| | In the summer of 1986, Ken Ribet proved that Fermat's Last Theorem is a special case of the [[Taniyama–Shimura Conjecture]]. | | In the summer of 1986, Ken Ribet proved that Fermat's Last Theorem is a special case of the [[Taniyama–Shimura Conjecture]]. |
| | In a series of lectures in 1993, mathematician [[Andrew Wiles]] announced a proof using techniques in algebraic geometry, relying on the nonconstructive [[Axiom of Choice]].<ref name="Occam">http://www.occampress.com/fermat.pdf Page 5</ref> A flaw was found before publication, and Wiles spent a year on fixing the flaw. Then, in September 1994, he and Richard Taylor announced a new version of the proof. However, criticism does continue on the internet.<ref name="Occam" /> Further criticism came from [[Marilyn vos Savant]], known for her very high [[IQ]] and commentary on [[mathematics]], in her column and book.<ref>Ask Marilyn ® by Marilyn vos Savant, Parade Magazine. November 21, 1993</ref><ref>''The World's Most Famous Math Problem: The Proof of Fermat's Last Theorem and Other Mathematical Mysteries'', Marilyn vos Savant. St. Martin's Griffin, 1993</ref> She questioned the use of [[Non-Euclidean geometry]] and the Axiom of Choice, among other points. She retracted her argument in a 1995 addendum to the book. | | In a series of lectures in 1993, mathematician [[Andrew Wiles]] announced a proof using techniques in algebraic geometry, relying on the nonconstructive [[Axiom of Choice]].<ref name="Occam">http://www.occampress.com/fermat.pdf Page 5</ref> A flaw was found before publication, and Wiles spent a year on fixing the flaw. Then, in September 1994, he and Richard Taylor announced a new version of the proof. However, criticism does continue on the internet.<ref name="Occam" /> Further criticism came from [[Marilyn vos Savant]], known for her very high [[IQ]] and commentary on [[mathematics]], in her column and book.<ref>Ask Marilyn ® by Marilyn vos Savant, Parade Magazine. November 21, 1993</ref><ref>''The World's Most Famous Math Problem: The Proof of Fermat's Last Theorem and Other Mathematical Mysteries'', Marilyn vos Savant. St. Martin's Griffin, 1993</ref> She questioned the use of [[Non-Euclidean geometry]] and the Axiom of Choice, among other points. She retracted her argument in a 1995 addendum to the book. |
| | The Wiles-Taylor proof also makes use of some [[Grothendieck]] tools in cohomological number theory that use an axiom beyond the standard [[Zermelo-Fraenkel|ZFC]] axioms. It is an open question whether these tools can be formalized into a ZFC proof.<ref>Colin Mclarty http://www.cwru.edu/artsci/phil/Proving_FLT.pdf</ref> | | The Wiles-Taylor proof also makes use of some [[Grothendieck]] tools in cohomological number theory that use an axiom beyond the standard [[Zermelo-Fraenkel|ZFC]] axioms. It is an open question whether these tools can be formalized into a ZFC proof.<ref>Colin Mclarty http://www.cwru.edu/artsci/phil/Proving_FLT.pdf</ref> |