Nonlinearity in living systems

Is the title of a recently published e-book by Frontiers in Applied Mathematics and Statistics. Here's an excerpt from the introductory editorial.

"The biological basis of physiological signals is incredibly complex. While many researches certainly appreciate molecular, cellular and systems approaches to unravel overall biological complexity, in the recent decades the interest for mathematical and computational characterization of structural and functional basis underlying biological phenomena gain wide popularity among scientists.[...] We witnessed wide range applications of nonlinear quantitative analysis that produced measures such as fractal dimension, power law scaling, Hurst exponent, Lyapunov exponent, approximate entropy, sample entropy, Lempel–Ziv complexity as well as other metric. [...] Also there is another more theoretical challenge of contemporary nonlinear signal measurements, especially including fractal-based methods. The question of choosing the right method and its possible adjustment in order for the results of the analysis to be as accurate as possible is the persistent problem.[...] We seek to bring together the recent practical and theoretical advances in the development and application of nonlinear methods or narrower fractal-based methods for characterizing the complex physiological systems at multiple levels of organization. [...]  A comprehensive understanding of advantages and disadvantages of each method, especially between its mathematical assumptions and real-world applicability, can help to find out what is at stake regarding the above aims and to direct us toward more fruitful application of nonlinear measures and statistics in physiology and biology in general."

Excerpts from this article in the ebook, "Measures and metrics of biological signals."


"With the growing complexity of the applied mathematical concepts, we are approaching some serious issues of foundations of Mathematics. Before that, let us mention that the symbol ∞ does not represent infinity uniquely since Cantor's discoveries in 1873, when he showed that arithmetical and geometric infinity, i.e., natural numbers and real line are different infinite quantities. As a consequence, infinity has been scaled in terms of pairwise different cardinal numbers. However, the size of this scale is enormous; it cannot be coded by any set. This was the creation of Set theory, and the beginning of the studies of foundations of Mathematics, which is probably never ending."

"We learned that Mathematical theories, packed around their axioms can be at the same level of logical certainty, while obviously impossible mixed together since with colliding axioms.[...] Let us just say that AC (Axiom of Choice) is very much needed in the foundations of Mathematics, but there are alternatives. [...] Some of the functions close to the above-examined fractals are complex enough to open the fundamental issue. [...] On the other hand, we can stay on the flat Earth and deal only with short approximation of the phenomena, avoiding entering the zone of the complex Mathematics and its fundamental issues. Yet, as proved by Goedel, we cannot escape the hot issues even remaining only in Arithmetic, nor in any theory containing its copy (like Geometry)."