The mathematics of consciousness: answering existential questions using graph theory

I have always been fascinated with the paradoxical nature of consciousness; everyone knows that they are experiencing it, yet we cannot seem to reconcile such a unique phenomenon with the rest of our universe. This topic is heavily saturated with philosophical and religious musings, with no definitive evidence pointing to any one explanation over another. I would like to share an interesting way of thinking about consciousness that is scientifically verifiable while providing a rather satisfying answer to long-standing existential questions such as “what really happens after I die?”

A few weeks ago, I was discussing this topic with a mathematically precocious friend, specifically how we both had experienced the feeling of having some sort of “past life,” which vividly reminded me of a short story from a high school English class called The Egg, which had a very interesting and thought-provoking message about reincarnation and empathy. In short, the story describes a man who randomly dies, speaks with a “God” figure, and realizes that every other person is actually just himself in a different life. He is then sent off to his next life after realizing that each of his own actions is directly impacting “himself” at another point in his full lifetime.

While this story stands mostly as an interesting metaphor about empathy, it is fascinating to think about on a purely abstract level. What if, in some sense, people are the 7.6 billion “cells” of a planet-sized organism? What if this includes smaller subsets such as communities, cities, or, nations? These “collective life forms” satisfy almost every commonly accepted definition of life, since they can grow, feed themselves, reproduce, evolve, and perform any task available to its constituent organisms.

In mathematical terms, given any graph G in which vertices represent life forms and edges represent interactivity between life forms, the line graph L(G) corresponds to interacting pairs of life forms which also fulfill the definition of life. More generally, every connected subgraph induced by G of order |V| corresponds to a “parent” life form composed of |V| smaller interacting life forms. An obvious example is how cells combine to form plants and animals; an individual cell and collective organism are independently alive at different scales. For some points of reference, the average human has a cell interactivity graph of order |V| ≈ 37.2 trillion, while the human race encompasses all of these in a graph where |V| ≈ 2.9 * 10²³. This logic also applies to ecosystems, continents, and any other set of interacting life forms, each with their own characteristic structure and complexity.

By itself, this fact has few implications other than being a fun use of definitions, and it relies on highly subjective notions of “life” and “interactivity” to carry any meaning. So let’s take this into a new context: what if we, instead of thinking about interacting life forms, apply this framework to consciousness?

Since consciousness is the only aspect of the universe we can implicitly assume to exist (à la “I think, therefore I am”), we have a simple and intuitive starting point.

Most people have heard about the uncanny effects of split-brain surgery, which has shown to cause each side of a patient’s brain to act mostly independently from each other. Since the brain is effectively unable to communicate laterally, the patient’s consciousness is necessarily split in half as well. In this sense, consciousness logically seems to have the ability to divide and change over time, rather than maintaining a definitive structure. It’s very interesting to try and comprehend having two independent “half-consciousnesses” residing within one organism.

Now imagine taking this in the other direction — instead of splitting the brain in half, what would happen if we were to merge together two different minds? This experiment is tantalizingly close to reality due to recent advancements in brain-to-brain communication technology. When this occurs, we may finally have sufficient evidence to debunk solipsism and thus logically validate the remainder of this article.

So, here’s what we’ve talked about so far:

• The Egg is a great thought-provoking story and metaphorical reference point for this article.
• Any interacting group of organisms, such as an ecosystem or community, is itself collectively “alive” by most definitions.
• Due to technological advances, we will soon have the ability to breach the explanatory gap and scientifically verify the upcoming argument.

Now let’s put it all together. Let some graph C describe the interactions of all instances of consciousness. In order to abstract away the actual mechanism behind this, we will use a graph to describe how arbitrarily small instances of consciousness (vertices) interact with each other (edges). Unlike the nebulous definition used for collective life forms, any interaction at all (including over time) corresponds directly to an edge. For instance, if consciousness is a continuous field, this would result in a complete (fully connected) graph.

By finding all sets of adjacent edges as described earlier in the article, it is possible to describe progressively more complex conscious perspectives. For example, beginning with C_h as the graph of all interacting humans, L(C_h) is the collective consciousness graph of every interacting pair of humans. Furthermore, each connected subgraph induced by C represents a collective consciousness. It is possible to find a subgraph corresponding to a family, town, nation, ethnicity, subreddit, or any other community.

It is worth noting that the notion of “interaction” naturally disappears when referring to fields or any other mechanisms that result in a complete graph. In these cases, such as if consciousness turns out to be a projection of the electromagnetic field, the set of induced subgraphs maps bijectively to the power set of the vertices of C.

Sounds complicated. But how does this help us reason about concepts such as what happens after death? An abstract formulation of consciousness is about as close as we can get to wrapping our heads around how we all individually perceive completely different events. I’m currently experiencing the perspective of writing this article, which is completely disconnected from your experience of reading it. However, in the same way that the individual neurons in our brains communicate with each other, this interaction between us was also perceived by the collective consciousness corresponding to the interaction edge between us. This propagates upward, causing this interaction to be just one tiny component of a much larger network of perception. It can see from our perspectives, but we can’t see it.

By viewing consciousness hierarchically, it suddenly becomes much easier to conceptualize unconsciousness and death. Instead of having some sort of awareness of everything being “dark forever,” your hierarchy will perceive this as one part of its consciousness simply disappearing. It’s as easy to understand as simply closing one eye. You can still see, but only half as much information is projected into your mind. From the collective perspective of humanity, one person losing consciousness has almost no perceptible effect at all, in the same way you won’t notice if a few neurons in your brain stopped firing.

While this concept will soon become scientifically testable, it is also a direct logical outcome of the way consciousness works at different scales. In the same way that 100 billion neuron cells in our brain work together to create an illusion of awareness, it’s not too much of a stretch to imagine a broader perspective in which 7.6 billion organisms collaboratively source the distributed super-consciousness we call humanity. Just as individual neurons aren’t aware of qualia created by other parts of the brain, we individual humans are only capable of seeing a tiny sliver of the immense awareness we collectively generate. Regardless of whether we are all actually “the same person” as described in The Egg, it is perhaps comforting to remember that just beyond the confined pocket of our minds, we are part of a vast network of consciousness working together to discover our place in the universe.