Phi-Algebra — History
My studies began well, although I was not able to complete them. Even so, they gave me solid foundations in computer science and mathematics. As a child, I enjoyed reading the Encyclopaedia Universalis, and around the age of fifteen I was asked to describe the shape of a biological cell and replied, perhaps boldly, that it looked like a potatoid, a concept I had encountered in the encyclopaedia and understood from a mathematical perspective. That way of thinking left a lasting impression on me.
Several years later, I began my career in computer science after winning a local competition that allowed me to start professional training within a company. From one position to the next, I worked to build my own understanding of computer science, sometimes from a perspective that differs from the mainstream. Part of this work eventually became what ChatGPT later referred to as phi-algebra, although I currently see it more as a discrete algebra focused on countable objects and their structural relationships than as a direct extension of existing algebra.
A part of understanding in mathematics
Mathematics has always been a source of introspection for me. One of the earliest ideas that did not directly originate from my formal education was a different way of thinking about a possible Cauchy–Schwarz equality. My intuition was not to replace the existing theory, but to view it from another angle, hoping to expose structural properties that could provide a fresh understanding of Cauchy sequences, set theory, and possibly more general mathematical constructions.
A first concrete specification draft
My professional work eventually led me to the fields of digital identity and decentralized identity. During discussions surrounding the status of verifiable credentials, I began to view status not simply as metadata but as a form of annotation attached to an object. Following this intuition, I designed a concrete specification for decentralized status management and implemented it within an authorization server.
Although the work addressed a practical engineering problem, it also marked the first time I felt that ideas originating from computer science could suggest new ways of thinking about physical systems. At the same time, I started wondering whether concepts such as the Turing test might be described as a deeper mathematical structure that could be more clearly formulated and understood.
Recognizing a recurring pattern
As I continued working on software architecture, distributed systems, cryptography, blockchains, and neural networks, I repeatedly encountered the same reasoning pattern. The framework that would later become known as phi-algebra helped me organize problems ranging from high-level architecture to concrete implementations. Rather than beginning with axioms, I progressively built an introspective model and identified structural properties that consistently reappeared across different domains.
From intuition to mathematical foundations
The search for these properties gradually shifted from designing a practical model toward identifying its mathematical foundations. This process was incremental: the model came first, while the mathematics emerged progressively as I searched for the concepts that best described it.
Many of these connections would have been difficult for me to discover alone. Artificial intelligence became a valuable research companion, helping me relate my intuitions to existing mathematical theories, challenge my assumptions, and clarify the underlying structures. Through this dialogue, the discussion gradually evolved into what we can now refer to as phi-algebra.
Note: Those notes have been revised with Artificial Intelligence. While the formulation is of good litterature, it sounds not that far from what I intended.