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. (Since 24 June 2021, the website has not been updated regularly.)
Non-Newtonian Calculus was reviewed by Ivor Grattan-Guinness in the journal Middlesex Math Notes.[101]
Excerpt: "There is enough here [in Non-Newtonian Calculus] to indicate that non-Newtonian calculi ... have considerable potential as alternative approaches to traditional problems. This very original piece of mathematics will surely expose a number of missed opportunities in the history of the subject."
Each of the following six books was reviewed in the journal Internationale Mathematische Nachrichten. [53]
1) Non-Newtonian Calculus [15]: Number 105, 1972.
Excerpt: "The possibilities opened up by the [non-Newtonian] calculi seem to be immense."
2) The First Nonlinear System of Differential and Integral Calculus [11]: Volumes 35-36, page 42, 1981.
3) The First Systems of Weighted Differential and Integral Calculus [9]: Volumes 35-36, page 40, 1981.
4) Meta-Calculus: Differential and Integral [7]: Volumes 35-36, page 140, 1981.
5) Bigeometric Calculus: A System with a Scale-Free Derivative [10]: Volumes 37-38, page 266, 1983.
6) Averages: A New Approach [8]: Volumes 37-38, page 266, 1983.
Non-Newtonian Calculus was reviewed by David Pearce MacAdam in the Journal of the Optical Society of America. [100]
Excerpt: "This [Non-Newtonian Calculus] is an exciting little book. ... The greatest value of these non-Newtonian calculi may prove to be their ability to yield simpler physical laws than the Newtonian calculus. Throughout, this book exhibits a clarity of vision characteristic of important mathematical creations. ... The authors have written this book for engineers and scientists, as well as for mathematicians. ... The writing is clear, concise, and very readable. No more than a working knowledge of [classical] calculus is assumed."
Each of the following five books was reviewed by Ralph P. Boas, Jr. in the journal Mathematical Reviews. [47]
1) The First Nonlinear System of Differential and Integral Calculus [11]: Mathematical Reviews, 1980.
2) The First Systems of Weighted Differential and Integral Calculus [9]: Mathematical Reviews, 1981.
3) Meta-Calculus: Differential and Integral [7]: Mathematical Reviews, 1982.
4) Bigeometric Calculus: A System with a Scale-Free Derivative [10]: Mathematical Reviews, 1984.
Excerpt: "It seems plausible that people who need to study functions from this point of view might well be able to formulate problems more clearly by using [bigeometric] calculus instead of [classical] calculus."
5) Averages: A New Approach [8]: Mathematical Reviews, 1984.
Non-Newtonian Calculus [15] was reviewed in the journal Mathematical Reviews in 1978. [47]
Non-Newtonian Calculus was reviewed in the magazine Choice. [41]
Non-Newtonian Calculus was reviewed in the journal American Mathematical Monthly. [48]
The First Nonlinear System of Differential And Integral Calculus [11], a book about the geometric calculus, was reviewed in the journal American Mathematical Monthly. [52]
Bigeometric Calculus: A System with a Scale-Free Derivative [10] was reviewed in Mathematics Magazine. [49]
Bigeometric Calculus: A System with a Scale-Free Derivative was reviewed in the journal The Mathematics Student. [58]
The article "An introduction to non-Newtonian calculus" [12] was reviewed by K. Strubecker in the journal Zentralblatt Math (Zbl 0418.26008) [43].
The article "A new approach to means of two positive numbers" [14] was reviewed in Zentralblatt Math (Zbl 0586.26014) [43].
Each of the following three books was reviewed by K. Strubecker in Zentralblatt Math [43].
1) Non-Newtonian Calculus [15]: Zbl 0228.26002.
2) The First Systems of Weighted Differential and Integral Calculus [9]: Zbl 0443.26005.
3) Meta-Calculus: Differential and Integral [7]: Zbl 0493.26001.
The article "A new approach to means of two positive numbers" [14] was reviewed in the journal ZDM (1986c.10787) [50].
Each of the following five books was reviewed in ZDM [50].
1) Non-Newtonian Calculus[15]: 1982a.00259.
2) The First Nonlinear System of Differential and Integral Calculus [11]: 1982a.00243.
3) The First Systems of Weighted Differential and Integral Calculus [9]: 1982a.00248.
4) Bigeometric Calculus: A System with a Scale-Free Derivative [10]: 19861.06868.
5) Averages: A New Approach [8]: 19861.06873.
Each of the following six books was reviewed in the journal Scientific Annals of Alexandru Ioan Cuza University of Iaşi: Mathematics Section. [55]
1) Non-Newtonian Calculus: Volumes 17-18, 1972.
2) The First Nonlinear System of Differential and Integral Calculus: Volumes 26-27, 1980.
3) The First Systems of Weighted Differential and Integral Calculus: Volumes 27-28, 1981.
4) Meta-Calculus: Differential and Integral: Volumes 28-29, 1982.
5) Bigeometric Calculus: A System with a Scale-Free Derivative: Volumes 29-30, 1983.
6) Averages: A New Approach: Volumes 29-30, 1983.
Each of the following three books was reviewed in the journal Nieuw Tijdschrift Voor Wiskunde. [57]
1) The First Nonlinear System of Differential And Integral Calculus: Volume 68, page 104, 1981.
2) The First Systems of Weighted Differential and Integral Calculus: Volumes 69-70, page 235, 1982.
3) Meta-Calculus: Differential and Integral: Volumes 69-70, page 236, 1982.
Each of the following two books was reviewed by Leo Barsotti in the journal Boletim da Sociedade Paranaense de Matemática. [54]
1) The First Nonlinear System of Differential and Integral Calculus: Volume 2, page 32, 1981.
2) The First Systems of Weighted Differential and Integral Calculus: Volume 2, pages 32-33, 1981.
Each of the following three books was reviewed in the journal L'Enseignement Mathématique. [59]
1) The First Nonlinear System of Differential and Integral Calculus: page 52, 1980.
2) Bigeometric Calculus: A System with a Scale-Free Derivative: page 83, 1982.
3) Averages: A New Approach: page 83, 1982.
Each of the following two books was reviewed by P. Wilker in the journal Revue de mathématique élémentaires. [92]
1) The First Nonlinear System of Differential and Integral Calculus, Volumes 37-40.
2) The First Systems of Weighted Differential and Integral Calculus: Volumes 37-40.
Non-Newtonian Calculus was reviewed by Otakar Zich in the journal Kybernetika. [45]
Non-Newtonian Calculus was reviewed in the journal Wissenschaftliche Zeitschrift: Mathematisch-Naturwissenschaftliche Reihe. [51]
Non-Newtonian Calculus was reviewed by M. Dutta in the Indian Journal of History of Science. [42]
Non-Newtonian Calculus was reviewed by Karel Berka in the journal Theory and Decision. [44]
Non-Newtonian Calculus was reviewed by David Preiss in the journal Aplikace Matematiky. [46]
Non-Newtonian Calculus was reviewed in the journal Physikalische Blätter. [62]
Non-Newtonian Calculus was reviewed in the journal "Scientia"; Rivista di Scienza. [63]
Non-Newtonian Calculus was reviewed in the journal Philosophia mathematica. [65]
Non-Newtonian Calculus was reviewed in the journal Revue du CETHEDEC. [68]
Non-Newtonian Calculus was reviewed in the journal Bollettino della Unione Matematica Italiana. [71]
Non-Newtonian Calculus was reviewed in the journal Cahiers du Centre d'Etudes de Recherche Opérationnelle. [72]
The First Nonlinear System of Differential and Integral Calculus was anonymously reviewed at abebooks.com in 2013. [168]
Excerpt: "In the spirit of non-Euclidean geometry ... , so [Michael Grossman] and his colleague Robert Katz have invented [non- Newtonian] calculus. ... Totally 'out there' in scope, it appears genuine and valid."
Non-Newtonian Calculus was reviewed at amazon.com by Steven Lesko in 2006. [170]
Excerpt: "A Top Mathematical Breakthrough: This humble, eloquent masterpiece gives a concise and readable introduction to a most powerful tool. ... Up there with the likes of Non-Euclidean geometry ..."
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.) [224]
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] (Professor Bashirov also presented his "Bigeometric Complex Calculus" at the special-session called "Mathematical Modeling and Computational PDE".) 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."
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