Fermat's Last Theorem, more can be proved more simply: Professor steers field toward a numbers-only proof

Mar 04, 2013

Fermat's Last Theorem—the idea that a certain simple equation had no solutions— went unsolved for nearly 350 years until Oxford mathematician Andrew Wiles created a proof in 1995. Now, Case Western Reserve University's Colin McLarty has shown the theorem can be proved more simply.

The theorem is called Pierre de Fermat's last because, of his many conjectures, it was the last and longest to be unverified.

In 1630, Fermat wrote in the margin of an old Greek mathematics book that he could demonstrate that no integers () can make the equation xn + yn = zn true if n is greater than 2.

He also wrote that he didn't have space in the margin to show the proof. Whether Fermat could prove his theorem or not is up to debate, but the problem became the most famous in mathematics. Generation after generation of mathematicians tried and failed to find a proof.

So, when Wiles broke through in 1995, "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."

McLarty is a Case Western Reserve philosophy professor who specializes in logic and earned his undergraduate degree in mathematics. He hasn't developed a proof for Fermat, but has shown that the theorem can be proved with much less set theory than Wiles used.

Wiles relied on his own deep insight into numbers and works of others—including Alexander Grothendieck—to devise his 110-page proof and subsequent corrections.

Grothendieck revolutionized numbers theory, rebuilding in the 1960s and 1970s. He used strong assumptions to support , including the idea of the existence of a universe of sets so large that standard set theory cannot prove they exist. Standard set theory is comprised of the most commonly used principals, or axioms, that mathematicians use.

McLarty calls Grothendieck's work "a toolkit," and showed, at the Joint Mathematics Meetings in San Diego in January, that only a small portion is needed to prove Fermat's Last Theorem.

"Most theorists are like race car drivers. They get the best out of the car but they don't build the whole car," McLarty said. "Grothendieck created a toolkit to build cars from scratch."

"Where Grothendieck used strong set theory I've shown he could do with only a fraction of it," McLarty said. "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 showed that all of Grothendieck's ideas, even the most abstract, can be justified using very little set theory—much less than standard set theory. Specifically, they can be justified using "finite order arithmetic." This uses numbers and sets of numbers and set of those and so on, but much less than standard set theory.

"I appreciate the wholeness of the foundation Grothedieck created," McLarty said. " I want to take the whole thing and make it more usable to practicing mathematicians."

Harvey Friedman, who famously earned his undergraduate, master's and PhD from MIT in three years and began teaching at Stanford University at age 18, calls the work a "clarifying first step," ScienceNews reported. Friedman, now an emeritus mathematics professor at Ohio State University, calls for McLarty's work to be extended to see if the theorem can be proved by numbers alone, with no sets involved.

"Fermat's Last is just about numbers, so it seems like we ought to be able to prove it by just talking about numbers," McLarty said. "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."

McLarty will talk more about that specific result at the Association for Symbolic Logic North American Annual Meeting in Waterloo, Ontario, May 8-11.

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User comments : 11

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Anda
1.5 / 5 (10) Mar 04, 2013
Had to read it two times:

So the work of two men (Grothendieck and Wiles) solved a 300 years old problem.

Now this man (undergraduate degree in mathematics) using their work says that it could be solved in a more simple way... but hasn't been able to do it yet! Common... that a joke?
Manitou
5 / 5 (3) Mar 04, 2013
@Anda, It's not a joke.
It's McLarty's conjecture based on Wiles's proof of Fermat's conjecture. Then Friedman made a conjecture based on Wiles's conjecture.
Or at least, that's the way I read it.
antialias_physorg
3.9 / 5 (9) Mar 04, 2013
Now this man (undergraduate degree in mathematics) using their work says that it could be solved in a more simple way... but hasn't been able to do it yet! Common... that a joke?

the way I read it what he heas shown is that Wiles used set-theoretical constructs that were overly broad. The actually relevant parts of set theory are fully contained in a much more limited...erm...set.
philw1776
2.7 / 5 (7) Mar 04, 2013
Yes.
I had done this a while ago in my spare time and was going to post my minimal set theory proof here, but the comments format is too short and lacks the needed symbology.
Whydening Gyre
1.8 / 5 (5) Mar 04, 2013
McLarty must be an associate of E. Andrulis... Unfortunately, he will spend the rest of his "almost" finding his answers...
Whydening Gyre
1.2 / 5 (5) Mar 04, 2013
Now this man (undergraduate degree in mathematics) using their work says that it could be solved in a more simple way... but hasn't been able to do it yet! Common... that a joke?

the way I read it what he heas shown is that Wiles used set-theoretical constructs that were overly broad. The actually relevant parts of set theory are fully contained in a much more limited...erm...set.

Ya. the set of - oops. einstein said sucess was achieved by not revealing everything...
Whydening Gyre
1 / 5 (5) Mar 04, 2013
Yes.
I had done this a while ago in my spare time and was going to post my minimal set theory proof here, but the comments format is too short and lacks the needed symbology.

Sheesh... sounds just like what Lemat said...
Whydening Gyre
2.3 / 5 (6) Mar 04, 2013
Sorry, Phil. Missed the sarcasm. My bad.
Job001
1 / 5 (5) Mar 05, 2013

We missed the point. The sarcasm should be toward our multi century blindness. Things are finite(quantum), not infinite(non-quantum). Mankind has been using the wrong math and creating infinity where there is none. All of the equations: gravity and coulombs charge forces and relativity are wrong(in the limit).

Force and interaction must of necessity be finite because in infinite regress the set cannot include quanta less than the smallest(Planck quanta, length, et al). Likewise mathematical sets including statistical Hamiltonian physics.
Whydening Gyre
1 / 5 (4) Mar 05, 2013

We missed the point. The sarcasm should be toward our multi century blindness. Things are finite(quantum), not infinite(non-quantum). Mankind has been using the wrong math and creating infinity where there is none. All of the equations: gravity and coulombs charge forces and relativity are wrong(in the limit).

Force and interaction must of necessity be finite because in infinite regress the set cannot include quanta less than the smallest(Planck quanta, length, et al). Likewise mathematical sets including statistical Hamiltonian physics.

Interesting dichotomy. However, should be obvious that the non-quantum is the reason the quantum exists at all...

Antialias mentioned that the only POSSIBLE closed system is the Universe itself. Logically, I cannot find this to be true, in that to be closed equals to not exist, in a manner of speaking.
I guess just another, bigger version of Lemat's Theorum...Go figure.
The Universe is the set of everything, including nothing..
How Fun!
Job001
2.3 / 5 (3) Mar 06, 2013
Gödel proved in 1930 that knowledge becomes paradoxical and incomplete within knowledge systems. Stepping out of mathematics and into labels Fermats last theorem solves quickly as follows:

The nth root of the sum of apples to the nth power and trees to the nth power cannot be rabbits.

http://en.wikiped...t_Gödel

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