22 October 2009

Truth, Beauty and Mathematics

Are mathematicians the architects of an unseen world? Sir Michael Atiyah, honorary professor at the University of Edinburgh (UK), lecturing at the TWAS 11th General Conference in Durban, South Africa, proposed that mathematicians, driven by a passion for beauty, discover truth along the way.

Atiyah, widely considered one of the world's most influential mathematicians, observed in his invited lecture on the second conference day that science and art, the two intellectual pillars of civilization, have been viewed as complementary and contrasting pursuits of humanity throughout history.

He notes that science has been viewed as an impersonal and objective human endeavour driven by a search for truth. Art, in contrast, is seen as a personal and subjective endeavour driven by a search for beauty. Where does mathematics fit in this intellectual dichotomy? For Atiyah, as the only discipline where absolute certainty can be achieved, maths is too often regarded as a sort of hyper-science – a "cold and forbidding" pursuit which rarely touches either its practitioners or observers.

Mathematicians, not surprisingly, do not want their discipline to be viewed in this simplistic and negative way. Instead, Atiyah suggests that the field of mathematics should be compared to that of architecture. He notes that, like architecture, mathematics enjoys its own design and structural principles, is elevated by lofty visions, relies on carefully engineered frameworks and materials, embraces graceful details meticulously shaped by its skilled practitioners, and often displays real purpose in the real world. In brief, Atiyah claims that both mathematics and architecture "have beauty and utility".

Yet, what is beauty? We can define it, says Atiyah, in terms of simplicity, originality and elegance. But, in the final analysis, Atiyah maintains, beauty yields to this simple definition: "You know it when you see it".

In mathematics, Atiyah says that beauty is on display, on a small scale, in a prime number – an integer that cannot be divided except by one and itself. On a large scale, mathematical beauty can be discerned in abstract symmetries – vast intellectual structures that carry implications which span across the discipline. Indeed, he says, abstract symmetries dominate the mathematical landscape much like the Cathedral of Notre Dame dominates the cityscape of the Ile de la Cité in Paris.

For mathematicians, it is the beauty of their discipline – its elegance and simplicity, its enduring foundations and its intricate and integrated frameworks – that drives their passion. Truth may emerge from their pursuits, says Atiyah, but it is beauty that explains their passion. In short, Atiyah says that mathematicians, in their work, "search for beauty and find truth along the way".

Thus, Atiyah is not surprised to find that a number of mathematicians have also been poets. He cites Omar Khayyam and Lewis Carroll, who are better known for their writing than their mathematics, and William Rowan Hamilton and James Clerk Maxwell, who are better known for their mathematics and physics than for their writing.

To help dispel the false dichotomy between science and art, and the prevailing notion among large segments of the public that mathematics is "cold and forbidding", Atiyah turns not to a mathematician but a poet – John Keats, who, in his "Ode on a Grecian Urn", famously wrote:

Beauty is truth, truth beauty – that is all
Ye know on earth, and all ye need to know.

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