[FOM] Re.. Question on the relevance of pragmatism to mathematical abstraction [nonconstructive numerical analysis]
Richard Haney
rfhaney at yahoo.com
Wed Nov 30 18:10:09 EST 2005
[Part of a multipart response: Nonconstructive numerical analysis]
Even the pragmatically motivated field of numerical analysis seems to
favor pockets of nonconstructive mathematics, which seem dubious as to
practical value. For example, the "regulators" (or "moduli of
convergence") in many asymptotic-behavior theorems for iterative
algorithms seem to be nonconstructive.
It seems to be fashionable to analyze computational algorithms so as to
provide some asymptotic result, and it seems the motivation is to
estimate the capability of the algorithm to produce a useful result in
a relatively short period of time. A typical theorem will be that some
algorithm's iterates converge quadratically, for example. And
quadratic convergence is typically regarded (or presumed) to be better
than asymptotically slower rates of convergence. But the (statement of
the) theorem provides no concrete (i.e., constructive) way of
determining when that quadratic behavior "will begin to appear" in some
sense. Obviously, by "prepending" a finite sequence of ((10^10)^10)^10
arbitrary values to such a sequence of iterates, one still gets a
quadratically convergent sequence. And a finite initial segment of any
sequence can be extremely misleading as to what the sequence will
eventually do. So obviously, the mere (nonconstructive) quadratic
convergence of the iterates of some algorithm is completely useless
from the strictly deductive point of view for the purposes of solving
concrete problems in the empirical world. (However, I suspect that
such nonconstructive theorems are popular in numerical analysis simply
because simple concrete examples of applications of such algorithms
often do in fact *seem* to converge quadratically in a fairly "small"
number of iterations. And this in turn suggests there may indeed be
theorems of a more constructive nature that could be proved where the
"when" (i.e., the regulator) of quadratic convergence is easily
determined from some sort of "computational complexity" of the
functions to which the algorithm is applied.)
But it seems that the typical technique in numerical analysis is not to
rely (entirely) on deduction for developing confidence in an algorithm
but simply to make somewhat plausible, ad hoc, intuitive assumptions as
needed, mix in some deductive analysis, and then test the
implementation empirically in a wide variety of conditions in order to
gain confidence that it will function as intended.
This seems to be what is invariably done for algorithms to numerically
solve differential equations -- an essential aspect of getting
spacecraft to Mars, for example.
Richard Haney
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