Mathematics Story

I want to know how God created this world.  I’m not interested in this or that phenomenon, in the spectrum of this or that element.  I want to know his thoughts, the rest are details.”  Albert Einstein.

An overwhelming intuition for Einstein was that there is an all-encompassing, intelligible, something, ‘out there’, some unified and unchanging reality behind the ever-changing particulars of everyday experience. This is what he was after, what he called the secrets of the “Old One“.

For Einstein the clues were to be found in the phenomena that are invariant, phenomena that are the same, regardless of manner of measurement, or relative position, or dynamic operation, or observer point of view.  He saw this in the speed of light, which was found to be the same to all observers, regardless of their own motion.  With this, space and time are relative, but space-time is not.  Einstein’s own great insight was that acceleration, inertia, and gravity are equivalent, and therefore, rather than a ‘force’ between two masses, gravity is inherent in all of mass and motion.  It is invariant, and so must be related to space-time, and so he derives his theory of general relativity:

Ruv– 1/2 guvR = 8πTuv

Matter tells space-time how to curve, and curved space-time tells matter how to move.” – John Wheeler.   “an entwined dance of space, time, matter, and energy” –  Brian Greene.    Einstein, Walter Isaacson, 2008

It is really a theory of what isn’t relative. Einstein preferred that it be called the theory of invariants.

It turns out there is a brilliant mathematics of invariants.  It is called group theory. It was invented by Evariste Galois, in France, in 1730. He was refused admission to the elite Ecole Polytechnique institute of mathematics, too advanced for their examiners to understand.   He died in a duel, at age . . . . 20 .

Galois wrote his theory on a mere sixty pages of personal notes, and in a famous letter to August Chevalier just prior to his duel.

My dear friend, In the theory of equations, I have investigated under which conditions the equations are solvable by a formula:  this has given me the opportunity to make this theory more profound, and to describe all the transformations possible on an equation even when it is not solvable by formula.”  The Equation that Couldn’t be Solved, Mario Livio, 2005

This theory is the mathematics of permutations and symmetries, which are patterns of geometry and number that remain unchanged during some defined operation. They are the invariants that mark the hidden unity and relations in disparate sets of phenomena. Imagine an unknown, multifaceted geometric object, unified, and complex, and dynamically changing.  Imagine its sides and corners are ink soaked. Next, imagine this object tumbling across a white sheet of paper.  The ink will create obscure and puzzling markings.  Group theory mathematics, when applied to these markings, will yield the clues to the configuration and dynamics of this mysterious object.

This theory may well be the most profound in all of mathematics.

Einstein stood on the shoulders of giants, . . . and on those of a 20 year old genius.




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