The Mathematical Saboteurs...

(Tim, this is funny, and oh so true...)
MATHEMATICAL VIRUSES*

David Hestenes

ABSTRACT.

The discovery of Mathematical Viruses is announced here for the first

 time. Such viruses are a serious threat to the general mental health of the mathematical

community. Several viruses inimical to the unity of mathematics are identified, and their

deleterious characteristics are described. A strong dose of geometric algebra and calculus

is the best medicine for both prevention and cure.

 

1. INTRODUCTION

Computer viruses have been prominent in the news lately. The increasingly widespread and

frequent communication among computers has facilitated the spread of computer viruses

to the point where viruses are seriously regarded as a threat to national security in the

United States.

The computer virus (CV) owes its name and perhaps its genesis to the biological virus

(BV). Like a BV, a CV cannot function by itself, but when attached to a host it replicates

repeatedly until it impairs the functioning of the host, sometimes to the point of disabling

the host altogether. Moreover, it is infectious, spreading from one host to another on

contact. The host of a CV is a computer program, to which the CV is attached as a

subroutine, replicating and spreading whenever the host program is run. Similarly, a BV

is a fragment of DNA containing instructions for self-replication which are activated when

the BV is in a living cell. Fortunately, antiviral agents can be developed to cure infected

computers as well as biological organisms.

My purpose here is to call your attention to another kind of virus | one which can

infect the mind | the mind of anyone doing mathematics, from young student to professional

mathematician. As I believe this is the first published paper to explicitly identify

such viruses, I take the liberty of naming and describing them as follows: A mathematical

virus (MV) is a preconception about the structure, function or method of mathematics

which impairs one's ability to do mathematics. Just as a CV is program which impairs the

operating system of a computer, an MV is an idea which impairs the conceptualization of

mathematics in the mind. Indeed, as one definition of \virus," Webster gives \something

that corrupts the mind or soul." Since the identification and classificalion of MVs has only

just begun, it would be premature to attempt a more precise definition. The better course

is to examine some specific viruses to form a firm empirical base for further study.

My first example is an easily recognized MV which is extremely virulent and as common

as the common cold. I call it the

coordinate virus, designating it by MV/C to denote genesis

and type, and characterizing it as follows:

MV/C: Coordinates are essential to calculations.

Physicists and engineers are especially susceptible to this virus, because most of their

textbooks are infected, and infected teachers pass it on to their students. Mathematicians

as a group are less susceptible, because many of them have been innoculated with a good

course in abstract algebra, though, as we shall see, some resistant strains have survived in

certain mathematical subspecialties.

The diagnosis and treatment of MV infections is still in its infancy, and it is especially

delicate when the infected host is not aware of his illness, as is usually the case. Diagnosis

of infection with the coordinate virus MV/C is comparatively easy, owing to the presence

of a well-defined syndrome of symptoms which I call

coordinitis: Typically, the infected

subject fails to distinguish the abstract vector concept from its matrix representation and

consequently has great difficulty conceiving and manipulating invariant functional relations

among vectors without expressing them in terms of coordinates; he is likely to regard the

real and complex numbers as more fundamental, or even \more real" than vectors. To

cite a specific example of such symptoms, one textbook by a distinguished physicist asserts

that \the vector calculus is like a folding ruler, before you can use it you have to unfold it"

(by which he meant, decompose vector formulas into components). In my own experience

of more than two decades teaching physics graduate students, I have observed that most

of them suffer from coordinitis and as many as 25% may be permanently crippled by the

disease.

Some may think that the coordinate virus is harmless or benign. After all, it is not

fatal. The afflicted can still limp along in their mathematical thinking. However, they are

condemned to a world of prosaic mathematical applications. They will never be able to

scale the Olympic heights to inhale the pristine air of abstract mathematics. Let there

be no mistake about the nature of the coordinate virus. There is nothing wrong with

using coordinates when they are appropriate. It is the insidious idea that coordinates

are somehow more fundamental or concrete than other mathematical objects that limit

conceptual capacity.

Though I would like to lay claim to the important discovery that mathematical viruses

exist, honesty compels me to admit that mathematicians must have known about them

all along. For as soon as a mathematician is introduced to the MV concept he begins

to notice viruses everywhere, and he is equally adept at naming them. Invariably, at

the mere utterance of the words \Bourbaki virus," a knowing smile breaks across the

mathematician's face, as if he is sure that we are both privy to some private indecency. I

will not attempt to describe the Bourbaki virus, for I doubt that the field of MV diagnostics

is sufficiently mature for the task. It is an important task, nonetheless, in view of evidence

that mathematics is still suffering from the ravages of a \Bourbaki epidemic."

While it may be suspected that many MVs are at large, in any attempts to detect and

neutralize them we must be alert to the dangers of misdiagnosis. One man's purported MV

may be another man's inspiration. Medical history is riddled with mistakes in diagnosis

and treatment. To avoid similar mistakes we must carefully establish suitable diagnostic

criteria; we cannot rely on mere hunches or opinions. The main burden of this paper is to

set down some specific criteria for the general mental health of mathematics and use these

criteria to identify several MVs by their deleterious effects.

(Continued: http://geocalc.clas.asu.edu/pdf-preAdobe8/MathViruses.pdf )

The Way It Is, Tim...

David Orlin Hestenes, Ph.D. (born 1933) is a physicist. For more than 30 years, he was employed in the Department of Physics and Astronomy of Arizona State University (ASU), where he retired with the rank of Research Professor and is now emeritus.

Hestenes has worked in mathematical and theoretical physics, [1] geometric algebra, neural networks, and cognitive research in science education. He is the prime mover behind the contemporary resurgence of interest in geometric algebras and in other offshoots of Clifford algebras, as ways of formalizing theoretical physics.

From 1976 to 1979, he was an Editorial Advisory Board Member (formerly called Associate Editor) of the American Journal of Physics. He is currently on the editorial board of the journal Foundations of Physics.

In 2002, the American Association of Physics Teachers awarded him its Oersted Medal for his notable contributions to the teaching of physics. He has been a Principal Investigator for NSF grants seeking to model instruction at both the high school and university levels.

http://en.wikipedia.org/wiki/David_Hestenes