torsdag 10 september 2009

The Resonable Ineffectiveness of Mathematics

                                                                The young Wigner.

In the previous post A Critical Analysis of the Ideology of Mathematics we made the observation that the foundation of school mathematics on all levels can be expressed in the words of Physics Nobel Laurate Eugene Wigner as: 
  • The unreasonable effectiveness of mathematics in the natural sciences. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. 
In his 1963 Nobel Lecture Events, Laws of Nature and Invariance Principles Wigner further explains:
  • Physics does not endeavor to explain nature, it only endeavors to explain the regularities in the behavior of objects, which are called the laws of nature. 
Acoording to Wigner mathematics is thus unreasonably effective as a language of expressing laws of nature interpreted as regularities of nature. 

But is this unreasonable? Is it not completely reasonable that analytical mathematics is effective in expressing regularities, like an elliptic orbit or harmonic oscillation? So if laws of nature express regularities it is fully reasonable that they can be expressed in the language of analytical mathematics.

But what is a law of nature? Is it really a regularity expressible by analytical mathematics as Wigner seems to claim?  

Let us take the same example as Wigner: Consider at planetary system governed by Newton's laws of motion, which no doubt are laws of nature. Is this all there is to say about planetary systems? No, it is not because the planetary motions are not included in Newton's laws. 

The motions result from letting the system evolve forward in time step by step according to Newton's laws from some initial state. In the simplest case of one planet orbiting a heavy sun the orbit is an ellipse, but with more than one planet the motion can be abitrarily complicated and not allow a representation in terms of elementary functions. 

Does this mean that there are no laws for the motion of a planetary system with many planets?
Of course not, but these laws are not explicit to us like Newton's laws, but hidden implicit and there is no golden rule how to find them and express them by elementary fucntions. 

We may compare with turbulent flow which is governed by Newton's laws but exhibits a very complex partly chaotic structure with a variety of interacting vortices on different scales.  But even a turbulent flow can exhibit some regularities in the form of certain meanvalues, which can be computationally predicted even if pointvalues vary chaotically, meanvalues like drag and lift.  However, there is no neat mathematical formula that expresses the drag and lift of a given body. Turbulent flow has to be computed step by step and there is no shortcut to regularity of solutions as in the case of  the elliptic orbit of one planet around a sun.

We are led to the conclusion analytical mathematics is not unresonably effective but rather resonably ineffective, while computational mathematics is resonably effective. 

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