Mathematics and Common Sense: What Is Their Relationship?
Mathematics is one of the greatest creations of the human
imagination. Common sense, as one dictionary puts it, is "native
good judgment" or "the set of general unexamined assumptions"; it
is, then, both time- and culture-dependent. In the prevailing
climate of opinion, it is only too easy to conclude that pure
imagination or pure reason holds sway in mathematics and that it
and common sense have little to do with one another.
The case that mathematics ignores common sense is easily made.
Consider, first, pure mathematics. Through the millennia,
mathematics has increased its stockpile of objects, statements,
paradoxes, crises whose existence derives from conflicts with what
had once been considered common sense. Think of the irrational and
imaginary numbers, of noncommutative entities, infinite sets,
functions with positive area that are zero almost everywhere. The
philosophers Berkeley and Hume (the latter in "Treatise on Human
Nature," Part 2, Book 1) both asserted vigorously that a straight
line cannot have an infinity of points on it, contradicting what we
routinely teach in geometry and analysis. We can hardly count to
5000 in an error-free fashion, and yet we are asked to believe in
the indubitability of hand-crafted mathematical proofs that are
5000 or more pages long. The list of places, past and present,
where common sense seems to have been superseded is extensive.
An examination of applied mathematics for its relation to
common sense might begin with those applications that relate to
social arrangements. Included here are things like money,
insurance, testing, various patterns for managing populations,
information collection and interpretation, prediction.
Take the current structure of airfares, for example. Does it
make sense that a two-way fare may cost less than a one-way fare?
Are the reasons for this well known to airline accountants, and
unknown to the public at large? Does a deeper form of common sense
underlie every deliberate violation of common sense?
As another simple but conflicted mathematization, think about
the red and green figures in traffic lights that are intended to
control pedestrian traffic. The operation of such signals is not
dependent on the moment-by-moment state of traffic. If you walk
when there is a green figure and don't pay attention to the ambient
traffic, you can easily get killed. If you pay attention to the
traffic, you can often walk in perfect safety when there is a red
figure.
Finally, look at those applications of mathematics that
are
essentially theoretical physics. The physicist Wolfgang Pauli,
famous for his exclusion principle, once remarked of a certain
proposed theory that it couldn't possibly be right because it
wasn't crazy enough. Are we dealing here with conformity to
reality or with the creation of new realities to which we learn to
conform and which then become the bases of a new generation of
common sense? A lively argument on this question has been going on
for some while.
The case for a positive relation between mathematics and
common sense may be a little harder to make, simply because of the
tendency of humans to ignore what stares them in the face. Part of
the downplaying of common sense arises from the belief
(counterproductive, in my opinion) that mathematics exists in a
Platonic world, divorced from the objects that inspire it and from
the people who create and judge it.
Mathematics exists embedded in a prior (not in the sense of
time) world of material objects and human artifacts, in the human
language and social arrangements in which it is pursued,
interpreted, and validated. Remove mathematics from this larger
world, and no piece of it can survive.
A generation ago, F.R. Leavis made this point for science
generally in his famous "two cultures" dispute with C.P. Snow.
Today, Bernhelm Booss-Bavnbek, an applied mathematician at Roskilde
University in Denmark, puts it dramatically: "Any of our pupils has
already solved his or her life's biggest math exercise before
entering school, namely, the handling of language and grammar."
Some people dream of an even stronger merger between
mathematics and common sense: a formalization of the latter in
terms of the former. Ernest S. Davis, a specialist in artificial
intelligence, has written:
Almost every type of intelligent task -- natural language
processing, planning, learning, high-level vision, expert-level
reasoning -- requires some degree of common sense reasoning to
carry out. The construction of a program with common sense is
arguably the central problem of Artificial Intelligence.
In my view, therefore, we should not deny the existence of
common sense in mathematics; it would be more useful to study ways
in which the tensions induced by the conflict of the two can be
productive. As an example of such fruitfulness, take the concept
of mathematical equality, symbolized by "=". Surely there is
hardly a more basic or fertile notion in mathematics. Yet it
stands in contradiction to common sense. The equality sign implies
an exactitude, a precision, an identity that are illusory in the
world as experienced.
Speaking much more technically, given the three statements a
= b, a > b, a < b, there is no effective method within certain
theories of computability for deciding which is true (Aberth,
Computable Analysis, page 50).
I can translate this dilemma into everyday practice with a
very simple example: In my own mathematical researches into matrix
theory, I often use MATLAB, a convenient commercial matrix
software package. (What I have to say is just as applicable to
other software packages.) MATLAB has the capability of making a
logical judgment as to whether an arithmetic statement is true (1)
or false (0). Further arithmetic computations can be based on such
0 or 1 outputs.
Now MATLAB (as normally employed) makes the following
judgments: The equation 1 + 10-15 = 1 is false. The equation 1 +
10-16 = 1 is true. Now, one might conclude from the last equation
that 10-16 = 0 would be true, but when MATLAB is queried about
this, it responds: False.
The point is this: The coding that yields these truth values
is part of the way in which the granularity of the floating-point
number system has been programmed. It is in contradiction to
normal arithmetic. Yet the code (or the chip) that does this
represents a mathematical system that has its own integrity, its
own kind of consistency, and its own range of utility.
MATLAB is not just a piece of software. It is a mathematical
structure and is as conceptual or as Platonic as anything else in
the mathematical world. Although it contradicts standard
arithmetic, it exists and is useful. It can be regarded as an
approximation to an absolute arithmetic ideal, but it does not have
to be. It is its own thing. Attempts to equate existence with
internal consistency have not captured the full essence of
mathematics. The claim of absolutism is the seductive siren song
that mathematics sings in a fuzzy world where common sense is time-
and culture-dependent and where the term is used simultaneously in
a variety of ways.
What led me to this topic was an invitation to give a plenary
lecture at a conference in Berlin entitled Mathematics (Education)
and Common Sense: The Challenge of Social Change and Technological
Development.
The conference was organized by the International Commission
for the Study and Improvement of Mathematics Education (CIEAEM).
Held July 23-29, 1995, at the Technical University of Berlin and
skillfully planned by Christine Keitel of the Free University,
Berlin, the conference attracted more than 150 participants from
more than 25 countries.
The center of gravity of the talks and workshops was much
closer to the lively experiments of day-to-day teaching in real
environments than to abstract discussions of philosophical issues
of the type I have attempted here. It soon became clear that the
common sense of mathematical education means one thing for students
in suburban New England and something entirely different for the
child vendors in Papua, New Guinea. A bit of what the three other
plenary speakers had to say gives some idea of the nature of the
conference.
Rijkje Dekker of the University of Amsterdam described some of
the work of the Freudenthal Institute. In particular, displaying
a paradoxically shadowed two-dimensional version of a real
three-dimensional object sitting in the sunshine, she discussed the
question of how much common sense and how much mathematics are
needed to analyze what the eye sees.
Since mathematical education and politics have been
inextricable for some time now, Juliana Szendrei of the Teacher
Training Institute of Budapest described the current educational
outlook in Hungary (the scene of the "Mathematical Miracle" that
spanned some 50 years at the turn of the last century). While
mathematical authors are no longer required to write that the
paradoxes of set theory reflect the contradictions of capitalism,
the new economics and its consequences have resulted in less money
for education and a de-emphasis of knowledge for its own sake in
favor of vocational training: "In real life, money supersedes all
values." (Sounds familiar, doesn't it?)
Alan Bishop of Monash University, Victoria, Australia, tackled
the contradictory pressures that come from new technology,
computers in schools, and the mathematics of different traditional
societies in, for example, aboriginal Australia, Africa, or
American Indian communities. Bishop's conclusion is that common
features of ethnomathematics and technology can, indeed, be found,
and that by focusing on those features, one can make appropriate
educational responses and discard less sensible ones.
To wrap up: Mathematics and its applications are amphibians
that live between common sense and the irrelevance of common sense;
they live between what is intuitive and what is counter-intuitive,
between the obvious and the esoteric, between what seems to be
rational and what seems to be "trans-rational" or magical
hocus-pocus.
The tension that exists between these pairs of opposites,
between the elements of mathematics that are stable and those that
are in flux, is the source of its creative strength. To foster a
critical attitude toward the existence of common sense in
mathematics and toward the ambiguous role it occupies is of prime
importance. The downplaying of common sense that has occurred in
recent decades has created an imbalance that is not productive and
can be dangerous. It ought to be reversed.
Philip J. Davis, professor emeritus of applied mathematics at
Brown University, is an independent writer, scholar, and lecturer.
He lives in Providence, Rhode Island, and can be reached at
AM188000@brownvm.brown.edu.
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