Voevodsky's unfinished project: Filling the gap between pure and applied mathematics

Reflexive structure of the conscious subject and the origin of language codes

The code paradigm in biological and social sciences arises to Aristotle. For conscious activity, Aristotle introduced the notion of reflexive self-awareness in sense perception. This reflexive process generates the codes that signify sensual perceptive events and constrain human behavior. Coding systems grow via the generation of hypertextual statements reflecting new meanings in the process defined by Marcello Barbieri as a codepoiesis. It results in the establishment of higher-level codes (metacodes) forming the semiotic screen that has a nature of the set of perceived objects internalized by the conscious subject in encoding the symbolic actions. The characteristic feature of the semiotic screen consists in its property of being shared between the communicating agents. A sufficient complexity of nervous system, through the appearance of mirror neurons that are fired both when a subject executes certain action and when he observes another subject performing a similar action, represents a prerequisite for the emergence of reflexive codes in evolution. The codes appearing as a result of reflexive awareness and establishing different sociotypes, span from the symbolic systems of art and music through the common language to the formal language of logic and mathematics. Social dynamics is based on the implementation of reflexive coding activity and results in the growth and decay of social systems and civilizations.

Overcoming the limits of natural computation in biological evolution toward the maximization of system efficiency

The goal-directedness of biological evolution is realized via the anticipatory achievement of the final state of the system that corresponds to the condition of its perfection in self-maintenance and in adaptability. In the course of individual development, a biological system maximizes its power via synergistic effects and becomes able to perform external work most efficiently. In this state, defined as stasis, robust self-maintaining configurations act as attractors resistant to external and internal perturbations. This corresponds to the local energy–time constraints that most efficiently fit the integral optimization of the whole system. In evolution, major evolutionary transitions that establish new states of stasis are achieved via codepoiesis, a process in which the undecided statements of existing coding systems form the basis for the evolutionary unfolding of the system by assigning new values to them. The genetic fixation of this macroevolutionary process leads to new programmes of individual development representing the process of natural computation. The phenomenon of complexification in evolution represents a metasystem transition that results in maximization of a system’s power and in the ability to increase external work performed by the system.

The drawbridge of nature: Evolutionary complexification as a generation and novel interpretation of coding systems

The phenomenon of evolutionary complexification corresponds to the generation of new coding systems (defined as а codepoiesis by Marcello Barbieri). The whole process of generating novel coding statements that substantiate organizational complexification leads to an expansion of the system that incorporates externality to support newly generated complex structures. During complexifying evolution, the values are assigned to the previously unproven statements via their encoding by using new codes or rearranging the old ones. In this perspective, living systems during evolution continuously realize the proof of Gödel's theorem. In the real physical world, this realization is grounded in the irreversible reduction of the fundamental uncertainty appearing in the self-referential process of internal measurement performed by living systems. It leads to the formation of reflexive loops that establish novel interrelations between the biosystem and the external world and provide a possibility of active anticipatory transformation of externality. We propose a metamathematical framework that can account for the underlying logic of codepoiesis, outline the basic principles of the generation of new coding systems, and describe main codepoietic events in the course of progressive biological evolution. The evolutionary complexification is viewed as a metasystem transition that results in the increase of external work by the system based on the division of labor between its components.

A mathematician's view of the unreasonable ineffectiveness of mathematics in biology

This paper discusses, from a mathematician's point of view, the thesis formulated by Israel Gelfand, one of the greatest mathematicians of the 20th century, and one of the pioneers of mathematical biology, about the unreasonable ineffectiveness of mathematics in biology as compared with the obvious success of mathematics in physics. The author discusses the limitations of the mainstream mathematics of today when it is used in biology. He suggests that some emerging directions in mathematics have potential to enhance the role of mathematics in biology.

Vladimir Voevodsky on mathematics and life

This is a brief overview of Vladimir Voevodsky's (1966-2017) intellectual and professional biography, which is partly based on the author's personal memories. Voevodsky's biologically-motivated mathematical works are considered in the context of his research in the Algebraic Geometry and in the Univalent Foundations of mathematics. Some biographical details, which are important for understanding Voevodsky's achievements and his personality, are provided.

Comments on Vladimir Voevodsky's biologically motivated works

Vladimir Voevodsky's contribution to Algebraic K-Theory is well known and was recognised by awarding him the Fields Medal. He became widely known in mathematical circles for his daring project of reforming the Foundations of Mathematics, a project he pursued until his untimely death. His ideas on the so-called "univalent" foundations of mathematics were taken up by a broad community of mathematicians and continue to be developed today. Less well known is that he devoted several years of his life to a biologically motivated project aimed at developing a mathematical theory of population dynamics. He never published anything about it. The aim of the article is to summarise his ideas on this project and to draw them to the attention of the scientific community.

Mathematics in biological reality: The emergence of natural computation in living systems

Mathematics is a powerful tool to express the computable part of the reality of the physical world. For living systems, mathematical relations emerge internally as an abstracting capacity in the course of development and adaptation to the external world. All living systems possess internal coding structures which represent their embedded description. They are anticipatory in the sense that the embedded description generates deterministic model of their behavior. If the model does not provide a correct result, they can evolve through the acquisition of new statements inside the embedded description that overcome limitations of the existing model. The newly generated statements acquire meaning in and from the changing environment. The growth of complexity, being a consequence of the internal active adaptation to externality performed by the systems, increases the amount of external work and generates the observed patterns of spatiotemporal structures of evolving systems. In living systems, the symbolic memory constraints are dynamic processes in themselves, co-evolving with the other components of biological systems. Separation of the symbolic memory and the dynamic laws (defined as the epistemic cut), required for self-replication of biological systems, forms the basis for their onto-epistemic relation to reality. In this regard, living systems possess their own internal abstracting capacity and invent mathematics. The digital structure of the genetic code is a manifestation of this mathematics.