I wrote this in a notebook a few weeks ago and it would be a good idea to explain myself. The lack of certainty and chaos in the Universe is obvious, yet so much of our modeling is based on deterministic thinking. This legacy of Newton pollutes so much of

our thinking that it is probably one of the greatest impediments to progress today. It is an “unknown known” that I wrote about a couple of days ago. I’m not going to go down the philosophical rabbit hold of “free will” or any of that. My point-of-view that this sort of discussion is fairly pointless, for all practical purposes we can never know our initial or boundary conditions well enough to treat our World as deterministic.

The assumption of an absolute determinism is the essential foundation of every scientific enquiry.

It is impossible to trap modern physics into predicting anything with perfect determinism because it deals with probabilities from the outset.

― Arthur Stanley Eddington

Newton’s laws were revelation, but also woefully incomplete. They spawned a revolution in science that we benefit from today, but their power was not fully realized until many revisions were made. Nonetheless, many scientists are still enthralled by the beauty and purity of his perfects laws. Examples abound and perhaps most acutely with the Navier-Stokes equations. I regard the Clay prize,

http://www.claymath.org/millenium-problems/navier–stokes-equation ,

associated with the proof of existence for the 3-D Navier-Stokes equations to be outright foolishness. It is the enshrinement of mathematics as divorced from reality. The reason is several fold: the equations as given are patently unphysical, and singularities are a relic of the same Newtonian thinking that defines the Universe as a clock simply moving forward with complete determinism.

This isn’t to say that the Navier-Stokes equations aren’t useful as approximations or for modeling under some circumstances, they demonstrably are. The problem is that they have taken on all too great a meaning, and we are asking questions of them they shouldn’t be answering. The specific unphysical aspect of the equations is the divergence free velocity field, which implies infinite sound speeds and divorces the equations from thermodynamics. This is strike one.

Strike two is the notion of needing a singularity to define its characteristic dissipative nature. This is a bit of a physical Mobius strip where the equations are simply showing their limits of applicability. By removing thermodynamics and sound waves from the system, the natural nonlinear mechanism for singularities goes out too. The observations of flow in nature indicate the presence of dissipation without specific dependence on the value for the physical viscosity, i.e., evidence of a singularity. The result is an enormous waste of time and energy.

… Nature almost surely operates by combining chance with necessity, randomness with determinism…

This is only one instance where determinism is hurting science. In general the modeling of the World or Universe proceeds along lines that implicitly expect determinism despite all the evidence to the contrary. A lot of the time the variation in behavior of a system is relatively small, or the nature of variability is reliable and can be captured in the constitutive laws. A problem that we are increasingly facing is the solution of systems where length and time scales where the variability exists are coming into the resolution of our codes. Continuing to promote the fallacy that the system is deterministic is simply wrong on the face of it.

In some cases the lack of deterministic evolution is obvious like with weather. There some of the issues are being attacked in a fairly head on manner. In other cases like continuum mechanics, the deterministic mindset is so pervasive that the old ways continue unabated. A bit of multiscale thinking is creeping in, but itself is polluted by determinism. Our models are getting to the point where grain structure and material structure can be directly modeling, yet we persist in using models defined when length and time scales were vastly larger. Our approach needs to change for progress to be made.

The uncertainty principle signaled an end to Laplace’s dream of a theory of science, a model of the universe that would be completely deterministic. We certainly cannot predict future events exactly if we cannot even measure the present state of the universe precisely!

We could still imagine that there is a set of laws that determine events completely for some supernatural being who, unlike us, could observe the present state of the universe without disturbing it. However, such models of the universe are not of much interest to us ordinary mortals. It seems better to employ the principle of economy known as Occam’s razor and cut out all the features of the theory that cannot be observed.