Can something be literally impossible to understand?

We are all familiar with concepts or ideas that are hard to understand, for example advanced math or a difficult puzzle, but what about ideas or concepts that are completely impossible for you, me, or anyone else, to understand. Not hard or even incredibly hard, but unequivocally impossible. And not ideas that are random or meaningless nonsense, but useful ideas that actually explain¹ some aspect of the universe we live in.

In other words: Are there incomprehensible concepts that absolutely cannot fit into our brains, but if we could understand them then we would find them useful?

Cats and doors

As I write this during the pandemic, my wife is in the office on a zoom call with the door closed. It’s not latched, a light push would open the door. Unfortunately, our cat doesn’t understand how to push the door open, so she is sitting in the hallway meowing loudly to be let in.

If the door were open just a crack then she would try to push through the crack which would inadvertently push the door open. She knows there is a way to get into the room and she understands the idea of trying to squeeze though a small opening, but the idea that she could push the door to create an opening just doesn’t work for her cat brain.

She’s young, and I think that, like most cats, she will eventually figure out how to push and pull doors open. But this is like cat-calculus to her. It needs to be demonstrated many times and only after a lot of practice, study, and many failed attempts will she finally master the art of door push-pulling. It’s a lot like the way a human needs to have calculus explained to them and then do a lot of study problems before they finally get it.

Puzzling rope games

People also have limitations. This video is about a man who had a rope puzzle for ten years during which he could never figure out the solution. If he’d been locked in a room and could only get out by solving the puzzle then he’d have been in big trouble. The puzzle looks pretty difficult and none of his friends could figure it out either. At some point, the host of a variety show heard about the man, went to interview him, and brought with him another man who is an expert in solving rope puzzles.

The puzzle expert they brought in specializes in rope puzzles. In fact he’s a world expert on rope puzzles and member of a Japanese association that is devoted to studying rope puzzles. The puzzle expert looked at the puzzle and very quickly saw the solution. He was even able to demonstrate in a very clear way how the puzzle could be solved. The solution turns out to be quite simple and the man who for ten years could not solve the puzzle was able to do it easily now that he’d been shown the solution.

The puzzle expert was able to solve the puzzle, not necessarily because he was smarter than everyone else, but because he’d studied many similar puzzles. He presumably has spent a lot of time with other members of the Japanese rope puzzle association talking about how to solve these puzzles. He is like an old cat that after years in a house with many doors has figured out how to head-butt doors open. If we gave him some other totally different type of puzzle then he’d probably be as puzzled as most other people, just like a door-opening cat would be stymied by a sliding pocket doors.

Two rope puzzles side by side. The left one is difficult, the right one is trivial.
Two rope puzzles side by side. The left one is difficult, the right one is trivial.

Not all rope puzzles are hard for people. The above picture shows a second puzzle that is just slightly, but critically, different from the hard one. If you look at the picture and notice the difference, then I’m betting that you also saw the solution to this trivially easy puzzle by just looking at it. In contrast, most people, even after watching the solution video, cannot visualize the solution to the hard puzzle. It sort of slips out of one’s brain as soon as they stop focusing on it.

Relative difficulty

I don’t think any cat could ever solve even the easy puzzle. You might, maybe be able to train it to perform a series of actions that result in the puzzle being solved, but even then the cat would have no understanding of what it was doing. (If you’re objecting that cats have no hands and that’s why they can’t solve the puzzle, then just replace it with some simple puzzle of big buttons or pushing blocks around such that a human would see it as trivial, and that a cat would be physically capable of solving it, but it would still be beyond the cat’s comprehension.)

If humans have a level of intelligence² that lets us easily solve problems that stump animals, why shouldn’t there be further levels of intelligence that see our problems as trivial? Is it possible that an artificial intelligence that we might build someday might look at that ten-year pending rope puzzle and trivially see the solution³ in the same way that we trivially see how to solve the easy puzzle? Would that mean that this hypothetical AI could conceive of ideas and concepts that would be simple for it yet still fundamentally impossible for us to understand in the same way that a cat will never understand knots?

Beyond human understanding

Ironically, I think that it’s sort of hard to get one’s head around the idea of an idea that you could never understand. It’s not something hard to understand or that you’d need to study for a long time, it’s something impossible to understand. It’s even more confusing if you also include that the idea is both useful and completely obvious by the metric of some other intelligence.

I really would like to say that we have logic and math⁴ that will allow us to eventually understand anything in our universe. Sadly, I think that would be incorrect. Yes, math can be used to define and manipulate ideas that we would otherwise find incomprehensible, but that does not really imply understanding.

For example, It’s not uncommon for engineers or scientists to work with vector spaces of a million or more dimensions, but does anyone really understand what it means to rotate in a one-million-dimensional space? Among other challenges, there are literally a million-minus-two different types of rotation in a one-million-dimensional space, each about hyperplanes of different codimensions. Good luck with that.

The other problem with wishfully thinking that our logic and math might be a universal solution to understanding any posible thing is that we came up these tools to explain the way we experience the universe. If something is absolutely beyond our comprehension, then the math and logic that we created to explain what we understand probably are not up to the task of explaining anything beyond what we understand.

Why would we expect otherwise?

This is the first article in a series. I invite you to read the second one.

Footnotes:

[1] I’m not talking about religious explanations, which would depend on faith. Faith is believing something despite there being no way to fully explain it or prove that it’s true. If it were possible for any level of intelligence to fully explain or prove something, then that thing would be mundane, not transcendent. In this article we are discussing our understanding of our mundane universe.

[2] It seems apparent that both human and cat intelligence might vary from one individual to the next. Whatever that variation might be, it is relativly small compared to the difference between between, for example, cats and humans. The smartest cat can’t do multiplication, just like the smartest human can’t solve large n-body problems in their head. It might also be worth keeping in mind that “intelligence” is not one-dimensional. Someone terrible with logic might be great at spatial reasoning. For that matter, a typical cat probably has a much more advanced understanding of smells than anyone reading this article.

[3] We already have programs that can solve these types of puzzles by brute-force search of the puzzle’s configuration space. For example, rapidly-exploring random trees (RRT) are great for that sort of thing. However, brute-force search is like a person, or monkey, who just tries doing things until they stumble into a solution. Algorithms such as RRT are just clever ways of organizing a brute-force search. That’s different from being able to conceptually see and understand the solution.

[4] Actually, logic is math.

About Me: James F. O’Brien is a Professor of Computer Science at the University of California, Berkeley. His research interests include graphics, computer animation, simulations of physical systems, machine learning, optimization, human perception, and the forensic analysis of images and video.

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Disclaimer: Any opinions expressed in this article are those of the author as a private individual. Nothing in this article should be interpreted as a statement made in relation to the author’s professional position with any institution.

All embedded images are Copyright 2022 by the author.

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Professor of Computer Science at UC Berkeley, Academy Award Winner, Company Founder, Advisor.

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