Tolerance

 

(What’s So Funny ’Bout) Peace, Love and Understanding

And each time I feel like this inside
There's one thing I wanna know
What's so funny 'bout peace, love and understanding? Oh
What's so funny 'bout peace love and understanding?

The universe is random, hyperbolic not flat, and understanding, err…tolerant.

If free will exists, then tolerance, the standard deviation from the mean of choice, also has to be almost 100%.

If, for example, before a choice you are unsaved, and after that choice you are saved, then each person is either 100% saved or 100% unsaved, and the most common choice is to be saved. Not everyone is saved at first, but eventually many more will be saved. Mathematically this describes an exponential distribution, λe^-λx, with a rate parameter, λ, choice, of 100%. The problem is that the mean of an exponential distribution is not the mode, the most common value. Also, the exponential distribution is not normal, it is highly skewed; and is undefined for any value less than 0.

It is possible to shift, translate, the exponential distribution along the x-axis to a new value, μ. For reasons that I hope will become obvious later, let's shift that exponential distribution to begin at 3. It remains skewed, and the most common value is not the mean. Also there are no values to the right of this new value μ.

So let's pretend  that the distribution is mirrored before this new value, and it effectively becomes a discontinuity at the new value. To the right of that discontinuity there is e.g. the United Star Ship Enterprise and a universe that favors System Optimality, Teamwork. To the left of that discontinuity there is a mirror, the Imperial Star Ship Enterprise, Mr. Spock has a goatee, and that universe favors a User Optimality, Every Man For Himself. This can be described mathematically as
(x<μ)*(λe^λx+1) +(x≥μ) *(λe^-λx), where again in the example μ = 3. The mode is equal to the mean which is equal to the median, and the skew is zero. However there is a discontinuity at μ. While a normal distribution also has a skew of zero, and the mean, median and mode are all equal to a value, the function is NOT a normal distribution because it does not satisfy the 68/95/99.7 rule. This would require that 99.7% of the values are within the mean plus or minus 3 standard deviations from the mean. This also causes the value at x=0 to be almost zero since the mean is equal to the median and between zero and the mean in a normal distribution will be 49.85% of the values, which is almost equal to 50%, the median. Since the mirrored exponential distribution at zero is not zero, it is NOT a normal distribution.

It also fails when it is turned into a Cumulative Distribution Function, CDF. It has a CDF of -1 until approximately μ-3*μ where it begins to increase to a CDF of 1, which it approximately reaches when x >μ+3*μ.  The ideal CDF should have a value of 0 when it is much less than the mean, and a value of 1 when it is much greater than the mean. The mirrored exponential distribution is NOT only not normal, it fails this CDF test, and it also is not consistent with observations. A distribution which is consistent with observations, and is considered to be normal, is a logistics distribution, e^-(x-μ)/s / (s*(1+ e^-(x-μ)/s)). This has a Cumulative Distribution Function of 1/ (1+ e^-(x-μ)/s). For the CDF to have a value of 0.5, the median, at μ, then s must be 0.5. Then the CDF has a value of zero when x is zero. Its mean, median, and mode are all μ. Also if the CDF has a value of 0.5, the median, when x = μ, and s must be 0.5, then σ, the standard deviation from the mean, must be s*π/√3, or 0.91. This is also identical to a form of the hyperbolic trigonometric function, ½*sech²(x-μ).

To put this in narrative, not mathematical terms, if there is free will, that is to say a choice can be made at any point, then tolerance must be much greater than 0, and in fact it must exactly be 0.91.  A choice is between nothing and something, (0,1), not between something and its opposite, ( -1, 1). Also it seems that the universe in which we live is hyperbolic, i.e. is not flat.