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\chapter{22}{Mathematics and Objectivity in Human Affairs}

	Consider the following extreme example of a desirable  future
state  of  affairs.   A super-smart high school student gets the idea
that the U.S. admit foreign tourists without visas to reduce red tape
and promote good will.

	He  sits  down  at his trusty computer console and asks for a
description of the present  policy  and  its  rationalization.   Back
comes  a  set  of  sentences in a formal but readable language.   The
rationalization is a pseudo-proof that the  policy  is  in  the  best
interests  of  the  country  and  is in accordance with the currently
accepted principles of justice. A pseudo-proof is like a proof except
that it contains steps wherein something that has been asserted to be
plausible is assumed.) SSHSS does not believe  the  pseudo-proof  and
after   much   labor   discovers  that  one  of  the  assumptions  or
pseudo-steps is not valid and succeeds in making a pseudo-proof  that
his  proposed  policy is better than the official one.   The computer
proof checker accepts his pseudo-proof, and  the  public  information
system  tells  him  that  the  relevant  official  is the head of the
Immigration Service.   Next  morning  the  head  of  the  Immigration
Service is informed by his console that someone he has never heard of
has proved that his policy in this area ought to  be  changed.   This
does  not  happen  very  often,  and he is rather annoyed because all
previous policy changes in this area have come as  a  result  of  the
work  of  his  staff.   Nevertheless,  he has to pay attention to the
proposal, because there is a policy that  if  a  government  official
ignores  a  computer  checked  pseudo-proof that his policy should be
changed for a month, the request for change goes up a  level  in  the
hierarchy.   Therefore,  he  has his staff examine the assumptions of
the argument carefully, and after a while everyone is convinced  that
the  new  assumptions are more plausible than the old, and the policy
is changed.

	This happy scenario is based on  the  future  achievement  of
several goals some of which are rather distant:

\item{1.} There  is  a  formalism in which facts about policies and
there effects can be expressed, and which allows  conclusions  to  be
drawn about the relative merit of different policies.

\item{2.} The  criteria that determine whether one state of affairs
is socially better than another are agreed upon and formalized  to  a
sufficient extent.

\item{3.} There  is  sufficient  public  confidence in the above to
cause government use of the formal methods.

\item{4.} The existing policies are formally described  and  formal
arguments justifying them are publicly available.

\item{5.} The  technique  for  manipulating the formalism is widely
understood by people who want to affect policy.

\noindent If these goals are met,  it  will  have  the  following  good
effects:

\item{1.} Anyone  who  feels offended by a policy even esthetically
can know precisely what it is and why it is thought to be correct.

\item{2.} If he can show formally that some other policy  would  be
better,  officials  will  pay  attention.  This depends on the formal
system acting as a filter so that the policy  making  officials  will
not  be overloaded with half-baked ideas.  On the other hand, getting
official attention will not depend on his status in society.

\noindent Let us compare this desired state of affairs with the present
state  of  affairs  in  our  society.    The degree of objectivity of
policies depends on the subject matter, and the number of people  who
can  affect the policy is greater, the greater the objectivity of the
matter.

\item{1.} The most  objective  are  is  the  body  of  theorems  of
mathematics. Anyone can submit a paper to a mathematical journal. The
referees of a paper are not supposed to pay attention to  the  status
of  the  writer  and often referee papers written by people they have
never heard of.  There are many journals, and if one rejects it,  the
author  can submit it to another.  Once a paper is published, it will
affect the mathematical ideas of  the  time.   Almost  all  published
mathematical  theorems  are  correct,  and  controversy  over whether
something has been proved is rare.  On  the  other  hand,  whether  a
mathematical  result  is important is not an objective affair, and it
certainly happens that important results are ignored for  some  time.
It  is  also  important  to note that the only equipment required for
mathematical work is paper and pencil and access to  a  library.   To
make  a  living doing mathematics requires an academic job, but there
are very few scandals where someone unable to  get  such  a  job  was
found  many  years  later  to  have  done  first class work which was
ignored.   There are a number of  success  stories  like  Ramanujan's
where  someone  in  an  obscure position was found to have done first
class work and brought into a  first  class  environment.   Moreover,
every year there are cases in which someone gets a full professorship
(ordinarily obtained in one's  thirties  or  forties)  in  his  early
twenties.

\noindent This   situation  is  not  a  consequence  of  some  superior
virtuousness of mathematicians.   Rather it is a consequence  of  the
objectivity of merit in mathematics.  It also exists in athletics and
in chess (Fischer became U.S.  champion at the  age  of  14  and  was
thereby recognized as an expert).

\item{2.} The situation is almost as good in physics and chemistry.
However, the possibility of verifying  an  idea  may  depend  on  the
facilities  for  making  experiments,  and  this  may  depend  on the
reputation of the person proposing the idea.

\item{3.} In engineering the matter is still more difficult because
the  ability to try out ideas is even more expensive.   Nevertheless,
there are large areas of engineering that are quite  uncontroversial,
because  it can be objectively calculated whether something will work
or not even if it is not so clear which of several methods that  will
work is the best.

\item{4.}Once we come to social ideas, the situation is much worse.
Getting an idea tried depends on  achieving  political  power.   Even
after   the  ideas  is  tried,  whether  it  is  any  good  is  still
controversial.   The proposer can claim that  it  is  someone  else's
fault that things worked out badly.

\noindent The  prevailing  opinion  is  probably that this situation is
inevitable.   In fact, some people proceed  from  the  difficulty  of
deciding  social  questions  to  claim  that engineering, physics and
mathematics are not objective either.  I would like  to  express  the
reverse  view:    economics,  sociology,  history,  and  politics are
possible, but difficult sciences.   They have  made  little  progress
except  for  economics,  and the future sciences in these fields will
ascribe little merit to present or  past  ideas  in  these  subjects.
However, the future will be better

	Another  necessary component of the objectification of social
decisions is the development of formal reasoning and its  application
to  non-mathematical  contexts.   This  problem has been clarified by
work in artificial intelligence and great progress can be expected in
the  next  decades.   Within  five  years  formal proof may become an
accepted tool form establishing the correctness of computer programs,
then  it will be extended to proving that computer systems meet their
specifications including  systems  that  interact  with  the  outside
world. Then it will become possible to determine when a social theory
actually predicts the  result  of  a  policy  and  to  determine  the
predicted  result.  This  will allow the social sciences to make much
more rapid progress than heretofore.
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\section{22.1}{Remarks}

\item{1.} The most extensive effort to make  a  general  scientific
theory  including philosophy and all the social sciences was Marxism.
It failed, but we have to try again and again until we  succeed.   If
10,000  years  goes  without  success,  it  may  be  time to give up.
Nevertheless, we have to remain skeptical about  the  claims  of  any
particular  attempt  and  avoid wishful thinking. The arguments for a
social  principle  of  complementarity   analogous   the   Heisenberg
principle  in  physics represent mere wishful thinking on the part of
obscurantists discouraged by the difficulties of social science  into
trying to prove such a science impossible.

\item{2.} Present  attempts  to  simulate  complex social events on
computers are almost all too simplified to be useful.  The formalisms
are  inadequate  to  express  the  kinds of knowledge people actually
have.

\item{3.}   Some  technical  ideas  relevant  to   this   goal   of
objectification  are  discussed  in  the  paper  ``Some  Philosophical
Problems from the Standpoint of Artificial Intelligence''  in  Machine
Intelligence 4, Edinburgh University Press 1969.

\item{4.}The foregoing is not presented as a complete argument.  It
is more aimed at encouraging others inclined to  think  in  this  way
than to convince the unconvinced.

\noindent To what extent is it possible to extend the objectivity
of mathematics to science, engineering and human affairs?

First: What do we mean by objectivity?  Consider that when someone
makes a mathematical assertion, he is expected to prove it.  The
mathematical literature publishes thousands of pages annually of
new mathematical assertions and their proofs.  Mathematical proof
is an extremely reliable process.  Less than one in a thousand
of published mathematical assertions are later found to be mistaken
or even occasion any controversy.  In popular fields of mathematics
where people check each others work controversy over what has been
proved is almost non-existent.  (There is plenty of controversy 
over what is useful, important, or beautiful in mathematics).

	This objectivity of mathematics has some important useful
consequences.  Namely, anyone who has an idea for improving
mankind's state of mathematical knowledge can write a paper
and submit it to a journal.  The process by which publication
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