What are Epistemological Assumptions?
Your epistemology is more like the whole set of all the basic assumptions of your view of the world.
The fundament on which you built your whole house of knowledge on.
The question of learning is now not whether to get a new couch, but whether you need to move into a completely new house and environment to get the point. If you haven’t seen it, you cannot imagine what it looks like. And if you haven’t been inside, you cannot imagine the subtleties and complexities of its interior. Then, you have to make wild guesses what could be behind things when you see them, and they look just so incredibly complex and mysterious.
The Beauty of Fractals
As I was a child computers were the new hip thing to do. So, at a way too young age we forced our minds into stuff we could barely get a hold on. But we went through this tragedy anyway, and made some stupid, but enlightening mistakes. None of them I would ever want to miss.
One day, a friend of mine showed up at the door with a new computer magazine that featured the Mandelbrot set. They gave it the funny name “Apfelmännchen” in German, which translates to “apple man”, which is funny as its inventor already has a German name that stands for a food named “almond bread”. As we looked at the algorithm, it was incredibly short, its core just being a simple loop of four lines, the rest a little bit of math to map it on the screen.
With this easy picture, the nice graphics it produced, and the simplicity of the algorithm, we were pretty damn sure this was something non-complicated we could get a grip on, that we could tackle. So in the first step, we copied the algorithm, which at that time meant retyping it from the print (that’s why we really liked short algorithms). The program produced the expected result. Then we started to play around with it, and the tragedy started.
The first thing was simple: Cycle the color palettes. This could be done. And we had a blast. So much complexity from so little, amazing. And the nice colors! But there was one problem: Zooming in to the interesting regions, we could follow the pixel on the screen line by line when calculating the image, and some of the more interesting ones took all night, or a day. All night wasn’t the problem, it could run when we were at school. But more than a day was a problem, as we had only one computer, and wanted to do other things with it also.
So we started to think about optimizing the algorithm. It looked so simple, it has to be able to be easily understood and optimized. We noticed, that it was some kind of computation that was hidden in a for loop. Looked like an iteration that is made to compute the index that is necessary for the coloring. But it appeared to start computing all the way back from way too early, no matter how far we zoomed into the picture. But within the frame we were looking at, so we thought, we don’t need all that information from back there because we don’t even want to look at it.
So then, we thought, can’t we start to calculate further down the line?
Now, you may say, that was really stupid. On the other hand, we were also tackling things that most of our peers would say was way too complicated. It shouldn’t have been a question of general intelligence. So why did we make this mistake, what was causing it? Anyway, we didn’t know, and thus we started to play around with it. Tried to find a way to understand this kind of formula, and how you could tweak those counters and variables to start calculating at the point we needed them to. And, oh well, were we persistent at our idiotic attempts. Today, you would maybe say, we were resilient. Or you might say we were too dumb to learn from our failure.
But then, a couple of days later, something amazing happened. It was just a short moment. There was this notion that … hey, this is a recursion. One value builds on the other. It is impossible to compute this without going all the way back. It just does not work. You need this … initial value.
Our whole way of thinking about it was transformed in an instant. We needed to experience this idiocy followed by a short moment of enlightenment. Noted and done, we immediately searched for other solutions. We’d need to store these initial values, but storing them for any reasonable zoom depth would mean to blow the memory of our tiny computer, and one had to compute them once anyway. Then, we looked for improvements in the computational implementation of the algorithm. It was all floating points, and thas was an expensive operation without coprocessors, which didn’t exist at the time.
Maybe, we thought, we could use integers. But the first attempt showed: the picture was all torn up. Then we combined two integers to get more precision. If we are a little bit more precise than was necessary, then we thought that should be ok. At the top level, the picture looked promising. Then, we zoomed in one step, and saw, that again, the picture was all torn up with artefacts. We just couldn’t figure out why. But then, after a similar while of consideration, a second Eureka moment happened. Apparently, in a recursion, all the small inconsistencies add up, and much earlier than we would assume from our usual calculations, we were running out of precision, because we had to calculate the sequence from the beginning every time. We didn’t know the term yet, but we had discovered a butterfly effect.
Electrical Engineering Expertise … not.
Now you may say this isn’t supposed to happen at the age of fourteen. But I want to make a point that it should. To put this in context: We learned about discrete-looped thinking long before our continuous-linear math was anywhere near practically useful. Sure, there was some algebra because we needed vectors to draw lines on the screen, but it was limited to the very basics.
Amazingly, I today still find it difficult to think in the subtleties of continuous-linear maths, particularly when the combinatory requirements get complicated. I can, however, spot fractal and self-similar structures or causes on the spot. In contrary to common belief, in today’s academic world, that does not make life easier. Many people didn’t go through our mandelbrot set disaster. Apparently, some transformation in thinking did not happen, because people were never required to do so. It’s not that this is something that makes you intelligent. There are a millions of definitely more intelligent people out there who are a ton better at math than I am. I was always very glad if I’d gotten a B, and not made some stupid mistake on the way. But it also does not happen out of nowhere. You have to tackle these problems, and get those mental bruises before you really grasp it.
As computer science students, we had an electrical engineering professor who knew that the subject was just an add on for most of us who were interested in information theory. So he split the exam in two halves: a direct current and an alternating current part. If you got the direct current part completely right, you got a D and passed, which was what most people aimed at. I guess I was the only student ever to hand in the alternating current part completely right and the direct current part completely wrong, and ended up somewhere between B and C. Thinking in those networks was something I was more used to, and it naturally attracted more interest. The linear stuff … I almost always forgot some minor, easy, boring step in the process that was nevertheless important for the end result. Other people, apparently, can do linear sequences way better than I do. I can do fractal patterns, and separate different types of epistemological assumptions from each other.
What happened was, that due to playing with chaos science our way of thinking was transformed. We had switched our epistemological assumptions from linear dimensions to discrete recursions and their emerging patterns. Maruyama would say we had transitioned from I-type to G-type thinking. And once this happened, it could not be undone. There is no way to un-think that step. With epistemological transitions one is not able to tackle more and more complicated relationships. It’s more like applying your logic in a new way, with new fundamental assumptions. Where one saw linearities, one now sees emergent complexity, attractors, repulsors, think of bifurcation szenarios and phase transitions, and the like. One has taken a new perspective, and the shift in perspective cannot be undone. One now sees two possibilities that are two explanations, and thinks about which one describes the problem more appropriately in a given context, instead of having to argue whether one is right or not.
Already being the weirdo, this did not contribute much to better mutual understanding, and the struggle still goes on today. One starts to be able to communicate better for a certain audience, but is left with a highly power law distributed minority one can communicate about these thoughts with. I really do hope more people read about chaos science and learn its patterns. Some day, the few basic functions and patterns may be part of standard curricula. One can teach this in high school. Maybe then we won’t produce so many scientific papers and theories that struggle with low linear correlation when they investigate something that builds on apparently looped and networked underlying structures. And maybe then we won’t be at war over Gods who have to explain everything that appears too complex to be explained by linear science. If you can imagine something as complex as conscience to emerge from a neuronal network, or a universe emerging from nothing, what, then, do you need spiritual explanations for? Scientific discovery appears to be the real mystery.
All images in this article are rendered from a one-line mathematical formula, based on recursion of sums of powers and trigonometrical functions, with computation depth coloring. No layering, no photoshopping.