Several questions are posed here, like What is evolution, really? or What is the difference between technological and living systems? How can we engineer living systems? What are the mechanisms for which nature transitions from distributed systems to autonomous systems and the other way around? How is the presence of an autonomous system changes the dynamics of the processes around him? How can autonomous systems achieve near error-free outcomes? How has nature been able to build such a scalable and robust architecture? How can some of these investigations help us develop new computational paradigms and new manufacturing architectures? Is there a way to generate living systems out of novel materials? How does life itself enter into the whole picture?

• These are the set of questions that we are trying to solve in the Galapagos Project.

Within the field of foundations of mathematics, a set of tools have been invented in the past one hundred years that remain underused in the sciences. Some of these include sheaf logics, paraconsistent logics, reverse mathematics, Homotopy type theory, Higher structures, measurement theory among others. Some of these tools could be used to develop theories of general observers, logic in quantum mechanics, hierarchies in biology, locally consistent models in ecosystem biology and neuroscience, and perhaps in other fields like economics and social sciences. Moreover, some of these tools are already in use, the tools coming from the field of automated reasoning in logic have been used in software and hardware verification and they are currently making a big impact in the mathematical community with proofs assistants like Lean, Metamath, and others. The study of these tools and their results is interesting in itself as well as its possible application outside of mathematics, like in blockchain technologies and in material computing verification.

• Some of the efforts in this line of inquiry are aligned with the empirical metamathematics and the multicomputation projects of the Wolfram Physics Project.

• Another line of effort in this topic involves material computing verification in a chemical compiler.

The concepts of entropy, energy, heat, and information have been puzzling scientists for centuries. Different frameworks have been proposed to understand these concepts, some of them are Classical Thermodynamics, Kinetic theory, Statistical Mechanics, Axiomatic Thermodynamics, Information Theory, Statistical Estimation, Optimal Transport & Information Geometry, Non-Equilibrium Thermodynamics, Resource Theory and recently Constructor Theory. Each of the frameworks has enlightened applications in chemistry, plasma physics, quantum mechanics, computation, pure mathematics, and even in applied fields like biology, statistics, and signal processing. Each perspective helps build a more complete picture of the nature of matter and information and its understanding is helping even in foundational questions like the nature of time.

• An effort in this line of inquiry is directed towards the formulation of the thermodynamics of universal constructors, and their consequences for the nature of computation, biology, and time.
• Another line of effort is to use the frameworks of description for thermodynamics for connection with other fields, like symplectic geometry, Information Geometry, Quantum Mechanics, and use it for applications in imaging and statistics.

(Draft)-Entropic Methods in Geometry, Imaging and Statistics Part 1: Statistics Theory