Anything one can ever learn about nature is based on contrast. “To be is to be the value of a variable (or to be some values of some variables)” [Boolos]. In a similar way, anything that can ever be documented as an essence of any learning process, can only be documented based on the intentional generation of contrasts like printing black letters on a white paper, generating magnetized regions on a magnetizable disc, displaying bright dots on a dark display, carving symbols into stone, sending a Morse code. In short, any storage of information is based on creating and maintaining contrasts with specific spatial patterns, any transmission of information is related to temporal patterns. As a first step towards a description of contrast, the basic equation can obviously be substracted from itself (just different indices are used):
\[\sum_{i=0}^{N_\Phi}\Phi_i-\sum_{j=0}^{N_\Phi}\Phi_j=0\]
In both cases the Phi_i can - and shall - be arranged in a monotonously decreasing order with Phi_0 being the largest value and Phi_N being the smallest value. The equation can then be rewritten as
\[\left(\Phi_{N_\Phi}+\sum_{i=0}^{N_\Phi-1}\Phi_i\right) -\left(\Phi_{0}+\sum_{i=1}^{N_\Phi}\Phi_i\right)=0\]
\[\left(\sum_{i=0}^{N_\Phi-1}\Phi_i\right) -\left(\sum_{j=1}^{N_\Phi}\Phi_j\right)=\Phi_{0}-\Phi_{N_\Phi}\]
Renaming j=i+1 yields
\[\left(\sum_{i=0}^{N_\Phi-1}\Phi_i\right) -\left(\sum_{i+1=1}^{N_\Phi}\Phi_{i+1}\right)=\Phi_{0}-\Phi_{N_\Phi}\]
\[\left(\sum_{i=0}^{N_\Phi-1}\Phi_i\right) -\left(\sum_{i=0}^{N_\Phi-1}\Phi_{i+1}\right)=\Phi_{0}-\Phi_{N_\Phi}\]
\[\sum_{i=0}^{N_\Phi-1}\left(\Phi_i-\Phi_{i+1}\right)=\Phi_{0}-\Phi_{N_\Phi}\]
\[\sum_{i=0}^{N_\Phi-1}\Delta\Phi_i=\Phi_{0}-\Phi_{N_\Phi}\]
As the Phi_i are monotonously decreasing, Phi_0 and Phi_N correspond to the maximum/minimum values Phi_max and Phi_min, respectively, yielding the “contrast equation”:
\[\sum_{i=0}^{N_\Phi-1}\frac{\Delta\Phi_i}{\Phi_{max}-\Phi_{min}
}=1\]
The difference between the maximum and minimum values is the largest difference which can occur in the system:
\[\Delta\Phi_{max}=\Phi_{max}-\Phi_{min}\]
It never can reach the value of 1 because both, Phi_max and
Phi_min, are considered as existing and thus both have non-zero values :
