Is there any mathematical approach for calculating a unique random number completely by hand on the paper without taking help from the computer in a way that:
1- Time would not be involved
2- No sensory data would be involved
3- That number can never ever be resulted using that approach again
For example: given that I have a flock of horses, all white, equally the same size, whose ages are unknown and I want to identify them by numbers in a way that no other white horse owned by anyone else would have that number (given all horses in the world are identified by numbers). also given that it's year 200 where there is no computer or calculator.
Is there any a pure mathematical solution for this problem?
From the conditions that you have, the answer is no. You cannot have such an algorithm. Suppose you have one machine only that gives these numbers. The machine knows which number it generated, so it has the notion of time (might not be linear, but it is monotonic). If you have more than one machine, say one that produces odd numbers, one that produces evens, each of them will have the notion of time (same reasoning as before). In addition, there is the sensory data condition that is contradicted (I can find out which machine I have).