FOM: Bishop-style constructive mathematics
Peter Schuster
pschust at rz.mathematik.uni-muenchen.de
Fri Dec 29 06:53:26 EST 2000
Let me add some points to Ayan Mahalanobis' FOM posting of Fri Dec 29 03:24 MET 2000.
First, Bishop-style mathematics of today, and as practised by Bridges, Richman,
and others, appears to be more general than Bishop's own theory. For example, now one
does not necessarily assume that one starts, like Bishop, from the real numbers as Cauchy
sequences of rational numbers with a fixed rate of convergence; one could work instead
with located Dedekind cuts, or even with an axiom system collecting the constructively
reasonable properties of any type of real numbers (cf. Bridges, Theor. Comp. Sci.,
vol. 219 (1999), pp. 95--109, see also http://www.elsevier.nl/locate/tcs).
This is what one could call the relative character of Bishop-style mathematics,
which is of course foreign to anyone who wants to put mathematics on a putatively firm
basis, especially to any constructivist strictly following Kronecker's and Brouwer's traces.
Secondly, as Ayan recalled, Bishop-style mathematics assumes less and so gets more
(generality): it turned out to be a common theory of classical, recursive, and intuitionistic
mathematics, and presumably also of computational settings such as Weihrauch's TTE (see Bridges,
loc. cit.). Bishop-style mathematics is therefore situated below or above all these models,
not beside them. [By the way, one could parallel this with the status Wittgenstein ascribes
to philosophy compared with the other sciences (Tractatus logico-philosophicus, 4.111).]
Although Bishop's original idea was not quite that of a theory of many models, he made some
conceptual choices very appropriate for that purpose. For example, he simply defined continuity
as uniform continuity on compact domains; recall that in recursive mathematics there is a
pointwise continuous function on [0,1] that is not uniformly continuous.
This is what one could call the ambiguous character of Bishop-style mathematics,
which is of course hard to accept by people exclusively devoting their professional life
to one of the models, especially by any (leisure-time) platonist who thinks that there
is only one mathematics.
Peter Schuster
Name: Peter M. Schuster
Position: Wissenschaftlicher Assistent
Instituition: University of Munich, Mathematics Department
Research interest: constructive mathematics
http://www.mathematik.uni-muenchen.de/~pschust
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