Counting and Adding-Up --> Scaling

When you add up random numbers you might get a Gaussian bell-shaped curve, with the peak at the average, and the spread denominated by their standard deviation. You'll get the same curve is you just choose ever second number, or if you added up an even member of the sequence and the subsequent odd number to get a new sequence.

There are other processes of adding-up, say weighting each number that characterizes a series of situations by the complexity of that situation, which, too, lead to distributions that scale nicely retaining their form. These so called automorphic forms (having the same shape) appear in understanding physical systems (the weight is the "Boltzmann factor," exp-Energy/Temperature), and mathematical number theoretical systems, where the weight is something like the number to the sth power, n^s, s being a negative number, the Riemann zeta function--whose transformation by something like a Fourier transform (the Mellin transform) exhibits that scaling symmetry.

In effect, counting leads to scaling.

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