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Sunday, July 7, 2013

The mathematics of beauty.

A mathematics of beauty and art. There is no such thing one would expect. Yet, Owen Jones (1809 - 1874), London architect, wrote the 'General Principles in the Arrangement of Form and Colour, in Architecture and the Decorative Arts ... ' in his book The Grammar of Ornament.

Proposition 4.
True beauty results from that repose which the mind feels when the eye, the intellect and the affections are satisfied from the absence of any want.

from Page PL. I

You'll find the other propositions at the link above in the Digital Copy of the 'The Grammar of Ornament'.

Tuesday, July 2, 2013

Mathematics: created or evolved?

Some ( if not most ) mathematicians think about mathematics as being 'created'. Paul Erdos said that there must be a book, The Book, containing the most important and most beautiful proofs. Who other than God could have written The Book? Einstein once said that 'God does not play dice' and Kronecker said that "God made the integers; all else is the work of man." There is still some creationism left among mathematicians. If there are several ways to attack a problem there is always one best, and that's the one described in The Book.

Where do our mathematical insights come from? Some say they are a form of Divine Intervention. Or is the entirety of mathematics somewhere hidden in our DNA? Evolution Theory versus Intelligent Design. We, the human species, are that part of the physical universe where it becomes self-aware. We ( the physical universe ) describe the behavior of the universe with the help of mathematics. In that sense mathematics evolved over the centuries. In the concept of a created mathematics, mathematics is there, for us humans, to be discovered. Is mathematics finite, created, timeless, deterministic and waiting for intelligent beings to be discovered, understood and applied? Or is mathematics human, a property of our species, a way to handle the complex reality around us. Unlikely as it seems there may be alternative bodies of mathematics possible.

Clearly, compared to the other sciences, mathematics evolves ( is discovered ) at a slow pace. But that's because mathematical truths are eternal ( they say ). The fundamental theorems are, but is the same true for the tools used by engineers and scientists in their daily work? Determinants have lost some of their appeal, quaternions may face the same destiny. Mathematics does change. But the idea that there is one ( created ) mathematics goes deep. You can't patent a mathematical idea for example.

Mathematicians lack a sense of urgency, except for their own careers or immortality maybe. If mathematics in its entirety could be owned by a company, would that ( have ) speed up the development of mathematics? And thus bring the solution of major planetary problems forward? What if there were competing bodies of mathematics, like operating systems, say Windows versus Linux? I think that that would have been possible in some alternate version of history. There -are- different ways of doing things in mathematics. Take geometry for example. In computer graphics you can choose between ( standard ) linear algebra, add quaternions if you like, or a portion of projective geometry so that you can compose translations and linear transformations with matrices, or: you can use geometric algebra which until not so long ago was either abondoned or exclusively used in quantum physics. My point being we may polish the mathematics until its ready for a place in The Book.

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Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)