Re: JSH: Math proofs

From: James Harris (jstevh_at_msn.com)
Date: 09/12/04 

Date: 12 Sep 2004 11:17:41 -0700

"W. Dale Hall" <mailtowd-hall@pacbell.net> wrote in message news:<59U0d.14646$QJ3.263@newssvr21.news.prodigy.com>...
> James Harris wrote:
> > jesse@phiwumbda.org (Jesse F. Hughes) wrote in message news:<871xhmefls.fsf@phiwumbda.org>...
> >
> >>magidin@math.berkeley.edu (Arturo Magidin) writes:
> >>
> >>
> >>>In article <87vfeze6cf.fsf@phiwumbda.org>,
> >>>Jesse F. Hughes <jesse@phiwumbda.org> wrote:
> >>>
> >>>
> >>>>Seems to me that we're not really avoiding assumptions in natural
> >>>>deduction so much as moving to a sequent-style calculus. I'm not sure
> >>>>that this does anything for us.
> >>>
> >>>Well, yes, sorry.
> >>>
> >>>A ->formal<- proof, defined as a finite sequence etc., will
> >>>necessarily begin with either a tautology or an axiom.
> >>
> >>I don't understand this claim either.
> >>
> >>There are formal systems of natural deduction. In these formal
> >>systems, a proof my begin with an assumption.
> >>
> >>I suspect that when you learned logic, you just didn't do natural
> >>deduction. If one wants to study logic, then natural deduction is an
> >>annoying formalization to use --- all those damn subproofs with their
> >>assumptions make for special cases in proving things. But natural
> >>deduction is a name for a class of related *formal* logics in which
> >>assumptions and assumption-discharging rules play a role.
> >>
> >>The sequent calculi are much handier for the logician and this could
> >>be what you learned. In that case, the transformation
> >>
> >> T,X |- Y
> >> --------
> >> T |- X -> Y
> >>
> >>(perhaps written T,X => Y |- T => X -> Y ) is an explicit rule of
> >>inference.
> >>
> >>
> >>[...]
> >>
> >>
> >>>The proof at issue originally here, the proofs contained in Wiles's
> >>>paper, are not formal proofs. And most of them do not "begin with a
> >>>true statement". Theorems 0.2 and 0.3, for example, begin with
> >>>assumptions, since they are proofs of A->B via the usual method method
> >>>of assuming A and deducing B, without ever invoking the Deduction
> >>>Metatheorem (which is unnecessary outside of formal logic, for the
> >>>most part). The proof of Theorem 5.2, that all semistable elliptic
> >>>curves over Q are modular, begins with "Suppose that E is a semistable
> >>>elliptic curve over Q", which is certainly not "a true statement".
> >>
> >>Indeed.
> >
> >
> >
> > Some poster claimed that this Hughes guy had all these posts with
> > proofs in them so I'm going back surveying what he's posted, and
> > noticed the statement above.
> >
>
> Which statement? That the proof of Theorem 5.2 begins with "Suppose that
> E is a semistable elliptic curve over Q"? The cryptic remark "Indeed"?
>

Actually I'm talking about his statement that the "proofs" do not
begin with a true statement.

My thought at the time was the he's questioning my statements about
math proofs, to whit, a math proof begins with a truth and proceeds by
logical steps to a conclusion which then must be true.

> > It's quite true that Wiles continually assumes modularity, and it's
> > kind of weird that no one thought to null test his work before because
> > his very paper just BEGS for it.
> >
>
> It's quite true that Wiles assumes (in several of the theorems of
> the paper under discussion) that a particular Galois representation
> is modular. It's just as true that you are attempting to claim to
> understand something that you don't. Wiles does *not* assume that
> a particular elliptic curve is modular. On the other hand, there are
> modular curves that are mentioned, for instance the curves X_0(N).
> He does not need to assume they are modular, since by their construction
> they *are* modular.
>

The null test is simple: assume the existence of a non-modular
elliptic curve and find some point in Wiles's work where that
assumption leads to a contradiction.

Math proofs begin with a truth and proceed by logical steps to a
conclusion which then must be true.

What I call a null test is to assume the opposite of the conclusion of
an argument and see if that assumption leads to a contradiction with
some logical step in that argument.

If it does not, then the argument is not a proof.

Proofs by contradiction are actually null tests.

All math proofs necessarily logically connect truths.

James Harris

Relevant Pages

  • Re: JSH: Curious, Wiless work, null test
    ... James Harris wrote: ... >>stone' to show that the given curve is modular. ... 0.3 describes sufficient conditions for an elliptic curve ... Galois representation, and quite another to deduce that your elliptic ...
    (sci.math)
  • Re: Null test, Wiless work
    ... > So far I've yet to see posters do anything than attack me!!! ... assume there is an elliptic curve out there that is not ... > modular, what makes you think Wiles's approach would have indicated ... > James Harris ...
    (sci.math)
  • Re: JSH: How easy? Wiless works flaw
    ... Given a modular form, one can use it in a straighforward way to ... an elliptic curve through that action. ... representations; ...
    (sci.math)
  • Re: JSH: Curious, Wiless work, null test
    ... but to a representation of the absolute ... representation rho to be induced by a modular form, ... The reason one discusses representations of the absolute Galois group ... One of the ways in which one can establish that an elliptic curve is ...
    (sci.math)
  • Re: JSH: Math proofs
    ... >>There are formal systems of natural deduction. ... E is a semistable elliptic curve over Q"? ... It's quite true that Wiles assumes (in several of the theorems of ... a particular elliptic curve is modular. ...
    (sci.math)
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