Uncertainty and randomness

Uncertainty is the lack of knowledge about the outcomes of future phenomena. Let us assume that any phenomenon can be described by a relationship between causes and effects, with a relation $y=f(x)$. Then the outcome $y$ depends on the relation $f$ and its inputs $x$. If we further assume that the relation is a function $f$, then each input give rise to exactly one unique outcome. So the only cause of variation of the outcomes, and thus uncertainty, are the inputs to $f$. Whenever $f$ is highly sensitive to a change in its inputs, we would notice with only mild variations in the inputs between successive trials of experiments, large variations in the output. We would denote such phenomena as random, because any possible outcome can be expected on each trail, independent of each other. This type of randomness follows from our practical inability to measure the inputs and calculate the outputs at very high accurracy. It is a subjective form of randomness. Rolling a dice is such an example. It is part of the world of gambling, random processes invented by humans to play games.

However for physical and biological phenomena, we normally do not observe such a high degree of randomness. It seems that the underlying functions are less sensitive to a variation of inputs, and produce regular patterns of outcomes. We can sometimes with a high degree of accuracy determine the future states, and reducing the amount of uncertainty almost complete. For example the calculation of planet trajectories. For other phenomena we can build reliable models to approximate the outcomes within a given bandwith. The study of these type of phenomena is based on statistical procedures.

We assumed that $f$ is a function. However the discovery that individual events on the quantum scale are objectively random, i.e. there are no hidden variables that causes these events, shows that this assumption no longer holds. The instant when a radioactive atom decays, or the path taken by a photon behind a half-silvered beam splitter are objectively random. Since individual events may very well have macroscopic consequences, the Universe is fundamentally unpredictable and open, not causally closed. God play dice.

For the study of phenomena it does not really matter whether the uncertainty is caused from subjective or objective randomness. The mathematically theory of randomness is called probability theory. Its theoretical development started with gambling theory. In the form of the famous correspondence between Fermat and Pascal. The evolution of probability theory was based more on intuition rather than mathematical axioms during its early development. In 1933, A. N.Kolmogorov provided an axiomatic basis for probability theory based on measure theory and it is now the universally accepted model.

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