## From Zeno’s paradoxes, through Andy Parker to the O-order

– “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”

– “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.”, i.e. “we can always fulfill part (here: 1/2) of the plan”,

– “If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless”

– “If everything that exists has a place, place too will have a place, and so on ad infinitum.”

– “There is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially”

– “Concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This…involves the conclusion that half a given time is equal to double that time.”

We observe the aspects of the minimal elements, or first steps regarding infinite descent have been among our thought for some time already. We tried to divide infinite amount of times to check what’s hidden there. But rather than the Holly Grails we have been doomed for reductio ad absurdum proofs exploiting the fact that we have the minimal element in our counting, namely number 1. But assuming that we would be able to find an object containing 1 unit, and all other numbers were integers, we would be still able to create a circle, or sphere by measuring the integer distance from one place, but would not be able to define the exact length of the circle. Not even the less exact number in rationals, because we assumed to have integers only. But extending our numbers to rationals, we would still not be able to describe the length of the circle, so given that we assume that such an object exists, we assume we should be able to describe its length, we have to include irrational numbers. But what does that really imply? This is a question for a lifelong analysis. What we know, taken from the BBC interview with Andy Parker, if we had the world composed of these smallest elements, we would be able to know what the minimal, further indivisible, size is. We would have many problems solved by the fact that we would never talk about infinity but rather about finity.

Given the irregularity of the problem of the existence of the minimal element, we shall re-consider the option to go back to roots and try to exploit it with reductio ad absurdum. We should first try to think about the rationale for a number system. What we currently do is, as I mentioned above, allow ourselves to include the existence of a number that is equal to the length of a circle, assuming that we can create a circle. What if not. If not, how would that be possible? Only elliptic shapes? Did we observe planets with exactly circular orbits? String theory wanders in those directions by imagining that the minimal object would have some finite length but an infinitely small width- but they again face the problem of infinity. On the contrary, assuming that we should allow for irrational numbers, what about complex numbers? Do we already observe that the notion of counting, i.e. the one understood that a state iteratively changes by adding one more element might be an illusion. The question is – what would then be the rationale for the number system? To describe the iteration (term from the previous post)?

Additionally, in a world build out of primes, we would firstly be able to model everything but the perfect circle. Crazy idea. Secondly, our system would be based on counting the lattice points (understand what primes mean, primes are coprime too).Thirdly, we would be allow for ressonance – i.e. interaction of two numbers with the same divisor (frequency), which could help us model the quantum entanglement. Given the infinity of primes, we’d have a lot of place to model the finite but constantly growing world. That would probably mean that we would have an infinite number of particles to model, but not necessarily now. The point is that we might live in the world built out of 10 first primes, or 11 first primes, or 1000000 first primes. The point is: the next prime may be now during the process of its creation, not aware of itself, but stored in the Book.

The question remains what is a moment? What is this iteration? What is motion? Motion is currently defined using the iteration. How does this iteration progress? What is its direction? For the sake of imagination training, lets focus on what we know. What we perceive is perceived by def., so we cannot fully trust it in determining these details. Given that iteration seems to work differently for different “places”, it is an informative feature rather than an ether like one, thus it has its plan (logic). If we now image a system built out of 1 million cogwheels created for making the Earth spin, we observe that we would notice Earth spinning only and describe its movement and the iteration would be exactly the thing that we see from our perspective. The experimental perspective will further eliminate contradictory claims that we support but we should also think of an automated approach to knowledge building- one exploiting the data structure. So we would not be able to see these cogwheels! The same for the iteration!

Assume now that the iteration also has its plan and every object has its plan, each plan is represented with the code. There is an interaction between the codes- we contain similar particles and therefore face the resonant effect. And this is why we are able to see ourselves.

Lets now think that we’d create a mechanism that would aim at creating a circle that is impossible to create (its length would have to exist in integers, but does not). What would that imply- that if we sent many people in different directions to make 100 steps, they would not be at the same distance from the starting point. And this mechanism would use only prime numbers for building this circle! How would this building look like? What else would such a world like to build?

But I am also thinking about the universe trying to create the next prime! Right now we would live in a world where a certain “length” does not exist and we still have to develop to have it created! So, we would live in a world described by $p_1,..,p_n$ and in the next step  $p_1,..,p_{n+1}$. How would such a world look like? How to ask to see through all prime glasses?