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Thursday, May 09, 2019

The Subtle Art of the Mathematical Conjecture | Quantized Columns - Quanta Magazine

Photo: Robbert Dijkgraaf
It’s an educated guess, not a proof. But a good conjecture will guide math forward, pointing the way into the mathematical unknown, argues Robbert Dijkgraaf, director and Leon Levy Professor at the Institute for Advanced Study in Princeton, New Jersey.

A conjecture creates a summit to be scaled, a potential vista from which mathematicians can see entirely new mathematical worlds.
Photo: Filip Hodas for Quanta Magazine

Mountain climbing is a beloved metaphor for mathematical research. The comparison is almost inevitable: The frozen world, the cold thin air and the implacable harshness of mountaineering reflect the unforgiving landscape of numbers, formulas and theorems. And just as a climber pits his abilities against an unyielding object — in his case, a sheer wall of stone — a mathematician often finds herself engaged in an individual battle of the human mind against rigid logic.

In mathematics, the role of these highest peaks is played by the great conjectures — sharply formulated statements that are most likely true but for which no conclusive proof has yet been found. These conjectures have deep roots and wide ramifications. The search for their solution guides a large part of mathematics. Eternal fame awaits those who conquer them first.

Remarkably, mathematics has elevated the formulation of a conjecture into high art. The most rigorous science cherishes the softest forms. A well-chosen but unproven statement can make its author world-famous, sometimes even more so than the person providing the ultimate proof. Poincaré’s conjecture remains Poincaré’s conjecture, even after Grigori Perelman proved that it is true. After all, Sir George Everest, the British surveyor general of India in the early 19th century, never climbed the mountain that today bears his name.

Like every art form, a great conjecture must meet a number of stringent criteria...

Finally, it is good to realize that the adventure does not always end with success. Just as a mountaineer can be confronted by an unsurpassable crevasse, mathematicians can fail, too. And if they fail, they fail absolutely. There is no such thing as a 99 percent proof. For two millennia, people tried to prove that Euclid’s fifth postulate — the notorious “parallel postulate” that states roughly that two parallel lines cannot cross — can be derived from the other four axioms of planar geometry. Then, at the beginning of the 19th century, mathematicians constructed explicit examples of non-Euclidian geometry, disproving the conjecture.
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Source: Quanta Magazine