Stochastics is very fundamentally based on Empiricism, and vice versa. Stochastics is very fundamentally a system of thought based on empirical measurement. And deep within the heart of this system lies the fundamental tenet that any measurement is "a" measurement, not "the" measurement.
Stochastics represents the acceptance that reality is a dynamic system, not a static Rationalist ideal. The concept of stochastics rests on the very fundamental certainty of uncertainty, of the knowledge derived from experience with experience, and the acceptance that any knowledge based on empirical measurements must ipso facto recognize that measurements represent a sample of reality, not reality itself.
Stochastics treats every measurement as part of a cloud of measurements that forms the fundamental fog of knowing. The concept of the bell curve merely illustrates that any measurement contains uncertainty which deviates with a fundamental expression that small deviations are in abundance while large deviations are rare. One therefore knows with relative certainty the deviation of a measurement only by repeated measurements. Thus there cannot be "the" measurement in the sense of a single knowable truth, but only "a" measurement, a single instance in a pattern that represents reality. The pattern being a stochastic bell-curve.
Stochastics exists as the empirical extension of Empiricism. Stochastics is derived from the experience of Empiricism, and vice versa. Empiricism is the fundamental belief that knowledge must reconcile with measurement. Measurement involves the fundamental counting of procedures conducted in a ritualistic manner. A measurement, any measurement, involves enacting a procedure that counts individual elements which are indistinguishable elements. One cannot distinguish one pound from another pound, or one inch from another inch, or any element of any measure from another element of that same measure.
As a consequence, the only way to ensure that elements are indistinguishable is to measure them. One must necessarily develop some procedure to measure the elements in such a manner that each procedure yields a means of ensuring that any element is indistinguishable from another. For example, any element used to measure weight must perfectly balance any other element on a balance scale. Any element used to measure distance must perfectly match a defined measure. Any "unit" of measure is an element that cannot be distinguished from other elements that are also "units" of the same measure. One therefore counts these units, these elements, in accordance with an established procedure to derive a measure.
Ideally, every unit of measure is indistinguishable from any other unit of the same measure. But this is a Rationalist concept. Empiricists know fundamentally that every unit of measure is different, but simply deviates within some established range in accordance with stochastics: that small deviations are in abundance while large deviations are rare. Thus the greater the number of elements in a measure the greater the chance that a rare large deviation will be included, but the less it will matter when averaged with the abundant elements with small deviations. Any single element may be one of the abundant small deviations, but may also be one of the rare large deviations. Therefore any measurement should be expressed in terms of many elements in order to assure, to ensure, that the measurement reflects a stochastic certainty of the uncertainty.
The mathematics of stochastics merely involves the bell-curve expression of what has been observed empirically: that small deviations are in abundance while large deviations are rare. The bell-curve, as such, is not stochastics, but rather expresses the experience of stochastics derived from empirical experience in practicing Empiricism. Stochastics is therefore nothing more or nothing less than the fundamental knowledge of what to expect when attempting to perceive reality.
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