Division by zero. Go ahead and laugh.

If 1/0 is defined a logical contradiction follows, to wit, 1 = 0. If you
like your systems to be consistent then forget about dividing by zero.

The only place were dividing by zero makes any sense is in an algebra
who only element is 0. Which is kind of useless.


Bob Kolker

 

How to make 1/x larger than N. Set pick x so that |x| < 1/N.

For any large N there exists an x for which |1/x| > N. Obviously the x
you choose is related to the given N.

Bob Kolker

 

The compactifier is not an element of R so it is the case that lim 1/x x
goes to 0 does not exist in R, which was all that was asserted.
Compactification of a system is a topological matter, not an algebraic
matter. The compactifiers do not have the same algebraic properties as
the normal members of the space. You can't have it both ways. Either you
make things nice topologically and loose algebraically or vice verss.
Either , but not both.

Bob Kolker

 

No contradiction follows. But please, by all means, feel free to exhibit
here how you think that 1 = 0 could be deduced after defining 1/0 in, say,
C*. If you have in mind what I suspect you do, then you will have made
an unwarranted assumption in order to get your contradiction. Namely, you
will have incorrectly assumed at some point that 0 has a multiplicative
inverse, thereby allowing factors of 0 to be cancelled. But of course
that's nonsense; 0 cannot have a multiplicative inverse, and so cancelling
factors of 0 would be unjustifiable.

Wrong.

David Cantrell

 

Right. That's what I said at the beginning: "That's true in R ..."

You snipped the part supplying possible context. But IIRC no particular
system was specified, thereby making the assertion vague.

Nor does 0, for that matter!

You're entitled to your opinion about that. I don't share it.

David Cantrell

 

Hi David

There is at least one other very different review of this book by a
professional mathematician:

http://www.maa.org/reviews/zero2.html

The point I wanted to make is that the discussions around Zero and
Infinity have a long history, perhaps as long as the history of
humankind itself.

I think it is legitimate to take up this discussion today, but one
should at least be aware of the existing background. Even if Seifes
book deserves the criticism by Gray, it could show the OP that perhaps
s/he is not the first person in the world to be troubled about these
issues.

Also, I feel that there are two kinds of answers that are possible:

- the answer that a professional mathematician gives (many examples
in this thread)

- a more historical and philosophical approach to the questions of
infinity and zero

Greetings
Matthias Kläy

 

Definition a/b = c if and only a = b*c. That is what division is.


assume 1/0 = x. This means 1 = 0*x (definition of division).
0*x = 0 (well known theorem for rings)
hence 1 = 0.

QED



If you have in mind what I suspect you do, then you will have made

Division means precisely multiplicative inverse. That is what it is.

My proof stands.

Bob Kolker

 

Not so. 0 + x = x for all x. 0 is the additive identity for the ring.

1 is the multiplicative identity for the group consisting of all
elements of the ring except 0.

0 is just what it ought to be.

I stated a mathematical fact, not a mathematical opinion.

Bob Kolker

 

Hi Matthias

Yes, that is indeed a very different review. I may have read it before,
and forgot about it. In any event, my opinion is much closer to Gray's.
If one wishes to read a book about zero, then I would suggest that
Kaplan's _The Nothing That Is_ is substantially better than Seife's
book.

I agree with everything you said below.

Best wishes,
David Cantrell

 

I agree with that, of course. But by your saying "Not so." you are claiming
that 0 has "the same algebraic properties as the normal members of the
space".

Let's look at two examples:

(1) x*y = x*z implies y = z
That's true for all "normal members of the space"; it is not true if x is 0.

(2) x has a multiplicative inverse
That's true for all "normal members of the space"; it is not true if x is 0.

So I can't imagine why you think that 0 has "the same algebraic properties
as the normal members of the space". 0 is absorptive under multiplication;
that's not "normal".

Fascinating. I must have missed the math course in which "nice" and "loose"
were rigourously defined.

David Cantrell

 

If 1/0 is defined a logical contradiction follows, to wit, 1 = 0. If you

Precisely.

So what do we do ? We declare it to be "undefined", or "does not exist".

What is really happening here is some type of disintegration of logical
consistency at the singularity which is created by attempting to divide 1/0.

Dont you find this amazing ? I am perplexed by this. I find it really
incredible. Yes, you can define things so that the whole question just goes
away. But, if you dare to attack it, there could be something to learn from
it.

 

You did'nt even bother to read what I said. Or if you did, you certainly
did'nt understand what I was saying.

Where did I say that one must allow division by zero ? I did'nt. I said that
it needs to be understood in the proper context instead of burying your head
in the sand and pretending that it "does not exist". Clearly it DOES exist,
otherwise we would not be having this conversation. Dont you agree ?

Undoubtedly. And I also qualified that by stating that it must be "isolated"
to prevent your consequences.

Seriously, this conversation is just a little bit more than highschool level
stuff.

You can, if you wish, define an operator, number of other object which could
be added to R such that arithmetic could be consistent even if you allow
division by zero.

For example, let @ be the object which represents inconsistency. Maybe @ is
a number. Maybe it's not. I dont know. Maybe you should get creative, figure
it out, then YOU can tell ME what it is.

If @ was contained in R, or maybe an R' or something, then you could say
that division is wellbehaved everywhere except that 1/0 = @. This can all be
done by definition. Sure seems more realistic that just stating that things
"do not exist" - as if -.

You are being very closed minded to this. I am not going to pursue this for
the next 5 years. I need quality feedback. If you want to criticize
professionally then please continue. If you want to ridicule then please
move on.

 

Now that was a good read !

Pretty funny too !

 

Very hilarious stuff !

 

I'm irritated by a physics which consists of hodgepodge mathematical models
which are basically things which are just pasted together from algebra.

You have two separate worlds. The abstract world of math, and the physical
universe. Math is traditionally exported from the abstract world into the
real world in the form of models. I think that physics is suffering from
this. Physics should be a mathematics in it's own right. And in thinking
about this, it seems that the best place to start is some type of
understanding of what a number is in the physical universe. Are there
numbers which really exist and are not merely abstractions ? I think so. And
I think that those numbers would be 1 and 0. You can have something, and
then you can have the abscence of something. You must be able to define
non-existence otherwise "everything" would exist. Clearly there are things
which do not exist.

If something exists you have 1 of that thing. You cannot have two, because
no two things can be identical. You can only have one of any item.

You can have zero of an item. This merely means that something does not
exist in the physical universe.

It is problematic to say that you have 1.5 inches, or 1/2 mile, or .75
liters. These things are quite impossible. Maybe you can come close, but
it's always just an illusion. You will never be able to obtain 2.5 inches of
material in the physical universe, it is physically impossible to obtain
_exactly_ 2.5 inches of material, for example.

So you have 1 and zero. Two numbers which seem to "exist very precisely" in
both the physical world and the abstract world. And this whole thing hinges
on the fact that no two physical objects can ever be identical.

If you acknowledge this as a fundamental axiom, you may be able to
reconstruct physics as an axiomatized system instead of a hodge podge of
algebraic contraptions.

 

It's not offered at all universities.

 

Fascinating. So according to _your_ definition:

Since 0 = 0*0, we must have 0/0 = 0. And
since 0 = 0*1, we must also have 0/0 = 1.
Therefore, since 0/0 equals both 0 and 1,
by transitivity, we have 0 = 1.

Seriously now: Of course I know the definition you intended to give. And
there's clearly nothing wrong with using that definition of division in
some systems. But you were responding to a statement concerning C*, for
heaven's sake! The definition you intended to give is therefore
inappropriate, not only to C* but to any system in which division of
nonzero quantities by 0 is defined.

No, that's obviously _not_ what it means in C*.

No, in C* and other systems in which division of nonzero quantities by 0 is
defined, division does not mean that.

It never even got off the floor!

David Cantrell

 

0 is quite normal. It is excluded from the multiplicative group.

0 has a property that other numbers does not have. 0*x = 0 for all x.
This follows from the distributive law of multiplication over addition.
It is the distributive law that "pastes" together the additive group and
the multiplicative group. 0*x = 0 essentially follows from the fact that

(a - a)* x = ax - ax = 0. But a - a = 0 (definition of additive inverse)
hence 0*x = 0. So blame zero's peculiarities on the distributive law.

This is the way the numbers we use work. If you want to deal with things
other than numbers, go ahead.

If you find having consistency a burden go play with systems that are
inconsistent.

Bob Kolker

 

We declare -division by 0- as undefined. 0 itself is a perfectly nice
number and it is special being the identity of the additive group.

Bob Kolker

There is no inconsistency. If division by 0 is allowed then 1 = 0 which
means everything is equal to 0.

The lesson to be learned is not to permit the rules of the system to
lead to inconsistency. That is not the least bit astounding. It is just
plain good sense.

Bob Kolker

 

In order to do that you have to eliminated logical implication. Which
would render mathematics totally useless. Do you really want to get rid
of implication and inference? Mathematics is all about implication and
infernce. It is all about drawing conclusions from the axioms of the
system. What you propose is the elimination of mathematics in the entirety.

I have told you why that is not possible if you want to do any
mathematics. Why bother with axioms if you can't draw conclusions from
the. Just write down isolated facts and forget any sort of logically
hierarchical system.

It isn't a number. It is isn't anything that makes logical sense. You
want a symbolic expression of inconsistency? Here it is: A and not-A
where A is any propositon. There is your @.

Bob Kolker