00100	            MATHEMATICS AND OBJECTIVITY IN HUMAN AFFAIRS
00200	
00300	
00400		Consider the following extreme example of a desirable  future
00500	state  of  affairs.   A super-smart high school student gets the idea
00600	that the U.S. admit foreign tourists without visas to reduce red tape
00700	and promote good will.
00800	
00900		He  sits  down  at his trusty computer console and asks for a
01000	description of the present  policy  and  its  rationalization.   Back
01100	comes  a  set  of  sentences in a formal but readable language.   The
01200	rationalization is a pseudo-proof that the  policy  is  in  the  best
01300	interests  of  the  country  and  is in accordance with the currently
01400	accepted principles of justice. A pseudo-proof is like a proof except
01500	that it contains steps wherein something that has been asserted to be
01600	plausible is assumed.) SSHSS does not believe  the  pseudo-proof  and
01700	after   much   labor   discovers  that  one  of  the  assumptions  or
01800	pseudo-steps is not valid and succeeds in making a pseudo-proof  that
01900	his  proposed  policy is better than the official one.   The computer
02000	proof checker accepts his pseudo-proof, and  the  public  information
02100	system  tells  him  that  the  relevant  official  is the head of the
02200	Immigration Service.   Next  morning  the  head  of  the  Immigration
02300	Service is informed by his console that someone he has never heard of
02400	has proved that his policy in this area ought to  be  changed.   This
02500	does  not  happen  very  often,  and he is rather annoyed because all
02600	previous policy changes in this area have come as  a  result  of  the
02700	work  of  his  staff.   Nevertheless,  he has to pay attention to the
02800	proposal, because there is a policy that  if  a  government  official
02900	ignores  a  computer  checked  pseudo-proof that his policy should be
03000	changed for a month, the request for change goes up a  level  in  the
03100	hierarchy.   Therefore,  he  has his staff examine the assumptions of
03200	the argument carefully, and after a while everyone is convinced  that
03300	the  new  assumptions are more plausible than the old, and the policy
03400	is changed.
03500	
03600		This happy scenario is based on  the  future  achievement  of
03700	several goals some of which are rather distant:
03800	
03900		1.  There  is  a  formalism in which facts about policies and
04000	there effects can be expressed, and which allows  conclusions  to  be
04100	drawn about the relative merit of different policies.
04200	
04300		2.  The  criteria that determine whether one state of affairs
04400	is socially better than another are agreed upon and formalized  to  a
04500	sufficient extent.
04600	
04700		3.  There  is  sufficient  public  confidence in the above to
04800	cause government use of the formal methods.
04900	
05000		4.  The existing policies are formally described  and  formal
05100	arguments justifying them are publicly available.
05200	
05300		5.  The  technique  for  manipulating the formalism is widely
05400	understood by people who want to affect policy.
05500	
05600		If these goals are met,  it  will  have  the  following  good
05700	effects:
05800	
05900		1.  Anyone  who  feels offended by a policy even esthetically
06000	can know precisely what it is and why it is thought to be correct.
06100	
06200		2.  If he can show formally that some other policy  would  be
06300	better,  officials  will  pay  attention.  This depends on the formal
06400	system acting as a filter so that the policy  making  officials  will
06500	not  be overloaded with half-baked ideas.  On the other hand, getting
06600	official attention will not depend on his status in society.
06700	
06800		Let us compare this desired state of affairs with the present
06900	state  of  affairs  in  our  society.    The degree of objectivity of
07000	policies depends on the subject matter, and the number of people  who
07100	can  affect the policy is greater, the greater the objectivity of the
07200	matter.
07300	
07400		1.  The most  objective  are  is  the  body  of  theorems  of
07500	mathematics. Anyone can submit a paper to a mathematical journal. The
07600	referees of a paper are not supposed to pay attention to  the  status
07700	of  the  writer  and often referee papers written by people they have
07800	never heard of.  There are many journals, and if one rejects it,  the
07900	author  can submit it to another.  Once a paper is published, it will
08000	affect the mathematical ideas of  the  time.   Almost  all  published
08100	mathematical  theorems  are  correct,  and  controversy  over whether
08200	something has been proved is rare.  On  the  other  hand,  whether  a
08300	mathematical  result  is important is not an objective affair, and it
08400	certainly happens that important results are ignored for  some  time.
08500	It  is  also  important  to note that the only equipment required for
08600	mathematical work is paper and pencil and access to  a  library.   To
08700	make  a  living doing mathematics requires an academic job, but there
08800	are very few scandals where someone unable to  get  such  a  job  was
08900	found  many  years  later  to  have  done  first class work which was
09000	ignored.   There are a number of  success  stories  like  Ramanujan's
09100	where  someone  in  an  obscure position was found to have done first
09200	class work and brought into a  first  class  environment.   Moreover,
09300	every year there are cases in which someone gets a full professorship
09400	(ordinarily obtained in one's  thirties  or  forties)  in  his  early
09500	twenties.
09600	
09700		This   situation  is  not  a  consequence  of  some  superior
09800	virtuousness of mathematicians.   Rather it is a consequence  of  the
09900	objectivity of merit in mathematics.  It also exists in athletics and
10000	in chess (Fischer became U.S.  champion at the  age  of  14  and  was
10100	thereby recognized as an expert).
10200	
10300		2.  The situation is almost as good in physics and chemistry.
10400	However, the possibility of verifying  an  idea  may  depend  on  the
10500	facilities  for  making  experiments,  and  this  may  depend  on the
10600	reputation of the person proposing the idea.
10700	
10800		3.  In engineering the matter is still more difficult because
10900	the  ability to try out ideas is even more expensive.   Nevertheless,
11000	there are large areas of engineering that are quite  uncontroversial,
11100	because  it can be objectively calculated whether something will work
11200	or not even if it is not so clear which of several methods that  will
11300	work is the best.
11400	
11500		4. Once we come to social ideas, the situation is much worse.
11600	Getting an idea tried depends on  achieving  political  power.   Even
11700	after   the  ideas  is  tried,  whether  it  is  any  good  is  still
11800	controversial.   The proposer can claim that  it  is  someone  else's
11900	fault that things worked out badly.
12000	
12100		The  prevailing  opinion  is  probably that this situation is
12200	inevitable.   In fact, some people proceed  from  the  difficulty  of
12300	deciding  social  questions  to  claim  that engineering, physics and
12400	mathematics are not objective either.  I would like  to  express  the
12500	reverse  view:    economics,  sociology,  history,  and  politics are
12600	possible, but difficult sciences.   They have  made  little  progress
12700	except  for  economics,  and the future sciences in these fields will
12800	ascribe little merit to present or  past  ideas  in  these  subjects.
12900	However, the future will be better
13000	
13100		Another  necessary component of the objectification of social
13200	decisions is the development of formal reasoning and its  application
13300	to  non-mathematical  contexts.   This  problem has been clarified by
13400	work in artificial intelligence and great progress can be expected in
13500	the  next  decades.   Within  five  years  formal proof may become an
13600	accepted tool form establishing the correctness of computer programs,
13700	then  it will be extended to proving that computer systems meet their
13800	specifications including  systems  that  interact  with  the  outside
13900	world. Then it will become possible to determine when a social theory
14000	actually predicts the  result  of  a  policy  and  to  determine  the
14100	predicted  result.  This  will allow the social sciences to make much
14200	more rapid progress than heretofore.
14300	
14400	Remarks:
14500	
14600		1.  The most extensive effort to make  a  general  scientific
14700	theory  including philosophy and all the social sciences was Marxism.
14800	It failed, but we have to try again and again until we  succeed.   If
14900	10,000  years  goes  without  success,  it  may  be  time to give up.
15000	Nevertheless, we have to remain skeptical about  the  claims  of  any
15100	particular  attempt  and  avoid wishful thinking. The arguments for a
15200	social  principle  of  complementarity   analogous   the   Heisenberg
15300	principle  in  physics represent mere wishful thinking on the part of
15400	obscurantists discouraged by the difficulties of social science  into
15500	trying to prove such a science impossible.
15600	
15700		2.  Present  attempts  to  simulate  complex social events on
15800	computers are almost all too simplified to be useful.  The formalisms
15900	are  inadequate  to  express  the  kinds of knowledge people actually
16000	have.
16100	
16200		3.    Some  technical  ideas  relevant  to   this   goal   of
16300	objectification  are  discussed  in  the  paper  "Some  Philosophical
16400	Problems from the Standpoint of Artificial Intelligence"  in  Machine
16500	Intelligence 4, Edinburgh University Press 1969.
16600	
16700		4. The foregoing is not presented as a complete argument.  It
16800	is more aimed at encouraging others inclined to  think  in  this  way
16900	than to convince the unconvinced.
17000		To what extent is it possible to extend the objectivity
17100	of mathematics to science, engineering and human affairs?
17200	
17300	First: What do we mean by objectivity?  Consider that when someone
17400	makes a mathematical assertion, he is expected to prove it.  The
17500	mathematical literature publishes thousands of pages annually of
17600	new mathematical assertions and their proofs.  Mathematical proof
17700	is an extremely reliable process.  Less than one in a thousand
17800	of published mathematical assertions are later found to be mistaken
17900	or even occasion any controversy.  In popular fields of mathematics
18000	where people check each others work controversy over what has been
18100	proved is almost non-existent.  (There is plenty of controversy 
18200	over what is useful, important, or beautiful in mathematics).
18300	
18400		This objectivity of mathematics has some important useful
18500	consequences.  Namely, anyone who has an idea for improving
18600	mankind's state of mathematical knowledge can write a paper
18700	and submit it to a journal.  The process by which publication