Mathematics has been designed to allow the accurate description of the perceived. The very perceived, as is, is already not accurate description of the given. For instance, if we have an apple and someone gives us two apples, we then have three apples. So, we know how much we have but lose information of the amount of apples that exist and don’t belong to us. The same thinking applies to geometry, ie. our position, our movement etc.
Our mathematics takes information based on the perceived and not the given. Thus, it is only as good as the perceived. Still, enlarging view will finally allow us capture potential contradictions and go beyond the perceived. I fail to predict enhancement in mathematics building based on the perceived, ie. any other than detecting contradictions in the vast code coverage through the use of automated computation is currently beyond me. If exists, would potentially require finding contradictions based on reccurence, with purely true (the given, not the perceived) edge (initial) conditions.
Now, lets take a closer look at frequencies that we use to model light waves. The more per second waves we have, the more visible change in light color (that is visible for eyes). Quantity (the amount of waves) influences quality (different entities, different waves). In real life, the more contradictions we find in any game strategy, the better we will play the game in the long run, even if we can’t find the best strategy analitically, hence the (enigmatically settled) proposal: quantity influences quality.
Now, again, as we cannot find the strategy for understanding the universe and our information is weak, we decide to find more information to eliminate contradictions. But, not only it is the linear increase in the amount of information we use to eliminate contradictions, but also the mathematical axioms that we may build that enable significantly faster development.
For instance, we first define distance as the extent of space between two objects. Then, we go deeper and define space and objects. Then, we notice that distance is not what we perceive and has a very specific character. In fact, it is a function of various other factors. Then, we treat those factors as eigenvalues and use them to describe distance. Then, we notice even more things. The more information, the more quality, in this sense.
Still, I would like to ask a philosophical question here. What should we do to omit information from the perceived and use the given information? Does the existence of the perceived imply the existence of the given? What would have to be the given? To what information would one refer had he forgotten about the perceived?
After closer investigation (in the context of Goedel incompleteness theorem), we have that we cannot decide about what’s outside the box having in mind only information vectors from the very box. That combined with the fact that every perceived results in the existence of the “rest” leads to the fact that we know we’re in the box (perceived) that is limited by information delivered (as far as we know it) by light waves (and restricted by the speed of light, ie. we cannot get all information that should travel since the universe birth, thus we cannot even think about having all the information about the outside; even if we had those information, that’d still be the perceived, but the thing is that we know we can’t- at least by the means we understamd) but we cannot say more about what’s outside the box. So many problems lead me to the crossroads where I need to tackle logic, ie. in the logic from our box, we cannot define what’s outside the box. For the logic to be improved, it’d probably should take into consideration more about the structure of the universe, ie. things we don’t understand. For the logic to be the perfect one, we’d have to find the equality between the given and the perceived. Still, as long as we don’t understand how to define the given, we only have different versions of the given, which allows development of technology rather than low level science, including number theory or logic.
And now, regarding the code coverage, Goedel explains how to attack thinking problems.
To develop the skill of correct thinking is in the first place to learn what you have to disregard. In order to go on, you have to know what to leave out: this is the essence of effective thinking. (1972, from chapter 1 – Gödel’s life)
This, of course, stems from the fact that we minimize the field of analysis (code coverage). Still, in order not to lose analytical quality we need to be sure what to leave out. In real life we’re almost incapable of finding the disregarded.
On the other side, how could we know what the given is- amongst the perceived we know to omit tons of invaluable information but have no key as how to access it. Lets now focus on a local decision. To be continued..