Quine on mathematization

Jurriaan Bendien bendien at tomaatnet.nl
Wed Jul 2 14:20:33 MDT 2003

Hi Les,

As regards the brief notes I wrote on the modelling of complex systems
(which you didn't reply to), I found a paasage in an essay by W.V. Quine,
"Success and Limits of mathematization", which may also be of interest to
others on the list.  Discussing the "perverse tendency to think of
mathematics primarily as an abstract or uninterpreted and only secondarily
as interpreted or applied, and then to philosophise about applications",
Quine concludes among other things:

"I have touched on the nature of mathematization, arguing that in its
primary form it develops within a science rather than being applied from the
outside. It is continuous with the growth of precision, and it blossoms at
last into algorithms and proof procedures. The most significant continuing
force for mathematization was measurement, because of the benefits of
concomitant variation. Finally I noted the danger of being seduced, by the
glitter of the algorithm, into mathematizing one's subject off-target. But I
should say something, still, about the famous formal limits to
mathematization that are intrinsic to the mathematics itself. Building on
Godel's work, Alonzo Church and Alan Turing showed in 1936 that
mathematization in the fullest sense is too much to ask even for so limited
a subject as elementary logic. They proved that there can be no complete
algorithm, no decision procedure, for the first-order predicate calculus.
There is, of course, a complete proof procedure for that calculus. However,
it follows from the Church-Turing theorem that there cannot even be a
complete proof procedure for nonprovability in that calculus. From this it
follows further that there cannot be a complete proof procedure for any
branch of mathematics in which proof procedures can be modelled. Elementary
number theory is already one such branch; hence Godel's original
incompleteness theorem. Besides these necessary internal limitations on
proof and algorithm, there is commonly also a voluntary one in the case of a
natural science. Mathematize as he will, and seek algorithms as he will, the
empirical scientist is not going to aspire to an algorithm or proof
procedure for the whole of his science; he would not want it if he could
have it. He will want rather to keep a large class of his sentences open to
the contingencies of future observation. It is only thus that his theory can
claim empirical import." (Theories and Things, p. 154-155).



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