As we learn more and more about a system, its complexity decreases and our understanding increases. As complexity decreases, the precision afforded by computational methods becomes more useful in modeling the system. For systems with little complexity and little uncertainty, closed-form mathematical expressions provide precise descriptions of the systems. For systems that are a little more complex, but for which significant data exist, model-free methods, such as artificial neural networks, provide a powerful and robust means to reduce some uncertainty through learning, based on patterns in the available data; unfortunately this learning is very shallow. For very complex systems where few numerical data exist and where only ambiguous or imprecise information may be available, fuzzy reasoning provides a way to understand system behavior by allowing us to interpolate approximately between observed input and output situations. Finally, for the most complex problems, there are required forms of learning due to induction, or combinations of deduction and induction, that are necessary for even a limited level of understanding. [...] The problem in making decisions under uncertainty is that the bulk of the information we have about the possible outcomes, about the value of new information, about the way the conditions change with time (dynamic), about the utility of each outcome-action pair, and about our preferences for each action is typically *vague*, ambiguous an otherwise fuzzy. In some situations the information may be robust enough so that we can characterize it with probability theory.

Ross, T. Fuzzy Logic with Engineering Applications (2004).

So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality.

Einstein, A. “Geometrie und Erfahrung” (1921).