Dimensional Thinking

a way of talking and representing data, or an inevitable consequence of neuronal wiring?

Are there certain ways that we are more inclined to make sense of things, to organise our knowledge? And if there are, where do these tendencies come from, are they merely of our historical moment or inescapably part of us? One such possible way of thinking is what I’d like to label “dimensional thinking.” And it is everywhere. It’s the talk of modes and scales, of varying along multiple spectra, of existing in spaces. It is all over the sciences and various other academic disciplines, it is the modus operandi of all that data science touches and inflects much AI-speak. Perhaps you will recognise it in other places not mentioned here.

What follows are like pieces of washed up sea glass or curious shells that I have collected while walking up and down the beach over the past while, and I have called around to your house with them in my hands to show you what I’ve found.

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The scaffolds we operate with(in)

We use dimensional thinking all the time. When caught up in big “what am I going to do with my life” questions, we might consider both what kind of work we find the most rewarding and meaningful, in conjunction with how comfortable we might want to be. A rough plot of four different kinds of life may come to mind*

*I am not, by the way, endorsing this way of thinking.

Although our focus will be more on its use in the academic world, dimensional thinking features colloquially in many turns of phrase. When we say that someone is “going off on a tangent,” there is an abstracted notion of departure from somewhere. It features when speculating about whether one might be “on the spectrum”, or whenever there is mention of a political or other spectrum, such as when someone (perhaps incoherently) describes themselves as “socially progressive but fiscally conservative.” In sum, it crops up whenever we are referring to general ways in which we think people vary. It features whenever we consider “if something is a difference in degree or in kind” and when we weigh up pros and cons. So dimensional thinking is something we have an intuition for and use in a variety of different circumstances.

But who cares? I’m curious because we live in a chronically hyper-specialised world, and I think it’s worth developing ways of thinking that aid sharing ideas and collaborating with others from different backgrounds. I’d like to explore whether something like dimensional thinking could serve as a common meta-language in an age of siloed complexity, and whether it should. Ultimately, I don’t think so, but we’ll present a good case in order to get there. Here are (what I think are) a few examples:

i) In the social sciences, dimensional thinking features in approaches informed by intersectionality. To describe the person in terms of their positionality or social location—their ethnicity, sex, gender, sexuality, class, religion, disability status, and so forth—is to use a multidimensional framework. Such a framework offers, very abstractly, ways to consider combinations and interactions to further nuance and address overlooked and manifold layered inequality and discrimination.¹ Dimensional thinking can also work at the level of populations as much as individuals. When we consider what variables contribute to political preferences and turn to various demographic factors, we are using dimensional thinking in our search for explanations.

ii) The concept of neurodiversity, too, seeks to give name to numerous ways in which people may differ. And while we may wish to argue that there is no such thing as a neurotypical individual but rather several dimensions along with we vary to differing degrees relative to others, drawing and revising categories can still help us understand unique ways of being-in-the-world, and in turn our personal and structural ignorances. Names are imperfect but give us something to organise ongoing conversation around.

iii) Let’s stay with neuro-speak for a minute. Dimensional thinking features in our attempts to understand neurodevelopmetnal conditions. When the brain is wiring up, variation occurs as a result of it being a noisy process,² and some variations reliably occur—to which categorisation manuals give us epilepsy, schizophrenia, and so forth (except the narrative is never that neat in practice, intra-category variation can question the validity of a cohesiveness of all constructs mentioned).

iv) Close to my heart is an example of applying dimensional thinking to understanding meditation. Appreciating the fact that there is no singular definitive type of “mindfulness,” or even “Buddhist mindfulness,” one group³ collated phenomenological reports to construct a multidimensional space of possible psychological states, areas of which comprise different styles of mindfulness practice. This shows that dimensional thinking can be applied not just to understanding people, personality, behaviour, but also different possible states of subjective experience:

Excerpted from Lutz et al. (2015)

v) In biology, many of us have been captivated by the discoveries and ideas of Michael Levin, whose research group created Xenobots (an artificial organism made of frog skin cells) and reprogrammed flatworms to have multiple heads. While animals have nervous systems that control behaviour so as to navigate “physical space,” he extends this way of talking to developmental processes as navigating “morphological space.” A genome ensures that a developing organism ends up at a specific point in morphological space (e.g. having two eyes, not four), but there may be multiple paths to that location. Mash up the face of a tadpole? No problem, it will reassemble. Blend up a hydra? It can reform. Displaced from its “trajectory in morphological space,” the organism reassembles.

Let’s summarise. Dimensional thinking is flexible. We pick our favourite phenomenon, natural kind, or noun to study. Doing this defines a figure from its background. When we grow unhappy with our binary description or someone tells us it’s too simplistic, it’s no problem. We can smudge the binary into a gradient or spectrum. And when we can’t describe our favourite phenomenon using a spectrum because that still feels too simplistic, we can add another dimension along which it can vary. Now we have a 2D plane, or 3D space, or N-dimensional space where we can represent many of the most important different possibilities our biological, economic, or ecological system, or person, can take on. When we plot changes in the multidimensional space over time, we see that some paths are taken more often than others (not every possibility is realised). Popular culture sifts out common patterns of progression in interests or behaviours in the “the X to Y pipeline” refrain. Limerence describes a certain regularity in our psychosocial world, so does homo sapiens in phylogenetic space, so does the Goldilocks zone of planetary habitability, so does the main sequence of star development, the phase diagrams of states of matter, metal alloys, and diagrams for the entirety of the universe. With dimensional thinking, we can work the rungs of nuance. It allows us to type or kind THINGS while eschewing Bad Discrete Categories and Simplistic Venn Diagram Thinking, giving us aggregates, fuzzy at the edges and which can bleed into one another, and without recourse to hand-waving vagary or loss of human intelligibility.

Dimensional thinking for discovery

One concern we might level at dimensional thinking is that it does not tell us how to choose these categories. In addition, sometimes we might not want to choose categories in the first place because we want a bias-minimised (scientific) discovery of what the borders are. Surely not everything we want to know can ever be expected to fit into neat measurable categories? But dimensional thinking has tools that promise being able to do just this. Categories not simply “given”? No problem. There are statistical methods for discovering regularities in data to aid in identifying how many dimensions are relevant for describing your favourite phenomena. This approach has proven popular in psychology and neuroscience, where prescientific folk-psychological categories like emotion, memory, imagination, may have no identifiable biological basis, and some want to know what the “natural” categories of minds are. For example, one study⁴ used such statistical methods to pick the best three dimensions we use when representing the mental states of others. Another study,⁵ this time in medicine, used PCA to create an abstract 20-dimensional “space of diseases,” which allowed them to discover new genetic variants that may contribute to similar diseases, diseases which previous taxonomies, influenced by arbitrary anatomical and cultural factors, may not necessarily have viewed as similar and drawn a link between. This highlights a powerful feature of these state-space models—similar things sit closer to each other in multi-dimensional space, which can allow us to draw connections between things that we previously might not have thought were closely related. Finally, although the language of “traits” and “variation” is an example of dimensional thinking, and it features strongly in psychology, biology, and medicine, academic philosophy the past few decades has been obsessed with discussing what is possible. When we consider the future, it is common to talk of a space of possibilities. And when reading a novel, we might speculate about the possible future trajectories of the characters. All of this, too, can be viewed as dimensional thinking.

It’s all very grandiose, but it can get better/worse, as we don’t even have to stick to thinking of these dimensions in strict 3D (Euclidean) space. Mathematicians have formalised dimensional thinking into what we could call topological thinking—the main concern of which is asking how many dimensions a system has, and whether the space is flat, curved, loops back on itself to form a ring, torus, or other weird shapes. Once recent popular overview of abstract mathematics⁶ sought to show that topological thinking can be applied to personality, baking, taste, colour, and many other things. Once you get a feel for Dimensional Thinking, it’s hard not to use it as a fun way of approaching any problem. And it does seem associated with rich theorising in various intellectual disciplines, to the point that sometimes I’ve felt that, at least in STEM, dimensional thinking is the destination many of our best explanations, theories, models, appear to be falling towards.

But I am concerned

And for a variety of reasons. But I first want to consider an annoying possibility—

Is this way of thinking inescapable?

I would not have seriously entertained this until neuroscientists started claiming that the neuronal scaffold used to construct a sense of space for navigation is the same scaffold we use to structure all relations⁷—abstract concepts, social relationships, etc. That the brain system used for navigation which constructs a sense of physical space and time (out place cells, border cells, object-vector cells, grid cells, and time cells), is the same structure used for organising all our learned knowledge.⁸ Concepts are created out of episodic experiences as their nuance is gradually shorn away, a process of turning episodic memories of what happened into semantic knowledge of the ways in which things happen (this is how one narrative goes, anyway). Regardless, the claim is that we use these scaffolds to makes sense of anything that can be related. Perhaps, therefore, there is a limit to what we can learn and conceive of, and all our thinking is built on foundations that make it “dimensional”, we can’t just “learn anything.” Innate structures have to be a certain way in order to afford us the ability to learn, and maybe it will turn out to have been overly Romantic to think that there are no limits to what it is possible to conceive of and that human experience should be defended from scientific explanation.⁹ But I think we are several centuries away, and that currently it is hubristic for neuroscientists and psychologists to assume we have discovered not just how we think, but the ways we could ever think. My argument against is not sophisticated: neuroscience is in its infancy, and we have a poor idea how to interpret what we find. The brain does not tell us how it works or how we should understand it. Unsurprisingly, we instead make sense of what we find using what’s already to hand. Many of these papers reference Kant (because we live in a cultural tradition strongly influenced by Kant’s philosophy), who argued that there are certain categories that must precede experience in order for experience to be possible, such as time (though it is worth noting is that the early Buddhist Abhidharma texts also contain lists of different “mental factors,” cetasika, that are necessary for experience to occur, and time and space also feature in vajrayana meditation texts too).

Before, our question could have been entirely normative: how should we organise our knowledge? Now the question shifts to being about whether our knowledge can only be certain ways because of particular preconditions that don’t actually feature in our thought, and we’ve caught onto this notion that we might find them in the brain. Now, we could easily swallow our tails speculating on whether these neurons are recapitulating properties of “the world”, and the reason we “think dimensionally” is because that’s just the way the world is — or whether “the world” appears this way to us (as unfolding in space and time) because cells wiring up this way support useful actions, no world-representing business needed. But even if there are limits to how we can organise our knowledge, and our thinking mirrors “something,” no one is ever going to descend from the stars and tell us we’ve gotten it right, and give us a pat on the back to say we’ve done it, great job, you’ve figured out the brain. I’ll explore this in a future post, for now, it means hubris regarding “how we think, according to neuroscience” can be put in the bin.

Dimensional thinking has become part of me and I don’t know if I want it

Even if this way of thinking is not “inescapable”, I have certainly automatised it and now find myself using it make sense of new things. I know I’m not alone (I have heard people describe what movies they like in the language PCA dimensions). What makes this such a pressing problem? For most people, whether or not this is the bedrock precondition of all thought does not matter. In many cases it is a highly useful heuristic. The very simplicity of a dimensional framing is powerful, making for a highly effective communicative device that can inspire practical action. We have limited ways of getting a handle on all the vastly complex issues in our living, evolving, inner and outer worlds—so what if we use simplistic heuristics like variables plotted in abstract spaces?

But for those of us regularly exposed to this way of representing (and I see these tools being used in increasingly diverse settings, and “graphical literacy” is itself now deemed an important 21st Century skill), it deserves some critical interrogation. We should wonder if this really is the only way we have of representing and communicating complexity in a compressible, human-accessible way. As with any powerful new technology, we run into the classic “when all you have is a hammer, everything looks like a nail” problem—the risk of applying the tools you know best, or the only ones you have, indiscriminately. Many disciplines when we gain exposure to them completely change the way we view the world, whether law or physics or architecture. There is a distinctive feeling of how one understood the world before learning it…and after. Some innocence has been removed, we suddenly see everything in a new light. And dimensional thinking may be one such perspective that comes with any discipline that uses statistics and quantitative data visualisation. But when we use these methods, it leaves a residue.

The main power of dimensional thinking is categorisation, it allows us to draw more nuanced lines. But we’re still in the business of drawing lines. And we should care about more things than drawing lines. Dimensional thinking has no thrust, it categorises, but does not itself explain why the categories are carved the way they are, or what to do with them. Now, it might seem unfair to level this criticism at dimensional thinking, to charge it with the responsibility of tackling every problem. But that is exactly what it promises. Let me present my point another way. What are the most significant developments in our thinking so far this 21st century? Big data is becoming the way of understanding and tackling problems. And Dimensional Thinking is, I believe, the ontology, the worldview, of big data. The ability to “understand” everything and anything with data is what it implicitly and explicitly promises. Now, we might counter that this is not the fault of dimensional thinking itself, but rather how it is used, the ends to which it is put, specifically when it is used with an attitude of or within a culture of “solutionism”—proposing technocratic fixes to all problems (in particular environmental and social issues). In this context, we have empowered dimensional thinking to corrupt whatever we touch with it, alluring us into thinking that we can even conceive of different possible futures, or understand people simplistically, when in actuality it can’t applied to everything. A lot of work has to happen in the background to use these statistical methods—they can only be applied to that which can be numerically represented, which means you have to find a way to operationalise your favourite phenomenon or noun into numbers, which may not always be appropriate. Perhaps then, when we temper a solutionist attitude with an awareness of when it is and is not appropriate to use dimensional thinking, we can avoid this uncritical wielding of our fancy machine learning tools, and all my intellectual arm flailing can cease with the simple observation that the problems discussed here are problems with how dimensional thinking is used, not it itself. But I’m not convinced that dimensional thinking can be let completely off the hook.

What worldview does dimensional thinking itself offer? A static one. “The past” for it is a thing that has already happened, a frozen corridor extending backwards and full of all sorts of things, rather than the ever-evolving ground on which we stand and continually recreate. The underlying ontological flavour remains its Western chimeric Christian-Enlightenment self. We retain the third person, the timeless and omniscient God’s-eye viewpoint: now of people and systems and everything ever possible. This is why I worry. It gives us the illusion of nuance and flexibility (and in many domains for operational tasks this may be perfectly satisfactory), but it is as if Western thought, in coming up against its limitations, squirming against accusations of all the trouble its essentialism has wrought, didn’t really change but instead went fractal, got more detailed but kept its frame.

And I have to remind myself, just because we can think of everything in a certain way (as information, as computation, as cognition, as social construction, as performative,¹⁰ and as clusters in multidimensional spaces), doesn’t mean we have to. Just because a way of organising knowledge seems to be effective in many areas has no bearing on whether our dimensional constructs and imagistic schemas “accurately represent something of the world.” They can simply be tools for organising data to inform action, and we don’t have to turn the tools into a worldview. And this is fine, but I think in order not to, we need to single out this way of thinking and challenge ourselves to ask—what comes after dimensional thinking? After we have poured so much effort into acquiring the ability to categorise the world into infinite pieces, what do we do with all these descriptions, all this descriptive power? What can we use instead when it doesn’t seem to work?¹¹ As a friend who read a draft of this remarked: “our best descriptions of what thought is at this moment might lead us to radically transform what thinking then becomes.” It might not be a hyper-cerebral “way of thinking” at all.

Anyway, time to go to bed or back to work or go outside.

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1. Crenshaw, K. (1989). Demarginalizing the Intersection of Race and Sex: A Black Feminist Critique of Antidiscrimination Doctrine, Feminist Theory and Antiracist Politics. The University of Chicago Legal Forum 140:139-167. Link.

2. Mitchell, K. (2020). Innate. Princeton University Press

3. Lutz, A., Jha, A. P., Dunne, J. D., & Saron, C. D. (2015). Investigating the phenomenological matrix of mindfulness-related practices from a neurocognitive perspective. The American psychologist, 70(7), 632–658. https://doi.org/10.1037/a0039585

4. Tamir, D. I., Thornton, M. A., Contreras, J. M., & Mitchell, J. P. (2016). Neural evidence that three dimensions organize mental state representation: Rationality, social impact, and valence. Proceedings of the National Academy of Sciences of the United States of America, 113(1), 194–199. https://doi.org/10.1073/pnas.1511905112. See also: Tamir, D. I., & Thornton, M. A. (2018). Modeling the predictive social mind. Trends in Cognitive Sciences, 22(3), 201–212. https://doi.org/10.1016/j.tics.2017.12.005

5. Jia, G., Li, Y., Zhong, X., Wang, K., Pividori, M., Alomairy, R., Esposito, A., Ltaief, H., Terao, C., Akiyama, M., Matsuda, K., Keyes, D. E., Im, H. K., Gojobori, T., Kamatani, Y., Kubo, M., Cox, N. J., Evans, J., Gao, X., & Rzhetsky, A. (2023). The high-dimensional space of human diseases built from diagnosis records and mapped to genetic loci. Nature computational science, 3(5), 403–417. https://doi.org/10.1038/s43588-023-00453-y

6. Beckman, M. (2021). Math without numbers. Allen Lane.

7. Whittington, J. C. R., Muller, T. H., Mark, S., Chen, G., Barry, C., Burgess, N., & Behrens, T. E. J. (2020). The Tolman-Eichenbaum Machine: Unifying Space and Relational Memory through Generalization in the Hippocampal Formation. Cell, 183(5), 1249–1263.e23. https://doi.org/10.1016/j.cell.2020.10.024 See also: Constantinescu, A. O., O'Reilly, J. X., & Behrens, T. E. J. (2016). Organizing conceptual knowledge in humans with a gridlike code. Science (New York, N.Y.), 352(6292), 1464–1468. https://doi.org/10.1126/science.aaf0941

8. Zador A. M. (2019). A critique of pure learning and what artificial neural networks can learn from animal brains. Nature communications, 10(1), 3770. https://doi.org/10.1038/s41467-019-11786-6

9. Sharf, R. (1998). Experience. In C. Taylor (Ed.), Critical Terms for Religious Studies. University of Chicago Press

10. Actually, this one is probably fine.

11. For example, a different way of thinking might be one which places primacy on growth and transformation and nothing ever being the same, of radical contingency (path dependence), where it is naive to think we can envisage a light cone of possibilities in the future, or assume that there are only certain ways we are capable of thinking.