Cantor’s Invisible Chemistry

A BEYOND BORDERS column by David Krakauer, President of the Santa Fe Institute

No matter how carefully and ingeniously you rifle and forage through the mathematical archives you will not find a trace of psychoanalysis or a single mention of the periodic table in Georg Cantor’s diagonal proof of infinite sets. And this is true despite the universality of Cantor’s findings and the fact that the idea of infinity pervades all of science, mathematics, logic, and even mythology.

As it is written in the Upanishads, “there is no joy in the finite, there is joy only in the infinite.” For complicated reasons that have nothing to do with the nucleus accumbens or dopamine receptors — the putative anatomical and neurochemical basis of pleasure — we understand perfectly what these Sanskrit inscriptions are getting at without invoking their cognitive infrastructure.

Just like Cantor’s proof, the Upanishads are understood within their own system of rules. We do not require for deeper understanding that the insights of logic and mythology be presented through any reductive potpourri of biological mechanisms. These might expand our under-standing of cognition but not the correctness of a proof or acumen of a metaphor.

The mysteries of the universe that complexity science seeks to explain are how widespread adaptive regularities emerge at multiple different levels and how each level comes to be best served by its own effective theory — from the theory of molecular interactions through to the theory of ecological stability. And furthermore, the way many of these theories share striking family resemblances by virtue of constraints of energy, time, and information. This is the fundamentally dual nature of complexity theory — recognizing the need for the autonomy of theories at different levels while at the same time exploring the common features of these theories.

Consider two profound representational frame-works — mathematics and natural language — that both use their own specific rules to explore and explain the worlds that they each represent. A paradox presented in natural language or mathematics is explained in terms of mathematics or language. Not in terms of psychology or the covalent bond. At the same time both math and natural language share the properties of syntax and semantics and conform to limitations of length, clarity, and comprehensibility. Thereby each serves the functions required by their contingent domain of application while possessing deep meta-theoretical affinities by virtue of shared structures, processes, and shared users (that is, human beings).

At a recent meeting hosted in Washington by the National Science Foundation and requested by the director of the NSF, France Cordova, and Kraston Blagoev from the division of physics, SFI convened a group of complexity researchers to summarize our current understanding of universality in complex systems.

Spanning pattern-formation, neuroscience, ecology, evolution, and collective computation, researchers reported on amazing regularities that apply across species and across niches and that can be understood by shared principles of scaling, evolution, information theory, and computation. While each area was presented without recourse to reductionism (that is, explanations dominated by interactions among microscopic constituents) common principles of entropy production, robust information encoding, convergent evolution, higher-order interactions, the control of networked components, and the efficient use of energy to store adaptive information, emerged as foundational principles in all complex systems.

There was a time not too many years ago when the idea of general theories of complexity seemed absurd. Early efforts that tended to over-generalize from toy models without strong empirical sup-port engendered skepticism in both the scientific community at large as well as among complexity scientists. The recent turn to strong empiricism has led to discoveries of startling regularity not dissimilar to those discoveries in the physical sciences made over the last few centuries.

This is a very exciting time in complexity science that promises not only to discover emergent laws of nature, but to explain why a diversity of approaches to understanding is required, why a grand unified theory is wrong-headed, and possibly to discover principled means of establishing connections across the full landscape of complexity theories.

From the Winter 2019–20 Edition of SFI Parallax. Read here.