**Here are a few excerpts from McHarris's Nature and Nonlinear Logic…**

**"We are simply not used to thinking in terms of nonlinear logic.** Sure, we can deal with simple systems. Take, for example, the deer population in Northern Michigan. A typical environment can only support so many deer. As the population increases, the deer overgraze, causing the forage to decrease, which eventually leads to less healthy, then fewer deer. With fewer deer, the vegetation recovers, which leads again to more deer. A straightforward example of feedback. We would not be surprised to find a herd oscillating between, say, fifty and twenty deer in alternate years. And, with climate change making Northern Michigan more verdant, we could easily understand a trend that saw the same herd increase and oscillate between larger numbers, such as eighty and thirty-two deer. It would merely be a case of different environmental parameters."

**"But suppose we performed a carefully controlled study** and discovered that a given herd oscillated over a longer timespan among four distinct numbers, or even eight numbers—or that the number of deer became completely unpredictable even though we were working under carefully controlled conditions? I wager that such an analysis would discourage most of even the hardiest researchers. This, however, is a relatively simple case of biological feedback, where nonlinear dynamics, then chaos sets in. And we shall see in the next section that it can be explained by a simple model. Chaos is the situation where seemingly complex, even inexplicable behavior results from simple systems."

**"Now, if this chaotic behavior were truly random,** there would be little point in following through with chaos theory. However, there is a definite, albeit subtle order in chaos.… But what does all of this mean in terms of free will? Simply this: Just as chaos theory can provide a bridge between determinism and statistical behavior, so can it provide a bridge between predestination and free will. Maps having self-affine infinite regression are ubiquitous in the universe, and these maps can be strictly deterministic—a given initial point inevitably leads to a definite final point. Nevertheless, because it is physically impossible to determine this initial point with the necessary infinite precision, one cannot work backward from the “predestined” final point to its defined cause. Mathematics can state things with certainty; physics cannot. In physics we are fond of stating, “In principle, it can be shown (even proven) that...” In principle, chaos theory shows this not to be true. Principle and practice represent two antipodal worlds—perhaps ne’er the twin shall meet."

If you want to read the entire essay in a PDF, click HERE.

**William McHarris**

**Who is Bill McHarris?**

With a B.A. from Oberlin College and a Ph.D. from the University of California, Berkeley, I went immediately to Michigan State University with a joint appointment between the Chemistry and Physics/Astronomy Departments. For some 35 years I was a practicing nuclear chemist/physicist at the National Superconducting Cyclotron Laboratory, first working primarily in experimental nuclear spectroscopy and progressing toward more theoretical topics such as the weak interaction. Some time ago I found chaos theory and chaos theory found me, and I have been working with it ever since, trying to fathom a possible connection with quantum mechanics. I am most recently Professor Emeritus of Chemistry and Physics/Astronomy at Michigan State University.

**Professor (b. 1937). B. A., 1959, Oberlin College; Ph.D., 1965, University of California, Berkeley. Nuclear Chemistry/Physics. Nuclear spectroscopy and reactions; exotic nuclei far from stability; mesons and nuclei, such as pion interactions/binding with nuclei; feedback from quark theory and elementary-particle physics into nuclear science and other fields. He is also a published composer and organist.**