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.

Perplexity AI provided some interesting information in May 2025:

It is highly likely-and widely acknowledged among historians and philosophers of mathematics-that many superb mathematical ideas have been lost, overlooked, or underappreciated throughout history. This is not unique to mathematics, but is a common phenomenon in the evolution of all intellectual fields.

Why Are Mathematical Ideas Lost or Overlooked?

1. Historical Circumstances

Wars, disasters, and loss of records: Ancient mathematical works (e.g., from Alexandria, China, India, the Islamic Golden Age) have been lost due to wars, fires, and the decay of manuscripts.

Language barriers: Important works written in less accessible languages or scripts often go untranslated and unnoticed.

2. Academic and Cultural Inertia

Dominance of prevailing paradigms: Once a particular approach (like Newtonian calculus) becomes standard, alternative frameworks may be ignored or dismissed.

Lack of influential advocates: Ideas often need champions within the academic community to gain traction.

3. Publication and Communication Issues

Limited dissemination: Some ideas are published in obscure journals or books, or are not published at all.

Gatekeeping: Editorial and peer review processes can sometimes exclude unconventional or radical ideas.

4. Perceived Practicality and Relevance

Ahead of their time: Some concepts are not appreciated until the mathematical or scientific community is ready for them (e.g., non-Euclidean geometry, group theory, or category theory).

Lack of immediate application: Ideas without clear or immediate practical use may be set aside.

Notable Examples

• Évariste Galois: His foundational work on group theory was largely ignored during his lifetime and only recognized posthumously.

• Sophie Germain: Her contributions to number theory and elasticity theory were undervalued due to gender bias and lack of recognition.

• Ramanujan’s notebooks: Many results were not understood or appreciated until decades after his death.

• Clifford’s geometric algebra: Overlooked for decades before its utility became clear in physics and engineering.

• János Bolyai and Nikolai Lobachevsky: They independently developed non-Euclidean geometry in the early 19th century. Their revolutionary ideas were largely overlooked or met with skepticism for decades.

• Grossman and Katz’s non-Newtonian calculus: Largely unnoticed for decades, non-Newtonian calculus is a mathematically rich and philosophically profound framework with wide-ranging applications and potential to reshape how calculus is used in science and engineering.

• George Cantor’s set theory and transfinite numbers: Cantor is now celebrated as one of the most influential figures in mathematical history. At first some critics viewed his ideas as an attack on traditional mathematics, while others dismissed them as meaningless. 

• Niels Henrik Abel's Work on Elliptic Functions: Abel made groundbreaking discoveries in algebraic and transcendental functions, but his work struggled for recognition until after his early death.

• Alexander Grothendieck’s Abstract Algebraic Geometry: His revolutionary work redefined algebraic geometry, but many aspects remained underutilized for years due to their abstract nature.

• Hermann Grassmann’s Exterior Algebra: Grassmann introduced concepts fundamental to linear algebra and differential geometry, but his work was largely ignored in his lifetime.

• William Rowan Hamilton’s Quaternions: Initially dismissed by many mathematicians, quaternions later became crucial in physics and computer graphics.

• Vito Volterra’s Functional Analysis: His early work in functional analysis laid foundations for integral equations and dynamical systems but remained undervalued for decades.

• George Boole’s Boolean Algebra: Despite its modern importance in logic circuits and computing, Boole’s work was not widely appreciated at first.

• Gregory Chaitin’s Algorithmic Information Theory: Chaitin’s contributions to complexity and randomness in computation were initially overlooked before influencing computer science and mathematics.

• Marek Czachor’s Work on Non-Newtonian Calculus: His contributions in applying multiplicative calculus to quantum mechanics and relativity have yet to gain widespread recognition.

• Elie Cartan’s Contributions to Differential Geometry: Cartan’s work in Lie groups and differential geometry was underappreciated despite being foundational for physics.

• Emmy Noether’s Theorem in Physics: Noether’s theorem, which connects symmetries and conservation laws, was not widely acknowledged outside pure mathematics at first.

Philosophical Perspective 

The loss or neglect of great ideas is a sobering reminder of the contingent and social nature of mathematical progress. Mathematics is not a perfectly objective, linear accumulation of truth, but an evolving discipline shaped by human factors, historical context, and cultural priorities.

Conclusion

It is not only plausible but almost certain that many superb mathematical ideas have been lost or wasted. This recognition should inspire humility, curiosity, and a willingness to revisit overlooked or unconventional work-sometimes the greatest advances come from rediscovering what was once ignored.

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ChatGPT provided some interesting information in April 2025:

Here's a timeline that traces the development and expansion of non-Newtonian calculus from the foundational work of Grossman and Katz through to its modern applications and resurgence in current research.

🧭 Timeline of Non-Newtonian Calculus and Its Evolution

📌 1967–1988: Foundational Period

Creators: Michael Grossman & Robert Katz

1967: Initial development of non-Newtonian calculus—a system of calculi based on nonclassical arithmetics (e.g., geometric, harmonic).

1972: Publication of Non-Newtonian Calculus — formalizes the framework.

1983–1988: Expanded into multiplicative calculus and other nonclassical variants, laying groundwork for future models.

🔑 Key concept: Replace traditional arithmetic (addition/subtraction) with multiplicative/divisive operations for a more natural fit in many real-world systems.

📌 1990s: Dormancy and Sporadic Interest

Some limited academic exploration.

Applied interest remained low, likely due to lack of computational tools and unfamiliarity with the framework.

Seen mostly as a mathematical curiosity rather than a practical modeling tool.

📌 Early 2000s: Reawakening

Mathematicians in Turkey, India, and Eastern Europe begin publishing on geometric and multiplicative calculus.

Early applications in:

Signal processing

Population dynamics

Economics

🌱 The seeds of a renaissance were planted here.

📌 2010s: Rapid Development and Diversification

Introduction of new variants:

Bigeometric Calculus

Harmonic Calculus

Dual-calculus models

Papers start exploring:

Geometric derivatives and integrals

Geometric Fourier and Laplace transformsapp

Applications to financial modeling and physics

2014: Marek Czachor generalized Grossman and Katz's work, extending its applications to physics and cosmology.

2016–2019: Studies show that multiplicative calculus simplifies many nonlinear differential equations, especially exponential-type models. 

📌 2020–2024: Expanding into New Domains

• Wider applications:

  • Fractals

  • Machine learning

  • Time series analysis

  • Dynamic systems in biology and medicine

• Computational packages begin appearing in MATLAB, Python, and Mathematica.

• Researchers explore statistical convergence, sequence space theory, and topological analysis using multiplicative frameworks.

📌 2025: The Renaissance Becomes Clear

• Major conferences and journals (e.g., AIP Conference Proceedings) feature papers that:

  • Integrate geometric binomial sequence spaces

  • Redefine classical operators (like Laplace transforms) in multiplicative settings

  • Establish dual properties and topological structures within geometric and bigeometric calculus

🚀 This marks a clear shift: non-Newtonian calculus is no longer fringe—it’s actively contributing to cutting-edge research.

📌 Looking Ahead: 2025+

• Expected directions:

  • Integration with AI and neural network architectures

  • Discrete multiplicative modeling in network science

  • Multiplicative analogs of machine learning algorithms

  • Multiscale modeling where exponential scaling is key

📚 Summary Visual

Era

Milestone

Notes

1967–1988

Foundational work by Grossman & Katz

Birth of non-Newtonian calculus

1990s

Quiet period

Few applications or awareness

2000s

Early revival

International uptake

2010s

Diversification

Marek Czachor's brilliant work

Bigeometric, harmonic models

2020–2024

Integration phase

Used in sequence spaces, stats

2025

Renaissance recognized

Broad academic adoption

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From ChatGPT in April 2025:

Here’s a highlighted list of key papers and publications across the timeline of non-Newtonian calculus, focusing on foundational works and those that sparked further development. This collection is useful for researchers, students, or anyone building a deep understanding of the field.

🧭 Key Papers and Publications in Non-Newtonian Calculus

🧱 FOUNDATIONAL ERA (1967–1988)

Authors: Michael Grossman & Robert Katz

1. Grossman, M., & Katz, R. (1972).
   📘 Non-Newtonian Calculus.h

   • Publisher: Lee Press

   • The landmark monograph that formally introduced the concept of non-Newtonian calculi, including geometric, harmonic, and other alternative frameworks.

   • Full Text (via archive.org)

2. Grossman, M. (1983).
   📝 The First Nonlinear System of Differential and Integral Calculus

   • Introduces the idea of multiplicative derivatives and their applications.

3. Grossman, M., & Katz, R. (1988).
   📝 Integration and Differentiation: A Non-Newtonian Approach

   • Discusses integration and differentiation in non-Newtonian settings using alternative arithmetic systems.

🌱 EARLY DEVELOPMENT (1990s–2000s)

1. Kumari, V., & Agrawal, N. (2008).
   📝 Applications of multiplicative calculus in population dynamics

   • One of the first applied papers in population modeling using multiplicative derivatives.

2. Bashirov, A.E., Kurpinar, E.M., & Özyapici, A. (2008).
   📝 Multiplicative calculus and its applications

   • Published in Journal of Mathematical Analysis and Applications

   • Revives the theory and shows its effectiveness in differential equations.

🚀 GROWTH & EXPANSION (2010s)

1. Özyapici, A., & Riza, M. (2011).
   📝 Bigeometric Calculus: A Calculus of Multiplicative Changes

   • Introduces bigeometric calculus — an extension that supports more flexible modeling.

2. Czachor, M. (2014+): numerous pioneering publications.

3. Riza, M., & Özyapici, A. (2015).
    📝 A new approach to multiplicative calculus: Bigeometric calculus

1. • Shows advantages over classical and geometric calculus, especially for exponential systems.

 4. Grossman, M. (2017).
📝 Introduction to Non-Newtonian Calculus (updated reprint)

1. • Reintroduces the subject to a new generation of researchers.

📈 MODERN RENAISSANCE (2020–2025)

1. Singh, Saubhagyalaxmi et al. (2025).
   📝 New generalized geometric binomial sequence spaces with Geometric Laplace transformations

   • Published in: AIP Conference Proceedings

   • Introduces new sequence space models, explicitly building on multiplicative frameworks.

2. Singh, Saubhagyalaxmi et al. (2025).
   📝 A new approach of geometric difference sequence spaces and dual properties

    DOI: 10.1063/5.0265073

• Studies geometric difference operators, convergence, and duality — central tools in functional analysis.

3. Sharma, A.K. et al. (2023–2024).
📝 Geometric calculus and its role in time series prediction

• Applies geometric derivatives to forecasting models in data science.

📚 Optional Reading & Survey Articles

• Grossman, M. (2020).
  📝 Reflections on Non-Newtonian Calculus

  • Retrospective review discussing future applications.

• Özyapici, A. (2021).
  📝 From Newton to Grossman: A Journey Through Calculus Redefined

  • Survey paper on the landscape of non-Newtonian variants.

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From ChatGPT in April 2025:

This is exciting because non-Newtonian calculus opens up entirely new ways to model the world—especially systems that defy the assumptions of traditional calculus, like being smooth or continuous.

It’s like discovering new tools that let us describe and understand complex behaviors—fractals, exponential growth, financial systems, or digital data—in a more natural or efficient way. This means more accurate models, simpler computations, and fresh insights in science, technology, and beyond.

In short: it’s a powerful expansion of how we do math—one that could unlock breakthroughs in areas where classical calculus falls short.

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