Autopoiesis, free energy, and mathematics

I have long been interested in the notion of autopoiesis introduced by Humberto Maturana and Francisco Varela in 1972. In short, autopoiesis is the model of living systems that sees every living system (from single cells to multicellular organisms) as individual unities whose living is the creation of themselves. Through the interaction of their components, they continuously regenerate and realize the processes that produce them. Living systems exist is a space determined by their structure. In this light, cognition became defined as the action or behavior that accomplishes this continual production of the system itself.

From my perspective, the notion of structural coupling which developed out of this framework, has the potential to contribute something important to a philosophy of mathematics. Two or more unities are structurally coupled when they enter into a relatedness that accomplishes their autopoiesis by virtue of ‘a history of recurrent interactions’ that leads to their ‘structural congruence.’ Also true is that every autopoietic system is closed, meaning that it lives only with respect to itself. Whether interactions happen within the internal components of a system, or with the medium in which the system exists, the system is only involved in its own continuous regeneration. The view of cognition proposed by Maturana requires that the nervous is just such a closed, autopoietic system, which also functions as a component of the organism that contains it.

Mathematician Yehuda Rav used these ideas to propose a philosophy of mathematics (which I referenced in a 2012 post). In an essay with the title Philosophical Problems of Mathematics in the Light of Evolutionary Epistemology, Rav writes:

Thus, Maturna (1980, p. 13) writes: “Living systems are cognitive systems, and living as a process is a process of cognition”. What I wish to stress here is that there is a continuum of cognitive mechanisms, from molecular cognition to cognitive acts of organisms, and that some of these fittings have become genetically fixed and are transmitted from generation to generation. Cognition is not a passive act on the part of an organism, but a dynamic process realized in and through action.

When we form a representation for possible action, the nervous system apparently treats this representation as if it were a sensory input, hence processes it by the same logico-operational schemes as when dealing with an environmental situation. From a different perspective, Maturana and Varela (1980, p. 131) express it this way: “all states of the nervous system are internal states, and the nervous system cannot make a distinction in its process of transformations between its internally and externally generated changes.”

Thus, the logical schemes in hypothetical representations are the same as the logical schemes in coordination of actions, schemes which have been tested through eons of evolution and which by now are genetically fixed.

As it is a fundamental property of the nervous system to function through recursive loops, any hypothetical representation which we form is dealt with by the same ‘logic’ of coordination as in dealing with real life situations. Starting from the elementary logico-mathematical schemes, a hierarchy is established. Under the impetus of socio-cultural factors, new mathematical concepts are progressively introduced, and each new layer fuses with the previous layers. In structuring new layers, the same cognitive mechanisms operate with respect to the previous layers as they operate with respect to an environmental input. …..The sense of reality which one experiences in dealing with mathematical concepts stems in part from the fact that in all our hypothetical reasonings, the object of our reasoning is treated by the nervous system by means of cognitive mechanisms which have evolved through interactions with external reality.

Mathematics is a singularly rich cognition pool of mankind from which schemes can be drawn for formulating theories which deal with phenomena which lie outside the range of daily experience, and hence for which ordinary language is inadequate.

Rav is imagining the development of mathematics as a feature of human cognition. But the perspective proposed by Maturana includes a theory of language. For Maturana, language is not a thing, and the essence of what we call language is not in the words or the grammar. Language happens as we live in the units that our coupling defines – through living systems, interlocked by structural congruences, that build unities. We are languaging beings the way we are breathing beings.

My experience with mathematics has suggested to me that, like words and grammar, the symbolic representation of mathematics is secondary to what mathematics is. Mathematics also seems to happen. And Maturana’s emphasis on autopoiesis and structural coupling has suggested to me that mathematics, like language, happens through the recursive coordination of behaviors. But perhaps unlike language, the relational dynamics that bring it about are somehow fed by the more fundamental structures in the physical world (both living and non-living), to which we are coupled, rather than by the features of the day-to-day experience that we share.

Conceptually, the view of biology proposed by Maturana is significantly different from main stream thinking in the biological sciences. One of the most important differences is the way living systems are each bounded by their individual autopoietic processes and, at the same time, nested within each other, infinitely extending living possibilities. In my opinion, this particular aspect of their thinking is the most promising in the sense that it is this aspect of their thinking that has the greatest potential to produce something new.

A recent article in Wired about the work of Karl Friston suggested to me that I might be right. Friston, a neuroscientist who has made important contributions to neuroimaging technology, is the author of an idea called the free energy principle. Free energy is the difference between the states a living system expects to be in, and the states that its sensors determine it to be in. Another way of saying it is that when free energy is minimized, surprise is minimized. For Friston, a biological system (Maturana’s unity) that resists disorder and dissolution (is autopoietic) will adhere to the free energy principle – “whether it’s a protozoan or a pro basketball team.”

Friston’s unities are separated by what are called Markov blankets.

Markov is the eponym of a concept called a Markov blanket, which in machine learning is essentially a shield that separates one set of variables from others in a layered, hierarchical system. The psychologist Christopher Frith—who has an h-index on par with Friston’s—once described a Markov blanket as “a cognitive version of a cell membrane, shielding states inside the blanket from states outside.

In Friston’s mind, the universe is made up of Markov blankets inside of Markov blankets. Each of us has a Markov blanket that keeps us apart from what is not us. And within us are blankets separating organs, which contain blankets separating cells, which contain blankets separating their organelles. The blankets define how biological things exist over time and behave distinctly from one another. Without them, we’re just hot gas dissipating into the ether.

The free energy principle is mathematical, and grounded in physics, Bayesian statistics, and biology. It involves action, or the living system’s response to surprise, in addition to the systems predictive abilities. This is one reason the theory has far-reaching potential for application. The audience that the free energy principle attracts is consistently expanding.

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