Re: The state-of-the-art in mathematics

mareg_at_mimosa.csv.warwick.ac.uk
Date: 01/09/05 

Date: Sun, 9 Jan 2005 11:52:45 +0000 (UTC)

In article <200501081520.j08FKGb18218@proapp.mathforum.org>,
        escultur36@hotmail.com ("E. E. Escultura") writes:
>On 04 Jan 2005, Robin Chapman wrote:
>>E. E. Escultura wrote:
>>
>>>
>>> I have constructed a counterexample to the Lebesgue theorem on the Riemann
>>> Integral, See Nonlinear Studies, Vol. 5, No. 2. Since this is published it
>>> can only be refuted through publication in a refereed jounrnal.
>>
>>No. Published material can be refuted by refuting it,
>>not necessarily by "publication in a refereed jounrnal" (sic).
>>
>>--
>>Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
>>"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
>>Francis Wheen, _How Mumbo-Jumbo Conquered the World_
>
>You are right here. But, refutation by other means such as internet forum,
>has limited impact because it the refereed journals are the official arbiter
>of scientific truth. Contributions including refutation are officially
>recorded in the data bases that monitor them.

It seems to be a common misconception that acceptance in refereed journals
is somehow the official arbiter of correctness. I don't suppose that is
remotely true for experimental sciences, and for mathematical results the
truth is more complicated.

Whether alleged proofs of high-profile results like Fermat's Last Theorem,
the Four Colour Theorem and the Classification of Finite Simple Groups are
generally accepted as being correct depends more than anything else on the
consensus of opinion among experts in the area. Belief in correctness by
the experts usually precedes publication. In some cases, like CFSG, some
of the experts may have reservations, and the consensus in that case is
something like "the result is almost certainly true, but it is not unlikely
that the current 'proof' contains nontrivial errors".

On the other hand, it is not particularly difficult to get erroneous (and
even in some cases completely nonsensical) results published in refereed
journals. The vast majority of papers accepted by mathematical journals
have been checked, not necessarily conscientiously, by a single referee.
Remember that the responsibility for errors in papers does not lie
ultimately with the referee, but with the author(s). This situation is
not particularly satisfactory, but the problem is less serious than you
might imagine, because most papers are of no lasting importance anyway.
As soon as a paper becomes of genuine interest to people other than the
authors, it will be read and checked more assiduously, and errors or
gaps in reasoning will generally be brought to light at that stage.
(That may be over optimistic - are there any instances where results
have been assumed correct and used in other proofs for long periods of time,
before serious errors have been found in them ?)

The same applies to refutations of published material, but that is more
straightforward. To refute an alleged proof, you can either point out a
a gap in the reasoning or, alternatively, exhibit a counterexample to
the claimed result. But you, Mr Escultura, have not refuted anything.
You have merely made unfounded assertions like "BT is a contradiction
in R^3". To show that something is a contradiction, you need to prove
logically, within the system you are working in, that it leads to a
contradiction, and you have not done that. You may prefer to remain
yourself within the confines of intuitionist or constructivist mathematics,
but that does not invalidate mainstream mathematics.

Derek Holt.

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