Sunday, January 20, 2008

309)Harmonious mathematical reasoning and the Universe in which we live, move and have our being; Quotes of Aga Khan IV.

This post is now part of the following collection of posts:

454)A Collection of Posts on Symmetry in Nature, as a Product of the Human Mind, Geometry and Harmonious Mathematical Reasoning; Quotes of Aga Khan IV

"In this context, would it not also be relevant to consider how, above all, it has been the Qur'anic notion of the universe as an expression of Allah's will and creation that has inspired, in diverse Muslim communities, generations of artists, scientists and philosophers? Scientific pursuits, philosophic inquiry and artistic endeavour are all seen as the response of the faithful to the recurring call of the Qur'an to ponder the creation as a way to understand Allah's benevolent majesty. As Sura al-Baqara proclaims: 'Wherever you turn, there is the face of Allah'.The famous verse of 'light' in the Qur'an, the Ayat al-Nur, whose first line is rendered here in the mural behind me, inspires among Muslims a reflection on the sacred, the transcendent. It hints at a cosmos full of signs and symbols that evoke the perfection of Allah's creation and mercy." (Aga Khan IV, IIS Convocation, October 19th 2003, London, U.K.)

"The Qur’an itself repeatedly recommends Muslims to become better educated in order better to understand God’s creation"(Closing Address by His Highness Aga Khan IV at the "Musée-Musées" Round Table Louvre Museum, Paris, France, October 17th 2007)

144)Harmonious mathematical reasoning and the Universe in which we live, move and have our being.

It is true that the universe in which we live, move and have our being is fashioned according to the principles of harmonious mathematical reasoning. In scientific, architectural as well as spiritual endeavour, this is true for many discoveries and creations. The Al-Hambra castle and gardens built in Islamic Spain are a major tourist attraction and many people are amazed at the positive and tranquilising psychological and psychic effect the buildings and structures inside seem to have on them. A recent documentary showed how the building was built according to precise structural and mathematical harmonics where the arc drawn from the diagonal of one rectangular piece served to become an anchor that defined an adjacent rectangular piece, and so on and so forth such that all the rectangular pieces used to build the structure, from the smallest to the largest pieces, were shown to have progressive and corresponding areas starting from the square root of 2 and ranging to the square roots of 3, 4, 5, 6, and so forth. The pure mathematical harmony and the symmetrical structures and calligraphy underlying the Al-Hambra is what makes it such a highly-visited tourist attraction and what gives many visitors their sense of well-being, pleasure and tranquility when they are within the structure.

Different structures in living systems also seem to follow certain ingrained mathematical patterns in their development, eg, the whorls and shapes of petals and other structures in flowers, etc. Isaac Newton had to develop his own type of mathematics(calculus) in order to properly describe his laws of motion. Mathematical symmetry depicted in Islamic art and architecture are symbols of transcendent divine knowledge.

During the 20th century there were many examples where discoveries in nature and the universe were predicted mathematically first before the actual physical discovery was made. Albert Einstein was a theoretical physicist whose momentous discoveries in special and general relativity were made creating and solving mathematical equations, not doing experiments in the field. It was only after he put forward his theories that other people performed experiments to confirm his genius in what he was saying with his mathematical equations. In the early to mid-20th century a scientist studying the different particles and forces of matter that make up the universe predicted mathematically the presence of a new particle in nature that was only discovered in the late 1970s after the technological development of particle accelerators or "atom smashers" energetic enough to show the existence of this particle.

The structure and function of the universe, indeed, follows the rules of harmonious mathematical reasoning and there is clear evidence that early Islamic philosophers and scientists recognised this and made it a significant part in their quest for knowledge of all types. A good example of this would be the group of mostly Ismaili Muslim thinkers from around 750 AD, called the Ikhwan Al-Safa(Brethern of Purity) who compiled an Encyclopedia(Rasail) of 52 volumes describing every type of knowledge, from wordly to spiritual, that was available to man at that time. The mathematical and life sciences feature prominently in this compilation of books.

The following article from the Economist shows how the mathematical elucidation of a structure may once again, in the future, describe how our universe operates at its most sub-microscopic level:

Higher mathematics

Truth and Lies

Mar 22nd 2007
From The Economist print edition

Mapping the most complex known mathematical object

FOR more than a century mathematicians have known about Lie groups. These are families of shapes named after Sophus Lie, a Norwegian mathematician who discovered them. There are four “simple” families of Lie groups and five others—this being mathematics—that are not quite so simple.

The simplest member of the simplest Lie group is the circle, which looks the same however it is rotated. Its higher-dimensional cousin, the sphere, has the same properties, only more so, and is thus the second-simplest of the same family. The five non-simple groups—dubbed “exceptional” in their complexity and symmetry—are harder to envisage and, for almost 120 years, the details of the most intricate of these have lain beyond reach. This week a group of mathematicians led by Jeffrey Adams of the University of Maryland announced that they had completed a map of the largest and most complicated one, a structure known to mathematicians as E8.

Lie groups have two defining features: surface and symmetry. A sphere has two surface dimensions. In other words, any place on its surface is defined by just two numbers, the longitude and the latitude. But it has three dimensions when it comes to symmetries. A sphere can spin on an axis that runs, say, from north to south, or on each of two axes placed at right angles to this. E8 is rather more difficult to visualise. Its “surface” has 57 dimensions—that is, it takes 57 co-ordinates to define a point on it, and it has 248 axes of symmetry.

Grappling with such a structure is as tricky as it sounds. But Dr Adams's team decided to have a go. They want to create an atlas of maps of the Lie groups. This involves making a description in the form of a matrix for each structure. (A matrix is a multi-dimensional array of numbers, such as that found in a sudoku puzzle.)

Dr Adams and his colleagues began by writing a computer program that would generate such matrices, a task that took them more than three years. It transpired that they needed 453,060 points to describe E8 but that they also needed to express the relationship between each of these points. That meant they had to devise a matrix with 453,060 rows and the same number of columns. In total this gives 205 billion entries. To complicate things further, many of these entries were not merely numbers but polynomials—sequences in which a given number is raised to a series of different powers, for example its square and its cube.

Processing such a vast quantity of data was beyond the capacity of even modern supercomputers, so the team were forced to tinker with the problem to make it tractable. This tinkering led them to a piece of ancient maths known as the Chinese remainder theorem.

This theorem is contained in a book written in the late third-century AD by a mathematician called Sun Tzu (not to be confused with the military strategist of the same name). It is used to simplify large calculations by breaking them down into many smaller ones, the results of which can then be recombined to generate the answer to the original question.

One problem addressed in the original book concerns counting soldiers. Sun Tzu's solution was that the soldiers should first split into groups of three, then groups of five, then groups of seven, with the number unable to join a group (in other words, the remainder) being noted each time. The three remainders can then be used to calculate how many soldiers are present. For example, if two were left over from the groups of three, three left over from the groups of five and two left over from the groups of seven, there would have been 23 soldiers in the unit (or possibly 233, but the difference should be obvious to even the stupidest commanding officer).

The researchers worked out how to use the remainder theorem to bring their calculation within the capacity of a supercomputer called Sage, which spent more than three days crunching the numbers to generate the map of E8. Not content with letting the supercomputer do all the arithmetic, the mathematicians simultaneously jotted down some calculations of their own on the back of an envelope. They worked out that if each entry in the matrix were written on paper that was one inch square, the answer would cover an area the size of Manhattan.

And the point is:

Apart from the satisfaction of mapping E8 at long last, mathematicians are pleased because the structure keeps popping up in another branch of intellectual endeavour: string theory. This purports to be the best explanation of the universe beyond the Standard Model of physics that describes all known particles and forces, but which is generally acknowledged to be incomplete. String theory requires that the universe has many more dimensions than those that are obvious, but that most of these extra dimensions are too small to be discerned with today's equipment. One of the ways in which they can be hidden involves E8, so having a mathematical map of its structure could be handy. Cheaper, too, than building a particle accelerator the size of the solar system.

Easy Nash aka easynash

The Qur'an itself repeatedly recommends Muslims to become better educated in order better to understand God's creation: Aga Khan IV(2007)
The Quran tells us that signs of Allah's Sovereignty are found in the contemplation of His Creation: Aga Khan IV(2007)
This notion of the capacity of the human intellect to understand and to admire the creation of Allah will bring you happiness in your everyday lives: Aga Khan IV(2007)
Islam, eminently logical, placing the greatest emphasis on knowledge, purports to understand God's creation: Aga Khan IV(2006)