The best books on new mathematical approaches to study complex systems

This book by G. Edelman played an important role in the development of our mathematical MES theory for complex "living" systems. Our specific application of MES to neuro-cognitive systems, named MENS, represents a kind of mathematical translation of Edelman’s book into Category Theory. 

Specifically, leveraging the categorical concept of a 'colimit,' we expand upon Edelman's principle of the "degeneracy of the neural code" by introducing a form of non-isomorphic redundancy termed the Multiplicity Principle (MP), wherein the system admits multifaceted components. Subsequently, we establish a significant result: if an MES adheres to the MP, the system is reliant not on pure reductionism but rather on an "emergentist-reductionism" as defined by the philosopher and physicist Mario Bunge.

We appreciate this book because it helped us to introduce the concept of a ‘hierarchical category,’ which is necessary to describe our MES. We accomplished this by translating Koestler's concept of a "hierarchy of holons," where a holon embodies a 'hybrid nature' akin to a two-faced Janus.

Technically, a hierarchical category organizes objects into numbered levels (0 to m). An object at level n is dual-faced: 'simple' compared to levels above n, but 'complex' compared to levels < n, this object being the "colimit" (or combination) of linked objects < n. Within a hierarchical category, we compute the 'complexity order' for each object. The category aligns with pure reductionism if it lacks objects with a complexity order greater than 1.

Thom's work on the "physics of meaningful forms," also known as Semiophysics, deeply influenced our approach to modeling structural changes in Memory-Evolutive Systems (MES). Indeed, in ‘natural’ systems, he classifies these changes as one of four standard changes—Birth, Death, Confluence, and Scission—which offer valuable insights into how components and interactions evolve over time within these systems. Integrating Thom's framework has enriched our understanding of MES dynamics. 

In MES, this led us to formulate the "Complexification Theorem," which characterizes the resulting category following structural changes: Birth or Death corresponds to adding or removing components, while Confluence leads to the formation of a colimit for a specific pattern of interconnected components, and Scission results in the decomposition of a given colimit.

The first three books contributed to the development of new categorical notions for our MES theory. Our fourth book takes a different turn, narrating the tale of an austere order of intellectuals secluded in Castalia, where they essentially engage in the cultivation and play of the Glass Bead Game. This game represents an abstract synthesis of all arts and sciences, stimulating creativity.

From what is said about the rules of the game, and in terms of Margareth Boden (well known for her research on creativity and A.I. ), this game satisfies Boden’s rules for combinatory and exploratory creativity but not for transformational one. We posed the mathematical challenge of modifying the game's rules to enable transformational creativity and, utilizing MES; we gave conditions for this to become possible.

The last book on our list is a recent addition, distinguished by its illustrative format and contemporary content; in particular. Structured uniquely, it comprises three parts (plus a foreword). 

The first section, titled "The Innovator’s Odyssey," narrates the journey of a young innovator designer grappling with bureaucratic hurdles in organizational settings. Amidst these challenges, he embarks on a quest for a "mathesis singularis"–a unique mathematical framework to aid innovation management. This quest leads him to discover our MES book.

The second part features an insightful interview with the author, offering valuable insights into the book's creation. Lastly, the third section includes a short Mathematical Appendix (for further exploration).