FOM: social construction?

[Lincoln Wallen]

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   From: martind at cs.berkeley.edu (Martin Davis)
   Cc: fom at math.psu.edu, csilver at sophia.smith.edu
   Date: Sun, 22 Mar 1998 16:52:36 -0800

   At 10:16 PM 3/22/98 GMT, Lincoln Wallen wrote:

   >   I take this relativistic point of view as moving to deny the possibility of
   >   establishing objective truth.
   >
   >   Martin
   >
   >I see nothing relativistic in the view whatsoever!  We must be
   >thinking about different ideas.  Are you sure you are not thinking
   >about social constructivism?  
   >

   Sorry! I seem to have misunderstood. I was reacting to:

   > that there is no position
   >"outside" the activity one is studying from which to gain a more
   >direct understanding of the structure and truth of the activity.
   >Hence it is the anthropologists who must act ethnically and seek to
   >understand *as the participants themselves understand*, not in some
   >other way.  

   I took this to imply that there is no TRUTH outside of this internal
   understanding. This is how I have understood Levi-Strauss. It has led to
   people equating the belief systems of tribal peoples with the findings of
   "Western" science.

I see.  In which case the ideas of Levi-Strauss and those I have been
paraphrasing probably do have some common ancestry but have diverged
significantly.

   I don't have the least quarrel with you said about scientific methodology;
   I'm not repeating to keep this message reasonably brief. 

   >One way of articulating what is eternal about the truths mathematical
   >practice gives us access to is to seek to understand this word
   >"eternal" through the structure of our practice.

   I just can't understand what this is intended to mean. The analogy with the
   "practice" of designing experiments doesn't help.

OK.  Words like "eternal" are used in different spheres of activity to
mean different things.  Religion, physical science etc.  What is its
meaning in mathematics?  Atemporal?  Invariant in time?  Physical
theories are clearly not invariant in time, though we have the sense
that certain relationships between phenomena that they are trying to
describe, are so invariant.  I think that the reproducibility of
empirical observation is a key part of the definition of what sort of
invariance we seek to build into our physical theories.  To achieve
such reproducibility we constrain experimental behaviour (and accounts
of such) to certain forms which, by trail and error, and reflection,
we have developed to its current state.  The fact that it works (in
fact the *way* that it works) is a testament to our ability to perceive
the regularities of nature and design our practices with such
regularities in mind.

   >  Is this saying
   >anything more than logicians have said in trying to isolate the
   >elements of logical consequence?  I don't think so.  But there is more
   >to understand.  That is all.

   What? Logical consequence, as it occurs in mathematical proof, is perfectly
   well understood. It is formally defined in the context of the well-defined
   rules of the predicate calculus. Of course in practice mathematicians can
   and should omit most of the tiny steps. When a doubter question that
   transition between two steps, the mathematician provides "more details".
   Between professionals, it would never get that far, but, in principle, this
   process could be continued all of the way to inferences in predicate calculus. 

No, sorry.  Not more about logical consequence.  More about how the
situation you attest to above is *actually* brought about.  As you
say, professionals do not formalise.  But the property of
formalisability is conveyed by argument.  (This is different from
conveying truth of course, though obviously related.)  How is this
actually achieved?  We would agree, presumably, that Leibnitz et al,
the early analysts, were doing mathematics, but the emergence of the
real numbers, set theory etc, from these investigations is part of
this activity of formalising.  Or might you feel it better to relegate
that work to pre-mathematics and raise post set-theoretic mathematics
to a privileged position...?  Likewise how was it possible to do
mathematics prior to the formulation of the pred. calc.?  The answer
is that the pred. calc. is a theory of the regularities in our
(mathematical) argumentative behaviour.  Once we are in possession of
such a theory we can, and you have just done so, use it as part
justification for our practices.  NSA is surely an extreme example
where reflection on these regularities gives us access to a way of
explaining in what way Leibnitz et al can be considered to be arguing
soundly (I stretch a point of course).

Mathematicians in practice talk of many things and appeal to many
forms of argument which have more to do with the character of the
subject matter at hand than any idealised notion of l.c.  This is
especially true of applied mathematics.  L.c. does give us a reference
theory against which we can describe the mechanisms we actually use.

There are also distinctions between practice and accounts of practice,
and it is not clear to me at his time whether article/proofs (as
published) are better thought of as accounts of practice, or part of
the central product of that practice.  It could be both.  That is,
that mathematical activity may be, as a matter of foundation,
activity/thinking/writing etc which sustains a certain "internal"
relationship between the practice and certain types of account of that
practice.  (I am not sure what I mean by "internal" here, please bear
with me; I say it in contrast to an "external" relationship we see in
physical science between theory and the world via experiment.)

But this is moving onto a different point...so I'll stop.

Lincoln Wallen