THE FALLACY IN THE FALLACY
A philosophical essay on the plausibility of dividing by zero
By Jeryl W. Lafon
DATAMATION, Nov. 15, 1971
It happens to virtually all computer programmers.
If it has not yet happened to you, the odds dictate that someday it will.
Your program has been running smoothly for weeks, months, maybe even years.
But suddenly one day, out pops a tell-tale message on the operator's console:
ZERO DIVISOR!
Have you ever noticed, when the type-out results from a limitation in the
hardware or software, the message comes out in the form of some obscure code,
like "ERR NO. X-15-P2A"? (When you look up the meaning of Error No.
X-15-P2A, it probably says something like "UNDERFLOW EXCEEDS FIXED-POINT BOUND
IN MIDDLE CURTATE OF REGISTER Q" which is a deliberately ambiguous way of
admitting that Register Q is not adequate to handle numbers beyond the range
of +-999. But when the message results from some microscopic oversight on the
part of the programmer, the message comes out in plain English--DIVISION BY
ZERO--so that all of the computer operators will know you goofed. ("Old
Berkenheimer has been dividing by zero again--har,har,har!")
If you are lucky, the software has been designed to ignore the zero
divisor and give you a correct answer in spite of the error message. If you
are not lucky, you will have to modify your program to test for a zero
divisor, and branch around the offending instruction in case the zero divisor
occurs.
My question is, how much longer do we intend to put up with this
indignity? Since it might be asking a bit much of the compiler writers,
software experts, and hardware designers to handle such problems
automatically, I am formally prepared to advance a plucky new hypothesis that
could do away with this zero-divisor nonsense forever.
The mathematicians assure us that division by zero is not permitted. The
mathematicians say (and not without cause) that when we divide by zero,
mysterious things happen. Examples they are fond of using to illustrate the
problem frequently start out by asking the student to assume that x=y. From
this, a series of apparently innocent equations are developed (e.g.
x(squared)=xy, or x(cubed)=y(cubed)), which end up by offering seemingly
incontrovertible proof that 2=1 (or similar anomaly). After the student has
been given time to ponder the enigma, the mathematicians will explain that the
fallacy lies in some step where both members of the equation were
surreptitiously divided by x-y, which must be zero since x and y are equal.
But it seems to me that there is a fallacy in the fallacy. (One of my
computer associates, Mr. L. Eisenzimmer, refers to this as a nested
fallacy.) Granted that the mathematicians have been operating under this
premise for quite a while now, I nevertheless resent being told that I am not
PERMITTED to divide by zero, on such tenuous grounds as displayed in the usual
examples. Therefore, I will attempt to show that dividing by zero is logical,
desireable, and practical.
In order to clarify the ensuing discussion, I am introducing a new word
to describe such concepts as zero and nothing. The word is pragmadox. A
pragmadox (pragmatic + paradox) is defined as a concept which is inherently
meaningless or self-contradictory, but which nevertheless, has practical
application. Imaginary numbers (e.g. the square root of -1) and geometrical
points are classical examples of a pragmadox. Geometrical points, for
instance, have been defined as locations in space, without size or shape. Now
if a point has no dimensions, it cannot be properly said to exist, except as a
figment of the imagination. But without such concepts we could never have
reached the moon.
The word "nothing" is another example of a pragmadox. Nothing can only
be defined with respect to something--i.e. as the opposite of something, the
absence of something, or as that which does not exist. Yet even nothing must
seemingly be something, in order to be an "opposite" or a "that which." As the
poet Wallace Stevens once put it, "There is not nothing; no, no, never
nothing." Well, maybe there isn't and maybe there is. But in either case, we
definitely need the concept.
Okay. If we select a quantity, x, and refuse to divide it by anything,
the quantity (I think any competent mathematician will agree) remains the
same. In slightly more mathematical terms,
x(not divided by anything)=x -or-
x(divided by nothing)=x -or-
x/ =x
Now if zero may be considered to be the mathematical equivalent of
nothing, I see no great harm in expressing the foregoing bit of philosophy in
an equation of the form, x/0=x. But here, some mathematician will pounce
gleefully on the fact that x/1 (is also)=x, and will accuse me of saying, in
effect, that zero equals 1. (These mathematicians really know how to hurt a
guy.) Well, in a peculiar sort of way, that may be exactly what I am
saying--as I will attempt to demonstrate in a moment.
First, however, let's examine the problem from a slightly different
angle. When we divide a number, x, into another number, y, we are trying to
determine the minimum number of x's contained in y, without exceeding the
value of y. If the numerator, y, happens to be zero, and the denominator
happens to be 2, we are asking how many pairs of units are needed to equal
zero. And the answer, of course, is no pairs. We need a total absence of
units, and the answer is therefore zero. Conversely, when we divide a
positive number by zero, we are asking how many non-units are needed in order
to equal or closely approximate a given number of units. But it seems
intuitively obvious that if Dick had no apples, Jane had no apples, and Spot
had no apples,--in fact, if everybody in the universe had no apples and pooled
them all together--there still wouldn't be enough non-apples to make even one
tiny fraction of a real apple. The answer, therefore, is "no" amount of
non-apples (or non-units (or zeros)). Thus the true quotient of x/0 is zero.
This answer may come as a shock to some readers. Being conditioned to
the idea that the smaller the divisor the larger the quotient, such readers
might suppose the quotient of x/0 to be of infinite magnitude --which is, in
fact, what the calculus textbooks attempt to teach. But as will be seen
later, this is true only under limited conditions--one of which is that the
dividend, x, also be of infinite magnitude.
Readers who are familiar with the surface of the Moebius strip (or the
shape of Klein's bottle) shouldn't have too much difficulty in grasping the
reasons underlying my theory that the quotient of x/0=0. (In fact, anyone who
finishes reading the article may find that his mind has been bent into
approximately the same shape.) The average computer programmer may find the
idea easier to grasp if he associates it mentally with a wrap-around computer
memory. And, for my mathematician friend, I will express the concept
symbolically as follows:
WRAP-AROUND INFINITY
(also courtesy of Mr. L. Eisenzimmer) Absolute value(x/a) approaches
infinity as a approaches 0; and x/a=0 when a=0
But there is still a third way of approaching the problem of x/0, and I'm
sure my mathematician won't overlook it. He may wish to interject, at this
point, that when we divide x into y we are actually attempting to determine
how many times we can subtract x from y without going negative. "Surely you
can see," he will argue, "that we can subtract zero from any positive number
an infinite number of times without going negative?"
Yes, I can see that. The trouble is, I can also see the foolishness of
it. To my way of thinking, the question is not how many times we can
subtract, but rather how many times we can subtract successfully. And I
submit that we do not add or subtract successfully unless we succeed in
increasing or decreasing the original quantity. For example, when we add 5 to
zero we have done something meaningful, because we have altered the original
amount. But if we attempt to add zero to 5, we accomplish nothing. (We can
alleviate the embarrassment of this dilemma by saying that we are adding zero
"and" 5, rather than zero "to" 5.) The mathematician, however, adds zero to 5
with a flourish, smacks his lips in satisfaction, and deludes himself that he
has obtained a constructive result. In actuality, he has merely gone through
an exercise in futility, and obtained an inevitable result. If he has done
anything constructive at all, it is to demonstrate the utter impossibility of
adding zero to anything. Therefore, although we subtract zero from x an
infinite number of times, we subtract successfully exactly zero times (the
true quotient).
If my mathematician is still around, he will probably want to ask me how
I propose to reconcile my original proposition (x/0=x) with the statement I
just made (x/0=0). In order to bridge this seemingly impossible chasm, I must
touch briefly on a subject which has gone too-long neglected--namely, the
relativity of numbers. Obviously, numbers are relative, and the usual
practice is to define them as either positive or negative with respect to
zero. But we showed earlier that the word "nothing" can only be defined with
respect to "something", and the same is true here--i.e., zero itself can only
be defined with respect to some other number (or numbers). If our hypothesis
is correct that the true quotient of x/0 is zero, then the immediate problem
is to isolate the relative value of zero on the imaginary mathematical scale
(Cartesian horizontal axis). Since we know that zero lies exactly halfway
between +n and -n, we can express the relative quotient of n/0 by the
following equation:
(n/0)relative=[n-(-n)]/2=(n+n)/2=2n/2=(tilde)n
In other words, by halving the difference between +n and -n, we have
found that the relative quotient of n/0 is a neutral n--i.e., it lies n units
in a negative direction from +n and n units in a positive direction from -n.
(I have stressed the neutrality of n in this case by using the Spanish letter
(tilde)n, which is doubly appropriate because the neutral n was discovered in
New Mexico.) The practical application of this pragmadox is manifested in the
fact that it satisfies the mathematician's craving for a unique result--i.e.,
it is not the same n that we would have obtained if we had divided n by 1
instead of by zero.
But my mathematician loves consistent results as well as unique results,
and he won't overlook the apparent fact that my answer still doesn't check.
He will be quick to point out that if my neutral n had a value, say, of 5,
then 5 zeroes wouldn't make 5, and zero fives wouldn't make 5 either. Well, I
absolutely agree that zero fives wouldn't make five, but I'm not so sure about
the first proposition. If we start out with one zero, then multiply that zero
by 5, it seems fairly reasonable to me that we should end up with five zeroes.
In fact, I am gripped by an urge to place a string of five zeroes right here
on the printed page, then ask my mathematician to count them for himself and
see if they don't add up to 5. His immediate response, naturally, would be:
"Ah, but that is mere word-trickery. You are treating zeroes as if they were
units, which isn't cricket at all." (Back to the old 0=1 pragmadox.)
Very well. For the time being, I'm prepared to let my mathematician have
his way. We will treat zeroes strictly as non-units, and we will assume that
there is no distinction in magnitude between 1 non-unit and 5 non-units. (To
do otherwise would be to equate non-units with negative numbers.) Under these
restrictions, I confess that my answer doesn't check. I can only say, by way
of defense, that when my mathematician has a value, x, and doesn't divide it
by anything (i.e., divides it by nothing), he is left with a value of x. And
if then he divides that x by 1, he is still left with a value of x. But do I
run around accusing him of saying that 1 is equal to nothing?!? It would seem
that my neutral x, as a quotient for x/0, is valid for all practical purposes,
since it is basically the same answer that my mathematician gets when he
doesn't divide x by zero.
In any case, if x is the relative quotient of x/0, the true quotient may
be expressed by taking the algebraic sum of +x and -x, then dividing by 2 in
order to obtain the average:
(x/0)true=[x+(-x)]/2=(x-x)/2=0/2=0
But here again my mathematician will attempt to pounce, tearing his hair
and screaming that, in the first place, x/0 (can't be)=0, because 0/x (is
also) =0, and in the second place, how can x/0 be equal to x and zero at the
same time (why don't I make up my mind, etc.), and in the third place, even if
five zeroes do add up to 5, zero zeroes certainly wouldn't, because zero time
zero is ZERO! (You know how these mathematicians always get in a lather about
everything.)
Okay, In spite of the fact that this particular mathematician has been
harrassing me ever since I began the article, I've grown somewhat attached to
him. I think he is a good fellow at heart, and it gives me no great pleasure
to stick another pin in his balloon. But I must gently point out that zero
times zero, at least from a semantic point of view, does NOT equal zero. When
we say that we have zero zeroes, we are actually saying that we have no
non-units. And an absence of non-units implies the presence of an indefinite
number of units. (In this case, my answer doesn't exactly check, but it
doesn't exactly not check, either.)
My mathematician is not going to be happy about this at all. But please
remember that we agreed to play the game according to his own rules. It was
he who insisted that we treat zeroes as non-units. In fact, I think this
conclusively proves that it is the mathematician who has furtively been
treating zeroes as units.
And at long last we have reached the crux of the matter. The old
nitty-gritty. The fallacy in the fallacy. Mathematicians have, for lo these
many years, been harboring a mental image of zero as a non-unit, while
simultaneously attempting to treat it as though it were a unit--a neutral
unit, to be sure, but nevertheless as a unit. Well, we pays our nickel and we
takes our choice. We are free to regard zero as a kind of neutral
pseudo-unit, or we may treat it as a non-unit. But not both. If we elect to
treat zeroes as non-units, we promptly deprive them of whatever neutrality
they might have had, and they become essentially negative in character.
(Hence the term non-unit or no-thing or nothing.) Therefore, we cannot apply
the same rules to a non-unit that we apply to true units, and expect the
non-unit to meekly conform. As the mathematicians are fond of saying (or were
up to now), we simply cannot mix apples with oranges.
Now for a quick analytical summary of everything we've postulated:
1. If we treat zeroes as pseudo-units, then n/0=n. (This is safe
because, as previously noted, it is the same result that mathematicians get
when they refuse to divide the number, n, by zero.)
2. If we treat zeroes as non-units, then n/0=0.
3. If we treat zeroes as pseudo-units, n x 0 =n. But we cannon mix
pseudo-units with true units any easier than we can mix non-units with true
units; therefore, to avoid confusion and stay on the safe side, we must
express the product of zero and n as zero with zero in this case being
understood as representing n pseudo-units, distinguished from true units and
non-units.
4. If we treat zeroes as non-units, then n x 0 = 0, provided n is not
equal to zero; otherwise, the product is indeterminate.
5. The same reasoning applies when we divide zero by zero--i.e., the
answer is 1 (necessarily expressed as zero) if we treat zeroes as
pseudo-units, and indeterminate if we treat zeroes as non-units.
Conclusions: Plainly, we computer people are going to be in serious
trouble if the mathematicians persist in regarding zeroes as non-units. We
have already seen that multiplying one non-unit by another non-unit generates
an indeterminate number of real units. There is nothing implausible about
this, but it is equivalent to making something out of nothing, and we
certainly don't want to be accused of that. Therefore, the only sane course
of action is to treat zeroes as pseudo-units, whereby we common folk can more
or less follow the conventional rules of mathematics.
Yes, that is the only path to follow, short of giving zero back to the
Arabs; and I heartily recommend that we follow it.
(Unless, of course, there is a fallacy in the (fallacy in the fallacy).)
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Mr. Lafon is a management analyst for the Bureau of Indian affairs. He
was previously ADP coordinator for the Albuquerque district of the Corps
of Engineers. He has had 10 years of experience in data processing and now
specializes in DP standards and procedures. (1971)