http://www.bbc.co.uk/berkshire/content/articles/2006/12/12/nullity_061212_feature.shtml
'Nullity is a number, and that makes a difference' By Ollie Williams
University of Reading academic Dr James Anderson insists his theory of
nullity is 'revolutionary', despite criticism from all quarters
following a video report in which he suggested a new means of dividing
by zero.
Here, Dr Anderson answers some of the comments we received following
last week's video report.
"Nullity is a fixed number with value 0/0," Dr Anderson tells us. "It is
not undefined, it is not indeterminate.
"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."
We invited Dr Anderson to the BBC in order to follow up the 1,000-plus
comments and criticisms we received in the wake of an article published
on 6 December 2006, entitled: '1200-year-old problem "easy"'.
What is nullity?
"To be quite precise: I am saying that the number 0/0 is a number. It is
a fixed number, not an undefined number or anything like that. That is
different to what goes on currently in computing, and in mathematics."
Has your theory been peer-reviewed?
"The work was developed over ten years, it's been peer reviewed and
reported in seminars in mathematics and computing departments in the UK,
and it's been reported at a learned society.
"The work has been proved consistent twice, by hand, by me, and has been
checked at another university by computer.
"The work arises from computing. There were problems that couldn't be
solved using the existing floating point numbers. It's been a long
process of solving a practical problem.
"It's now in the form where it can be submitted to mathematics journals
- it has been a long journey from computer science to mathematics and
that's entirely normal."
Have any of the comments received caused you to rethink your theory?
"I have examined all of the comments and over a hundred counter-proofs
to my work.
"Each was incorrect except one, which challenged a clause in equation 10
of the analysis paper. That challenge is entirely correct but it does
not change the substance of what I said in public, and I had a second
published theorem which establishes the result as well.
"As far as I can see the work is sound, it is computing work, and now it
can be developed as mathematical work. It is entirely normal for the
work to be controversial when it is developed in other subjects before
it is accepted by mathematicians."
Isn't this just NaN ('not a number'), a device in use for decades, under
a different name?
"NaN is, as it says, not a number. Nullity is a number - that makes a
difference. It is a paradigm shift in the way you think. If you think of
0/0 as a fixed number it changes the way that you do calculus.
"It is true that the IEEE float standard defines NaN, which deals with
exceptional cases on the basis that 0/0 and various other things are
undefined or indeterminate.
"That's difficult, the IEEE standard defines some unusual behaviour. For
example NaN is not equal to NaN - that makes sense if you think NaN is
an indeterminate value, but not if NaN is a fixed number.
"It's much more natural for programmers to think of variables as being
equal if they're identical.
"With IEEE float it is not entirely clear what you're supposed to do
with NaN as an argument to a function. Are you required to return NaN as
a result or may you return something else?
"That is perfectly clear with my arithmetic - nullity is just a number,
you can use it in arguments, you can return any value you like. The
semantics is simpler, it's clearer and easier for programmers to handle."
What can you achieve with nullity that you can't with an error message
on a calculator?
"Nullity has a precise arithmetical value. The trans-real arithmetic is
total, and complete, and contains real arithmetic as a sub-set.
"You can calculate values with nullity and those are meaningful. The
arithmetic is simpler than IEEE-float.
"Trans-real numbers I have defined to be the real numbers augmented with
plus infinity, minus infinity, and nullity.
"What I have done is to take algorithms from arithmetic that happen to
work for division by zero, collected them together, developed them as
algorithms, proved that they're consistent, then axiomatising it and
proving it by computer."
Can you express nullity in binary?
"There are many, many ways of coding these numbers in binary, and I've
done it. If anyone doubts me I can hit them over the head with a
computer that does it."
Why do you think so many mathematicians have so much trouble accepting
your theory?
"I say that 0/0 is a fixed number and mathematicians are entirely used
to thinking of it as undefined. I take a different stance and I believe
I can maintain that stance. I'm prepared to step into the mathematical
arena and argue my case there.
"If you are used to thinking of programming in terms of the real numbers
then you will only be able to think of nullity as an exceptional state.
That's undeniably true and is the way most programmers and
mathematicians think.
"If, however, you make the paradigm shift and accept that 0/0 is a fixed
number, then new equations become possible."
--
Ronsen
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