Find a formula of the exponential growth of the number of scientific problems appearing in harmonically developing society.
As the science is developing the more and more number of scientific problems K(t) is being solved. If the problem is not dead-end, then at its solution a “tree of new problems” appears, at least – two problems.
Thus, the rate of the natural objective growth of new urgent problems dK(t)/dt, where t – is a time, is in average proportional to their current amount K(t):
dK(t)/d(t) = a(t) K(t) (1)
where a(t) – is a coefficient of proportionality, characteristic for each epoch t. This coefficient defines the rate of the relative growth of the amount of problems in harmonically developing civilization:
а = а (t) = {1/К(t)} * [dK(t)/dt] (2)
If the human civilization immediately starts to solve new problems, then their general number increases by the exponential law.
1. Problem 1. Find a formula of the exponential growth of the number of appearing scientific problems in a harmonically developing society.
Solution. Let’s integrate the expression (1), for it let’s rewrite it in the following form:
Let’s transfer to a definite integral over the time period from t0 (start) till t (finish):
what is required to prove.
From (3) let’s transfer to the periods of doubling Tп of the urgent scientific problems for the moment t0 (start epoch) till the moment t (future epoch). That is let’s solve the problem for the limiting case of the minimal requirements to the developing society:
K(t) = 2K(t0) (4)
2. Find a formula for the period of doubling Tп of urgent scientific problems in limiting minimally developing society.
The period of doubling Тп of the urgent scientific problems is a time of development of society, during which the number of new problems, being solved by the society, increases twice.
2. Problem 2. Find a formula for the period of doubling of the urgent scientific problems Tп in the limiting minimal developing society.
Solution. By the definition from the formula (3) we get:
For the time equal to the period of doubling Tп = t0 – t the condition a(t) = a = const is true, that is why from (5):
Tп = (ln2)/а (6)
or in the general case of the dependence а = a(t):
Tп(t)= (ln2)/a(t), (7)
3. Find a formula for the average life time τ(t) of the old problem which is being solved from the time moment t0.
Now let’s generalize the problem and find the average lifetime of the scientific problem that is the average periods of solution of the urgent problem being considered by the developing society.
Average life time τп of a scientific problem – is a time during which one old problem is solved and “g” new problems appear, which are genetically connected to the old one.
3. Problem 3. Find a formula for the average lifetime τп(t) of the old problem being solved from the moment of time t0.
Solution. Let w(t) – is a probability of that the problem, existing in the moment of time t0, still exist in the time moment t. Then the quantity dw – is a probability of its solution (disappearance):
dw = - awdt (9)
The scheme of probabilities distribution
Integrating the expression (9), we get:
w(t) = w(t0) * exp[-at] (10)
Since at t = t0 the problem still reliably existed (was not solved, does not disappear), then w(t0) = 1 (equals reliability)
w(t) = exp(-at) (11)
From the expression (11) one can find the average life time of the old problem according to the definition of the mathematical average:
Comparing the expression (6) with the expression (12), we get:
