FOM: Improved: Hersh Fast & Loose?&Feferman's questions

[Robert S Tragesser]

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