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== Claim of a Proof ==

== Claim of a Proof ==

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 Is There a "Simple" Proof of Fermat's Last Theorem, Page 5 (occampress.com)]</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 Is There a “Simple” Proof of

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 -  Is There a “Simple” Proof of

Fermat's Last Theorem? Part (1) Introduction and Several New Approaches by Peter Schorer (cwru.edu)]</ref>

Fermat's Last Theorem? Part (1) Introduction and Several New Approaches by Peter Schorer (cwru.edu) [http://www.cwru.edu/artsci/phil/Proving_FLT.pdf]</ref>

Unlike other mathematical breakthroughs, this claimed proof of 1993 has facilitated little, if any, insights or simplifications since then.

Unlike other mathematical breakthroughs, this claimed proof of 1993 has facilitated little, if any, insights or simplifications since then.