Mathematics at Scale and Computability
"Gardeners are acutely attentive to the deep patterns of natural order, but are also aware that there will always be much lying beyond their vision." Finite and Infinite Games
The pure joy enveloped me when I was flipping the pages of a math book while sitting on the bench. My eyes constantly jumped over definitions, theorems, examples, remarks, and back to the definitions. I was so excited to look over all the objects that were defined in terms of fuzzy mathematics. My immediate desire, after I closed that book down, was to come back to it again as soon as possible. I was driven to learn and read those concepts more in depth without any rush. Who knew that all that was just a dream. The only reason I remembered it was because I was triggered by a couple of words from the Youtube video which I watched early that morning. A big question was raised in the video: about the possibility of developing a mathematical theory of consciousness. At this exact moment, I immediately got thrown back to my dream where I held the Book which had partial answers to it.
Then the discussion in the video switched to talking about measurements. At that moment, my thoughts immediately escaped to wonder about a scientific method in general. The main power of science is that whatever model we have created we are able to check its validity at any time and place almost with no restrictions (mostly on macro level). However, this scientific thinking is missing out on a completely different variety of phenomena that we can’t simply measure and encode with the goal of reproducing the results. First, everything is perceived differently depending on the scale of how we interact with the objects. For example, when we ask ourselves if some process is continuous or discrete, we should ask ourselves “On which scale should we perceive it?” If we are going to look at a simple car traffic at normal speed, then we will think and observe this traffic as a discrete process. However, as soon as we increase the movements of the cars 10x, then all the machines will suddenly merge into one continuous flow.
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In the same way, we can only truly understand and record certain nature phenomena only upon an infinite exposure to it. For example, let us imagine that we have a box in front of us that contains some red and black balls inside. I will tell you a little secret that it contains 4 red and 6 black balls; 10 balls total. But, let’s say that you forgot immediately our secret, and want to figure out how many balls are in the box. However, there is one catch. You can’t look inside the box and you are allowed to take only 4 balls at the time. Assume that you made your first attempt and got all 4 red balls. The probability of this event is really small and equal to about 0.476% and might set you on a wrong thinking path that all balls in the box are red. To figure out the real state of things, we need to do more tries. Then the more and more information we will learn about the distribution of the balls. After repeating these many-many times, we can exactly reconstruct how many red and black balls were originally in the box. However, imagine now that we will take the amount of black balls and make it significantly larger than the number of the red balls. It will force us to increase the number of experiments even more to figure out how many red balls we had originally.
Every possible event around us that entertains our brain consists of an uncountable number of similar boxes with their own balls. However, we have not that many tries to pick up the balls from the box during our lifetime to figure out the actual ratio of the balls in the box. The only way to answer this question is if we encounter a small box with not that many balls inside it and with enough tries to get their true distribution. In the same way, factual science can record, reproduce and study only those phenomena which are computationally possible. It would be silly of us to assume that a couple of billion brains can decode and completely understand how the universe works. We will probably never know, but taking this journey and getting as close as possible to glimpses of understanding is still worth it. Nothing is better in this life as to be driven by pure curiosity while being submerged in pure love along the way.