[FOM] re harvey re my "effective number theorists" (II)

Gabriel Stolzenberg gstolzen at math.bu.edu
Sat Apr 15 23:33:05 EDT 2006

   This is part II of my reply to Harvey's reply to my message,
"effective number theorists."

   Again, it begins with a quote from my message, followed by
Harvey's question about it and then my answer to his question.

> > However, it remains to be checked whether number theorists mean
> >the same thing by these words that Harvey does.  They are, after
> > all, classical mathematicians and "effective algorithm" is a term
> > used by classical mathematicians not constructivists.

> I don't know what you have in mind for a possible difference between
> how I use the relevant words and how number theorists use the relevant
> words. I would like to hear.

   Sorry, I was trying to be tactful and ended up being misleading.
They may indeed use words like 'algorithm' a different way, e.g.,
as in classical recursion theory (although I think this is unlikely).
But the main difference I have in mind is that you call something a
construction (say, a bound or algorithm) only if it is one and you say
that there is none only if there is none---whereas I would need to
know more before I could be confident that number theorists do too.

   My reason is that number theorists are classical mathematicians and
classical math is rich in cases where algorithms are said to have been
defined when not only have they not been but cannot be.  And it also
is rich in false claims of the form, "We've proved that it exists but
there is no construction."

   I'll mention just a few cases.

   In an elementary analysis text, the authors define "an algorithm"
for adding infinite decimals.  But there isn't any such algorithm.
(Because there isn't any for deciding, for all infinite decimals, p
and q, whether p + q < 2, in which case, the whole number part of its
expansion is 1, or not < 2, in which case, it's 2.)

   In a calculus book, Peter Lax gave what he said was an algorithm
(informal computer program) for getting intermediate values under the
assumption of the IVT.  But there is no such algorithm.  He included a
"step" that cannot in general be carried out.

   In a preprint I once received, a prominent philosopher wrote,

    "As long as we accept the correctness of Newton's Law of Gravity,
    for example, we are committed to the statement that the evolution
    of an N-body system will be in accordance with the solutions to
    the appropriate system of differential equations; and it is to
    this day quite unknown whether the solutions to these equations
    are recursively calculable even when N = 3."

   The view he expresses here had been folklore since the beginning
of the last century and, for all I know, still is.  However, I wrote
back explaining that he was talking here about a 1st order ODE that,
sufficiently close to the initial conditions (close enough to stay
away from collisions and then some), satisfies a Lipshitz condition.
Hence, we're talking about Picard's method, which is a construction
par excellance.  (In this case, the solutions are even analytic.)

   The philosopher then removed the statement from his paper.

   With best regards,


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