FOM: Improved: Hersh Fast & Loose?&Feferman's questions
Robert S Tragesser
RTragesser at compuserve.com
Sat Jan 3 07:20:20 EST 1998
Need we be protected against all doubt-throwing
skeptical terrorists in order to claim "certainty"?
In my first Re:H.Fast and Loose? I assumed
that the reader had plowed through my posting
that inspired Hersh to object. I should not have
assumed that.
Here is how things stand.
If I have understood him, Hersh is happy to
characterize sciences by "reproducibility of
results and consensus" and mathematics in
particular as being distinguished among the
sciences by its being somehow very generous in
reproducibility and consensus. I certainly think
it is unrealistic to characterize science in terms of
consensus, and I think that it is possible for a
mathematician to be right and deep and
indefinitely marginalized by the mathematical
community.
But I am after bigger game:
I think that Hersh's _is_ a "fast and loose"
characterization of mathematics (though I'm not
happy with the accuracy of 'fast and loose').
A more ponderous and tighter
characterization would propose an answer -- a
direct answer -- to Feferman's questions: What
is distinctive about the verification structure of
mathematics? What is distinctive about the
conceptual content of mathematics?
I think that Hersh is so bent on giving us a
post-Lapsarian human (all too human) vision
(perhaps in the spirit of Hannah Arendt assuring
us that evil is so human that we are all equally
capable of it?) of mathematics that he has
overlooked the distinctive verification structure
drives mathematics and which distinguishes it
from what to a mathematician must appear the
squalid slums of empirical -- not to mention
physical (see Kac's "The Pernicious Influence of
Mathematics on Physics", and Martin Krieger's
_Doing Physics_) -- thought.
By way of suggesting to Hersh that there is
after all something distinctive about
mathematical thought -- that it is capable of an
impressive sort of certainty even if it is not
protected by Gardol against all doubt-throwing
skeptical terrorists -- and more deeply by way of
suggesting a direction in which one might
foresee an answer to Feferman's questions, I
presented a two-stage conception of how
mathematics comes about:
Stage 1. Some witty person notices that a
number of practical problems can be solve
definitively in one's head rather than by trial and
error. They [isn't this in the end the preferred
way of evading the sickly "Should I say he or
she dilemma?"] begin to collect such problems
and their solutions and the considerations which
make it clear that the given solutions are indeed
_the_ solutions. And perhaps they go on to
invent problems which have this character but
which are of no practical importance (I object
strongly to those who keep the mathematical
baby too snugged to Mother Nature and her
reputedly discerning powers of selection -- like
Ian Hacking suspects of language, I think that
the better part of language and mathematics
come from them playing on their own
unsupervised by Big Mamma).
Stage 2. Mathematics proper begins when that
witty person notices of some of the
considerations by whay they see that this is
definitely the solution to that are cogent beyond
the concrete details of the particular problem.
That they have in hand a solution to either a
more general class of problems or to a more
abstract problem. MATHEMATICS PROPER
BEGINS WITH THE
CHARACTERIZATIONAL PROBLEM: the
problem of characterizing either the more
abstract problem or the more general problem.
(I call this Berkeleyian abstraction because for
Berkeley it was the only legitimate sort of
"abstraction").
I gave as an (admittedly cheap) example,
the problem of determining the least number of
fruitfly one must collect at random in order to be
certain that one has two of the same sex. Then I
observe that one can solve this a priori by
considering all possible 2-combinations of M, F;
and then all possible 3-combinations of M, F.
Then one can see that somehow in its
essence the problem is not about fruitfly and sex.
mathematics begins by trying to characterize the
more general/abstract problem. (And one can
use a demonstration as it were isomorphic to the
demonstration in the fruitfly-sex case.)
This suggests trying out this thesis: that
what is distinctive about mathematical concepts
is that we can somehow canvass all the
possibilities a priori; and whatever there is about
the cncpets that enable us to do this makes for a
priori demonstrations. . .is the ground of the
verification structure distinctive to mathematics.
It was worth considering how someone who
was mathematically blind might go about solving
the fruitfly-sex problem.
Notice that they woukld have to go through
a stage of collecting singleton fruitfly and
elaborately learning that they weren't getting two
frutifly of the same sex. And they could not
notice in advance or afterwards that they weren't
getting two because they were only choosing one
firefly. Nor could they have found 1
combinations, 2 combinations, 3 combinations.
. .the sensible order in which to proceed in their
quest for a practical/empirical solution to the
problem, for they would first have to
experimentally determine that combinations of 2
fruitfly are smaller than combinations of 27
fruitfly, and so on.
Mathematical thought and understanding, I
suggest, is Gardol enough to protect us from the
doubt-bombs of the skeptical terrorists. . .
rbrt tragesser
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