Fuzzy Techniques, Laplace Indeterminacy Principle, and Maximum Entropy Approach Explain Lindy Effect and Help Avoid Meaningless Infinities in Physics

Abstract

In many real-life situations, the only information that we have about some quantity S is a lower bound $T<S$. In such a situation, what is a reasonable estimate for S? For example, we know that a company has survived for T years, and based on this information, we want to predict for how long it will continue surviving. At first glance, this is a type of a problem to which we can apply the usual fuzzy methodology—but unfortunately, a straightforward use of this methodology leads to a counter-intuitive infinite estimate for S. There is an empirical formula for such estimation—known as Lindy Effect and first proposed by Benoit Mandelbrot—according to which the appropriate estimate for S is proportional to T: $S=C\cdot T$, where, with some confidence, the constant C is equal to 2. In this paper, we show that a deeper analysis of the situation enables fuzzy methodology to lead to a finite estimate for S, moreover, to an estimate which is in perfect accordance with the empirical Lindy Effect. Interestingly, a similar idea can help in physics, where also, in some problems, straightforward computations lead to physically meaningless infinite values.