As I have investigated all of the things in science and mathematics that get my attention, I have developed an impression of mathematics that, philosophically, seems most consistent with Humberto Maturana’s biology of language. Maturana outlines his perspective in great detail in an essay by the same name that appeared in 1978 in the text Psychology and Biology of Language and Thought, edited by George Miller and Elizabeth Lenneberg.
Language must arise as a result of something else that does not require denotation for its establishment, but that gives rise to language with all its implications as a trivial necessary result. This fundamental process is ontogenic structural coupling, which results in the establishment of a consensual domain.
In a piece that appeared in Cybernetics and Human Knowing, Maturana says the following:
Living in language, doing all the things that we do in language, however abstract they may seem, does not violate our structural determinism in general, nor our condition as structure determined systems. As I observed our languaging behavior and the behavior of other animals, I realized that the central aspect of languaging was the flow in living together in recursive coordinations of behaviors or doings, and that notions of communication and symbolization are secondary to actually existing in language.
Language cannot be understood as a biological phenomenon if we do not take seriously our operation as structure determined systems. If we do not do so we remain trapped in the belief that language is a system of communication and thinking with representations (symbolizations) of an independent reality that contains us as its primary constitutive feature. And if we do not understand language as a biological phenomenon we shall remain in the mystery of self-consciousness through believing that this somehow reveals an intrinsic cosmic duality, and we shall not be able to understand ourselves as the self-conscious transitory beings that we are.
I see at least two important ideas that are needed to bear the weight of Maturana’s perspective. The first of these is the dynamic interaction of unities:
…we human beings exist in structural coupling with other living and not living entities that compose the biosphere in the dimensions in which we are components of the biosphere, and we operate in language as our manner of being as we live in the present, in the flow of our interactions, in our domains of structural coupling.
The other is the shift away from questions about ‘being’ to an inquiry into ‘doing.’
As a result of this fundamental conceptual change, my central theme as a biologist (and philosopher) became the explanation of the experience of cognition rather than reality, because reality is an explanatory notion invented to explain the experience of cognition (see Maturana, 1980).
While there is no mention of mathematics in any of the Maturana pieces that I’ve looked at, it seems to me that the evidence is growing that mathematics happens, like language, and that we live in it, much like the way Maturana describes our “living in language.” And I suspect that what we might not yet understand is the extent to which we share it, as it exists in manifold forms throughout nature. Brain processing can look very mathematical (the abstractions in visual processing and learning, the Bayesian models of learning, etc.). Foraging patterns, eye movement patterns and the patterns in how we search for words all look the same. And, of course, our perception of the mathematical nature of the universe itself continues to enable an almost incomprehensible expansion of what we can know.
In a paper on How Humberto Maturana’s Biology of Cognition Can Revive the Language Sciences, Alexander Kravchenko takes note of some of the resistance to Maturana’s perspective but concludes with the following remarks:
One of the most important consequences of adopting the biology of language is the relational turn in approaching the mind/language problem. Much of what an organism does and experiences is centered not on the organism but on events in its relational experiential domain, one that crosses the boundary of skin and skull. In its endeavor to answer the question “How does the brain compute the mind?” the neural theory of language overlooks the incoherence of the proposition that mind is a complex computational function of the brain. In the biology of cognition there is no such thing as “the mind” in the operation of the nervous system, and “the mind” is nothing but an explanatory notion: “language, self-consciousness and mindedness are different forms of existing in the relational domain in which a living being lives, not manners of operation of the nervous system” (Maturana, Mpodozis & Letelier 1995: 25). (emphasis added)
It is this relational idea that can have a significant impact on a philosophy of mathematics, as it will inevitably locate mathematics both in and around us.