
                          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).)

--------------

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)
