On January 1^{st} 2011 a new field for research was born. It was the begin of the EC FP7 project INBIOSA, Integral BIOmathics Support Action (**Grant ****№****. 269961)****.** Since then over 100 scientists from a number of disciplines worldwide have been exploring a substantial set of theoretical frameworks, known as *Integral* *Biomathics. *Their goal is to devise a ** comprehensible theory of life**. Our research follows in the footsteps of the Macy conferences (1946-1953) in cybernetics [1], The Waddington circle of conferences on theoretical biology (1965-1968) [2,3], the club of Rome on the predicament of mankind [4-5] and the club of Budapest projects, as well as the initiatives of The General Evolution Research Group and The Lindisfarne Association. In what follows, we delineate a non-exhaustive list of

**key themes for research**which we would like to pursue.

**Integral Biomathics**

**Foundations for a Biology-Driven Mathematics and Computation**

The following two directives comprise the research scope of Integral Biomathics (IB):

*I. Living Systems Modeling*

- Develop new realistic mathematical models tailored for living systems as adequate integration and development of a variety fields in mathematics (e.g. algebraic topology and geometry, geometric and cohomological algebra, functional analysis and calculus, differential geometry, fibred spaces and their non-linear connections as images, chaos and bifurcation theories, different kinds of logic, etc.), streamlined and channeled by an overarching formal theory aiming at understanding biological systems and designing synthetic ones.
- Develop new simulation, visualization and creativity support techniques and tools for these novel mathematical models of the living based on the same principles.

*II. Steps towards a “New Integral Science” *

The essence of typical questions to be addressed in IB is about how to take account of the (possibly fuzzy) interactions between discrete and continuous phenomena, leading to the emergence of complexity.

- Design an original general system of abstractions within the biological domain that can be
*relationally*examined.It should support multiple complementary mathematical approaches to phenomena that can be brought into dialogical juxtaposition. - Define ways of identifying the biological properties that are as unique to such complex conglomerations as ‘volume’ and ‘temperature’ is to a set of molecules, or the ‘flexible redundancy’ property ubiquitous in biological systems, called degeneracy or multiplicity. The goal of this quest is the articulation of an
*evolutionary mathematics*that deals with the emergence of organization from non-random selection among replicating variations within complex populations of processes.

What we are interested in here are not only space-time scale invariant properties of living organisms, but also their *cardinal* and yet still inherently “biological” properties that may *differ* across the space-time scale. Our view of emergence includes both the *emergence* of more complex objects as aggregates of patterns of interacting lower level objects, and the emergence of complex interactions between them, which take place at the higher levels from the global structure of the lower levels, but cannot be locally observable through lower level components and processes. We also understand emergence as a product of a system functioning over time in multiple dimensions/scales, falling in relation to the unfolding of its larger environment.

**1. Approach: Constructivist and Innovative Mathematical Cross-Disciplinary Models **

The main activities that will be addressed here are:

- Develop dynamic models of biochemical and biophysical systems accounting for multiple scales and time frames as they relate to new forms of dynamic modeling and physical mapping/scanning systems. Analyze how scales themselves can be of emergent character in a process-oriented world.
- Develop convergent theoretical syntheses of adequate mathematical and computational concepts and methods, brought into dynamic relation with each other. Such a relational mathematics and computation are expected to model both the dynamics of the system in a local neighborhood with its specific temporality, and at a global level of the system emerging from the possibly conflicting relations between these local dynamics, through a kind of communication and negotiation/adjustment between near and far neighborhoods.
- Construct models of “hybrid” systems presenting a combination/juxtaposition of continuous, as well as discrete time changes accounting for their relational, numerical and geometrical aspects. To analyze biological problems, the mathematical challenge is how to combine these different domains, which are generally studied separately in orthodox mathematics.

As category theory unifies many mathematical domains and is also at the frontier with logic and computer science, it can be used in models formally describing natural phenomena. Its derivatives such as grupoid and sheaf theories can be used to restructure and organize the classical domains of numerical analysis, differential calculus, geometry and the more modern ones of topology, cohomology, dynamical systems, field theory, fuzzy sets, chaos theory, self-organized criticality etc. in a new, biology-driven way. Category theory should itself be enriched and made more flexible by addition of more structure and interdependence between the compositional elements and processes, for instance by introducing statistical categories and various types of causalities. Categorical models are well equipped to analyze the problem of emergence, going further than Robert Rosen’s notion of entailment, up to the emergence of higher cognitive processes, and allowing to incorporate 1^{st} person approaches (Topological Psychology). They can also provide multiple perspectives related to the problem of “class identity” and material space/time flow.

The working “algorithm” to realize this approach might be defined as follows:

- Investigate phenomena in living systems by trying to describe them using the above (integrated) formal toolset to deliver an evolving model.
- At the point where the model does not match the experimental results, switch to or develop new formal means to reflect and explain these peculiarities, thus advancing the model to a next stage.
- Focus on both objects and processes and on their interactions.

This method should not be understood as strictly formal. In other words, the “match” with experimental results could be verified by means of computer programs, or only require pencil and paper. On the other hand, there are also *negative *mathematical proofs (limitation results), e.g. by logical deduction, predicate calculus, or even gedankenexperiments involving visualization tools (geometry, animation, etc.).

**2. Focus and Implementation: Integration and Development over Multiple Scales**

Integral Biomathics is a cross-disciplinary approach, involving both internalist and externalist mathematical biology and biological mathematics and computation. Its goal is to build the foundations of a “unified theory” of life, all throughout emergence and evolution and involving both “natural” and “artificial” living systems. The first step towards its completion was made: the fusion between a multivalent situation and context aware computational logic, the *Wandering Logic Intelligence* (WLI, Simeonov, 2002), and a multi-scale dynamic category theory, *Memory Evolutive Systems* (MES, Ehresmann and Vanbremeersch, 2007), *WLIMES* (Ehresmann and Simeonov, 2012; Simeonov and Ehresmann, 2017; Simeonov, 2017). The next step is the integration, reconfiguration and adaptation of adequate mathematical and computational theories and abstractions, such as e.g. Turing oracle machines, as well as heuristics and a broad range of simulation, visualization and other creative support techniques capable of dealing with phenomena and data that cannot be handled by formalisms only. This will allow interrogation marks/interfaces between its constituents and build bridges to other disciplines, while ensuring long-term sustainability of development on this formal base.

The operative framework of Integral Biomathics is defined as a multi-perspective approach to knowledge production: observation of new phenomena / incorporation of new forms of entailment-generating-technology (e.g. scanning methodologies) as well as modeling approaches -> articulate convergent theoretical synthesis across divergent fields -> integrate multiple mathematical formalisms under one relational umbrella -> *develop integrated mathematical models accounting for multi-scale structures and multi-temporal dynamics* -> study the dynamic relation between emergent phenomena and predictive phenomena -> justify initial theoretical approaches via computational modeling -> develop empirical demonstration and verification -> articulate a falsifiable theoretical foundation for practical studies. This gives us a panoramic view of the system with all its structures, dynamics and functionality:

- Enable the use of information from different areas of discourse to examine how low-level processes “percolate up” and relate to higher levels, and how human scale behavioral processes may enable 1
^{st}and 3^{rd}person comparative relations. - Define concrete approaches to discrete computational methodologies (functioning at different scales) to capture change over time from a series of different multi-modal observational perspectives. Define systems that can also present coherent integrated high-level processes that relate to the lower level processes. This is about an integration of the computational aspect and its material underpinning.

The longer-range objective of Integral Biomathics envisions the step-*wise* replacement of oracles by a more elaborated theory of life involving other modeling tools. This approach remains mathematics-based and biology-driven. In particular, contemporary biology and physics do* not* address the following questions/goals:

- Are the currently existing scientific/mathematical/computational theories sufficient, such that meta-level Turing Oracle Machines (TOM) could be replaced by models within these existing theories?
- Are the current theories insufficient in the sense that no amount of additional data is going to replace some of the oracles in our models?
- Can we postulate/conjecture that even if (2) holds, a theory (or a set of compatible and/or complementary theories) able to replace oracles by models can be conceived/unveiled? In other words, can we imply that decision making and judgments lie
*within*the theory? - What is missing on the way to creating a Unified Theory of Life and Consciousness?
- How to create a “Tree of Life” (or perhaps a universe of multiple and simultaneous worlds), a living ontology of facts, axiomes, propositions and theories, in biology, physics and science as a whole?
- Can biology be associated with the emergence of decoherence in quantum mechanics? How could the Turing’s oracles be naturalized in the framework of quantum physics?