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%object[ess,jmc]	Mathematical objectivity and the right of initiative
\centerline{\bf MATHEMATICAL OBJECTIVITY AND THE RIGHT OF INITIATIVE}
\medskip
	Everyone has the experience of thinking some government or
business policy is wrong and ought to be changed and feeling frustrated
by his inability to communicate his objections or even to learn what
the rationale of the policy is or who is in charge.
I propose some possible partial remedies.
These ideas are somewhat farther out than most of these essays, because
the necessary science is still undeveloped.  I include it, because it
involves my own field of research.
\medskip

\centerline{\bf A happy scenario}

	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 from the Immigration Service computer in
Washington expressed in a formal but readable language.  The
rationalization is a non-monotonic 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 or wherein the known
phenomena of a certain kind are assumed to be all there are.
Non-monotonicity will be explained later.)

	Our student does not believe the pseudo-proof and after much
labor discovers that one of the assumptions is not valid or that
there is an important consideration not taken into account 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 the
policy ought to be changed.  This does not happen 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, because there is a rule 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.

	The example of the student is non-typical, because such
challenges to policy will more likely come as a result of professional
work by employees of businesses and other institutions affected by the
policy.  I chose it to emphasize the fact that if the information is
universally available and the criteria for evaluation are sufficiently
objective and are implemented by computer, then anyone can play.

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

	1.  There is a formal language in which facts about policies and
their effects can be expressed, and which allows conclusions to be
drawn about the relative merit of different policies.

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

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

	4.  The existing policies are formally described and formal
arguments justifying them are publicly available.

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

	6.  The arguments are checkable by a computer program.

	If these goals are met, the effects will be good.  Anyone who
feels offended by a policy, even esthetically, can know precisely what it
is and why it is thought to be correct.

	Further, if he can show formally that some other policy would be
better, officials will pay attention.  This depends on the computer
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.

\medskip
\centerline{\bf Objectivity}
	
	Let us compare this desired state of affairs with the present
state of affairs in our society, in which people have great trouble
agreeing about deciding what to do.  The main reason this is so is that
theories about how the world works are tools in struggles for political
power.  For example, arguing that the death penalty deters crime helps
put conservatives in political office, and arguing that it doesn't
helps put liberals in office.  As long as the social sciences are weak
and subject to wishful thinking, social science arguments will often be
subordinated to politics --- even by social scientists.  Contrast this
lack of objectivity with the fact that both liberals and conservatives
accept the engineering investigations that determined that Challenger
exploded because the O-rings between the solid rocket booster
segments failed.

	Marxists used to say that Marxism provides a basis for
objective political decisions, and some of them still say it.  Jewish,
Christian and Islamic fundamentalists say that their sacred books
provide a sure guide for these decisions. 

	The most common belief among Westerners, whether they be
intellectuals, academics, or politicians is that there is an
intrinsic difference between political decisions involving human
conflicts of interest and purposes and engineering decisions.  There
is a recent trend towards denying the possibility of making even
engineering decisions objectively, on the grounds that they always
involve conflicting interests. 

	My own position is that political decisions can be made just
as objectively as engineering decisions, but not yet, because the
necessary science has not been developed and is not immediately about
to be.  Marxism represented an attempt to wish such a science into
existence.  Once Marxism had a scientific style even though Marxists often
reached wrong conclusions because its theories were inadequate.
However, its objective character was not strong enough to withstand
the demands put on it by seekers of political power, and its practice
entirely lost its original proto-scientific character.  A new
attempt to base political practice on a science might suffer the same
fate if the science was too weak to provide truly objective answers
by widely understood methods and if power struggles strained
objectivity too much. 

	Marxism 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.

	Returning to our high school student wishing to influence
policy, his chances today are small even if he can find out what the
present policy is.  He still won't be able to find its rationale
written down in a form subject to challenge.  His chances of
influence are much greater in other areas of endeavor.  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.

	The most objective area 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.  Once a
paper is published, it will affect the mathematical ideas of the time.

	When someone makes a mathematical assertion, he is expected to
prove it.  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.  There is plenty
of controversy over what is useful, important, or beautiful in mathematics.
It certainly happens that important results are ignored for some time.

	This objectivity of mathematics has 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.  Important contributions have been made by very young people
and quickly recognized.

	It is 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 that of the early 20th century Indian
mathematician Ramanujan, where someone in an obscure position was found
to have done first class work and brought into a first class environment.

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

	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.  However, in the theoretical branches
of these subjects, the outsider has a good chance.  Recall that Einstein
was an examiner in the Swiss patent office and entirely unknown
when he sent his paper proposing the theory of relativity to
{\it Annalen der Physik}, then the world's leading physics journal.

	In engineering the matter is 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.
\medskip
\centerline{\bf Non-monotonic logic}

	The goal of making policy decisions objective goes back to
Leibniz at the end of the 17th century.  He advanced the idea that
eventually people will say ``Let us calculate'' instead of arguing
about policy.  The development of mathematical logic, and this is
the relevant science, has been slow.  To explain mathematical logic
would take us too far afield, but its essential idea is that the
allowed formulas should be precisely described and checkable by computer
and so should the allowed reasoning steps.  Leibniz didn't get very
far, and real mathematical logic dates from George Boole's {\it Laws
of Thought} written in the 1840s.  Mathematical logic reached a mature
level in the 1920s, and subsequent developments relevant to making
policy arguments mathematical have been mainly technical.
Unfortunately,  mathematical logic as it developed until the late 1970s
is inadequate for arguments in the social sciences or about the common
sense world, and it has been applied mainly to studying the foundations
of mathematics rather than to realizing Leibniz's goal.

	Research in artificial intelligence has led, since the middle
1970s, to what's called non-monotonic logic, and this provides new hope
for dealing with the common sense world and the world of social science.
Since the subject is less than ten years old, what can be done isn't
entirely clear, and maybe my hopes for it will prove illusory.

	The usual systems of mathematical logic are monotonic in the
following sense.  If a conclusion follows from a set $A$ of facts,
then it will also follow from any collection $B$ of facts that include
all the facts in $A$.  We shall see why methods of reasoning that
aren't monotonic are wanted.

	Monotonic logic and present mathematical theories work well
when the phenomena to be taken into account can be delimited at the
beginning.  This delimitation of the theory is not carried out within
the theory, but in the surrounding English prose.  For example, in
physics the law discovered by Galileo that the distance an object will
fall is governed by the formula $s = {1\over 2}gt↑2$ is subject to
the condition that nothing interrupt the fall of the object during the
time considered, but the formula says nothing about that.  Whoever
proposes to use the formula must worry about that outside of the
mathematical theory itself.

	Artificial intelligence is concerned with making computer
programs with common sense that reason about such matters.  If this
is to be done, the computer program must be able to reason about what
might interrupt the body's fall.  When we contemplate putting such
facts about falling bodies in the computer for programs to use we
encounter the difficulty that no one can list in advance all the
phenomena that may interrupt a body's fall.

	The solution seems to involve supplementing the reasoning
methods of logic with a rigorous version of the principle
philosophers call Ockham's razor after a 14th century philosopher
William of Ockham who said, ``Do not multiply entities beyond
necessity''.  In the case of falling objects, it amounts to
concluding the object will fall until stopped by one of the
interruptions we are taking into account.  If we have failed to
consider a relevant fact we will still get a conclusion, but it may be
wrong.  Human reasoning takes this into account and it seems that
intelligent computer programs will also have to reach conclusions on
the basis of collections of facts that may not include all that is
relevant.  Thus artificial intelligence will also have to risk error,
though not necessarily every kind of error that humans make.

	Social science and its application to human affairs involve
non-monotonic reasoning to a much greater degree than physical science,
because if the theories are to be useful they must include the processes
that allow limiting the facts taken into account in some situations,
while going beyond these limitations in other situations.  I believe
that formalizing non-monotonic reasoning will make possible social
science theories that are more powerful, more realistic, more explicit
about what phenomena they are taking into account, and more capable of
being supplemented by additional facts.

	In conclusion, I think there is hope for achieving
increased objectivity in human affairs.  This will provide
increased scope to people with good ideas talents rather than the
skills of a politician or courtier.
\vfill\eject\end