March 4, 2013
Clever Philosopher Offers Elegant New Approach To 350-Year-Old Math Riddle
redOrbit Staff & Wire Reports - Your Universe Online
In 1637, the French lawyer and part-time mathematician Pierre de Fermat put forward a simple and elegant numerical riddle that would puzzle and confound math geeks for 358 years. Known as Fermat´s Last Theorem, or simply Fermat´s conjecture, the theorem states no whole, positive numbers can make the equation xn + yn = zn true when ℠n´ is greater than 2.
Scribbling haphazardly in the margin of an old Greek math book, Fermat famously wrote he simply didn´t have enough space in the margins of the book to show his proof, leaving generations of mathematicians and theorists to speculate as to whether or not he truly had a valid proof for his own theorem.
Using a roundabout, backdoor approach that involved dizzyingly complex number theory, the Oxford mathematician Andrew Wiles demonstrated conclusively in 1995 that Monsieur de Fermat was, in fact, correct: The equation is unsolvable.
Now, however, Colin McLarty, a professor of philosophy and mathematics at Case Western Reserve University claims there´s a far simpler way to prove Fermat´s theorem — one that doesn´t involve complex mathematical wizardry with names like “modularity theory” and “epsilon conjectures.”
According to McLarty, the mathematics community almost didn´t know how to react when Professor Wiles solved one it´s most loved and hated riddles of all time.
"It was just shocking to a lot of us that it could be proved," McLarty said. "And we thought, 'Now what?' There was no new most famous problem."
Drawing upon his training in logic and philosophy, as well as a degree in mathematics, McLarty says he has demonstrated the correctness of Fermat´s Last Theorem without a mathematical proof and with far less abstract and circuitous theory than that used by Wiles nearly two decades earlier.
In his mind-boggling 110-page mathematical proof, Professor Wiles relied on his rich and intricate knowledge of mathematical theory to show Fermat was right. In particular, he leaned heavily the work of the revolutionary German mathematician Alexander Grothendieck, one of the key figures behind the development of modern algebraic geometry.
McLarty says Grothendieck provided a sort of "toolkit" that was useful for certain mathematical problems. However, at the annual Joint Mathematics Meetings in January, he gave a lecture in which he insisted only a few of those tools are actually needed to prove Fermat´s conjecture.
"Most number theorists are like race car drivers. They get the best out of the car but they don't build the whole car. Grothendieck created a toolkit to build cars from scratch," McLarty explained.
"[But] where Grothendieck used strong set theory I've shown he could do with only a fraction of it," he continued. "I use finite-order arithmetic, where all sets are built from numbers in just a few steps.
"You don't need sets of sets of numbers, which Grothendieck used in his toolkit and Andrew Wiles used to prove the theorem in the 90s."
McLarty demonstrated even the most complex and abstract of Grothendieck's ideas can be justified using very little set theory. What is known as ℠standard set theory´ is simply the collection of the most commonly used principals, or axioms, used by practicing mathematicians. Grothendieck´s work included the notion of the existence of a universe of number sets so large standard set theory could not even prove they exist.
In McLarty´s vastly simplified approach to Fermat´s problem, he says all mathematicians need is basic ℠finite order arithmetic,´ which uses even fewer sets of numbers than standard set theory.
"I appreciate the wholeness of the foundation Grothedieck created," McLarty said. "[But] I want to take the whole thing and make it more usable to practicing mathematicians."
The famous mathematician Harvey Friedman, who as a teenage prodigy earned his undergraduate, master's and PhD from MIT in three years and went straight to teaching at Stanford University at age 18, called McLarty´s work a "clarifying first step."
Friedman says mathematician´s should now get to work extending McLarty´s approach to see whether the theorem can be proven using only numbers without sets.
"Fermat's Last Theorem is just about numbers, so it seems like we ought to be able to prove it by just talking about numbers," McLarty added. "I believe that can be done, but it will require many new insights into numbers. It will be very hard. Harvey sees my work as a preliminary step to that, and I agree it is."