From Mathematical Gnostics to Machine Gnostics: A History¶

Introduction¶

Mathematical Gnostics is a unique approach to understanding data uncertainty—one that challenges conventional statistical thinking. Rather than viewing uncertainty as indeterministic or purely random, Mathematical Gnostics treats every data point as the result of measurable, material causes, even if not all influencing factors are directly observable. This philosophy underpins the Machine Gnostics library, which brings these ideas into the realm of modern data analysis and machine learning.

This page traces the origins of Mathematical Gnostics, its sources of inspiration, the birth of Machine Gnostics, and the impact these ideas are having today.

Historical Background¶

The origins of Mathematical Gnostics trace back to the early 1980s, when Pavel Kovanic, a researcher at the Institute of Information Theory and Automation of the Czechoslovak Academy of Sciences in Prague, first published his pioneering work. His approach was met with skepticism by mainstream statisticians, as it departed radically from accepted paradigms. Nevertheless, the theory continued to develop and gain traction through both theoretical advances and practical applications.

Key milestones in the development of Mathematical Gnostics include:

As a new paradigm of data variability, mathematical gnostics has been developing since the end of the 1970s and applied in a number of research projects.
The first printed publications in 1984 and presentation at an international conference.
Practical applications in economics and financial analysis, leading to a series of books and publications.
International collaborations, notably with American professor M.B. Humber, resulting in a co-authored book (unfinished due to his passing) and further dissemination of the theory.
Early private implementations of gnostic software in several languages, including BASIC, C, S-PLUS, and limited C++ versions.
The first free/public software implementation in R, which may still be in use today.
A significant step was taken by Z. Wagner, who developed an independent version of gnostic software in the Octave language. This software was first applied to both offline and fully automatic online analysis of particle size distribution of atmospheric aerosol, and then used within the TTSM team at the Institute of Chemical Processes of the Czech Academy of Sciences. Although it had very limited user documentation, it supported research-level applications and contributed to a series of articles published in top journals.
Application of gnostic methods in environmental and medical research, including participation in major European Union projects.

For more information on the history and development of mathematical gnostics, visit: math-gnostics.eu

Sources of Inspiration¶

Unlike traditional statistical methods, which rely on large data samples and probabilistic assumptions, Mathematical Gnostics is rooted in the laws of nature and the analysis of individual data items—even in small samples. This non-statistical approach treats data uncertainty as a consequence of real, measurable conditions, not as mere randomness.

The development of Mathematical Gnostics was inspired by both the limitations of classical statistics and the foundational principles of several scientific disciplines:

Theory of General Systems: The gnostic cycle of observation and feedback, emphasizing knowledge (from the Greek “gnosis”) rather than a priori assumptions.
Theory of Measurement: H. von Helmholtz’s work on quantification and the mathematical structure of measurement.
Geometry: The use of non-Euclidean geometries (Riemannian, Minkowskian) to model uncertainty and variability.
Relativistic Physics: Insights from Einstein’s mechanics, connecting data variability to the movement of relativistic particles.
Thermodynamics: The original thermodynamic concept of entropy, as introduced by R. Clausius, applied to data uncertainty.
Electromagnetism: J.C. Maxwell’s theories, including the concept of Maxwell’s demon, linking entropy and information.
Matrix Algebra: The manipulation of data structures and operations, as formalized in modern algebra and implemented in computational tools.

These inspirations, drawn from the natural sciences, provide a robust and universal foundation for understanding data uncertainty—making Mathematical Gnostics a powerful alternative to classical statistics.

“Let data speak for themselves.” This guiding principle reflects the core philosophy of Mathematical Gnostics: to extract maximum information from data, relying on the data values themselves, and to model uncertainty in a way that is consistent with the laws of nature.

The Birth of Machine Gnostics¶

The realization that the robust, nature-inspired foundation of Mathematical Gnostics could serve as the basis for a new generation of artificial intelligence and machine learning was a key motivation for the creation of Machine Gnostics. In 2022, Dr. Nirmal Parmar began exploring the integration of mathematical gnostics with modern machine learning, seeking to create models and algorithms that are both assumption-free and deeply aligned with the laws of nature.

This work led to the birth of Machine Gnostics: an open source project dedicated to providing a unified framework for data analysis models, machine learning models, and—looking ahead—a future deep learning framework. Machine Gnostics aims to empower researchers, engineers, and practitioners with tools that combine the rigor of mathematical gnostics with the flexibility and power of contemporary AI.

By making Machine Gnostics open source, the project invites collaboration and innovation from the global community, ensuring that the theory and its applications continue to evolve and serve a wide range of scientific and practical needs.

The impact of this integration is already being felt, and the vision for Machine Gnostics continues to grow.

Impact and Vision¶

The journey from Mathematical Gnostics to Machine Gnostics marks a significant shift in how we approach data uncertainty, analysis, and artificial intelligence. By grounding our methods in the laws of nature and embracing a non-statistical, axiomatic foundation, we have opened new avenues for robust, assumption-free data science. Machine Gnostics is already empowering researchers and practitioners to extract deeper insights from data—whether in small samples, complex systems, or real-world applications where traditional statistics fall short.

Looking ahead, the vision for Machine Gnostics is ambitious: to become a universal framework for data analysis, machine learning, and, ultimately, deep learning. By remaining open source and community-driven, Machine Gnostics invites collaboration, innovation, and critical feedback from scientists, engineers, and thinkers worldwide. Together, we can continue to push the boundaries of what is possible in data-driven discovery.

Testimonials¶

Dr. Zdeněk Wagner¶

'Listening to the data'

New discoveries and paradigms usually arrive when there is a need to get a solution to an unsolvable problem. They have their time to come. Many years ago I had to solve a complex task, regression of high pressure vapour-liquid equilibria (HPVLE). It poses a lot of difficulties with convergence because at the mixture critical point some derivatives are infinite, with slightly modified interaction parameters in the mixing rules the equations have no solution, and, last but not least, values measured by different authors are not always in agreement and thus oscillations in the iterative algorithm can occur. The statistical tools did not work for me. At that time I visited a seminar and met Dr. Pavel Kovanic who presented his novel paradigm of uncertainty, mathematical gnostics (MG). We discussed almost the whole night because it looked promising, especially with the key rule, let the data speak for themselves, i.e. do not force them to have a particular distribution of measurement errors but get the shape of the distribution function from the data. We met uncountable number of time after the seminar and I received full support from him. I first tried MG on simple tasks, I made my first three programs (the next was always a little more complex and more versatile than the preceding one) and found that it really can solve the tasks better.

It was not easy to persuade statisticians that MG is a valid tool especially because HPVLE was still difficult. And another need came. A colleague who dealt with atmospheric aerosols had twenty thousand data sets and did not know how to analyze them. He told me that he would believe MG if I analyze the data and get useful results. And after some time I told him that there is something strange every day approximately half an hour before midnight and half an hour after midnight. And his response was: “Oh, these bad guys!” The data were measured at the top of a mountain by several groups and many instruments. They agreed that they would leave the cars below the mountain in order not to take samples of the exhaust gases. People from one group had to change the filters in the instrument exactly at the midnight and they decided to go up to the top by cars because nobody sees them. And analysis of the data by MG found it. Since then I have analyzed many millions of data sets for the aerosol community.

Yet another need came when we were in danger of losing a job. At that time my new colleague, Dr. Magdalena Bendová, got interest in MG. We decided that we should show that we have special knowledge. Being inspired by Nassim Nicholas Taleb we turned MG into our “black swan”. Explaining her this new paradigm helped me to realize what the reviewers would understand. Together we managed to promote MG so that it is now accepted by reviewers of good scientific journals. MG then became our main tool for data analysis and recently Magdalena's PhD student, Dr. Nirmal Parmar, opens a new horizon for novel applications. And I still learn how to teach this subject.

Dr. Magdalena Bendovà¶

'My Journey with Mathematical Gnostics'

When I joined the Eduard Hála Laboratory of Thermodynamics in 2003, I had no idea that something like the theory of Mathematical Gnostics even existed. It was only when I began discussing how to analyze the phase equilibrium data I had just measured with my colleague, Dr. Zdeněk Wagner, that the path revealed itself. He suggested trying a robust regression method to estimate parameters of the thermodynamic models I proposed to use, using MG. That led to our first paper together and to my introduction to both the method and its author, Dr. Pavel Kovanic.

In that first publication, Zdeněk tried his best to present the robust regression along a gnostic influence function in the simplest way possible, just enough so that the reviewers that were used to using statistical methods wouldn’t get too mad at us. Years have passed, and such precautions are no longer necessary.

Over time, I learned how to use the custom-made scripts that Zdeněk had coded; to analyze my data, to critically assess it, and to detect outliers. This led to many more papers, each allowing us to push the boundaries of MG (and of the referees’ comprehension) further. For my part, I know I have only scratched the surface of this remarkable theory. Yet I remain convinced that it is one of the most powerful tools in data analysis. Not a rival to statistics, but rather a complementary approach that fills the gaps where traditional statistical methods fall short.

I’m especially proud that our Ph.D. student, Nirmal Parmar, who has since soared toward his own horizons, embraced this theory and made it his own. He’s now exploring how to combine machine learning with mathematical gnostics. It won’t be smooth sailing. The theory is complex and the project ambitious. But I am confident that, with more successful implementations, it will reach a broader audience and gain recognition in both the scientific and industrial communities for its profound usefulness.

Acknowledgments¶

Acknowledgments

The development of Mathematical Gnostics and its evolution into Machine Gnostics would not have been possible without the dedication and insight of many individuals. I would like to express my deepest gratitude to:

-Dr. Pavel Kovanic (1942–2023), for his foundational work and vision in creating Mathematical Gnostics. His legacy continues to inspire this project and the broader scientific community. His guidance and pioneering spirit remain a guiding light for all who build upon his work.

-Dr. Magdalena Bendová, my PhD supervisor, for her guidance, encouragement, and support throughout my research journey.

-Dr. Zdeněk Wagner, my expert supervisor, whose expertise in Mathematical Gnostics and his development of the Octave software for data analysis were instrumental in my understanding of the field. His mentorship inspired me to extend these ideas further and integrate them with machine learning and artificial intelligence.

I am also grateful to all colleagues, collaborators, and students who have contributed ideas, feedback, and encouragement along the way. The open source community, with its spirit of sharing and innovation, continues to inspire the ongoing growth of this project.

Dr. Nirmal Parmar

If you are interested in contributing, collaborating, or simply learning more, we welcome you to join us on this journey.

Explore the documentation, try the tools, and help shape the future of Machine Gnostics.