Reductionism, Holism, and Abstraction in Model Building and Thinking

Reductionism permeates how we view the world because it is so intuitive. When we explicitly construct models to understand the world or solve a problem, we tend to formulate abstractions or break down problems into constituent and simpler parts. Implicit in this reductionist approach is that there even exists “constituent” parts. Even the notion of intuition itself implicitly assumes there are various “moving parts” of a system that are interconnected in some way, and through a creative subconscious process we discover those connections that weren’t apparent or considered before.

I will attempt to explore what reductionism means, and consider where the reductionist approach is ill-defined or at least limited. We will see how the reductionist approach can lead to more complexity at best, or may simply be the wrong approach since it is ill-defined. The examples I consider will draw primarily from physics, but will touch other fields such as biology.

Photo by Hello I'm Nik 🇬🇧 on Unsplash

Atomic View of the World

The great physicist Richard Feynman pondered what was the most important aspect of science to preserve if we were forced to forget everything else.

If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis that all things are made of atoms — little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.
— Richard Feynman

This atomic view of the world implies reductionism, but let’s consider a very simple physical system where reductionism fails, a two-electron system.

Consider one electron. Now consider another electron. Together they form a two-electron system. It’s intuitive to consider them as two distinct objects much like two tennis balls or two baseballs are distinct objects. We say the two-electron system can be reduced to two independent electrons, and also that two independent electrons can be brought together to form a two-electron system. Nothing counterintuitive or mysterious here. So far, so good.

However, if the electrons become related through the mysterious quantum mechanical phenomenon called entanglement, they can no longer be viewed as two distinct electrons making up a two-electron system, but rather a single holistic object that cannot be viewed in terms of two constituent electrons.

To maintain focus and limit the scope of this post, let me just say that entanglement is not just mere theory or a thought experiment, but has been observed through experiments yielding empirical data. Just this phenomenon of studying the entanglement of a two-electron system (or more precisely, a system of two particles with two levels each) is an entire deep field itself without even considering multiple particles and the superstructure of quantum mechanics. I’ll just reference this article from a reputable journal, Physical Review B, as representative of empirical and experimental findings of theory.
Vasilyev, et al — Spin-entanglement between two freely propagating electrons: Experiment and theory

Thus, even in one of the most simplest of physical systems, a two-electron system, the notion of reductionism fails. So now we have an entangled electron pair that was previously two unentangled and distinct electrons, but that cannot be viewed as a system of two electrons, but rather as a single holistic object.

So what is more fundamental: a single electron, two unentangled electrons, or an entangled pair of electrons?

Clearly and intuitively, a single electron is more fundamental than two unentangled electrons since each electron can be brought together to form the unentangled electron pair, and the unentangled two electron system can be reduced to constituent electrons.

However, a single electron may not be viewed as more fundamental than an entangled pair that was previously two distinct fundamental entities before the entanglement, because that entangled electron pair cannot be reduced into constituent electrons.

The empirically observed physical phenomenon of entanglement precludes the notion of a strict hierarchy based on reductionism.

Hierarchical Levels of Abstraction are Ill-Defined

Now because it is incorrect and ill-defined to definitively say whether a single electron or an entangled pair of electrons is more fundamental, it calls into question the very existence of levels of abstraction from both a point of epistemological, relating to subjective knowledge only, convenience in model building as well as ontological truth, related to the actual nature of a physical system in this case not tied to subjective knowledge. In other words, not only can we not have hierarchical levels because we do not have the knowledge or sophistication in model building to do so, but it is fundamentally impossible as a characteristic of reality.

(As an aside, when getting into abstractions in science and philosophy, we are also bumping up against the limitations of words. Words in every day usage have great utility, which interestingly derives from their imprecision since there’s an assumed context by all parties. However, it’s this very imprecision of words that become a formidable barrier when discussing very nuanced issues in science and philosophy, so we must be hyper vigilant about implied meaning, tacit assumptions, and connotations, even political/philosophical world views, such as whether science should be done through the lens of Logical Positivism, whether something is Science or not, or what the very criteria of to judge whether it is Science or not!)

To complement the specific example of a two-electron system given above, another example would be that a system of a large number of atoms that can be described by quantum mechanics individually via a wave equation would be better suited to be described by statistical mechanics, as in the statistical properties of a large number of particle.

The usefulness of statistical mechanics may be a matter of convenience in that we can in principle calculate the wave functions on the order of 10²⁵ particles if we only had enough computational power, storage capacity, and outliers in probability distributions. However, it may be less a matter of epistemological convenience and more of ontological truth in the transition from statistical mechanics to a cleaner macroscopic conception of a system via thermodynamics. Here, the notion of individual particles are dissolved in favor of macro-level abstractions such as temperature/heat/energy, volume, and pressure. The bridging abstraction here is the notion of entropy, which is the level of disorder or the number of ways to arrange the microscopic states of a system, which itself requires another large conceptual leap itself with many assumptions.

This points to the popular conception of a hierarchy of levels going up in spatial extent: elementary particle physics yields to atomic physics, which yields to chemistry, which yields to genetics, which yields to biology, which yields to ecology, economics, geopolitics. The implicit views here are that the abstractions are a matter of model building convenience, as opposed to ontological reality.

In all these cases, there’s an assumed scaling parameter, whether it’s scaling in spatial extent, or in complexity and number of constituent entities. This scaling parameter implies that a lower-level abstraction could theoretically suffice to describe a higher level of abstraction if only we had the computational power and information to do so. Going the opposite way, we say that a system can be decomposed via reductionism to lower-level “constituent entities”, if only we were to intentionally leave the convenience of models that work well at the higher level.

But even the two-electron system shows that we cannot easily move up and down levels of abstraction because there actually is no strict hierarchy even in principle and since it is ill-defined, and this is not just as a matter of convenience of building models. One cannot describe psychology, or economics, or geopolitics in terms of atoms, not even in terms of individual people.

It appears the only way to do so would be via a model that matches the complexity of reality itself, as in approaching a one-to-one emulation instead of a reduced and modeled system.

Holism and Emergent Behavior

So now that we’ve considered just a tiny subset of where reductionism and hierarchical levels of abstraction are ill-defined, what can be said about holism? Or the related phenomena of holism and emergent behavior? I’d like to consider three examples: water waves, the human brain and consciousness and memory, and cellular automata and computational irreducibility.

When water waves form and travel through the ocean, the water molecules constituting the waves do not actually travel through the ocean for the entire journey. These molecules instead serve as a medium for which the very real mechanism of waves for energy transmission occurs through water particles. There’s very precise and accurate mathematics that describe this emergent phenomenon of waves, even though they are not necessarily composed of the constituent molecules, but rather through the molecules and their coordinated motion in a confined spatial area.

In many ways, this points to the idea of consciousness or even something more humble as memory in the human brain, which we turn to now.

What does it mean that there is consciousness?

I take the strictly physicalist/materialist view that the mind does not exist as separate from the physical world, but as an emergent phenomenon of the computations underlying it. Consciousness is an emergent property, and indeed it’s what computations “feel” like as an irreducible computation. However, just because consciousness fundamentally comes from computations in neurons does not mean that we can use the computation of any one neuron or set of neurons to explain consciousness, since emergent behavior is more than the sum of its parts.

Now let’s take a step back and consider something far less mysterious than consciousness, memory. Many psychological experiments have shown that memory is not necessarily localized to specific regions in the brain and rather, but rather encoded holistically.

Neither consciousness nor the more mundane human memory can be reduced to the constituent neurons processing that consciousness or encoding that memory. Thus, the idea of half a consciousness or half of an accessible subjective memory are ill-defined, forget more precisely like 37.8% conscious or memorable. Yet another example where the lower constituent neurons cannot be combined to form consciousness, nor consciousness able to be decomposed into constituent building blocks of consciousness or physical neurons.

It’s this irreducibility that newer disciplines involving Cellular Automata and concepts as the Principle of Computational Irreducibility have appeared. Here, simple rules can lead to complex unpredictable behavior. Even precisely and clearly defining the simple rules in the Cellular automata cannot lead to predicting emergent and complex behavior.

Furthermore, the only way to get to the end of a computation of a program based on those simple rules is to go through all the intermediate computations. There is no way to compress it and no short cuts, an idea known as the “Principle of Computational Irreducibility”. It’s in effect a type of holism both in computational time steps and in terms of simple rules, even if simple rules cause emergent properties. This of course in the information and computation domain, and not in the spatial extent domain like the other examples here.

Implications for Model Building and Thinking

So what does this mean for understanding the world or building models to solve problems? The main takeaway is that we must consider the correct level of abstraction and be very careful in jumping through levels of abstractions since:

  • Appeals to lower levels of abstractions is untenable because reductionism may be ill-defined, as in the two-electron case
  • Appeals to higher levels of abstraction is untenable because holistic and emergent properties may arise that cannot be described via lower level of abstraction

This has implications in modeling and thinking about:

  • Physical systems such as: entangled electron pairs, water waves, or large number of particles giving rise to thermodynamics.
  • Information processing such as: cellular automata running a computationally irreducible program, or neurons encoding information giving rise to retrievable information, neurons processing information that may give rise to emergent consciousness.
  • Human behavior such as: how individuals form groups that ultimately form larger groups such as nation-states or coalitions of them describable by geopolitics.

AI/full-stack software engineer | trader/investor/entrepreneur | physics phd

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