questfortruth
Well-Known Member
INTRODUCTION
Colin McLarty, ``What does it take to prove Fermat's last theorem?
Grothendieck and the logic of number theory,''
Bulletin of Symbolic Logic 16(3), 359-377 (2010).
MY CONTRIBUTION
\begin{abstract}
I argue, that a good paper with good proof can contain weak spots.
So the proofs for God do. And so even the proof of Fermat's theorem does.
Therefore, if you demand the strong proof for Fermat -- you will fail,
just like demanding the strong proof for God. Valid proof is allowed to be weak,
but it is better to be rather short weak proof, than over 100 pages long weak proof.
Remaining over 300 years an unsolved problem, the recently
proven Fermat's Last Theorem has stimulated the
development of algebraic number theory in the 19th century
and the proof of the modularity theorem in the 20th century.
I am presenting new insights on the numbers and on the Theorem.
\end{abstract}
\section{Is Wiles's proof weak or strong one?}
Fermat's Last Theorem is one of
the most popular theorems in mathematics. Its condition is
formulated simply, at the ``school'' arithmetic level, but many
mathematicians have been looking for the proof of the theorem
for more than three hundred years. Proved 1994 by Andrew Wiles
(proof published 1995).
But some parts of the work contain stronger arguments than
other parts. And this means that there are the weakest points
and the strongest points. And this is a normal phenomenon in a
research work.
Humankind is not yet done with Fermat's Last Theorem. The
Dr.~McLarty's paper~\cite{Mc} explores the set-theoretic assumptions
used in the currently published proof of Fermat's Last Theorem,
how these assumptions figure in the methods Wiles uses, and the
currently known prospects for a proof using weaker assumptions.
``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 said. ``I believe that can be done, but it will require
many new insights into numbers. It will be very hard.'' I am
presenting several new insights here.
Can a research paper that has weaknesses be logically correct?
Does a weak spot in the paper asks from a reader a tiny bit of
trust, faith, hope, and love, a
bit of scientific courage and determination? A bit of risk?
A bit of real life.
\section{G\"odel}
INTRODUCTION
MY CONTRIBUTION
Due to the Incompleteness Theorems of G\"odel~\cite{Godel} one
can say, that some true conjectures do not have valid proofs.
One could think it also about the attempts to prove the
Fermat's Last Theorem outside of the set theory methods, but I was
lucky.
At school task-books, all problems have a solution. On the other hand,
the universities are teaching problems without found solutions, [but
with an unknown, hidden answer, e.g.\ we do not know is the Twin
Prime Conjecture trueor false, but there is definite answer to it].
While joyful discovering solutions
to the Riemann, Goldbach, ABC Hypotheses, Fermat's Last Theorem,
I cannot find a solution to the Twin Prime Conjecture. Maybe the
twin primes don't have a solution: they have an answer that cannot
be reached by a logical path.
This fact was first discovered by G\"odel.
My demonstration of G\"odel's incompleteness theorem:
Some hypotheses have a limited number of solutions, for example, two.
But we don't know that. Then, having found these two solutions, we
will look for the third. Then the probability of finding the third
solution is less than 100 \% even using unlimited resources and
research time. The probability of finding solutions is
not dictated by the order of finding solutions, therefore the
probability of finding the very first solution is also less than 100 \%.
Since there are problems where the probability of finding a solution
is less than 100 \%, then there are also problems where the number of
possible solutions is zero [because of definition of probability].
Colin McLarty, ``What does it take to prove Fermat's last theorem?
Grothendieck and the logic of number theory,''
Bulletin of Symbolic Logic 16(3), 359-377 (2010).
MY CONTRIBUTION
\begin{abstract}
I argue, that a good paper with good proof can contain weak spots.
So the proofs for God do. And so even the proof of Fermat's theorem does.
Therefore, if you demand the strong proof for Fermat -- you will fail,
just like demanding the strong proof for God. Valid proof is allowed to be weak,
but it is better to be rather short weak proof, than over 100 pages long weak proof.
Remaining over 300 years an unsolved problem, the recently
proven Fermat's Last Theorem has stimulated the
development of algebraic number theory in the 19th century
and the proof of the modularity theorem in the 20th century.
I am presenting new insights on the numbers and on the Theorem.
\end{abstract}
\section{Is Wiles's proof weak or strong one?}
Fermat's Last Theorem is one of
the most popular theorems in mathematics. Its condition is
formulated simply, at the ``school'' arithmetic level, but many
mathematicians have been looking for the proof of the theorem
for more than three hundred years. Proved 1994 by Andrew Wiles
(proof published 1995).
But some parts of the work contain stronger arguments than
other parts. And this means that there are the weakest points
and the strongest points. And this is a normal phenomenon in a
research work.
Humankind is not yet done with Fermat's Last Theorem. The
Dr.~McLarty's paper~\cite{Mc} explores the set-theoretic assumptions
used in the currently published proof of Fermat's Last Theorem,
how these assumptions figure in the methods Wiles uses, and the
currently known prospects for a proof using weaker assumptions.
``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 said. ``I believe that can be done, but it will require
many new insights into numbers. It will be very hard.'' I am
presenting several new insights here.
Can a research paper that has weaknesses be logically correct?
Does a weak spot in the paper asks from a reader a tiny bit of
trust, faith, hope, and love, a
bit of scientific courage and determination? A bit of risk?
A bit of real life.
\section{G\"odel}
INTRODUCTION
MY CONTRIBUTION
Due to the Incompleteness Theorems of G\"odel~\cite{Godel} one
can say, that some true conjectures do not have valid proofs.
One could think it also about the attempts to prove the
Fermat's Last Theorem outside of the set theory methods, but I was
lucky.
At school task-books, all problems have a solution. On the other hand,
the universities are teaching problems without found solutions, [but
with an unknown, hidden answer, e.g.\ we do not know is the Twin
Prime Conjecture trueor false, but there is definite answer to it].
While joyful discovering solutions
to the Riemann, Goldbach, ABC Hypotheses, Fermat's Last Theorem,
I cannot find a solution to the Twin Prime Conjecture. Maybe the
twin primes don't have a solution: they have an answer that cannot
be reached by a logical path.
This fact was first discovered by G\"odel.
My demonstration of G\"odel's incompleteness theorem:
Some hypotheses have a limited number of solutions, for example, two.
But we don't know that. Then, having found these two solutions, we
will look for the third. Then the probability of finding the third
solution is less than 100 \% even using unlimited resources and
research time. The probability of finding solutions is
not dictated by the order of finding solutions, therefore the
probability of finding the very first solution is also less than 100 \%.
Since there are problems where the probability of finding a solution
is less than 100 \%, then there are also problems where the number of
possible solutions is zero [because of definition of probability].
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