general theory of non-Newtonian calculus, and heuristic guides for  application. Subsequently, with Jane Grossman, they wrote several other  books/articles on non-Newtonian calculus, and on related matters such as weighted calculus, meta-calculus, averages, and means. (Please see  items [7 - 15, 34, 35] in the References.)

Michael Grossman and Robert Katz initiated non-Newtonian calculus on 14 July 1967. Prior to that day, non-Newtonian calculus was unknown to them, and (apparently) to everyone else. In Non-Newtonian Calculus (1972), Grossman and Katz included the following paragraph (page 82):

"However, since we have nowhere seen a discussion of even one specific non-Newtonian calculus, and since we have not found a notion that encompasses the *-average, we are inclined to the view that the non-Newtonian calculi have not been known and recognized heretofore. But only the mathematical community can decide that."


NOTE. In 2014, Marek Czachor (Gdańsk University of Technology in Poland), independently created non-Newtonian calculus, and generalized the earlier work of Grossman and Katz.

NOTE. For more information about the history of non-Newtonian calculus, please see Appendix 3 of this website.

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 Internet, each of the books can be read and  downloaded, free of charge, at HathiTrust, Google Books, and the Digital Public Library of America.   

NOTE. Non-Newtonian calculus was recommended as a featured topic for the 21st-century college-mathematics-curriculum, in the keynote speech at the 27th International Conference on Technology in Collegiate Mathematics (ICTCM) on 13 March 2015. The keynote speaker was the mathematics-educator Eric Gaze. His speech is entitled "Complexity, Computation, and Quantitative Reasoning: A Mathematics Curriculum for the 21st Century". (The conference was sponsored by Pearson PLC, the largest education company and the largest book publisher in the world; and the Electronic Proceedings of the conference were hosted by Math Archives (archives.math.utk.edu) with partial support provided by the National Science Foundation.)  Please see item [224] in the Referencessection.

NOTE. A special-session (mini-symposium) called "Non-Newtonian Calculus" was  held at the 17th International Conference on Computational and  Mathematical Methods in Science and Engineering (CMMSE), 4-8 July 2017,  at Rota, Cadiz - Spain. The 2017 CMMSE Scientific Committee  included over 50 distinguished scholars in a variety of fields, and  CMMSE conferences are attended by numerous scholars worldwide. The  special-session on non-Newtonian calculus was organized by Fernando  Córdova-Lepe and Marco Mora, both from Universidad Católica del Maule in Chile. Included among the papers delivered at the special session on non- Newtonian calculus: "Introductory elements for the development of a multiplicative statistic" by Carol Pavez Rojas, Fernando Córdova-Lepe, and Karina Vilches Ponce (all from Universidad Católica del Maule in Chile); "Linearity and its algebra in the bi- geometrical context" (i.e., multiplicative linear-algebra) by Fernando Córdova-Lepe, Rodrigo del Valle, and Karina Vilches Ponce (all from Universidad Católica del Maule in Chile); and "Bigeometric Complex Calculus" by Agamirza E. Bashirov and Sajedeh Norozpour (both from Eastern Mediterranean University in North Cyprus). "Bigeometric Complex Calculus" was also presented at the special- session "Mathematical Modeling and Computational PDE". [324, 383, 384, 385] From a  translation (from Spanish) of the conference- announcement:

     "Non-Newtonian Calculus (NNC) ... has been increasing its development  through the recoding of the multiplicative world (from the point of view of the standard calculation) as an essentially linear domain, and  therein lies the nucleus of importance. Many advances and applications  in science, engineering and mathematics are appearing more frequently.  

     "This mini symposium will be one of the first international meetings of a dispersed scientific community that has worked or is working on this  topic and annoting a mark in the history of the NNC. Taking into account the novelty of the subject, all topics related to NNC (theory and  applications) are welcome."

.