Wolfgang WoppererNotesEssays

From Predictive Processing to Topological Thinking: Prolegomena to a Future Paradigm

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The heart of a scientific paradigmUnderstood as “accepted examples of actual scientific practice – examples which include law, theory, application, and instrumentation together – [that] provide models from which spring particular coherent traditions of scientific research”. (Kuhn 1970, 10)

 is a practice (“how understanding is being done”), its foundation an ontology (“what there is to understand”). Any future paradigm should reflect our best knowledge about how biological and social systems actually understand the world instead of phantasising about how understanding could or should work.

According to current researchSummarised, e.g., in Clark (2013) and Seth (2021).

, understanding is generating, using and validating predictive models of the world. A predictive model can be implemented in various waysImportantly, it can be an implicit model, i.e. a non-conceptual, embodied representation of a system’s environment.

, depending on the system that has it: In bacteria, it is implemented in hard-wired chemical reactions (“chemotaxis”); in humans, it is a simulation running on our brain (“controlled hallucination”Seth (2021), 17

); in civilisations, it is their cultural technēHaslanger (2019)

 (“social construction”).

Conventional ontologies don’t reflect this emerging understanding of understanding: They are based on our manifest image of the world“[T]he framework in terms of which man came to be aware of himself as man-in-the-world” (Sellars 1963, 7). Sellars describes its objects as follows: “Thus we are approaching an answer to the question, ‘what are the basic objects of the manifest image?’ when we say that it includes persons, animals, lower forms of life and ‘merely material’ things, like rivers and stones.” (ibid., 9) See Ladyman & Ross (2007, 3–5) for a thorough critique of ontologies derived from this image.

, abstracting from it only to a certain degree – they posit things, events or processes that exist in space and time and that interact causally as we experience everyday things to interact in space and time. But our manifest image is the product of our brains’ simulations and of social construction – it cannot claim to have anything to say about what there really is to understand. Its features are an explanandum, not an explanans. For a more reliable starting point we must, again, look to science.

The best and most current physics tells us that space and time, and a fortiori everything that we imagine to exist in space and time (things, events, processes), cannot be fundamental to reality.Nima Arkani-Hamed makes a very convincing case for this in his public lectures, e.g. in Arkani-Hamed (2017).

 They emerge from something deeper, from what there really is.

Thus an adequate ontology has to discard space and time (and with it things, events and processes) and aim to describe this deeper reality. To do so, we can start with the basic fact that there is difference in the world – that “something can be distinguished from everything else”Friston (2019), 4, who develops a formally rigorous framework based on an (albeit implicit) ontology similar to the one outlined here. The same point is made by René Thom (1975, 1): “Whatever is the ultimate nature of reality […], it is indisputable that our universe is not chaos. We perceive beings, objects, things to which we give names. These beings or things are forms or structures endowed with a degree of stability […].”

. We call such distinguishable somethings systems: groups of interdependent items forming a coherent whole, defined by their boundary, i.e. by what is and what is not part of them. Each item that is part of a system can itself be understood as a system. This means that our ontology will be about a hierarchy of systems, up to the world as the system of all systems. And since we want our ontology to be as parsimonious as possibleThere are long-running debates in academic philosophy whether parsimony should be an unqualified optimisation target for ontologies. I believe it should – if it is combined with a simultaneous optimisation for productivity.

, it will be about nothing else.

Note that this ontology doesn’t privilege any single level of description – there is no fundamental reality, no “true” system (apart from the world as a whole). It’s systems all the way down – and up: An elementary particle is just as real as a biological cell, a dog as real as a family, a culture as the climate. What will be privileged are parsimonious descriptions: accounts describing systems on the informationally most efficient level. In this ontology, maximum parsimony goes formidably well with maximum diversity of entities.This is very similar to Don Ross‘s “Rainforest Realism” (Ross 2000), which is a realist interpretation of Daniel Dennett’s “real patterns” (Dennett 1991), based on information-theoretic efficiency. Another parallel is Hoel’s (2017) work on how higher-level causation emerges from microphysical reality due to an increase in effective information.

Now, since we have discarded space and time as not fundamental, systems in this abstract and fundamental sense cannot be said to exist in space and time. But how and where do they exist?

The most parsimonious way to think about this is to just assume what our best models of systems assume. From physics to forecasting our most advanced models are captured in some form of mathematics, so they strictly only assume what they need to run their mathematics: a state spaceMore technically, we “infer just that fundamental structure and ontology that is required by the dynamical laws” (North 2013, 188). An example for this is a Hilbert space in which equations describe vectors or operators.

. Systems thus minimally exist as sets of states in a state space. State spaces are topological spaces, or more precisely manifolds, of which the states are the constituent points. So, extrapolating from what our models need to work, what there really is are systems existing as structures in topological spaces.

What we really want to understand are patterns of system behaviour. How would such an ontology help us do that?

For any system, exhibiting patterns of behaviour just means tending to be in certain states. These patterns thus form “basins of attraction” in the system’s state space: topological structures defined by its attractors, i.e. by topological singularities that shape the topological space around them and define possible state trajectories through it.See Abraham & Shaw (1992) for an excellent visual and DeLanda (2002) for a very clear conceptual introduction to these ideas.

 As systems exist in exactly the states their attractors define, we can say they are these attractor-defined topological structures.Juarrero (1999, 156): “A system’s identity is captured in the signature probability distribution of its dynamics.”

 Understanding patterns of behaviour, then, is topologically describing the defining attractors.This is what Manuel DeLanda calls topological thinking, the term in the essay’s title. His explication of Deleuze’s dynamical process ontology (DeLanda 2002) is based on ideas similar to the ones outlined here. Topological thinking is being adopted in a wide range of disciplines, albeit mostly without explicit ontological reflection. See for example the shift towards geometric descriptions in fundamental physics (from Hamiltonian mechanics to the so-called amplituhedron and “positive geometries”), Joscha Bach’s automata-generated “global fractal”, Karl Friston’s Free Energy Principle and its vector gradients, the manifold hypothesis in machine learning, attractor landscapes from molecular biology and genetics to climate and earth system science, and Dave Snowden’s “energy gradients”.

With this, we have come full circle:

Topological structures are not only something very abstract and far removed from our Manifest Image, but also something we can actually imagine. Our brain has evolved to perceive the world to exist in space and time (a fact that needs to be explained), and it can construct topological structures as abstractions from space and time that still retain a quasi-visual quality.

Thus we can use topology to describe highly abstract and complex facts about the world in a very efficient and explicit, but even more importantly concrete and tangible way. We can use our ontology to generate not only less familiar, but also more accessible models.

So if we accept how we really understand things, understanding what there really is means describing the world at a very high level of abstraction in a very tangible way: describing systems in topological terms.

This should be the practice at the heart of our next paradigm.Methods to be derived from this core practice could range from mathematical modelling and visualisation techniques through scale-free abstractions like Markov blankets, gradient flows and self-organised criticality to de-familiarisation strategies and the induction of psychosis-neighbouring states “at the edge of chaos” that enhance pattern recognition. Possible containers for these practices range from hard science through the facilitation of distributed cognition to the arts and subcultural experimentation.

References

Acknowledgements

I am grateful to Glenda Eyoang and Gregor Groß for comments on drafts of this essay.