I wonder, if there are no people around to eat cake, would it even exist?

He simply said, that this was his definition without stating any background work or proof. 0/0 is horse shit! He then used 6th grade math and poor logic to get an expression of 0/0 (which is his horse shit nullity concept), and he claims to have solved a 1200 year problem? What an ass-wipe…moron!

“I think it foolishly arrogant for Dr. Anderson to proclaim it as “solving a 1200 year problem” and to describe it even indirectly as a “paradigm shift.”

Yes, I agree! I don’t know what the back story was on this BBC item. There may well be some justifiable reason for going into schools and drumming up publicity with outrageous claims … trying to get kids interested in doing maths at A-level, maybe? But why get the press there? Or maybe he’s just an outrageous self-publicist.

“If both of these limits [the limit from the left, and the one from the right] are equal to l then this can be referred to as the limit of f(x) at p. Conversely, if they are not both equal to l then the limit, as such, does not exist.”

Also, lim (x -> 1) of sqrt(x) is a unique result, as it turns out. The answer is 1. If you take sqrt(x) where 0 ≤ x ≤ 1, sqrt(x) > x, and gets closer and closer to 1 as x gets closer and closer to 1.

A better example would be lim (x -> 0) 1/x, which is +∞ when approached from the right, and -∞ when approached from the left. In this case the limit isn’t non-unique, it is just nonexistent.

However, these are sort of tangential to the debate at hand! I apologize.

And I admit, I am much less opposed to the idea of “nullity” as I was at first, after having given it some though, although I think it foolishly arrogant for Dr. Anderson to proclaim it as “solving a 1200 year problem” and to describe it even indirectly as a “paradigm shift.”

“What about lim (x -> 1) of sqrt(x)? Not a unique result, surely?”

That is a good point. I think it is the case that the limit is said to exist only if the limit is the same as the number is approached from the left as it is when approached from the right, but I honestly am not 100% sure on that.

I’ve been thinking more about nullity, and as far as “messing up” the real numbers goes, it does so to a lesser extent than I first though. Aside from potential limit issues (jury is still out on that one), most other arithmetic seems to work well enough as it did before.

There are still issues, though. I’ve seen it written (I would give an exact reference if I knew it offhand, I think it was in the comments to one of the BBC articles) that 0 = 0^1 = 0^(2-1) = 0^2(0^(-1)) = 0(1/0) = 0/0 = nullity. There’s probably a way of defining things to get around this (perhaps he already has done so), but it’s a problem if left unresolved.

” … none of the old things on the smaller set break.”

That’s a good point. I need to to go back to read again what Anderson said to see if that’s a fair criticism, but at first glance what you say is a fair punch.

However, just picking up on one point. can I ask about your assertion:

“In the real numbers, I can take the limit of a function as its variable approaches some real number. Real analysis tells me that this will yield a unique result.”

What about lim (x -> 1) of sqrt(x)? Not a unique result, surely?

In any case isn’t 0/0 different; it could be regarded as the result of two limits and the result depends on how the limits progress – which one ‘gets there first’?

I came across this article by Philip Dorrell the other day:

www.1729.com/blog/ZeroDividedByZero.html

which considers a possible application of the nullity idea, but I haven’t had time to work through it all. However the motivation is interesting even if in the end I don’t feel his conclusion is supportive of Anderson’s contention.

In adding the negative numbers to the positive numbers, all the things you can do with the positive numbers still work. For example, a positive integer to the power of a positive integer will still result in a positive integer, even when the negative integers are included.

When you add the rational numbers to the integers, any things you can do with integers still work. For example, multiplying two integers together will still give an integer result. New things are allowed, such as division (dividing a number x by a number n isn’t allowable in the integers, because it’s the same as multiplying x by 1/n and 1/n isn’t an integer), but none of the old things on the smaller set break.

When you add the complex numbers to the real numbers, all the things you can do with real numbers still work. It now allows you to talk about the square root of a negative number, but all the normal arithmetic of real numbers is unaffected.

Let’s take the reals. In the real numbers, I can take the limit of a function as its variable approaches some real number. Real analysis tells me that this will yield a unique result.

Let’s look at the limit as x approaches 0 (from the right) of x * (1/x). With a little simplification, we can get x*(1/x) = x/x, and the limit of that as x approaches 0 from the right is 1.

If we throw nullity into the mix, we have the limit as x approaches 0 from the right of x * (1/x) = 0 * ∞ = nullity (0*∞ = nullity, according to Dr. Anderson).

So, we have the limit equals 1, but it also equals nullity. We’re doing legal operations only on real numbers, but ending up with a different result than if we had done it over the reals without nullity (the limit in that case would be unique).

So, adding nullity to the reals isn’t comparable to any of your examples. While it adds new things (although I would argue it doesn’t add anything new, since as far as I can see if nullity appears at all in an equation the result ends up being nullity), it breaks old things. None of the other cases you said broke anything.

- Andrew

“I think I used division-by-zero to prove that 1 = 0.”

There are various algebraic tricks to “prove” 1 = 0 or 1 = 2 which usually involve doing something “illegal” along the way, like dividing by zero. But that kind of anomaly suggests that definitions ought to be extended or changed to avoid the inconsistency. It would be interesting to see if algebra based on Anderson’s arithmetic still allows these anomalous “proofs”.

Post #81. Back in high school, I was taught that the square root of -1 was undefined. And so it is if you are only allowed to use real numbers. But once you _define_ a symbol (conventionally i or j) to denote sqrt(-1), the number line (or spectrum as timos calls it) becomes the complex number plane. And lot of new things are possible.

-2/-2 =1
-1/-1=1

1/1=1
2/2=1

Now zero(0) is a number exactly between -1 and 1, therefore 0/0=1.
As is any number divided by itself.
So, nullity is voidity.

Thus you can invent a type of PossiblyInfiniteReal and define division for that type as:
if divisor is infinite then 0 else divident/divisor

You should probably either not define other operators or make them fail if the value is infinite.

I can think of various places where this might be useful and avoid having a conditional operation.

However it’s not the same as a mathematic theorem – at least, not a useful one. And it probably isn’t something to claim as a scientific breakthrough.

[On a semi-related topic - see if you can work out the maths behind the progress bar as Internet Explorer loads a page!]

www.bookofparagon.com/Mathematics/PerspexMachineVIII.pdf

Quote (#75):

“If Anderson’s stuff is correct ( – IF – I’m not defending his results here, only trying to say that he’s not an idiot for giving it a go) – then uses will be found.”

if X (not equal to) 0 then
print(1/X)
else
print(”Nan”)

Start with the set of whole positive numbers and zero.

What is the solution to the equation x + 1 = 0? (Notice that all the coefficients belong to the set of whole positive numbers and zero). This equation cannot be solved using the set of whole positive numbers and zero. To find a solution we have to invent negative numbers. The solution is x = -1.

Now we have the set of positive and negative numbers and zero.

What is the solution of the equation 2x – 3 = 0? (Notice that all the coefficients belong to the set of whole positive and negative numbers and zero). This equation cannot be solved using the set of whole positive and negative numbers and zero. To find a solution we have to invent fractions (which can conveniently be expressed as decimals). The solution is x = -1.5

Now we have the set of positive and negative numbers, fractions and zero.

What is the solution of the equation x(squared) + 1 = 0? (Notice that all the coefficients belong to the set of whole positive and negative numbers, fractions and zero). This equation cannot be solved using the set of whole positive and negative numbers, fractions and zero. To find a solution we have to invent ‘imaginary’ (prefer: complex) numbers. The solution is x = +- j (or +- i if you prefer)

Now we have the set of all complex numbers.

What is the solution of the equation ax + 1 = 0 where a is an arbitrary complex number?

Oh! Shit! we can’t do that because there is a singularity when a = 0. So for once, we’ll just give up. We won’t challenge our way of thinking about numbers at all. Just give up.

The point is that at every stage you can make a (spurious) case for simply giving up:

Negative numbers? what kind of talk is this? Numbers are for counting things. I can have one apple, two apples, or no apples. If I haven’t got any apples, I haven’t got any apples. It doesn’t make sense to say I haven’t got 2 apples. Harrumph! The square root of minus one? Can’t be done, squire. Anyway, what’s the point? Square roots are for calculating the size of a square when you know its area. No-one would ever want to find the square root of minus one: that would mean a square with a negative area! Hahahaha!

If Anderson’s stuff is correct ( – IF I’m not defending his results here, only trying to say that he’s not an idiot for giving it a go) – then uses will be found.

And by the way, using a few lines of code like:

if X 0 then
print(1/X)
else
print(“Nan”)

is no more satisfactory an argument against the concept than

if X >= 0 then
print(sqrt(X))
else
print(“No way, squire”)

is an argument against complex numbers.