Individuals, Information, and the lenses of mathematics

First, I would like to apologize for neglecting Mathematics Rising in recent months. Changes to the classes I’ve been teaching at UT Dallas, (that were necessitated by Covid-19) consumed so much of my time that it became difficult for me to do much more than teach my classes and take care of my family. I’m hoping to be able to do better. So let me begin today.

It happens often that Quanta Magazine brings news of profound and novel approaches to any number of new scientific questions. And these reports often make a very positive contribution to the perspective I have been trying to nurture at Mathematics Rising. In July, Jordana Cepelewicz wrote an article with the title: What Is an Individual? Biology Seeks Clues in Information Theory. Two of the words in this title quickly got my attention – individual, and information. The word individual got my attention because the significance of ‘the individual’ plays an important role in so many things including politics and religion, and information because once mathematics was used to define information, it has become a mathematical lens that is enormously useful to so many questions in science, including quantum mechanics and theories of consciousness. Cepelewicz makes the pithy remark that nature has “a sloppy disregard for boundaries,” taking note of the M.O. of viruses, bacteria, insect colonies or “superorganisms,” and the myriad varieties of symbiotic composites that live in our world. “Even humans,” she says, “contain at least as many bacterial cells as ‘self’ cells.”

To emphasize the value of clarifying what we mean by ‘individual,’ she writes:

Ecologists need to recognize individuals when disentangling the complex symbioses and relationships that define a community. Evolutionary biologists, who study natural selection and how it chooses individuals for reproductive success, need to figure out what constitutes the individual being selected.

The same applies in fields of biology dealing with more abstract concepts of the individual — entities that emerge as distinct patterns within larger schemes of behavior or activity. Molecular biologists must pinpoint which genes out of many thousands interact as a discrete network to produce a given trait. Neuroscientists must determine when clusters of neurons in the brain act as one cohesive entity to represent a stimulus.

“In a way, [biology] is a science of individuality,” said Melanie Mitchell, a computer scientist at the Santa Fe Institute.

Biology, many agree, has been under-theorized, but this is no doubt changing. The Stanford Encyclopedia of Philosophy has an entry on Biological Individuals from which one can see the development of conceptual frameworks to address the question.

The Quanta Magazine article is based on the work of David Krakauer, an evolutionary theorist and president of the Santa Fe Institute, and Jessica Flack who studies collective behavior and collective computation (also based at the Santa Fe Institute). They created a group tasked with finding a new working definition of the ‘individual.’

At the core of that working definition was the idea that an individual should not be considered in spatial terms but in temporal ones: as something that persists stably but dynamically through time. “It’s a different way of thinking about individuals,” said Mitchell, who was not involved in the work. “As kind of a verb, instead of a noun.”

How do you create this view? What can we use to see things this way? The lens they chose is information theory.

Krakauer and Flack, in collaboration with colleagues such as Nihat Ay of the Max Planck Institute for Mathematics in the Sciences, realized that they’d need to turn to information theory to formalize their principle of the individual “as kind of a verb.” To them, an individual was an aggregate that “preserved a measure of temporal integrity,” propagating a close-to-maximal amount of information forward in time.

Their formalism begins with propositions:

  • Individuality can exist at any level of biological organization (sub cellular to social)
  • individuality can be nested (one individual within another)
  • and individuality exists on a continuum meaning systems can have quantifiable degrees of individuality.

The last of these might translate the question of whether a virus is alive or not, into the question, how living is a virus. In other words, where does it lie on the continuum of individuals.

The abstract of their paper is very clear about what this model hopes to accomplish:

Despite the near universal assumption of individuality in biology, there is little agreement about what individuals are and few rigorous quantitative methods for their identification. Here, we propose that individuals are aggregates that preserve a measure of temporal integrity, i.e., “propagate” information from their past into their futures. We formalize this idea using information theory and graphical models. This mathematical formulation yields three principled and distinct forms of individuality—an organismal, a colonial, and a driven form—each of which varies in the degree of environmental dependence and inherited information. This approach can be thought of as a Gestalt approach to evolution where selection makes figure-ground (agent–environment) distinctions using suitable information-theoretic lenses. A benefit of the approach is that it expands the scope of allowable individuals to include adaptive aggregations in systems that are multi-scale, highly distributed, and do not necessarily have physical boundaries such as cell walls or clonal somatic tissue. Such individuals might be visible to selection but hard to detect by observers without suitable measurement principles. The information theory of individuality allows for the identification of individuals at all levels of organization from molecular to cultural and provides a basis for testing assumptions about the natural scales of a system and argues for the importance of uncertainty reduction through coarse-graining in adaptive systems.

(Coarse-graining is a simplification of the details in a system that is as true to the system as the details themselves – the way that temperature represents the average speed of particles in a system)

There is no doubt that this will be a lucrative diversion from the noun-like way we have identified living things. It is, as I see it, one of many efforts in a broad scheme related to the provocative ideas brought to light by biologists Francisco Varela and Humberto Maturana in the 1980s. Rather than list the properties of living things, Maturana and Varela observed that living things are characterized by the fact that they are continually self-producing – not reproducing but self-producing. The cell, for example, what we have long thought of as the fundamental living thing, is a network of processes that are organized as a unity, where the interaction of these processes continuously and directly realizes the unity itself. The cell is what the cell does. They called this process autopoiesis, its Greek roots meaning self (auto) and produce (poiesis). The being and the doing of an autopoietic system cannot be separated. With nested autopoietic systems (or nested unities), Maturana and Varela are imagining individuals in much the same way as Krakauer and Flack. I wrote about Maturana and Varela in a post called Autopoiesis, free energy and mathematics. In that post I introduced a related idea: Karl Frist’s Free Energy Principle, where all kinds of systems are understood in autopoietic terms. What’s missing in Maturana’s theory is the formalism that makes it possible to investigate the consequences of this kind of modeling. Information theory provides this for Krakauer and Flack. Probabilistic programming and machine learning provide it for Karl Frist. Maturana and Frist are both referenced at the end of Krakauer and Flack’s paper.

For me, these discussions always bring to mind the thoughts I had when I read Thomas Mann’s The Magic Mountain. I read the novel in the absence of any commentary about it, and I remember feeling an unexpected affection for the story’s protagonist, Hans Castorp. Mann began writing the story in 1912, but he completed it after World War I, in 1924. Unexpectedly restricted to a tuberculosis infirmary high in the Alps, a young Castorp is an innocent and eager explorer of biology and medicine. Encouraged by the cold and by his solitude, he rested and read from the library of physicians, fully self-training in the language and images of biology. I remember thinking that Castorp’s look at science and medicine was guileless, without the prejudices created by the pragmatism of fixing things, or the desire for useful knowledge. And, I thought, that contemporary ideas in medicine and biology, that are built on these early observations, and that we tend to think of as just true, were not the only ideas that could grow from the kinds of insights to which Castorp was privileged. Here are just a couple of his reflections:

This body, then, which hovered before him, this individual and living I, was a monstrous multiplicity of breathing and self-nourishing individuals, which through organic conformation and adaptation to special ends, had parted to such an extent with their essential individuality, their freedom and living immediacy, had so much become anatomic elements that the functions of some had become limited to sensibility…

What then was life? …It was the existence of the actually impossible-to-exist, of a half-sweet, half-painful balancing, or scarcely balancing, in this restricted and feverish process of decay and renewal, upon the point of existence. It was not matter and it was not spirit, but something between the two, a phenomenon conveyed by matter, like the rainbow on the waterfall, or like flame.

Maturana and Varela, Frist, Krakauer and Flack, are just some of the explorers that now confirm my hunch that there’s never just one way to see.

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