Depth in a Surface-Level World

Using derivatives and linear combinations to model consciousness and expression

The architect Aaron Vartiainen spoke in a recent video about how much assumption is implicit in our perception. He argues that all interaction in the world happens on the level of surfaces. Sight is only point-to-point contact between the eye’s lens and the object’s outermost layer of atoms. Everything we hear is first projected onto the eardrum’s flat head. We don’t perceive each other as flesh and bone beings, but as closed surfaces of skin. We can imagine that the aggregate of our sensory information, produced as it is by the interaction between surfaces, fills out an almost echolocative map of the surfaces of our surroundings. Prior knowledge and assumption complete the map and fill out the volumes.

This framework of using essentially [n-1]-dimensional inputs (surfaces) to infer qualities about n-dimensional objects is how our perception of the world is constructed from sensory input. I suggest it can be applied in a wider context to provide insight on how our perception of self and relation to others is constructed from more complex forms of input.

To apply the framework well, it is useful to build a simple mathematical intuition for the relationship between an object’s boundary and mass. We can imagine a function f(r) = πr² where, as r increases, f(r) reflects the mass of an increasingly large circle. In trying to find the boundary of that circle, we might realize it equals the derivative of f(r), 2πr. If we consider how f(r) changes over progressively smaller intervals in r, it becomes clear why the derivative is equal to the circumference. Bringing this intuition over to our 3 spatial dimensions, the same holds true. Define a function s(r) = 4/3πr³, the mass of a sphere. In the same way the circumference represents the incremental difference in f(r) for arbitrarily small changes in r, s’(r) is equivalent to the surface area of the sphere: 4πr².

So when we perceive a ball by its surface, we are observing the derivative of the ball’s ‘function’ as it acts on 3-d space. More precisely, the concrete sensory data we receive reflects the value of the derivative when r equals the size of the ball’s radius. This derivative relationship between boundary and mass only exists in radial symmetry, as boundary is uniform w/r/t mass. However, we can generalize the concept of perception through a derivative in a lower dimension to non-uniform objects. By derivative, I mean a local projection of a higher dimensional system onto a lower dimensional boundary. When I observe a lamp, a tree, the moon, I am seeing its internal behavior projected onto a 2-dimensional edge. Our brains essentially do the work of integrating that edge over the total space of the object to perceive it in 3 dimensions.

Now we can apply this idea to objects of higher dimensions, namely human interaction. For a ball, only 3 variables are needed to encode its position in space. Adding a fourth, time, allows us to track the total activity of the ball in the universe. Consider trying to instead model the ‘state’ or total activity of the human consciousness. We need a vector of arbitrarily high dimension, because there are an arbitrary number of variables that can influence a conscious state. Precisely, there are many possible combinations of neurotransmitters and neuroactivity that can affect the consciousness, which we can loosely categorize into ‘dimensions’ like hunger, lust, physical discomfort, or melancholy. This means we can model consciousness at time t using a state vector [v_t] reflecting the weights of all those dimensions. For example, if I haven’t eaten all day, my conscious experience would be disproportionately affected by hunger, represented by a higher weight on the hunger variable in the state vector.

Of course, the current state of consciousness does not reflect its total activity. Namely, it does not tell us how those weights should change over time. To address this, we can use a combination of two personality matrices: W, an internal dynamics matrix, and A, an attention matrix. W displaying the aggregate personality of the conscious, and the rate of change of the weights of [v]. If hunger is high, irritability might have a high growth rate. A encodes how the consciousness interprets and weights a new experience. We can denote a new experience by the vector [u], with some base weights reflecting a general embedding for each event. For example, the event of eating a donut might be associated with a negative weight in hunger and a positive weight in pleasure. A transforms that vector [u] into how the experience is perceived by our consciousness. If the hunger weight is high in [v], A gives special weight to hunger and the negative hunger weight of [u] is magnified. Essentially, a donut provides more hunger relief to a hungry person than a full one. If W updates [v] and A encodes [u] into its conscious perception, the sum effect of the two matrices W and A is to represent the ‘personality’ of the consciousness, how it adjusts and reacts to time and stimuli.

Each second, we can say, [v_t] is updated by the equation [v_t+1] = W[v_t] + A[u]. This is indeed a very crude model. It might be more precise to model the parameters of consciousness in matrix layers. Factors like hunger, pain, and fatigue would have high weights in the short term, while self-fulfillment, career, and romance take effect on a more subtle timeframe. However, our simple model allows us to appreciate both the complexity of the constantly evolving consciousness as well as what is lost in interaction.

Analogous to how we perceive the derivative of the physical body of a person in seeing them, we observe the derivative of their conscious state in conversation with them. In talking to someone, we don’t see their changing state weights, [v], but the derivative of that state function, projected onto the lower dimensions human expression is capable of. Think of trying to explain yourself while emotionally distressed; your language and expression breaks down, you’re almost hysteric. You’re never able to truly express through any combination of gestures and sounds your internal feelings. However, in trying to express yourself, the emotional weights on [v] get projected onto the visible edge of the consciousness—that is, expression. Seeing how your expression degrades and becomes erratic when subjected to those emotions, your audience is able to integrate that derivative over their understanding of your conscious space to infer the degree of extremity in your feelings.

Consider how much information is lost in every interaction between two people. First, the highly complex emotions and incentives provoking each statement are compressed into your expressive space, reduced from the degree you can feel them to the degree you can convey them. Then the statement is further compressed by the attention matrix, A, of the person receiving the statement, and the meaning is distorted by the bias of the receiver. The statement then has to be decoded by the receiver. This last step is essentially the work of ‘assumption’ Vartiainen described, or the integration of lower-dimensional input over the perceived space of the object, which allows for perceptual constructs to be formed.

It is worth noting that in 3-dimensional spatial perception, the integration step is mostly trivial. This is because from just the three positional variables, we have a fair estimate of the r, or integrating space, of the object. This means that by virtue of perceiving an object, we know enough about its spatial properties to use our derivative input to make reasonable assumptions about the object as a whole.

The evident issue is that this property does not hold up on the scale of conscious minds. Upon meeting a person for the first time, I don’t by any means have enough information about the personality of their conscious (i.e. their personality matrices) to make reasonable conclusions about them as a person. With high-priority indicators like fatigue and joy, it is somewhat easy to draw conclusions. I can usually clearly see if a stranger is tired or excited. But with deeper-level weights on the consciousness like self-perception, insecurities, or ambitions, it becomes increasingly hard to predict qualities of the consciousness from a basic derivative.

How do we define the process of coming to know someone? The more interactions we have with a person, the more states of [v] we are able to witness them in, the better assumptions we can make about their personality’s ‘mass’, or integrative space. All this means is that we gain a more intimate knowledge of the evolving weights on their personality matrices, and are thus better able to do the decoding work of integrating expressive inputs. It is worth asking further questions about if we are ever able to take enough varied vantage points of an object to escape our own personality biases and make the most reasonable assumptions.

It is perhaps a scary or lonely concept to be inherently trapped within one’s own conscious experience. We are subject ourselves to the shifting weights of our [v], and are only able to interpret the world through the biased lenses of our personality matrices. Vartiainen spoke of a world where life and meaning are composed of derivatives acting on derivatives, where the complex systems inhabiting the universe are never able to interact in their purest form. However, through mathematical intuition and integrative thought processes, we are able to provide a meaningful framework for how our understanding of the nature of those derivatives can in fact reveal truer truths about our world.