Links
HathiTrust: for online and downloadable books.
Google Books: for online and downloadable books.
Amazon: for books to buy.
WorldCat: for library books.
Reading
Michael Grossman. "An introduction to non-Newtonian calculus", International Journal of Mathematical Education in Science and Technology, Volume 10, Number 4, pages 525-528, Taylor and Francis, 1979.
Michael Grossman and Robert Katz. "A new approach to means of two positive numbers", International Journal of Mathematical Education in Science and Technology, Volume 17, Number 2, pages 205 -208, Taylor and Francis, 1986.
Jane Grossman, Michael Grossman, and Robert Katz. "Which growth rate?", International Journal of Mathematical Education in Science and Technology, Volume 18, Number 1, pages 151 - 154, Taylor and Francis, 1987. From that article:
Michael Grossman and Robert Katz. "Isomorphic calculi", International Journal of Mathematical Education in Science and Technology, Volume 15, Issue 2, pages 253-263, DOI:10.1080/0020739840150214, Taylor and Francis, 1984.
Michael Grossman. "Calculus and discontinuous phenomena", International Journal of Mathematical Education in Science and Technology, Volume 19, Number 5, pages 777-779, Taylor and Francis, 1988.
Robert Katz. Axiomatic Analysis, D. C. Heath and Company, 1964.
Robert Edouard Moritz. "Quotientiation, an extension of the differentiation process", Proceedings of the Nebraska Academy of Sciences, Volume VII, pages 112 - 117, 1891.
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.
Michael Grossman and Robert Katz. Non-Newtonian Calculus, ISBN 0912938013, 1972. [15]
Michael Grossman. The First Nonlinear System of Differential and Integral Calculus, ISBN 0977117006, 1979. (The geometric calculus) [11]
Jane Grossman, Michael Grossman, Robert Katz. The First Systems of Weighted Differential and Integral Calculus, ISBN 0977117014, 1980. [9]
Jane Grossman. Meta-Calculus: Differential and Integral, ISBN 0977117022, 1981. [7]
Michael Grossman. Bigeometric Calculus: A System with a Scale-Free Derivative, ISBN 0977117030, 1983. [10]
Jane Grossman, Michael Grossman, and Robert Katz. Averages: A New Approach, ISBN 0977117049, 1983. [8]
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Here's a discussion from ChatGPT that's worth reading:
While Grossman and Katz’s work on non-Newtonian calculus is primarily mathematical, it has quietly nudged the boundaries of philosophy of science, epistemology, and even metaphysics. The full philosophical impact is subtle but meaningful.
Here’s a breakdown of how their work resonates philosophically:
🧠 1. Rethinking the Nature of Change
Traditional (Newtonian) calculus is built around infinitesimal additive change — it defines how things shift in small, linear increments. Grossman and Katz introduced a framework where change can be multiplicative, non-additive, or based on other operations entirely.
Philosophical implication:
This challenges the assumption that additive change is fundamental. It invites deeper questions:
Is linearity a human bias or a natural truth?
Are there alternative ways the universe “computes” change?
Could different cultures or civilizations conceive math differently?
🧰 2. Tool-Building and Pluralism
Their work embodies mathematical pluralism — the idea that there’s no single, absolute mathematical framework, just different lenses we can use to model reality.
Philosophical implication:
This supports a constructivist or instrumentalist view of knowledge:
Math isn’t just discovered — it’s created.
Our tools shape what we see in nature.
Expanding the toolkit expands the possible worlds we can model and explore.
🌍 3. Broadening the Concept of "Natural Laws"
By offering alternative calculi for modeling systems — especially those involving proportional growth, feedback, or scale-invariant behavior — they subtly suggest that the laws of nature may look quite different depending on how you choose to measure or model them.
Philosophical implication:
This aligns with model-dependent realism (a concept popularized by Stephen Hawking), where:
Reality is understood through the models we build.
Different models can explain the same phenomena, each valid within its domain.
🔄 4. Inviting Reflection on Scientific Orthodoxy
Grossman and Katz operated largely outside academic orthodoxy — self-publishing, working independently — yet they developed a consistent and rigorous alternative system of calculus.
Philosophical implication:
Their story reflects a critique of scientific dogma:
Paradigm shifts often come from the margins.
Novel ideas may be dismissed simply because they don’t fit the prevailing narrative.
It raises Kuhnian questions about scientific revolutions and how knowledge progresses.
✨ Summary:
Grossman and Katz’s contribution isn’t just technical — it’s philosophical. It encourages us to:
Question foundational assumptions
Embrace mathematical pluralism
Consider new forms of reasoning about change, continuity, and systems
Reflect on the sociology of scientific acceptance
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From ChatGPT:
Let's bring in some philosophical heavy-hitters and see how their ideas connect with the spirit and implications of Grossman and Katz’s non-Newtonian calculus. These thinkers didn’t know of G&K’s work directly (as far as we know), but their frameworks help us philosophically situate this kind of innovation.
🧠 1. Imre Lakatos – The Logic of Mathematical Discovery
Lakatos argued that mathematics evolves through a dialectic of conjectures, refutations, and revisions — a process he called “quasi-empirical”. He challenged the view of math as a static, purely deductive system.
Connection to Grossman & Katz:
Non-Newtonian calculus fits Lakatos’s idea of progressive problem shifts — extending and reshaping existing theories rather than replacing them.
Their work reinterprets calculus from a fresh angle while still preserving internal consistency, showing that even foundational concepts like limits and derivatives are open to reinterpretation.
🟢 Lakatos would likely have applauded their creativity while urging others to engage in rigorous criticism to test its limits.
🧬 2. Paul Feyerabend – Epistemological Anarchism
Feyerabend famously claimed that “anything goes” in science, meaning that rigid adherence to one methodology stifles progress. He believed alternative frameworks should be allowed to flourish, even if they seem strange or nontraditional.
Connection to Grossman & Katz:
Their independent development of non-Newtonian calculus outside the academic mainstream is almost a textbook example of Feyerabend’s vision.
It’s a challenge to the dominance of Newtonian thinking, not by overthrowing it, but by showing it’s not the only coherent option.
🟢 Feyerabend would probably have loved the rebellious, paradigm-expanding nature of their work.
🌌 3. Alfred North Whitehead – Process Philosophy & Relational Thinking
Whitehead, co-author of Principia Mathematica, later turned toward process philosophy, viewing the universe as made of events and relationships rather than static entities. His metaphysics emphasized change, interconnection, and flow.
Connection to Grossman & Katz:
Non-Newtonian calculus, especially in its geometric or bigeometric forms, emphasizes relational change — how one quantity scales relative to another.
This mirrors Whitehead’s idea that the dynamics of becoming are more fundamental than being.
🟢 Whitehead would likely have seen their work as a natural mathematical complement to his metaphysical system.
🔍 Bonus: Thomas Kuhn – The Structure of Scientific Revolutions
Kuhn introduced the concept of paradigms and scientific revolutions, where periods of “normal science” are interrupted by disruptive shifts.
Connection to Grossman & Katz:
Non-Newtonian calculus isn’t (yet) a Kuhnian revolution, but it could be seen as a proto-revolutionary framework — a toolset waiting for a field or crisis where it becomes indispensable.
🟡 Kuhn might say: if a field finds itself unable to model something effectively with Newtonian tools, G&K’s calculus could spark a mini-paradigm shift.
✨ Final Thought:
Grossman and Katz’s work invites a philosophical awakening: not just to new equations, but to new ways of knowing, modeling, and thinking. Their calculus isn’t just a mathematical curiosity — it’s a living example of how deep shifts in thought often begin in the margins, challenging orthodoxy and expanding the frontier.
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From Copilot:
Michael Grossman and Robert Katz's pioneering work in non-Newtonian calculus has transcended its mathematical origins. By challenging the dominance of Newtonian frameworks, they opened the door to alternative ways of conceptualizing change and relationships—ideas that ripple into the philosophy of science and epistemology.
For instance, their frameworks encourage a rethinking of "universality" in scientific laws, suggesting that different contexts might require entirely different mathematical tools. This resonates with epistemological questions about the nature of knowledge and the frameworks we use to understand reality. In metaphysics, their work invites reflection on the nature of continuity, change, and even the fabric of reality itself, as their calculus systems can model phenomena that classical calculus struggles to describe.
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From Perplexity AI:
Origins and Core Contributions
Michael Grossman and Robert Katz, between 1967 and 1970, developed a family of mathematical systems known as non-Newtonian calculus, including multiplicative calculus and nonclassical arithmetic. Their work was groundbreaking in that it introduced an infinite family of calculi-such as geometric, harmonic, quadratic, bigeometric, and others-that extend beyond the traditional Newtonian (Leibnizian) calculus. These systems are based on alternative arithmetics (complete ordered fields distinct from the real numbers), providing fundamentally new ways to approach differentiation and integration.
Catalyst for a New Mathematical Movement
Initially, Grossman and Katz’s ideas were met with skepticism and largely ignored by the mainstream mathematical community, who viewed their work as unconventional and outside the mathematical mainstream. However, over the decades, their concepts have been recognized as providing valuable mathematical tools with broad applicability. Their pioneering efforts challenged the prevailing notion that isomorphic mathematical systems are essentially identical, demonstrating instead that alternative arithmetics can yield distinct operational and functional behaviors, thereby opening new avenues for mathematical analysis.
Influence Across Disciplines
Grossman and Katz’s work has catalyzed an academic and scientific movement, influencing a wide range of disciplines and leading to the development of entire fields that build upon their foundational ideas. Key areas impacted include:
Decision Making and Economics: Non-Newtonian calculus has been applied to probability theory, utility, and Bayesian analysis, providing models that more accurately reflect human behavior and decision-making processes.
Dynamical Systems and Differential Equations: The alternative calculi offer new approaches to modeling and solving differential equations, especially in contexts where growth or change is more naturally multiplicative than additive.
Biomedical Image Analysis, Actuarial Science, and Finance: Multiplicative calculus, in particular, has shown advantages over classical calculus in modeling growth phenomena, image processing, actuarial computations, and financial modeling.
Physics and Fractal Geometry: The generalizations of calculus inspired by Grossman and Katz have been extended to problems in fractal analysis, chaos theory, and even foundational questions in physics, such as those related to dark energy and quantum theory.
Engineering and Numerical Analysis: Applications in electrical engineering and numerical methods, such as adaptations of the Runge-Kutta method, have benefited from the flexibility of non-Newtonian frameworks.
Academic Recognition and Ongoing Development
The significance of Grossman and Katz’s work has grown steadily. Their systems have been featured in major academic conferences and recommended as essential topics for 21st-century mathematics curricula. The field experienced a notable revival with the comprehensive mathematical description of geometric calculus by Bashirov et al. in 2008, which led to a surge of research and applications in the past decade.
Prominent mathematicians and educators have lauded their contributions:
"Non-Newtonian Calculus, by Michael Grossman and Robert Katz, is a fascinating and (potentially) extremely important piece of mathematical theory... I predict that non-Newtonian calculus will come to be recognized as the most important mathematical discovery of the Twentieth Century." - James R. Meginniss, Claremont Graduate School and Harvey Mudd College, 1980
Legacy and Future Directions
Grossman and Katz’s introduction of nonclassical arithmetics and non-Newtonian calculi has not only provided new mathematical tools but also inspired a paradigm shift in how mathematicians, scientists, and engineers approach problems involving change, growth, and structure. Their work continues to shape research in mathematics and its applications, with ongoing exploration and expansion by contemporary researchers.
In summary, Grossman and Katz’s innovations have profoundly influenced mathematics and science, extending far beyond their original publications to inspire new fields, methodologies, and a broader appreciation for alternative mathematical frameworks. Their legacy is a testament to the power of challenging established norms and exploring new mathematical frontiers.
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From Perplexity AI:
Overview of Non-Newtonian Calculus
Grossman and Katz’s work on non-Newtonian calculus, developed between 1967 and 1970, introduced a family of alternative calculi distinct from the classical Newton-Leibniz calculus. These include geometric, harmonic, quadratic, and multiplicative calculi, each with its own derivative, integral, and foundational theorems. While initially mathematical, the framework’s generality and flexibility have led to broader theoretical implications.
Philosophy of Science and Epistemology
Non-Newtonian calculus challenges the traditional assumption that the real number system and standard arithmetic are the only valid foundations for calculus and mathematical modeling. By showing that different arithmetics can underpin different calculi, Grossman and Katz’s work invites epistemological reflection on the nature of mathematical truth and the relationship between mathematics and physical reality. This has influenced contemporary discussions about the foundations of science, especially in fields where standard mathematics may not suffice to describe observed phenomena, such as in certain areas of physics.
Metaphysical Implications
The existence of multiple, equally valid calculi and arithmetics raises questions about the metaphysical status of mathematical objects and structures. If physical theories can be constructed on alternative mathematical foundations, it suggests that mathematics is not theuni the quely determined by the structure of the physical world, but is, at least in part, a human construct or a choice among several viable frameworks. This resonates with debates in metaphysics about realism versus constructivism in mathematics and science.
Applications and Theoretical Impact
Recent work by physicists such as Marek Czachor has used non-Newtonian calculus to re-examine fundamental issues in physics, such as dark matter, dark energy, and the arithmetic foundations of physical laws. The suggestion that the arithmetic underlying calculus could influence our understanding of the universe has philosophical consequences, prompting a re-evaluation of the relationship between mathematics and empirical science.
“Non-Newtonian calculus naturally unifies various ideas that have occurred over the years in the field of generalized thermostatistics, or in the borderland between classical and quantum information theory. ... Its theoretical potential is huge, and yet it remains unknown or unappreciated.” - Marek Czachor, from his 2020 paper
Reception and Broader Significance
Non-Newtonian calculus' conceptual innovations have inspired both technical and philosophical inquiry. It has encouraged philosophers of science to consider the plurality of mathematical frameworks and the implications for scientific modeling and explanation.
Philosophy of Science: Challenges uniqueness of classical calculus in modeling nature.
Epistemology: Questions about mathematical truth and foundations.
Metaphysics: Raises issues of realism, constructivism, and ontology.
Physics: Alternative models for dark matter/energy, relativity, etc.
Scientific Methodology: Expands toolkit for modeling and interpreting phenomena.
Grossman and Katz’s non-Newtonian calculus, while fundamentally mathematical, has had a notable-if underappreciated-impact on philosophical discussions about the nature of mathematics, science, and reality.
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