# Division by zero. Go ahead and laugh.

Discussion in 'Undergraduate Math' started by Lefty, Oct 7, 2004.

1. ### robert j. kolkerGuest

You are right. I reject inconsistency root and branch. If tha makes me
closed minded, then so be it. Logical inconsistency is the death of
science and mathematics, both of which I care for a great deal.

I am being both professional and charitable. Serious advocacy of
inconsistency is either a bad joke (on your part) or a clear symptom of
inanity. Which is it?

Bob Kolker

robert j. kolker, Oct 9, 2004

2. ### robert j. kolkerGuest

FOr that reason 0/0 is not permitted either. It is not equal to any
particular value therefore is not defined. Get it straight. Dividing by
zero in any circumstance leads to inconsistency which is why we don't
divide by zero.

0/0 is not a number.

Bob Kolker

robert j. kolker, Oct 9, 2004

3. ### David W. CantrellGuest

It is clearly permitted according to the definition you gave.
It might be better if YOU got it straight! I was using the definition YOU
gave.
No, it doesn't. And I see that you conveniently snipped the end of the
message in which you attempted to exhibit such an inconsistency, but failed
to do so.
Of course, if it did lead to inconsistency, we wouldn't have systems such
as C* in which division of nonzero quantities by zero is defined.
What I said above was based on YOUR definition.

David Cantrell

David W. Cantrell, Oct 9, 2004
4. ### robert j. kolkerGuest

O.K. Lets fix the definiton. No division by zero period. That answers
your objects AND it prevents an inconsistency. Win win all around.

Bob Kolker

robert j. kolker, Oct 9, 2004
5. ### The Ghost In The MachineGuest

In sci.math, Lefty
<>
wrote
Arithmetic does not "fall apart" if one allows division by the
arithmetic identity (0); it merely mutates.

Of course, the classical definition of "real number" precludes 1/0, 0/0,
and -1/0, but one can easily extend the numbers, although one
may lose some of the more interesting properties.

Call this class of numbers the "zero-divisible reals" (not to be
confused with "the extended reals", which only adds +oo and -oo,
as I understand it).

The zero-divisible reals might add three elements.

1/0, which is greater than any existing real number.
-1/0, which is less than any existing real number.
0/0, which is a very weird thing, as you'll see.

Now one has to define:

a + 1/0, a - 1/0, a * 1/0, a / (1/0) for any real a.
1/0 + a, 1/0 - a, 1/0 * a, (1/0) / a, for any real a (*including* 0).
etc.

One could get into some interesting if slightly useless territory here.
For example, is 2/0 > 1/0, leading to a class of "real infinities"?
Or is 2/0 = 1/0, the more or less standard extension?

Where did you want to go with this?
And why would we want to do that?

Zero is a natural consequence of the definition of subtraction. Briefly
put, if a = b + c for integers a, b, c > 0, then c = a - b and
b = a - c. That defines "subtraction" if a > b and a > c; the next
question is what happens if a < b and a < c, or perhaps defining
the quantity "a - a". After a little work one gets -- zero, with
its properties, including the problematic one of x * 0 = 0 for any
integer, rational, algebraic, real, or complex x. Since d = e * f
implies e = d/f and d = e/f for any positive integers d, e, f (or,
for that matter, many other numbers), 1/0 cannot be defined in the
standard reals -- x * 0 = 1 is unsolvable! -- and 0/0 is meaningless
in the standard reals, since x * 0 = 0 for all standard real numbers x.

One can't even be sure whether 0/0 > 0, < 0, or = 0.
You are welcome of course to define 1/0 and 0/0 as you see fit, bearing
in mind that the theorems of arithmetic may require some adjustment.
For example, one can also attempt to define 1/0 * 0 = 0.

It may depend on how one approaches this problem. For example,

1/x * x = 1
1/x^2 * x = 1/x
1/x * x^2 = x

for all x != 0.

So if one takes the limit as x approaches 0, one can either get 1, 1/0,
or 0, when one multiplies 1/0 by 0. At best, this is rather problematic.
I would suggest that you define your arithmetic results given the
three elements 1/0, -1/0, and 0/0, and proceed accordingly (and
*very* carefully). You may get an inkling of some of the difficulties
involved.

For example, does x = 1/0 solve the equation (1/0) * 0 = 1? If so,
then one has to define a * 1/0 for real a > 0 carefully, lest one
misapply the cancellation rule with a * 1/0 = b * 1/0 and get a = b,
even though a != b initially. Fortunately, there's a precedent; if
a * c = b * c and c = 0, then there's no drawable conclusion regarding
a and b; perhaps c = 1/0 could be added to that exclusion set.

Good luck, but I'm not all that hopeful.
Sorry, I'm not Catholic.

The Ghost In The Machine, Oct 9, 2004
6. ### David W. CantrellGuest

Hmm. I guess that the definition of "normal" in this context must have been
presented in the same math course in which "nice" and "loose" were defined.
Sorry I missed that course.
I wouldn't use the word "blame". Zero's peculiar, but that's just its
nature. But if something's peculiar, it's abnormal. (Do we have another
inconsistency here: 0 is both normal and abnormal ?)
I don't.
I never have and never will.

David Cantrell

David W. Cantrell, Oct 9, 2004
7. ### LeftyGuest

That's fine and things work very nicely if you do. I'm very happy that
algebra is consistent because this special provision that (1/0 DNE) makes it
so.

If there were no inconsistency then there would be no debate. The
inconsistency is traditionally resolved by setting 1/0 as DNE. I think that
there are other more _rigorous_ ways to resolve this which will also leave
arithmetic intact, and simultaneously acknowledge the existence of the

Fine. You can use an arithmetic where 1/0 does not exist and it is
consistent. Fine. But, if someone else modifies the definition so that
arithmetic remains consistent and the singularity at 0 can also be handled
algebraically, then what would you say to that ? One definition is better or
worse than another - and for what reason ?

I am saying that the definition of setting 1/0 as DNE creates an arithmetic
which is consistent, but this definition is rather arbitrary, because you do
have the option to attempt to assign a symbology which could treat the
singularity. Why would one be preferable to the other ?

Lefty, Oct 9, 2004
8. ### LeftyGuest

entirety.

Absolutely not. Sincerely, this would be impossible and undesirable. Yet,
perhaps this is a popular misconception, and a major motivation to stay away

A paradox does not undermine mathematics. It only makes it more fascinating
in my opinion.

I am suggesting the exact opposite of what you are saying. I am suggesting
an operator which is defined in such a way that it leaves arithmetic intact.
I think that this must be possible.

Correct. In fact, I dont know what it is except to say that it is a
contradiction. It exists as such, and there should be a means to handle it
algebraically within the framework of algebra. It would probably need a
symbol of some kind, and maybe some other gadgetry. I do not know.

It is a beautiful thing if only you could see it as such. Math is the only
science where one can construct absolute truths. Does it seem odd that
everything should explode at a singularity ? I find it absolutely
captivating, and I hope you will continue to argue against me because it may
encourage more contemplation.

Something is happening - something fundamental and strange, and it does'nt
seem one bit elegant to amputate zero (for 1/x) and toss it out by stating
"it does not exist".

Lefty, Oct 9, 2004
9. ### LeftyGuest

You are being very closed minded to this. I am not going to pursue this
for

Bob, I'm really sorry if I have offended you. This is not my intention, nor
do I intend to undermine math. I wish that you could believe me when I say
that I am trying to enhance the scope of mathematics constructively. I wish
I could prove this to you that I am not being disingenuous. I have no
ulterior motives, and again I apologize if it seems that I have seemed to
make any such overtures.

I fully respect your feelings in the above. I know that mathematicians are
_the_hardest_working_ and the most scientifically _honest_ people who have
ever existed. I know this.

Think about it for a moment. Even if you discovered a logical inconsistency
in the general model somewhere - what do you do with the enormous logical
structures which have been meticulously erected for centuries ? Throw them
in the trash ? The works of Gauss ? Euler ? No! I do not think that there is
ANYTHING which could render these works invalid. This is impossible!

There is no way to shake the structures which exist. Even if mathematics
were succeptible somehow to space/time, the structures which have been
observed and proven are all still valid.

My aim is to demonstrate somehow that math and physics reflect each other in
yet another way - the existence of paradoxes. Singularities. Whatever they
are, whatever you call them, I think that logic can break down "locally" in
math and also in physics. The universe wont collapse, and neither shall
math.

I only wish we could formally axiomatize physics. Actually, what I mean by
this is that the physical universe itself be axiomatizable to create a
"non-abstract mathematics", a math of real objects which is non-abstract. I
dont think that I'll ever succeed at this, and perhaps this will only
illustrate how truly insane I might be. But I'll make you a promise. I
promise to not harrass people to try to force them to believe me. Nobody
will ever notice this material, and no-one will ever care.

Lefty, Oct 9, 2004
10. ### LeftyGuest

If you find having consistency a burden

Imagine for a moment that there is some strange thing out in space
somewhere. Could be a black hole, - whatever. Imagine for one moment that
this object has the very real physical attribute that logic breaks down or
is altered when things go near it.

Do mathematicians have any tools to describe this thing ? Can you model it
mathematically ? Or, would you have to conclude that it does not exist
because there are no algebraic tools available to model what you are
observing in the physical world ?

Lefty, Oct 9, 2004
11. ### robert j. kolkerGuest

Good. Now do the following. Produce an algebraic system with the
following properties.

1. The elements form a ring with respect to addition.

2. The elements form a multiplicative group (that means 0 has a
multiplicative inverse).

3. The distributive low of multiplication over addition holds.

4. The system is consistent. No logical contradictions.

5. The system is non-trivial, i.e. it has more than one element.

6. You can use an instantiation of the system to do your income tax.

Get back to us when you figure out how to do it.

Bob Kolker

robert j. kolker, Oct 9, 2004
12. ### robert j. kolkerGuest

Forbidding division by 0 is precisely admiting the singularity.
You think so? Fill the room with your brilliance. Do it.

Bob Kolker

robert j. kolker, Oct 9, 2004
13. ### robert j. kolkerGuest

A contradiction i.e. a proposition of the form A and not-A destroys
mathematics. It enables -any- propostion to be proved. This is a well
know excercise in any logic 101 course. A and not-A has the value false
and a truth table excercise show that FALSE implies P for -any-
proposition P. If FALSE (i.e. A and not-A) were provable any proposition
follows. Such a calamity renders mathematics unusable except for finite
systems for which models exist. Good bye calculus. Goodbye real and
complex variables. Goodbye number theory. Since physics is based
squarely on the theory of real and complex variables, goodby physics.
Do it then. Let us see it and show that it is consistent. No
consistency, no mathematics.

Bob Kolker

robert j. kolker, Oct 9, 2004
14. ### robert j. kolkerGuest

You mean by no two, that the objects are distinct. Hence any pair of
distinct objects are distinct. With which I agree wholeheartedly.

The contrary to the existence of two (or more) distinct objects as that
there is either one object or no objects. Which would you prefer?

Do you believe there is only one object and we are deluded in seeing
distinct object? If so, our two eyes, which are really one eye and our
ass and our stomach is deluding us badly.

Bob Kolker

robert j. kolker, Oct 9, 2004
15. ### David W. CantrellGuest

Thanks for the suggestion, but I'll pass.
What you request is clearly impossible, as we are both well aware.

David Cantrell

David W. Cantrell, Oct 9, 2004
16. ### robert j. kolkerGuest

That there thing may be physical but our math cannot touch it. Physical
existence does not depend on our being able to describe whatever it is
mathematically. Our mathematics is limited by logical consistency. It is
entirely possible that a thing can exist such that we cannot comprehend
it by a logically consistent system of concepts.

Just because we have nimble three pound brains and we can build rocket
ships does not mean we are able to comprehend everything. There are more
things in heaven and earth than are dream't of in our philosophy.

Bob Kolker

robert j. kolker, Oct 9, 2004
17. ### LeftyGuest

I dont want to see any of these things going away, except perhaps the
physics. The models of physics I have seen are all essentially mathematical
contraptions which do not exctly match the physical world. Physics should be
a mathematics in it's own right, where the number system and operands match
the physical world _exactly_ . I dont know if this is possible, but I do
know that I have never seen anything like that.

I'm not saying that physics is wrong.

What I mean by this is that mathematics is pure abstraction. But physical
processes exist completely in the physical universe, with the exception of
the mind and maybe some other oddities somewhere which intersect both
worlds, the abstract world and the world of space/time. Physics as we know
it utilizes abstractions to model real things. This is a pulling together of
two separate worlds. And typical physics models _never_ describe things
completely, they always leave out information. I think that physics can be
axiomatized so that it need not borrow from the abstract world when building
models.

Maybe.

I will try to put something on paper and will post here if I can come up
with such a thing, trying hard not to sound too stupid, I know I've already
blown my credibility, but I certainly dont mind being criticized for being
wrong, it's all just experimentation.

It's probably folly.

But where would man be today if he were'nt such a fool ?

Lefty, Oct 9, 2004
18. ### LeftyGuest

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

Actually, I am trying to explain that the naturals do not _exactly_ describe
physical objects. If you count 5 imaginary marbles in your mind, then you
have counted 5 identical marbles. No problem. But if you count 5 real
marbles which you bought from the marble store, then your counting process
is inexact. No two marbles in space/time can be identical. You cannot count
to 5. You must count to 1, 5 separate times. Each marble is unique.

You have either 1 marble, or 0 marbles. Marble exists, or it does not. This
line of reasoning is _exact_ . It matches the universe with a precise
exactness which you dont ordinarily see in most models of real events.

It reminds me alot of some existentialist stuff, more philosophical garbage
you may say - but I think that the axiom I have given is a statement of a
number-theoretic nature. I come to this group because others know much more

The bulk of reality does exist as a single object. It is also composed of a
plurality of subunits known as "things".

I think that separate things do exist. There is 1 of each. For some things,
there are zero pieces available, these things are non-existent in the
universe of space/time.

Lefty, Oct 9, 2004
19. ### robert j. kolkerGuest

Exact matches are a physical impossibility. If for no other reason than
the uncertainty principle.
You can't get an -exact- match between experiment and theory. The
instruments used in the experiments are susceptable to thermal
perturbation and other environmental disturbances not to say anything
errors. Exact measurements of complementary observables with non zero
commutators is not possible courtesy of the uncertainty principle. All
experments come with error bars. If the prediction falls inside the
error bars the experiment supports the prediction.

Physics is NOT mathematics although mathematics is necessary to express
the quantitative relations of physical theory. Physics is -empirical-. A
physical theory is precisely as good as the predictions it makes.
Expermental fact trumps physical theory each time, every time. Theories
serve. Facts rule.
You just did.
Carnap and the logical positivists tried to axiomatize physics for three
decades. They failed. Reality is just too big to be fit into a
formalism. Physical theories will always be incomplete because we are
always learning New Stuff. When light from far reaches of the kosmos
reaches us it tells us about Newt Stuff we have literally never seen
before. There is not way we are ever going to be able to deduce physical
reality a priori.
Much better off. Paradoxes are the stigmata of sloppy workmanship and
fuzzy thinking. If our theories can not catch up to facts, that is NOT a
paradox. It is a deficiency of our thinking and understanding. We are
Mark 3 apes with three pound brains, mediocre senses and the gift of
gab. We are the smartest baddest apes in The Monkey House. That is what
we are.

Bob Kolker

God ever geometrizes. The rest of us pay taxes and commute to work -
Plato Rabinovitz

robert j. kolker, Oct 9, 2004
20. ### N. SilverGuest

If the models exactly matched the "physical world," then they would
be the "physical world." The point of having a model is that it is a
a stripped-down-to-essentials rendition of a physical process that
allows us to analyze and make mathematical predictions about the
process.
We want to construct a simple system that mirrors a more
complicated one. Then we can modify the model as we
use it to more accurately reflect the process. If we are
successful, our sequence of models will converge.
You don't have the authority to even think that.

N. Silver, Oct 9, 2004