fredag 16 september 2016

Mathematics as Magics 2

The idea of the unreasonable effectiveness of mathematics in Wigner's formulation is based on the analytical solution of the two-body problem given in Newton's Principia Mathematica showing that a single planet subject to the inverse square law of gravitation from a fixed sun, will move in an elliptic (or parabolic or hyperbolic) orbit.

Newton could thus confirm Kepler's laws from a single hypothesis of the inverse square law, with  Newton as mathematician thereby convincingly playing the role God! This gave mathematics a tremendous boost into the queen of sciences with immense (seemingly magical) power, which is the basic argument behind extensive compulsory school mathematics: Learn math and play God!

But if Newton was playing with one planet, God is playing with many planets and thus solves the N-body problem of N bodies moving under mutual gravitational attraction with N any number. But already the 3-body problem has resisted analytical solution since Newton, which can be seen to signify the unreasonable ineffectiveness of analytical mathematics in Hamming's formulation.

But the N-body problem can be solved by computational mathematics for N very large, which expresses a reasonable effectiveness of mathematics + computer.

You can explore the N-body problem in the following apps for young minds:
We learn that mathematics + computer (as NewMath) is powerful and should be taught as a subject of reasonable effectiveness, understanding that analytical mathematics alone may be unreasonably ineffective.