FOM: Schwartz,&SolFef's "lot from a little" maxim
Robert Tragesser
RTragesser at compuserve.com
Wed Jan 28 16:39:08 EST 1998
I was asked to say what I see in Jack
Schwartz's "The pernicious influence of mathematics
on physics" [WSieg&SF noted it orginally appeared on
pages 356-360 of:Logic, Methodology and Philosophy
of Science (Proceedings of the 1960
International Congress),edited by E. Nagel, P. Suppes,
and A. Tarski;Stanford University Press, 1962.)]
Here is one point(below), and this by way of
showcasing some perhaps not widely appreciated aspects of
Sol Feferman's "in mathematics we have a lot from a
little" maxim.
Of the many ways in which Sol exploits
this maxim (or rather the phenomenon), as for example
in the "Why a little bit goes a long way: Logical
foundations of scientifically applicable mathematics",
or (very interesting to a philosopher) his use in "What
rests on what?" to point out that
"The uninformed
common view -- that adopting one of the non-platonic
positions means pretty much giving up mathematics
as we know it -- needs to be drastically corrected,
and that should also no longer serve as the last-ditch
stand of set-theoretical realism",
a particularly edifying use occurs in
Sol Feferman's Working Foundations '91,
"I am in agreement with the constructivist position
as to the subjective source of basic mathematical
conceptions; but for me these are supposed to be
conceptions of certain kinds of ideal worlds,
including ones which are not countenanced
constructively (such as "platonistic" world of sets).
These worlds (or world pictures...) are presented
more or less directly to the imagination, from which
the basic principles are derived by examination.
All else (in each picture) is obtained by rational
reflection on, and from, basic concepts and
principles. IT MAY BE THAT AT THE OUTSET ONLY
RELATIVELY CRUDE FEATURES OF A WORLD-PICTURE CAN
BE DISCERNED IN THIS WAY. MY MAIN SLOGAN HERE IS
THAT NEVERTHELESS, _FOR MATHEMATICS_, A LITTLE
BIT GOES A LONG WAY."
When I first read this, I drew a thought or
conclusion from it which I don't think I've seen Sol
be explicit about (and he may not believe it -- but his
remarks on CH suggests he does). Before
I say what this thought is, I'll quote a neat line
from the Jack Schwartz essay:
"The phenomenon to be observed here [after discussing
the attractive but deceiving Birkhoff ergodic theorem]
is that of an involved mathematical argument hiding
the fact the we understand only poorly what it is
based on."
Notice that Sol observed that even though
we may capture only relatively crude features of
a (mathematical) world-picture, nevertheless, this
little, because we are in mathematics, may go a
long way.
Here the very charming and deceptive power
of mathematics to take a little and go a long way with
it may in fact be concealing one of two things
from us,
(1) how poorly we do understand the world-picture
at issue, or
(2) that the world-picture(d) itself is
crude and underdetermined, but this is concealed from us
by the wealth of mathematics that flows from the little
bit that we do understand or the little bit which is
decently determinate.
If someone does think (as Sol seems to think) that
such world-pictures are products of our imagination (I like to
say, after Russell, logical imagination), then this
together with the "lot from a little" wisdom would give reasonable
grounds to suspect that (for example) the world-picture SET
may very well be insufficiently determinate to settle CH (and
yet at the same time generate a lot of sublime mathematics) and
give an explanation of why it is that some have such terrific
faith in the determinateness of SET (viz., they are impressed --
and deceived -- by all that sublime mathematics SET does
support -- but are deceived or carried away because they have
not appreciated the dangers lurking in the phenomenon of, in
mathematics, a little going a long way.)
robert tragesser
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