Commentary by Philip J. Davis (originally appeared in SIAM News, Nov. 1995)
Mathematics Awareness Week 1996
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 The list of places, past and present, where common sense seems to have been superseded is extensive.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.