Is Zero Divided by Zero Really Solved After 1200 Years?

[Alkatran]

There is now a section of him responding to objections:

http://www.bbc.co.uk/berkshire/content/articles/2006/12/12/nullity_061212_feature.shtml

That changes the way you do mathematics - that statement is revolutionary. It remains to be seen whether it is correct or not, whether people accept it or not, but that's my position.

Wait, he's not sure if it's correct or not?! How can you be teaching mathematical ideas when you're not sure they're correct??

 

[cristo]

Wait, he's not sure if it's correct or not?! How can you be teaching mathematical ideas when you're not sure they're correct??

Let alone teaching them to school kids!

 

[Rach3]

Wait, he's not sure if it's correct or not?! How can you be teaching mathematical ideas when you're not sure they're correct??

Well, it's "only a theory". You know, like evolutionary theory, intelligent design theory, algebraic topology theory, or TimeCube.

 

[EmilK]

Well, it's "only a theory". You know, like evolutionary theory, intelligent design theory, algebraic topology theory, or TimeCube.

I'd like to refer you to an excellent entry by ZapperZ that can be found here :)

 

[Alkatran]

What really, really bothers me about this story is the comments.

People giving obviously wrong inconsistency proofs, people saying "yeah, well people said imaginary numbers didn't exist", people saying 1/0 is equal to infinity, just people making it clear they don't understand division in general.

How hard is it people? a/b = c if and only c is the unique solution to a = bc.

I also wonder why the sets {x | 1 = 0*x} (the empty set) and the set {x | 0 = 0*x} (the set of all real numbers, given that we're working with the reals). They pretty clearly show why you can't divide by 0. Sort of hard to justify giving the empty set or the set of all reals numeric values!

 

[unit_circle]

Well, it's "only a theory". You know, like evolutionary theory, intelligent design theory, algebraic topology theory, or TimeCube.

I'd like to refer you to an excellent entry by ZapperZ that can be found ot.com/2006/10/imagination-without-knowledge-is_16.html]here[/url] :)

I think you forgot to turn on your saracasm detector, EmilK. Rach3 was jesting when he compared evolution to the TimeCube (at least I hope so, Rach3). It's funny the TimeCube was mentioned, because when I told my brother about this Anderson crackpot, he said "Division by 0? Isn't that TimeCube territory?"

 

[Hurkyl]

How hard is it people? a/b = c if and only c is the unique solution to a = bc.

It's easy to prove someone wrong if you define all the symbols to mean something different than the other person intends. "Transreal" division is not the same thing as division in a field.

 

[Alkatran]

It's easy to prove someone wrong if you define all the symbols to mean something different than the other person intends. "Transreal" division is not the same thing as division in a field.

I wasn't making a point against his system in that post, I was complaining about the public's lack of knowledge on how division is defined. Otherwise, excellent point.

 

[Sleek]

If your heart pacemaker divides by zero, you're dead

Hurray to exception handling :P.

 

[Karlisbad]

This seems to me a big nonsense similar to the "hoax" by Peter Lynds and his "supposed" phylosophical solution to Zeno Paradoxes..

By the way i think that the problem with 0/0 was studied and made rigorous with infinitesimals and so on, although i would like to know what "impact" this would have with derivatives since for h-->0 the quotient:

$$\frac{f(x+h)-f(x)}{h}$$ is just 0/0 and hence a "nullity" , if this can be applied to some math derivatives and integral could benefit from this number.

 

[AKG]

This seems to me a big nonsense similar to the "hoax" by Peter Lynds and his "supposed" phylosophical solution to Zeno Paradoxes..

By the way i think that the problem with 0/0 was studied and made rigorous with infinitesimals and so on, although i would like to know what "impact" this would have with derivatives since for h-->0 the quotient:

$$\frac{f(x+h)-f(x)}{h}$$ is just 0/0 and hence a "nullity" , if this can be applied to some math derivatives and integral could benefit from this number.

This isn't true. For h-->0, this is the L such that for all e > 0, there exist d > 0 such that, well, you know the rest. There's a specific meaning for "h-->0" which is different from h=0.

 

[Hurkyl]

On a related note, division is discontinuous anywhere where the denominator is zero. So, even if you use the transreals, you still cannot do any of the naïve things you would like to do with the limit of f/g when g goes to zero.

 

[Cyrus]

I don't know about what he is doing mathematically, but he says in the BBC audio clip that his use of nullity helps solve computer problems and that he has a working chip that does math with this concept that is not easily done without it.

He says very clearly that his concept of nullity is not mathematical, but computer science related, something to do with NaN not= NaN.

That being said, maybe we should wait and see if this actually does provide any use before bashing him in the head? Because I read a lot of people here complaining about his use of Nulity for MATHEMATICAL concepts, which he clearly says is NOT what nullity is for.

 

[FrogPad]

I don't know about what he is doing mathematically, but he says in the BBC audio clip that his use of nullity helps solve computer problems and that he has a working chip that does math with this concept that is not easily done without it.

He says very clearly that his concept of nullity is not mathematical, but computer science related, something to do with NaN not= NaN.

That being said, maybe we should wait and see if this actually does provide any use before bashing him in the head? Because I read a lot of people here complaining about his use of Nulity for MATHEMATICAL concepts, which he clearly says is NOT what nullity is for.

So were those children computer science students?

 

[Cyrus]

What does that have to do with anything?

 

[FrogPad]

Because I read a lot of people here complaining about his use of Nulity for MATHEMATICAL concepts, which he clearly says is NOT what nullity is for.

All I'm saying is his presentation (probably more for the BBC) was in a MATHEMATICAL environment.

 

[NoTime]

I don't know about what he is doing mathematically, but he says in the BBC audio clip that his use of nullity helps solve computer problems and that he has a working chip that does math with this concept that is not easily done without it.

He says very clearly that his concept of nullity is not mathematical, but computer science related, something to do with NaN not= NaN.

That being said, maybe we should wait and see if this actually does provide any use before bashing him in the head? Because I read a lot of people here complaining about his use of Nulity for MATHEMATICAL concepts, which he clearly says is NOT what nullity is for.

He discovered the database NULL field?
Or the NULL pointer?

 

[Cyrus]

How should I know? I am just repeating what he said on the radio interview. Someone linked it, you can listen to it for yourself.

I said something about NaN not = NaN. I don't know if this has to due with the database Null field or the Null pointer. He said he got it to work on a chip, and that it solves formulas that are otherwise more difficult using standard algorithms.

No where, did he say this concept was to be used for mathematical proofs.

 

[FrogPad]

How should I know? I am just repeating what he said on the radio interview. Someone linked it, you can listen to it for yourself.

I said something about NaN not = NaN. I don't know if this has to due with the database Null field or the Null pointer. He said he got it to work on a chip, and that it solves formulas that are otherwise more difficult using standard algorithms.

No where, did he say this concept was to be used for mathematical proofs.

Here's a video:
http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/divide_zero_feature.shtml

Here's a quote:
"That changes the way you do mathematics - that statement is revolutionary. It remains to be seen whether it is correct or not, whether people accept it or not, but that's my position."
-http://www.bbc.co.uk/berkshire/content/articles/2006/12/12/nullity_061212_feature.shtml"

 

[Hurkyl]

Because I read a lot of people here complaining about his use of Nulity for MATHEMATICAL concepts, which he clearly says is NOT what nullity is for.

Very few people are criticizing his technical work; it's almost entirely directed at the comments he makes. And some of them are mathematical. For example, taken from the link in the OP:

Dr James Anderson, from the University of Reading's computer science department, says his new theorem solves an extremely important problem - the problem of nothing.​

For example, take this quote from Perspex Machine VIII: Axioms of Transreal Arithmetic

For example, contemporary real analysis recognizes two special limits $\infty = 1/0 = k/0$ when $k > 0$ and $-\infty = -1/0 = -k/0$ when $k < 0$.​

where he makes one of the classic freshman mistakes. (he goes on to say he finds it disturbing that it doesn't include a number for k/0 when k=0)


And his problems aren't limited to the mathematical domain. Again, going back to the link in the original thread:

"Imagine you're landing on an aeroplane and the automatic pilot's working," he suggests. "If it divides by zero and the computer stops working - you're in big trouble. If your heart pacemaker divides by zero, you're dead."​

which is, of course, absurd. Computers divide by zero all the time. (I just did it on my computer in the course of writing this message, just to prove a point) And it demonstrates an utter lack of understanding of fault-tolerant programming.


In summary, (almost) nobody's complaining about his arithmetic system -- everyone's criticising his remarks about other things. Even if we assume he's doing good work with his Perspex machine, that doesn't mean he's not a crackpot.

 

[Kurdt]

I'm sure a similar stink was kicked up when Dirac found it useful to use his delta function in quantum mechanics. Perhaps if this is useful in the field of computer science then its implication and justification in a purely mathematical sense will have to be investigated as was Dirac's delta function. I suppose this strikes a chord because it is portrayed clumsily by the media.

 

[DeadWolfe]

But what on Eath IS the use of this concept supposed to be (aside from the obvious application to pacemakers)?

 

[Hurkyl]

I'm sure a similar stink was kicked up when Dirac found it useful to use his delta function in quantum mechanics. Perhaps if this is useful in the field of computer science then its implication and justification in a purely mathematical sense will have to be investigated as was Dirac's delta function. I suppose this strikes a chord because it is portrayed clumsily by the media.

I would be surprised. TMK, when Dirac introduced the delta function, he solved problems with it. And, TMK, Dirac demonstrated that he had a grasp of his subject.

Dr. Anderson, on the other hand, doesn't seem to have used transreal arithmetic to do anything. But that hasn't stopped him deeming his own work revolutionary, and contacting the media so that the whole world will know how smart he his. (And, in the process, making plenty of very basic mistakes about other fields... and his own)

 

[Hurkyl]

Comapre Dr. Anderson's work with, say,

http://www.math.su.se/~jesper/research/wheels/wheels.pdf

or, if you just want the arithmetic,

http://www.math.su.se/~jesper/research/wheels/wheelsshort/wheelsshort.pdf

 

[Kurdt]

I haven't had a proper look at his work as I've been busy but I do like to keep an open mind. I'll save my opinion for then.

 

[DeadWolfe]

Hmm. At his website he has books where he has solved the mind.body problem.

In the preface he says it's completely mathematical, but with no equations. Instead it has "visions", so that you can see consciousness.

Yeah, he's definately not a crackpot.

 

[ScaleMaster]

I am by far no math guru, and I am sorry for bringing up a topic that ended almost 2 months ago, but I just recently discovered this whole "nullity" bs,

But wouldn't his idea also indicate that numbers have an "end" ?

Infinity is defined as 1/x as x gets closer and closer to 0

------------
example

1/1 = 1
1/.1 = 10
1/.00001 = 100000
1/.000000000000000000001 = 1.0 × 10^21
etc, etc
------------

he is going off the concept that 1/0 is infinity and is using it as, "the final number" when infinity is not a number, it's a concept.

So he is basically saying that numbers have a stopping point.

 

[d_leet]

Infinity is defined as 1/x as x gets closer and closer to 0

It is?

/*extar characters*/

 

[ScaleMaster]

That's the way I was always taught it, it makes sense to me, and it works. It's also what Anderson was using in his video when he said that 1/0 is infinity. and -(1/0) is negative infinity.

 

[d_leet]

That's the way I was always taught it, it makes sense to me, and it works. It's also what Anderson was using in his video when he said that 1/0 is infinity. and -(1/0) is negative infinity.

It makes sense, yes, but I don't think it's the definition of infinity.

 

[ScaleMaster]

well that's the assumption that he is also going by. Which is where he messed up.

 

[Hurkyl]

I am by far no math guru, and I am sorry for bringing up a topic that ended almost 2 months ago, but I just recently discovered this whole "nullity" bs,

The number system he created is perfectly valid. It's his attempts at publicizing it that's junk.

But wouldn't his idea also indicate that numbers have an "end" ?

Kinda sorta. If you take the nonnullity "transreal" numbers, they certainly form a closed interval, $[ -\infty, +\infty]$ with a well-defined first and last element. (Of course, his construction says absolutely nothing about the real numbers)

However, the set of all transreals is not linearly ordered, so I don't think it makes sense to say that has an end.

Infinity is defined as 1/x as x gets closer and closer to 0

Infinity is defined if and only if you are working in a system that defines it, and it is defined exactly how that system defines it. "1/x as x gets closer and closer to 0" is not a well-formed statement, so I should hope that no system defines infinity in that manner.

I imagine what you meant to say is that infinity is defined as
$$\infty := \lim_{x \rightarrow 0} \frac{1}{x}.$$
That's not how it's defined in the projective reals, but it's a true statement there. There is no object called "infinity" in the extended reals (though "positive infinity" is often called "infinity" as shorthand). In fact, that limit does not exist in the extended reals: it has both $+\infty$ and $-\infty$ as limit points.

And, of course, this limit doesn't exist in the reals.

he is going off the concept that 1/0 is infinity and is using it as, "the final number" when infinity is not a number, it's a concept.

To reiterate, infinity is exactly what it's defined to be. For example, there are "extended real numbers" named $+\infty$ and $-\infty$, and there is a "projective real number" named $\infty$.

 

[Alkatran]

Infinity is defined as 1/x as x gets closer and closer to 0

In general, saying some statement S equals infinity is just a shorthand way of saying for any number you pick, it can be shown S is greater than that number. 1/x as x gets closer to 0 (FROM THE RIGHT) happens to satisfy this.