What does the universe of human knowledge look like? A neat disk with the frontier at the edge? Or a porous hairball with the frontier sprinkled throughout?
How should we depict the universe of human knowledge? Does a graphical representation even make sense? One answer is the disk pictured below, sometimes used in orientation sessions with beginning doctoral students:
The disk of knowledge. Borrowed from Matthew Might (1).
The full disk represents all human knowledge. You begin your life at the center of the disk and travel outwards as you learn. The blue part is elementary school. The green part is high school. The pink stuff represents your undergrad years. Then as you specialize in grad school, through coursework and mentored study, you travel a narrow path towards the outer edge — the frontier.
And if you can add a bump, as depicted below, extending human knowledge just a bit, then you get a PhD:
The disk of knowledge is a nice graphical model, capturing some important aspects of the PhD experience. But my own experience does not fit. I started in robotics in 1976, when I joined the MIT AI Lab’s PhD program. Two years later I finished my master’s thesis. After fixing some errors caught by the reviewers and polishing it a bit, it was published. That paper, sole-authored, is my most-cited publication ever. Yet I was only halfway to the rim of the disk of knowledge — where would I put the bump?
Clearly, we need a different graphical model. What would be fun? A torus? The donut of human knowledge sounds cool. Or, would a fractal work? We tend to look back and romanticize the early days of our field, when an earlier generation laid the foundation. But what if that perspective shifts in time and repeats with every generation? Perhaps the frontier that we see is similar to the frontier seen by our forebears. And perhaps someday our intellectual descendants will view us as the founders of their field, or sub-field, or sub-sub-field. When you move your gaze to the new frontier, and zoom in, maybe it looks just like the old frontier. Self-similar, like a fractal. But I cannot figure out how to construct that fractal. So, instead of a fractal, how about a hairball?
The hairball of human knowledge
The disk of knowledge, and the doctoral student’s path to that circular frontier, is a useful way to envision your task when you enter grad school. But I don’t think it was ever intended as a precise model of human knowledge.
How can we develop a more precise model? Let’s start with the intellectual path of an individual, progressing in the universe of ideas. My own intellectual path feels like a rather chaotic meandering. Sometimes I forget things and have to re-learn them; sometimes I get confused and learn things I have to unlearn; and sometimes I circle around an idea but never get there, never sure whether it was a legitimate idea to begin with. So, I envision my own path not as an arrow or a nice slim bulge, but as a crazy path like this:
Now, to define the sum of all human knowledge, perhaps we could construct the intellectual paths for all humans that ever existed, and then superimpose them, like this:
It is not a disk; it is closer to a hairball. (I am told this rendering looks more like a Miro painting or a Calder mobile. I would be flattered, except I was going for a hairball.) Where is the frontier? It is hard to say. Some of those curves are about to double back towards known ideas, and some of them will hit new territory — previously unknown ideas. We can define the frontier as the ideas that are about to be discovered. It isn’t a continuous boundary by any means — there is no boundary.
A more careful construction
In the process of writing this, my mind wandered (how topical!) to the idea of a more principled hairball construction. Here’s how I proceeded. First, does it make sense at all to construct a path of an individual’s mental development? Since I am a computationalist, I believe that any state of mind can be well-approximated by the state of a computer, and therefore can be represented as a string of symbols in some formal language. So the universe of mind-states corresponds to a subset of that very high-dimensional space of symbol-strings, and the cognitive path of a mind corresponds to a path through that space. So, yes, we can define the cognitive path of a mind. But can we project that path to a space of ideas?
What is an idea? I cannot define it precisely, but maybe we do not have to. We only require that for every idea and mind-state, we could in principle say whether that mind-state includes that idea. Does that mind know that idea? So an idea can be very broad — facts, prejudices, techniques, skills, utter bogosities — whatever you like.
Now, let each idea be represented as a point in the plane, and let your mind’s cognitive path pass through all the ideas you ever absorbed, in order.
What if you forget an idea? What if you absorb several ideas simultaneously? I don’t see a good way to address those issues. What about the ideas you are born with — innate ideas shared across the species? We could capture that if we wanted — start the curve before you are born, back when minds first evolved. Each path would include evolution as well as individual development.
So, yes, we can express the cognitive path of each human through the space of ideas. Let’s make up a simple world and see what we get. Let’s assume there are 20 minds, and 20 ideas. Every mind starts with zero ideas, and picks up ideas one at a time. Some of the ideas are learned in school, according to a strict curriculum, and some ideas are hit on randomly outside of school.
Specifically, let’s say the 20 minds all attend school on odd days and play randomly on even days. In school they learn their ABCs: A on day 1, B on day 3, etc. The other ideas are labelled X1 through X10. The probability of a mind absorbing one of the X ideas on any given day is 1% for X1, 2% for X2, on up to 10% for X10.
I got ChatGPT 5.2 to simulate the world and record all the cognitive paths. It used some nifty tricks to project all the paths in idea space to a fairly compact, approximately constant-density region of the plane.
Below are four plots: the universe of all 20 ideas; the 20 cognitive paths through idea space; the growth of the hairball with time; and the growth of the set of “known” ideas over time.
Where is the frontier? We can answer that question only in retrospect: at any given time, the unknown soon-to-be-discovered ideas define the evolving frontier of knowledge.
There are many aspects of the real world not captured in my simulated universe, including the forgetting, simultaneous ideas, and evolution issues described above. But perhaps most notably, there are dependencies among ideas, which I captured in the ABC curriculum only in a simplistic way. Further refinement is left as an exercise for you and your favorite AI.
A beautiful mess
What conclusion might we draw from this exercise? The hairball works! Human knowledge and intellectual progress is a hairball! We spend our intellectual lives in that hairball. It is a mess. There are orderly portions, of course, but on the whole, it is a mess, and the frontier is particularly messy. Living in that mess is a challenge but also a beautiful experience, especially if you can make a little progress sorting things out or extending the frontier.
Acknowledgments
Thanks to Cindy Gelb and Merrick Furst.
The drawings of the disk of knowledge were borrowed from:
Matthew Might, “The illustrated guide to a Ph.D.” 2010.
http://matt.might.net/articles/phd-school-in-pictures/
Might be interesting to color-code papers as original, confirming, or refuting…
Everything we *know* is supposed to have been confirmed, right?
I didn’t mean “know” that way. I meant it like, “You know that idea that the earth is hollow, and there is another civilization down there?” “Yeah.” “Totally bogus, right?”
Since the advent of LLMs I have thought about human knowledge as a hyperdimensional manifold that we traverse, that all knowledge we could have exists across our shared collective experience exists on this manifold defined by our brains and the world we experience. I’d like to think the tsne projection of all of those thoughts looks like the hairball you defined. Every time we discover something new it’s because we have discovered new basis vectors by which we can explore and walk along. If every synapse is a dimension, then we are traversing a 1T dimensional vector. I really appreciated this read as it helped concretize some of my thinking about the subject too.
This actually makes me feel better about my own messy learning progress. Especially on living in a mess is a beautiful experience…
Funny, I always thought of learning as a stochastic process – evidenced by my myriad ‘inventions’ I think are original, which I often find out have existed for decades. When you mention the hairball and your own learning process, I felt a bit vindicated. How many other people must have genius, groundbreaking thoughts which are dismissed as “ah, someone’s surely thought of this” and vice-versa.
When I think too hard about this subject, I tend to get too ‘meta’ and start thinking about how everything is probabilistic (or is it!? – also rhetorical)
Great read!