A number defines the quantity of objects, i.e. the power of the set of objects. To discuss about a prime, we therefore have to talk about sets of objects. As for a notion of the set, we observe that if we talk about a specific set, then we change the relation of objects within this set and all other objects outside the set, and here we need to assume that this relation does not influence choosing a set.
If such a relation itself would be able to influence the choice of a set, then what would define the choice of relations, and sets of relations etc. After all there might be a predefined structure defining how the data is to be aggregated. In this sense, the idea behind decomposing the information of quantity into a multidimensional vector using prime numbers is appealing.
The very decomposition, though, is not about geometry itself, as geometry, as such, but about spreading the information into a vector of information while preserving the character of the quantity-based information within the number. So now, again, back to the clue, i.e. the character of the information hidden in the number.
Number contains information about quantity. There is 1-1 connection between the quantity of elements and the number. However, the objects need to be of the same type. Going deeper into the objects, we observe that the objects consists of yet another objects, and also that the connection is not a “set” type of connection, but rather a more sophisticated structure. That shows that the counting of the objects may not be as simple as using the integers.
Then, we notice that there have been introduced numerous number systems and number types, like complex or rational numbers. Those numbers have certain characteristics and ought to describe 1-1 the relation that is hidden in the real world. Still, in order to achieve that we would have to understand what is there to be modelled, i.e. what type of information do we need to address.
Rational numbers address the idea that the line starting from point A and ending in point B might consist of infinite amount of points. So, the idea that there is this infitnity between two points. But then we would need to have this infinity somehow tackled.
Complex numbers address the idea that we might use the unit i that does not exist among the integers to look at the numbers on 2d plane. There, we also notice that all of the points on the circle have the distance to the center point, i.e. (0,0). Such an approach could be used to put the numbers into infinite number of planes, for all of the planes we could have circles, and then generalized circles for multidimensional spaces.
Still, is it this notion of having the same distance from a certain point that should shape creating the rationale for a number? Does it allow enough connection between these numbers? If not, have we thought about a notion of number where the numbers would be very strongly interconnected, i.e. interconnected in a way that would resemeble some sort of multidimensional structure, as if we were looking at the crystal structure, but multidimensional.
Now, lets focus that we number the elements by “incrementing” this generalized notion of quantity, i.e. counting along the goemetries of objects consisting of yet another objects etc. How could such numbers look like?
Now, think again about prime numbers. If you multiply a number by a prime (a prime that is not a divisor of the very number) then you somehow save the information about that object and give it a specific character. That could be used for handling a object that has some structure, i.e. using the notion of prime we might model the geometrical (not for eyes) structure of objects and connections between objects. In this sense, objects with common divisors should be connected.
To even talk about such ordering, we would have to make it starting from low-level, i.e. a particle level. Still, it is very likely that our first attempts would be not good enough, which means we would have to have the numbering scalable and allowing changes within its structure. For such a modelling we could use a generalized notion of graphs, assigning primes to relation types and nodes. In the final round we would have it with primes only and could forget about the notion of graph.
This new enhanced notion of graph could be created with prime numbers only. A few questions that arise involve:
– are we capable of making it fully scalable and allowing changes to its structure?
– how would then look counting? ie. adding objects? it would probably have to look like addition in chemistry, ie. adding H2 to 02 would give water in some sense?
We observe that having infinite number of prime allows us to infinitely develop this system. Building this system must be more like sketching and constant improvement rather than making everything depict the universe perfect from the beginning. It has been addressed in one of the previous posts, though, that having low-level-based strategies for the understanding of universe functioning is unlikely to bring the true results. Low-level strategies are more interconnected, having graphs in mind, with low-level adaptations, such as algorithms and higher level are more into physics and end-products, more with engineering and experimentation. Connecting two dots: one of the very low mathematical level, lets say, a prime, and one of very high abstraction, like universe expansion is unlikely to be true and the mathematical-level theorem would just give a physical level background, and, as such, might also be often developed w/o that much of low level abstraction.
New ideas of how to connect dots of high level in compact systems, but of high parameter space, involve a lot of decomposition and machine learning and still require a lot from the mankind. Don’t be too proud, humans.