We all have “dimentia”

12 min read

Dimentia = dimension + dementia
You have dimentia, and that’s a good thing. If you know it. Let me explain.

Dimensions

Our minds are unequipped to apprehend a space with more than 3 dimensions. What would it be like to live in 4 dimensions? Despite our ability to painfully build a few basic intuitions, the question makes no sense. What would a tree or a dog look like? Too hard. Let’s take a simpler case: the cylinder. What does a cylinder look like in 4D? It turns out it could be three different things, called the cubinder, the spherinder, and the duocylinder. Following the same principle, did you imagine the different kinds of dogs that could exist in 4D? If you’re like me, what your mind actually generates when you think of a 4D-dog is just a 3D-dog in a 4D space. Our intuition is powerless, it fails absolutely. For instance, in 4D, two planes can intersect in a point. How is your intuition doing?

But 4D is child’s play. What about 1000D space? It looks like we don’t need to build intuition for such an absurd thing, but it turns out that feature spaces in machine learning are easily that size, if not orders of magnitude bigger. I write “size” but I mean dimension. We even lack the words. When we meet these spaces in computer science, we are naturally drawn to applying our 3D intuition to them, but that would be a huge mistake. A mistake so common that it has a name: the curse of dimensionality.

But 1000D space is child’s play, because it is Euclidean. Even in 1000D, you can draw a 2D triangle and it behaves like you expect. In 2D hyperbolic geometry, which is not Euclidean, the triangle breaks your intuition. If you try to build the biggest triangle you can, you end up with one delimited by 3 parallel lines (you read me well) and even so, its volume is finite (idem). In fact, in hyperbolic space, there is a maximum area for the triangle, and any polygon more generally. And of course, you could have a 1000D hyperbolic plane. And even weirder things. Networks are non-Euclidean spaces as well, and hyperbolic space is remarkably useful to network analysis and visualization. Building intuition for network topology is actually so hard that we generally fail to account for the fact that depending on the network, the challenge might be completely different. A lattice may behave like a 2D plane while a large, real-world, scale-free network may behave like a high-dimensional, heterogeneous, hyperbolic space. Our intuition has been left far behind at this point.

To get an empirical taste of 4D space, take a look at 4D Toys and the making of Miegakure, both by Marc ten Bosch. Also take a look at the upcoming 4D Golf and Hyperbolica for an experience of non-Euclidean spaces, both by CodeParade.

Complexity

I believe that complexity refers to the horizon of our understanding. It looks like complexity can be defined in itself, as an empirical feature that we may one day get to understand; but that is an illusion. Complexity is generally described as entanglement. But there are infinitely many entanglements, and just a few disentangled things. The things we see as disentangled are those we can parse, and the rest we call complex. It follows from a simple alternative. Either anything in existence can fit within our limited understanding, or not. The extraordinary claim of our entitlement to understand everything lacks extraordinary evidence; it lacks evidence at all; and it ultimately boils down to main character syndrome. Once again, we fail to realize that we place ourselves as the center of the cosmos.

Entanglement is tricky because it looks so material. Entanglement is knotiness, and everyone knows a knot, even though there are infinitely many kinds. But entanglement hides a more subtle form of resistance than the frustration of undoing shoelaces. It resists our general strategy to know: divide and conquer. The frustration of complexity comes in its own peculiar flavor, where the phenomenon to know disappears as we divide it, forcing us to hold all the parts at once in our mind. Networks are a good example, because their raison d’être is to care about relations (the links) instead of substance (the nodes). If you can study the social structure of a group after the properties of its members, then you don’t need network analysis; Excel suffices. But the social structure lies in the relations, and your phenomenon disappears if you divide the group into its members, precisely because you lose the relations. The phenomenon lies in the relations. Divide-and-conquer is lossy, and complexity makes you pay that loss back with interests. Entanglement is, in essence, resistance to division. It may come as a surprise, but complexity is not complicated. That’s because our ability to know is not that complicated, and complexity is entirely defined by it.

Our limitation is the fundamental reason why we divide things to know them. We divide into chunks we can apprehend. We trade time for space, which is the fundamental operation of analysis, the essence of computing. We absorb in multiple times what we can’t take in one. And we have been quite successful at dividing anything. The problem does not lie in the size of the chunks, which we can always divide; the problem is the price of dividing. Division being lossy, when you divide in two you generate a third chunk: the loss, the cost of the cut. This additional chunk is not always small. When it is big, we call the phenomenon complex. Complex phenomena resist by overmultiplying parts as you divide them. Your time too is limited.

You unpack phenomena for a living and you believe that complexity is real, independent of your subjectivity. When a phenomenon resists you, you find more clever ways to divide, and you conquer. You don’t see complexity primarily as entanglement, but as emergence. The shape of dunes is not written in the laws of wind, or in the grains of sand. It emerges from the complex system of sand and wind. The shape of dunes is written in the laws of complexity. Dividing works for a while, because complexity is a ladder. The first steps only challenge the most obvious ways of dividing, and you find clever alternatives. It’s not just the wind and the sand, it’s also the system. The system is the cost of the cut, the supplemental chunk. You divide the entanglement into its pieces plus their relations, and you see that extra chunk as just another chunk. But higher up the ladder of complexity, cutting the supplemental chunk also comes at a price. You need to account for the meta-relations between the parts and their relations. You need to account for the meta-system of the parts and the system-as-a-part. You need an instrument to analyze the results of your analysis. And so on. The self-serving postulate that everything has its laws becomes a liability. Either complexity forces you to accept that knowing is relative to your limited ability to describe phenomena as systems, and therefore subjective, or it forces you to postulate the existence of unknowable laws, which makes you leave empiricism for pseudoscience.

Remember that Leibniz believed that we could one day prove all of mathematics, and that Gödel killed that dream.

Dementia

Facing multi-dimensional spaces, or the topology of complex networks, is facing the abyss of our own finitude. Our instinct makes us look elsewhere. The sense of loss and helplessness that comes with the defeat of our understanding is generally deemed unproductive. We take what we get, we leave what we don’t, we forget it exists, and we turn our back to the abyss. It is simply more practical to delude ourselves into thinking of any space on the basis of the 3D space. This leads to making mistakes on the way, but we fix them once we get there. We give them a name, like “curse of dimensionality”, think of them as a problem, and find solutions. Divide and conquer, always. That’s all we can do. That has always worked, right?

I’m interested in the possibility that loss and helplessness can be productive. My primary motivation is to overcome collective denial about complexity. I see a potential problem in the fact that we, human beings, have basically the same limitations. Intelligence is not like wealth, where some one-percenters get provided with orders of magnitude more than the rest. What makes our geniuses geniuses is not their ability to think thousands time faster or memorize near infinite knowledge. If we were computers, none of us would be supercomputers; all of us would be consumer laptops. This makes it so easy to mistake our own homogeneity for a law of nature. I glimpse the possibility that we have collectively agreed to pretend that anything that we can’t divide and conquer simply does not exist. That is why I’m up for trying another path. Maybe, living with a higher sense of loss and helplessness could help us see what we might be missing otherwise.

I take seriously the eventuality that our confidence gets in the way of the science we do. I am looking for ways to temper that confidence, to reopen the doors to things prematurely understood. I try to factor in the possibility that we might be unknowingly blind. How would you know that you’re missing some senses? Missing compared to what?

This leads me to dementia, and acquired disabilities in general. Some people are forced to experience the loss of some of their abilities. I don’t know much about this, beyond having friends in that situation, and having experienced a depressive episode myself. Here I’m just sketching a way forward. I think that we should draw inspiration from people who have experienced that kind of loss. I am particularly interested in dementia, because I believe that the affected person may experience various degrees of awareness of their own condition. I wonder: how do you become aware of your own impairment? How does this awareness change your relation to the world? How can you overcome your limitations? I want to harness the psychedelic nature of dementia and repurpose it to help us touch the walls of the human thought box, so that we can some day break free of it.

Dimentia, with a “i”.

If you have ever tried seriously to learn about the fourth dimension, you have heard about Flatland, the 1884 novel by Edwin Abbott. The main point of the book is to have you experience the 3D world from the standpoint of a character living in the 2D world, Flatland. You realize that the 2D world is quite different from our own, and that the 3D world cannot look the same to a 2D person than to us 3D folks. The main protagonist, Square, sees his flat world as a line, the same way we see our 3D world as a plane, even though in both cases we can have a sense of depth. Being pulled out of Flatland into the third dimension does not change Square’s visual system; he still sees the world as a line, even though what it sees gets different. He gets to experience the existence of the 3D world, but it does not make it much easier to understand.

An episode about Flatland in Randall Munroe’s webcomic XKCD.

Square has what one could call dimentia: dementia about dimensions. He experiences to some extent his own inability to apprehend the third dimension. Flatland, the book, is famous for this precise reason: it gives you a practical idea of the problems to face experiencing the fourth dimension. It makes you understand why you may never fully get it, and yet what kind of work you would have to do to compensate for your own limitations.

The word dimentia comes from Johanna Drucker. I was last week in Dagstuhl for a seminar about Visualization and the Humanities; I will say more about it in a few upcoming posts. I had a short presentation to make, and during the questions, I made the same point as above about drawing inspiration from people with dementia to understand complex spaces. As I went back to my seat, Johanna gave me a post-it with dimentia written on it. I understood that it referred to some a previous work but I could not find a reference to it, it may not be published. Dimentia is also a common misspelling of dementia, it has an entry in Wikitionary.

The dimentia post-it passed to me by Johanna Drucker.

The point of dimentia is that we all have it. Dimentia is a natural condition of the human being. If we had the opportunity to meet a being vastly superior to us (alien, AI, angel, pick your favorite) then maybe we would give a name to all the things they have and we don’t. Then it may become clear that we have dimentia. And it would crush our main character syndrome. Until then, we are inclined to believe that we’re perfectly fine people, made in the image of God. That’s why we need a bit of creative help to think outside the box. Let’s take it seriously that we have dimentia, and start wondering how to factor in our inability to deal with a number of things beyond our cognitive abilities.

You have dimentia, so does everyone else, and it’s a great thing as long as you are aware of it. I suspect that we have more to gain by studying understood things as if they were incomprehensible than by studying incomprehensible things as if we were entitled to understand them. Because at the end of the day, I am not so confident that we actually understand what we believe we do. At least for the topology of complex networks, it seems to me that we have left a number of unknown unknowns on the side of the road.

PS: Thank you Johanna!



Cite this blog post
Mathieu Jacomy (2023, September 25). We all have “dimentia” Reticular. Retrieved April 25, 2024, from https://reticular.hypotheses.org/7955

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