As an infinite number of people have emailed in to tell me over the past 15 minutes, there’s a maths professor in Reading who reckons he’s been teaching schoolchildren to divide by zero. I’m not saying this is necessarily unbridled nonsense, but it’s interesting, for starters, that this spectacular breakthrough has only been picked up by a local TV newshound, and is being peer reviewed by schoolchildren.

It mainly seems to involve saying that the answer to a sum where you divide by zero is “not a number”, which is, as several people have pointed out, “not a breakthrough”. Computers can tell when they’re about to get a divide by zero error, and they are generally programmed to catch it. Even Excel has this feature. Planes do not drop out of the sky. Pacemakers do not stop firing. Anyway, not entirely my field but some entertaining commentary coming in. The BBC news story is archived below (they do have a tendency to change their stories after they appear on badscience…):

www.bbc.co.uk/berkshire/content/articles/2006/12/06/divide_zero_feature.shtml

**1200-year-old problem ‘easy’
**

*Schoolchildren in Caversham have become the first in the country to learn about a new number – ‘nullity’ – which solves maths problems neither Newton nor Pythagoras could conquer.*

*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.* *“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.”*

Watch a video report from **BBC South Today’s Ben Moore**, then let Dr Anderson talk you through his theory in simple steps on the whiteboard:

Video: Dividing by zero: **Ben Moore reports **>

Video: Dr Anderson’s theory in detail >

[you can play Realplayer video without installing Realplayer: www.codecguide.com/about_real.htm ]

*Computers simply cannot divide by zero. Try it on your calculator and you’ll get an error message.*

But Dr Anderson has come up with a theory that proposes a new number – ‘nullity’ – which sits outside the conventional number line (stretching from negative infinity, through zero, to positive infinity).

**‘Quite cool’**

The theory of nullity is set to make all kinds of sums possible that, previously, scientists and computers couldn’t work around.

“We’ve just solved a problem that hasn’t been solved for twelve hundred years – and it’s that easy,” proclaims Dr Anderson having demonstrated his solution on a whiteboard at Highdown School, in Emmer Green.

Highdown pupils: ‘confusing at first’

“It was confusing at first, but I think I’ve got it. Just about,” said one pupil.

“We’re the first schoolkids to be able to do it – that’s quite cool,” added another.

*Despite being a problem tackled by the famous mathematicians Newton and Pythagoras without success, it seems the Year 10 children at Highdown now know their nullity.
*

**Ian Jackson writes:
**

*I haven’t done a detailed comparison but this marvellous new theory*

which is going to revolutionise mathematics and computation bears a

striking resemblance to the theory underlying IEEE floating point

specification. IEEE floating point already contains a value `NaN’ aka

`Not a Number’, which is used (for example) for the result of division

by zero, and programs can choose to continue processing rather than be

interrupted (with the NaNs spreading through the computation).

which is going to revolutionise mathematics and computation bears a

striking resemblance to the theory underlying IEEE floating point

specification. IEEE floating point already contains a value `NaN’ aka

`Not a Number’, which is used (for example) for the result of division

by zero, and programs can choose to continue processing rather than be

interrupted (with the NaNs spreading through the computation).

Reading University’s staff page for him is this:

with a bunch of hyped-sounding stories under `In the News’.

*But I wanted to read about this marvellous new mathematics, which
wasn’t listed there, so I looked at what is described as
“Dr. Anderson’s personal web page”*

www.bookofparagon.com

which is really quite nutty.

which is really quite nutty.

*At least there’s a link to the marvellous division by zero `paper':
* www.bookofparagon.com/Mathematics/PerspexMachineVIII.pdf

And meanwhile…

And meanwhile…

if you’re interested, division by zero does seem to have attracted eccentrics around the world over the years, most notably the excellent Theodore Rout (“And the law is their set of dividing and multiplying by zero… as long as they maintain their incorrect dividing and multiplying by zero, then they enable me to cause things to cease to exist, and that is why I have the power to do so”). Court transcripts for Rout below, they’re fantastic. And a prize for anyone who can get me the audio of the Rout hearing!

www.austlii.edu.au/au/other/hca/transcripts/2002/C4/1.html

Update 8/12/06

The BBC page now has this extra entry:

UPDATED: 11:50 GMT, 8 December 2006

“*Given the, er, light-hearted mathematical debate Dr Anderson’s theory has generated, we’re delighted to announce he will join us on Tuesday 12 December to answer questions and discuss some of the criticisms levelled against his theory of ‘nullity’. You will be able to hear in more detail from Dr Anderson on this page later on Tuesday. Many thanks for your comments.*”

Some great commentary below, the bottom line seems to be that this is not particularly new, that planes don’t drop out of the sky through divide by zero errors, that it’s a bit odd that he’s teaching this to children, and that it’s a bit odd that the BBC made it into a TV story.

For my own part, I would say I don’t think it’s very surprising that there are people like this academic out there who are, let’s say, very enthusiastic about their work. What is odd, to me, is not one man with a very slightly unusual take on their own idea: what is odd is a reporter, editor, producer, newsroom, team, cameraman, soundman, TV channel, web editor, web copy writer, and so on, all thinking it’s a good idea to cover a brilliant new scientific breakthrough whilst clearly knowing nothing about the context. Maths isn’t that hard, you could even make a call to a mathematician about it.

In fact, I would go so far as to say that it’s all very well for the BBC to think they’re being balanced and clever getting Dr Anderson back in to answer queries about his theory on Tuesday, but that rather skips the issue, and shines the spotlight quite unfairly on him (he looks like a very alright bloke to me).

I think from reading the commentary here and elsewhere that a lot of people might feel it’s reporter Ben Moore, and the rest of his doubtless extensive team, the people who drove the story, who we’d want to see answering the questions from the mathematicians.

## kim said,

December 7, 2006 at 11:16 am

Isn’t Reading also the home of Kevin Warwick, the cybernetics guy who also comes up with some, er, unusual ideas?

## Allan said,

December 7, 2006 at 11:39 am

“Amongst other things, the paper explains why the standard model of arithmetic is not valid.”

Strewth. Give the guy a Fields Medal!

## moopet said,

December 7, 2006 at 11:47 am

I can’t see the videos because I don’t have a realmedia player installed (and that’s how it’s going to stay) but this sounds far too much like just assigning a name to something to kludge it to work. Either you then ignore it for the rest of the calculations or it comes back to haunt you and “proves” that 1 = 0. But what do I know, I failed this guy’s class a decade ago

## Idolator said,

December 7, 2006 at 11:52 am

We do that in measure theory and the like all the time. There’s an entire set of algebras that can do division by zero. Indeed, there are many such sets of algebras. In general, the only thing they don’t like is âˆž-âˆž (infinity minus infinity).

## DrJon said,

December 7, 2006 at 11:54 am

In Computer Science we have a thing called type theory, which underpins the whole field. In type theory (and associated logics) there is a thing called “bottom” _|_, which in languages such as Haskell is rendered as “undefined”. en.wikipedia.org/wiki/Bottom_type. Ask a mathematician or a theoretical computer scientist why this guy is a jerk. Teaching kids set theory or (even better) type theory may be a good thing, but to feed them this rubbish is just going to cause them problems.

## superburger said,

December 7, 2006 at 11:54 am

“The theory of nullity is set to make all kinds of sums possible that, previously, scientists and computers couldnâ€™t work around.”

They can, and do. If you are writing code and are worried about a dev by zero type problem, then just include an if-else loop to catch it.

Presumably now we can divide by zero, Dr Anderson can tell us what Tan 0 evaluates to.

## rekso said,

December 7, 2006 at 12:10 pm

The paper is both trivial and eccentric, but it’s just bad science in the ordinary sense (rather than the bonkers pseudoscience that we’re used to seeing here) and contains no obvious errors. The fact that it’s been covered in the media is a rather bizarre touch.

## dbhb said,

December 7, 2006 at 12:26 pm

In fairness, the actual paper does discuss IEEE floating point arithmetic and NaN, but claims that this ‘transreal arithmetic’ is better.

Of course I’m no mathematician so I really can’t judge the validity of the work, but the guy and his ‘papers’ certainly have a whiff of PT Barnum about them, which should probably give anybody pause.

## fraggle said,

December 7, 2006 at 12:52 pm

My brother has lectures with this guy and apparently he spends all his time talking about how he’s come up with a way to divide by zero. The guy is clearly a quack. If you look at his website, the whole thing honestly reminds me of “Time Cube” (google if you havent seen it already).

www.bookofparagon.com/

Check out the “Perspex Machine” paper:

www.bookofparagon.com/Books/Perspex/Perspex.pdf

Some quotes:

The problem of how minds relate to bodies has puzzled philosophers for centuries and will, no doubt, continue to pose riddles for millennia. In the mean time, this book sets out one solution to the mind-body problem. It explains how a machine that operates on the perspective simplex, or perspex, is both a mind and a body. In particular, it shows how to use the perspex machine to build a robot that has a mind.

Glossaries usually define uncommon words in terms of common ones, but this creates a vicious circle of defining words in terms of words. This glossary attempts to break out of this vicious circle by defining words in terms of perspexes so that a developer can relate words to the physical components of a perspex robot that give the words meaning.

If this manoeuvre is successful it opens up the possibility of a virtuous circle in which everything â€“ words, robots, humans, and all physical things â€“ are perspexes. Hence, explanations and the things to be explained are all things of the same kind â€“ perspexes. In particular, there is no explanatory gap between a thing and its explanation, there is just a web of perspexes carrying the one into the other.

## GMcP said,

December 7, 2006 at 12:54 pm

“Quite simply, and shockingly, the success of IEEE floating-point arithmetic demonstrates that real arithmetic is not fit for purpose.”

I wouldn’t count on it.

## beebhack said,

December 7, 2006 at 12:59 pm

Surely critical software doesn’t fail if something provokes a divide-by-zero condition — it just handles it? Sounds dramatic; I wonder why it wasn’t challenged.

## fraggle said,

December 7, 2006 at 1:16 pm

The whole thing about saving airplanes from crashing is pretty ridiculous. If you have a look at his papers, nullity + a = nullity, so it’s about as useful as NaN is. The planes still fall out of the sky.

The real question is: how on earth did he get permission to teach his pet maths theory to school kids? I seriously doubt that this stuff has been properly peer reviewed by real mathematicians.

## Ithika said,

December 7, 2006 at 1:30 pm

What a ridiculous joke… what is it about cranks with crazy ideas trying to bypass the proper procedures and go straight to teaching in the classroom?

## SciencePunk said,

December 7, 2006 at 1:35 pm

Without having looked too hard at this, I’d agree that it’s valid but useless. It’s no different from Excel’s “#DIV/0!” handler as far as I can see.

## dolfinack said,

December 7, 2006 at 1:35 pm

Poor kiddiewinkles.

## BrickWall said,

December 7, 2006 at 1:41 pm

If the school can find time outside the over prescriptive national currciulum surely it can find something more useful, inspiring and informative than this?

## RobW said,

December 7, 2006 at 2:02 pm

Lets run through his maths for a minute:

1/0 = infinity

then

0 = 0^1*0^(-1) = 0*(1/0)

but as he states 1/0 = infinity therefore we have 1 = 0*infinity therefore nullity = 1.

idiot

## RobW said,

December 7, 2006 at 2:08 pm

sorry the middle line should read nullity = 0^1*0^(-1) = 0*(1/0) = 0*infinity = 1

i’m an idiot too

## RobW said,

December 7, 2006 at 2:13 pm

Infact I have just realised this is the WORST bit of maths ever as he claims nullity is not on the number line and therefore 1 is no longer on the numberline which messes everything up. This does all rely on 1/0 = infinity which any decent mathematician will tell you is a load of bull as nothing = infinity as infinity is not a defined number. Functions can tend to infinity, for example

f(x) = 1/x as x tends to zero

f(x) tends to infinity.

but it does not equal infinity.

Stop this man teaching maths immediately

## coracle said,

December 7, 2006 at 2:47 pm

Reading entries like this I am reminded of one of Terry Pratchett’s Diskworld books. A fairly old one about a female wizard I believe.

In it, having been to an immensly complex lecture on magic two wizards discuss what they have heard. One of them remarks to the other:

It’s not that I know more now, but that I am more aware of my ignorance.to which the other replies

Exactly, we’re ignorent in more intelligent ways.(paraphrased from memory)

Which, reading the above discussio, is precisely how I feel.

## rekso said,

December 7, 2006 at 2:55 pm

RobW: I don’t think it’s true in his system that 0*infinity = 1. He does indeed have (as an axiom) that infinity = 1/0, and multiplying by zero gives

0*infinity = (1/0) * 0

but you’re not allowed to cancel the zeros, precisely because 0/0 is nullity and not 1.

It’s certainly a crazy system, but I’m not convinced it’s actually inconsistent.

## typical said,

December 7, 2006 at 2:59 pm

RobW, you’re assuming that because 1/0 = âˆž, then âˆž * 0 = 1. But he’s already defined his way out of that (A16)… notice that his paper does not include the equality property of multiplication (it hints at the equality property of addition, in T57, though only in one direction).

## conejo said,

December 7, 2006 at 3:00 pm

Well now, I am neither a mathematician nor have I read the whole of the paper (c’mon life’s very short) so I might be about to make a fool of myself but I think maybe y’all are a bit quick to condemn.

First, he acknowledges the IEEE definition of NaN (and implicitly the Excel â€œ#DIV/0!â€ error handler. These are, after all, merely low-level traps to preempt a calculation whose outcome is undefined – they do not constitute a theoretical framework to handle division by zero in a systematic and consistent way.

Second, the fact that some of his ‘transreal arithmetic’ does not sit comfortably with commonly understood real arithmetic is no barrier either: neither does complex arithmetic. If you’re not comfortable with the concept of sqrt(-1) then it would be easy to throw stones at people who use complex arithmetic. Introducing (phi) to denote 0/0 is no more outrageous than introducing i (or j if you prefer) to denote sqrt(-1). The question is: given a set of axiomatic elements and operators can a consistent arithmetic be built? According to his discussion section of the paper, the answer is yes.

Third, would it be useful? I don’t know, but now he poses the question, I am surprised I hadn’t thought of the oddness in assuming that, say, 2/0 is regarded to be the same as 3/0. And if transreal arithmetic could be used to avoid equations screwing up because of singularities then … maybe yes.

Fourth, is he as much a quack or a crank as, say, the TimeCube stuff? Hard to say, actually unless you simply equate the complexity of mathematical notation with the in-your-face-let’s-use-bold-and-coloured-font-styles approach of cranky web sites. Yes, the guy makes large claims (“it solves the problem of how minds relate to bodies”) and is maybe a bit of a self promoter but since when was that a crime? As far as I can be bothered to look, he publishes in respectable journals and at respectable conferences.

I doubt very much that he should be teaching this stuff to kids just yet though.

## Mojo said,

December 7, 2006 at 3:17 pm

Doesn’t he realise that Theodore J. Rout has already claimed copyright in the correct way of multiplying and dividing by zero? Unfortunately he chose to try to do this by challenging the result of an election on the grounds that multiplying the votes by zero would make them cease to exist. At least, I think that’s what he was doing…

www.austlii.edu.au/au/other/hca/transcripts/2002/C4/1.html

## rjmunro said,

December 7, 2006 at 3:17 pm

This method (defining a special value to represent 1/0, and continuing processing, rather than throwing an error) is old and not much use. However, a team at Sun Microsystems have been working on “Interval Arithmetic”, have a much better system. Interval arithmatic means that you always store the maximum and minimum of any value, so if a user puts 1.0 into a box, you store 0.95,1.05 for example. When you divide by an interval containing zero, the result is an interval between the largest number you can represent in the system and the largest negative number you can represent in the system, so say it was a 6 digit integer system, the number would be 999999,-999999. Note that the numbers are the other way round (largest number first).

## john souray said,

December 7, 2006 at 3:21 pm

The links given by Fraggle are well worth following up, though the comparison to the Time Cube is a bit unfair. The Time Cube is self-evidently deranged, whereas I think a lot of this stuff is much more interesting. (By which I’m not denying that it is deranged; just that it’s not quite so in-your-face).

Dr Anderson is clearly a fan of Alan Turing, and his “perspex machines” (a charming visual image of these suggests itself, but is entirely spurious, as “perspex” is a contraction of a “perspective simplex”, and not the convenient plastic substitute for glass) are clearly conceived as a development of Turing machibnes. In fact, they are “super-Turing” machines, by which, if I’ve understood right, Dr Anderson means they can do everything a Turing machine does, and then some more as well. Amongst the extra things a Perspex machine can do is, you won’t be surprised to learn, the ability to divide by zero.

www.bookofparagon.com/Mathematics/PerspexMachineVII.pdf

Now, I’m not qualified to evaluate any of this stuff. But speaking purely as a lay reader, I find one thing leaps of the page at me. If you follow the links through you’ll find a sort of summary paper that quotes Turing at great length. I know who Turing is, of course, and have some idea of why he’s important, but I’ve never read him directly. But even in these brief quoted extracts, it shines out that Turing is a beautiful lucid writer. He is dealing with with innovative concepts at a very high level of abstraction – the analysis of the abstract behaviour of machines (I think all machines, of which computational machines are considered as a subset? – but it’s not important if I’ve got it wrong). And though it’s not what you’d call easy going, you can immediately grasp both what he is getting at, and what the purpose is. In fact it’s quite inspirational: makes me want to look at some Turing in more detail, in its original context.

I think Dr Anderson would like to think like Turing, write like him, and be respected like him. But he simply lacks that ability. For me, I simply have no idea what he’s talking about, or why either. (Well, I know why division by zero might be a problem, but whenever I’ve come across it in computing I’ve just used the practical Error trap solution suggested above.)

There may be people somewhere for whom this stuff is meaningful, and I invite them to correct me, but really, what is one to make of a paragraph like this:

“In Section 4 we simplify the perspex machine by substituting an augmented Euclidean geometry for projective geometry. This leads to a general-linear perspex-machine which is very much easier to program than the original perspex-machine. We then show how to map the whole of perspex space into a unit cube. This allows us to onstruct a fractal of perspex machines with the cardinality of a real-numbered line and space. This fractal is the universal perspex machine. It can solve, in unit time, the halting problem for itself and for all perspex machines instantiated in real-numbered space, including all Turing machines. It can implement Turingâ€™s choice and oracle machines. We also cite a proposed experiment to test the physical reality of the perspex machineâ€™s model of time, but we make no claim that the physical

universe has the properties of the perspex machine. We leave it that the perspex machine provides an upper bound on the computational properties of any physical things, including manufactured computers and biological organisms, that have a

cardinality no greater than the real-number line. This view is consistent with some other views of the physical (un)realisability of super-Turing computers.”

Of course, it’s a bit unfair to wrench a paragraph like this from the middle of a paper, as maybe some of these concepts will have been defined earlier on. But no, with Dr Anderson’s papers, it’s like that all the way through, right from the opening line.

There is a familiar phenomenon with computing textbooks where you start reading them and everything seems not only simple and straightforward, but almost insultingly so, and then suddenly you turn over and think you must have turned two pages at once because you suddenly realise you’re not in Kansas any more. Dr Anderson’s papers are a bit like that, except right from the outset. You think you must have missed an introductory paragraph.

The tragic possibility occurs to me that Dr Anderson is engaged in a sort of solipsistic reverie. He may well have glimpsed some higher truth, but is simply incapable of communicating it in any meaningful or practically useful way. If Gene Ray, the Time Cube man, is like the man in the street, shouting incomprehensible imprecations at passing buses, then Dr Anderson is like the living refutation of Wittgenstein’s theory that a private language is impossible.

If you open up some of his video links (I recommend the one illustrating the Dijkstra Perspex) you’ll get some quite impressive effects. I’d recommend Dr Anderson actually builds this machine and submits it to next year’s Turner prize, complete with one of his research papers as an explanation of the artist’s “concerns”.

## sven said,

December 7, 2006 at 4:21 pm

My experience with the

Proceedings of the SPIEdates back a few years, but I recall that it is not peer-reviewed. You send in a paper you wish to present at a conference and if the session organiser thinks it’s ok, you get a talk and it is published as is in the Proceedings.## Despard said,

December 7, 2006 at 5:00 pm

coracle, #20: the book is Equal Rites, and you’re close, but the paragraph in question runs more like:

‘they both basked in that wondrous shared feeling of being vastly more ignorant than ordinary people, who were ignorant of only ordinary things’ (paraphrased again, I’m sure someone will come up with a direct quote sooner or later!)

## Chris Surridge said,

December 7, 2006 at 6:00 pm

Mark C. Chu-Carroll over at “Good Math, Bad Math” also has a great rant on this subject: “Nullity – the Nonsense Number“

## phayes said,

December 7, 2006 at 6:15 pm

Looks like a (partial) reinvention of the wheel:

www.math.su.se/~jesper/research/wheels/

## martino said,

December 7, 2006 at 6:43 pm

Hi to everyone in bad science this is my first pot…

A couple of points:

1. Computers can divide by zero without crashing planes

Apart from catching errors by testing before dividing by zero, afterwards through detecting NaNs or by exception handling , some languages can handle this quite seamlessly. For example in J 1 divided(%) by zero is infinity (_)

1 % 0

_

and infinity-infinity is indeterminate (_.) not 0

_ – _

_.

You can still work with infinity (and indeterminate, – infinity and so on)

3 + _

_

And this is just one example from one language. Seems to be very poor reporting not to have covered any of this (albeit J is a unusual langauge but it is not alone most matrix and functional languages handle this seamlessly)

Anyway dividing anything by zero is another way of saying subtract by nothing which is inifinity not nullity.

So no revolution in computing science forthcoming and no saving planes from crashing.

2. Cant comment on the math theory, may it is valid maybe not. Still right now it is at best a tentative and unaccepted theory and should have no place in a scool curriculum. So what is he doing teaching his currently unaccepted/unknown/novel theory only (and just) published in an non-peer reviewed journal to school children!! I think a schools inspector should be called in to investigate…

3. If there was any reason to report this surely it should been because a teacher was inflicting his unproven theory on school kids. Maybe the BBC will follow up with this theme. Should he be teaching them at all?

## martino said,

December 7, 2006 at 6:50 pm

Sorry made a mistake and can’t edit my comments I should have written

Anyway dividing anything by zero is another way of saying how many times can I subtract by nothing which is inifinity not nullity.

## abahachi said,

December 7, 2006 at 6:56 pm

Chu-Carroll’s summary is spot on. It’s not a number; you can call it NaN, or nullity, or Doris, and it’s still not going to produce any useful arithmetical result. Still, a salutary reminder that it’s not only Arts graduates who are capable of stringing a lot of impressive-looking words together to produce pretentious gibberish.

## Nebbish said,

December 7, 2006 at 7:08 pm

Why is he wasting time on this when he could be teaching non-standard analysis to school children? Anyone else remember NSA?

## Nebbish said,

December 7, 2006 at 7:26 pm

…and yet it’s the Physics department at Reading that gets closed down (right on, bit of politics etc.)

## RobertK said,

December 7, 2006 at 7:39 pm

If anyone looks at the paper, section 2.1 Axioms is probably the good bit.

It lists a bunch of things you can do to ‘Nullity’ and not change it. (Adding a, or multiplying by a constant, etc.) So once you get Nullity in a calculation, it seems that you’re stuck with it.

It also claims 1/0 = infinity as an axiom, which is probably what’s wrong with the whole thing.

If there are any pure maths types about, take a look at 3.2 (Hand Proofs) and see if it makes any sense. My guess is that it will all work, (_if_ you accept the axioms), but not do anything useful.

## inicholson said,

December 7, 2006 at 8:08 pm

My problem with this is the same as the one I have with Intelligent Design (I’m a biology teacher and i don’t use the fab videos that “Truth In Science” sent me). If this is a fantastic breakthrough in maths (biology) could we publish it in a few peer reviewed journals and have it accepted by a few mathematicians (biologists) BEFORE we start teaching it to children? They tend to believe everything we say (my classes believe kryptonite has fewer oxygen atoms than kryptonate because I told them for a joke).

## Filias Cupio said,

December 7, 2006 at 8:25 pm

john souray:

‘…clearly conceived as a development of Turing machibnes. In fact, they are â€œsuper-Turingâ€ machines, by which, if Iâ€™ve understood right, Dr Anderson means they can do everything a Turing machine does, and then some more as well.’

If true, this would be a remarkable advance. We’ve spent over 60 years looking for an algorithm-implementer which is more powerful than a Turing machine, without success. The Churchâ€“Turing thesis hypothesises that there is no such scheme/device. Introductory theoretical computer science is full of proofs that lambda calculus can simulate a Turing machine and vice versa, and a single tape Turing machine can simulate a multi-tape Turing machine etc.

## phayes said,

December 7, 2006 at 8:35 pm

@Nebbish “Anyone else remember NSA?”

Yes – Abraham Robinson’s modernisation of the Newton & Leibniz infinitesimal approach by extending the reals to the hyperreals. It’s been claimed it’s easier to learn elementary calculus this way because it removes the necessity for all those convoluted Îµ-Î´ arguments in proofs. There’s a text book here:

www.math.wisc.edu/~keisler/calc.html

@inicholson

I don’t think this is even new (see my previous post), let alone a fantastic breakthrough and you are probably right about not teaching it to children: In that respect this reminds me of Wildberger’s claims about the benefits of teaching his rational trigonometry.

## Andrej Bauer said,

December 7, 2006 at 10:07 pm

I am a professional mathematician. I looked at the axioms and they make sense, and are probably right. However, these are far from being original. First, you can adjoin +infinity and -infinity to obtain something called “the extended real line, see en.wikipedia.org/wiki/Extended_real_number_line. Then you can adjoin a bottom element, see en.wikipedia.org/wiki/Bottom_element, to represent an undefined value. This is all standard and quite old. In fact, it is well known in domain theory, see en.wikipedia.org/wiki/Domain_theory, which deals with how to represent things we compute with, that adjoining just bottom to the reals is not a good idea. It is better to adjoin many so-called partial elements, which denote approximations to reals. Bottom is then just the trivial approximation which means something like “any real” or “undefined real”.

Since this blog is devoted also to bashin, err, constructively criticizing science journalism, let me just say that whoever reported on this failed to call a university professor to check whether it was really new. Any university professor would have told this reporter that there are many ways of dealing with division by zero, and that Mr. Anderson’s was just one of known ones.

## Twm said,

December 8, 2006 at 1:21 am

I’m really disappointed that he used the example or falling aeroplanes and failing pacemakers as if his new revelation would save lives.

CPUs produce the famous integer divide by zero exception which on most operating systems would mean that the current process/thread is terminated.

It is almost always a programmer bug, but if you expect a code fragment to occasionally produce a DBZ result, then you can legitimately handle this case. In fact the exception avoids having to come up with a representation of NaN in a machine word which would waste bits. Imagine the headaches of a 31bit operating system.

Integer division is what a lot of programmers have experienced as ‘sudden death’ but the bug in their code is fixed and they move on. Any sort of ‘crash’, be it divide by zero,, out of memory or disk corruption is most probably due to a lack of test coverage.

For floating point number however, IEEE FP may be implemented in software or hardware, but the common pattern is that the code continues to run on divide by zero, but the results of the calculation may not be what you expect.

You get some horrible ‘fixes’ to the problem such as adding 0.00000000001 to a variable to ensure it is never zero! Yuk.

a quick read of Design By contract should help rationalise how you would avoid checks all over the place for zero.

en.wikipedia.org/wiki/Design_by_contract

A crash is a good thing in a test environment- it is a sign that some problem has at least been detected and needs to be verified. The idea is to do enough testing to catch as many bugs as is required to make the product pass safety and quality control test.

If you divide by zero but didn’t mean to, then having the result come out as a nullity is not going to magically restore the logic in your code.

I’m sure that rounding errors silently introduced over time by iteration upon iteration of floating point arithmetic produce significantly more problems.

## TM said,

December 8, 2006 at 2:26 am

I dunno Ben, 0 certainly is a causing problem integer. It certainly wrecked havoc on the navy ship Yorktown when their computer system tried to divide by 0.

www.vnunet.com/articles/print/2100362

Also, I dont think it’s fair to equate the classroom of kids the group for peer review. I think his point is that dividing by 0 with using nullity is so simple kids can do it.

0 has always been problematic for mathematicians as well as philsophers. Aristotle killed the man who was trying to introduce it into math at that time. 0 crushed aristotles’ bivalent logic system. For more on this consult the book ‘Zero, history of a dangerous idea’

TM

## Andrew Clegg said,

December 8, 2006 at 9:40 am

The US Navy’s shoddy NT systems notwithstanding, there’s something fantastically 60s-sci-fi about divide-by-zero errors making planes drop out of the sky and people’s hearts stop.

“Everything I say is false” and the giant evil machine brain starts spewing thick black smoke etc…

Andrew.

## bootboy said,

December 8, 2006 at 12:48 pm

It’s definitely bad media reporting and the researcher in question appears to be more than happy to fan the flames of nonsense. I also think that teaching such systems to school children (or even undergraduates) is absurdly irresponsible. But it’s not bad science per se. Useless and a complete waste of time, perhaps, but not necessarily bad science.

As conejo remarks above, you can introduce imaginary entities into mathematical systems if you want. As long as your system is internally consistent and well defined, you can manipulate the various symbols according to the rules of your conceptual space, derive rules, theorems and so on. Mathematics is, after all, simply reasoning within abstract rule-bound spaces. This guy’s axioms look ok to me – no obvious internal inconsistencies and they have been presumably churned through a theorem proving bit of software to check them out to (he claims so anyway).

I mean complex numbers and their algebra are simply the result of somebody once saying “what if there was such a thing as the square root of -1, which obeyed various axioms that I’ll introduce” and working out what that particular space looked like – how you can manipulate those numbers and so on. It was only much later that people started to find things in the universe which could be mapped into this space, manipulated there, and mapped back while retaining their correspondance with the real universe throughout (e.g. current/phase of ac IIRC). String theory is another exercise in working out the rules of an abstract space which includes various imaginary entities, although the mapping between that space and any part of the real universe looks destined to remain forever speculative.

Having said all that, what marks out this algebra from those mentioned above is that this one is almost certainly destined to remain forever useless. The problem being that both “infinity” and “nullity” are not quantifiable. i.e. 2*inf = inf and 2*nul = nul. This means that as soon as you introduce either of them into an equation, they’re going to swallow everything else and reduce them to a single unquantified entity – pretty much exactly the same as the NaN does. I really don’t think there’s any way around this and I can’t possibly see it having any application in the real universe whatsoever.

Contrast this with complex numbers where 2i != i and string theory, where values in the various dimensions can be compared.

So, in sum, while the mathematics looks okay (I couldn’t be bothered working through it in detail), it’s almost certainly worthless. It is not surprising that it has been produced by somebody who also claims to have created a super-Turing machine which can solve the halting problem. Yeah. Right. Not too far off gibberish if you ask me.

## oneiros said,

December 8, 2006 at 1:09 pm

Update on BBC Berkshire:

Given the, er, light-hearted mathematical debate Dr Anderson’s theory has generated, we’re delighted to announce he will join us on Tuesday 12 December to answer questions and discuss some of the criticisms levelled against his theory of ‘nullity’.You will be able to hear in more detail from Dr Anderson on this page later on Tuesday.

Go get him! 😉

## martino said,

December 8, 2006 at 1:59 pm

john souray:

â€˜â€¦clearly conceived as a development of Turing machibnes. In fact, they are â€œsuper-Turingâ€ machines, by which, if Iâ€™ve understood right, Dr Anderson means they can do everything a Turing machine does, and then some more as well.â€™

Like solve the Halting Problem? Dont know what the meaning of a super-Turing machine could be unless it at least solves the Halting Problem. If he was claiming this then that would be some huge advance, somehow I dont think nullity gets you there except to nullity itself.

## killary45 said,

December 8, 2006 at 1:59 pm

The problem here is that Dr Anderson is computer scientist and not a mathematician. He is assisted in his research by two other computer people and chooses to publish his work in the journal of the Society of Photo-Optical Instrumentation Engineers rather than a specialist mathematical journal.

If he had consulted mathematicians or submitted his work for peer review to a maths journal he would not have embarrassed himself in public in this way. “Bootboy” has given a very clear explanation of how the system is neither new nor interesting nor useful. I am absolutely certain that no other mathematicians will find his work of any merit at all.

For Anderson to teach a partial and simplified version of his system to school children is ridiculous. What does he hope to achieve by this except confusion?

The BBC has been made to look foolish by this feature.

## martino said,

December 8, 2006 at 2:15 pm

I think I think we are confusing the content for the context.

I think the fundamental issue here is that this is very bad journalism:

a) Bad reporting – no facts checking

b) Bad editing – allowing this be published without fact checking

c) Bad journalism – surely the real story is that a teacher is allowed to teach his pet theory in school. Whether invented by him or not ,it is still unproven to date, regardless of it possibly being useful in the future or not. If we allow this why not let any teacher introduce thier favoutire theories in thier field – creationism, intelligent design, alternative medicine???

## Delster said,

December 8, 2006 at 3:22 pm

ok… my mathematics is not great but in the white board picture abouve he has zero divided by zero with nullility as the result.

Correct me if i’m wrong but would the answer not be 1 rather than 0 as zero is equal to zero and hence would be divisable by itself precisly once?

## conejo said,

December 8, 2006 at 4:07 pm

Delster, there’s a touch of Humpty Dumpty about this (and it’s no surprise that Lewis Carroll was a mathematician):

“When I use a word,’ Humpty Dumpty said,’ it means just what I choose it to mean, neither more nor less.’

‘The question is,’ said Alice, ‘whether you can make words mean so many different things.’

‘The question is,’ said Humpty Dumpty, ‘which is to be master – that’s all.’

In the usual sense of the word divide, we take it to be a representation of the everyday concept of division: I have one cake and I divide it into three pieces, giving me 3 pieces, each 1/3 of the whole. In that context it doesn’t make sense to ask what it means if I divide my cake into zero pieces. And if I haven’t got a cake at all, it makes even less sense to divide it into zero pieces. In this sense (both in every day (for dealing with cakes) and ‘ordinary’ arithmetic) it doesn’t make sense to divide any number of cakes (1,2,3, … or 0 or infinity ) into zero pieces – hence division by zero is undefined whatever the numerator is.

But we are in fact masters of the words, like HD, so we can redefine division to mean something else if we want, as long as we don’t try to relate it to cakes. Or at least we only try to relate it to cakes as a special case. This is exactly what happened to the concept of division (and all the other arithmetic operators) when they were extended to deal with complex numbers.

Sorry to keep banging on about complex numbers but I think the analogy is a good one.

## cribbins said,

December 8, 2006 at 4:47 pm

Martino – Yes, Anderson claims his universal perspex machine can solve the halting problem for itself, all perspex machinges and all Turing machines:

www.bookofparagon.com/Mathematics/PerspexMachineVII.pdf

How this sits with Church’s thesis, I don’t know; it’s not my area.

## bootboy said,

December 8, 2006 at 5:21 pm

That perspex machine sounds like a curious beast. I read the introduction and was confronted with some truly bizzare claims:

“The perspex machine can model all physical things, including mind, to arbitrary accuracy”That’s a rather grandiose claim. My understanding of complexity is that the only 100% accurate model of a complex system is the system itself. But apparently the perspex machine can produce a model of the universe to any level of accuracy!

The following is even worse:

“We then develop the Walnut Cake Theorem that shows that, under very weakassumptions, numerical bounds generally tighten non-monotonically. The practical

consequence of this is that the performance of geometrical machines generally

improves non-monotonically, with repeated relapses in performance, and,

conversely, degrades non-monotonically, with repeated improvements in performance. As the human brain is a geometrical arrangement of neurons this

explains why human learning is non-monotonic, and why progressive brain illnesses

show repeated cycles of remission and relapse. It also explains why science

progresses via a sequence of non-monotonic paradigm shifts.”

Amazing. We go from numerical bounds (bounds on what? what is causing them to tighten?) to brains to the history of scientific thought, all explained by one machine. The first and the second concepts are linked by being “geometrical” whatever the hell is meant by that, the link to the third is mysterious.

Funnily enough, there does seem to be a tendency in mathematicians and computer scientists who spend too much time contemplating abstract spaces to lose all touch with reality and this looks like a really prime example.

## oneoffmanmental said,

December 8, 2006 at 5:51 pm

P=NP!!!!!

## Nebbish said,

December 8, 2006 at 8:22 pm

His book on perspex androids makes most stimulating reading, particularly the chapter on spirituality, which has questions at the end, such as:

“If you are a lay member of a Christian church, would you accept Holy Communion

from a woman? Chimpanzees differ from humans in less than 10% of

their DNA. Would you accept Holy Communion from a chimpanzee? Would it

make any difference to you if the chimpanzee were male or female? Would you

accept Holy Communion from a genetically engineered human that differed

from you in more than 10% of his or her DNA? Would you accept Holy Communion

from an android?”

www.bookofparagon.com/Books/Visions/Visions.pdf

## undergrad said,

December 9, 2006 at 12:05 am

this might be a silly comment, but this sounds like a sort of number dualism to me.

## jackpt said,

December 9, 2006 at 2:51 am

I hate commenting on things like this for fear of saying something inaccurate or wrong, but I believe this is one of those mathematical things like a new sphere that are difficult enough to comment on after some thought. My head hurts.

## ceec said,

December 9, 2006 at 4:02 pm

Thanks, Nebbish (post 54) for the extract. I like the implied spiritual descent from men to women to chimps. Maybe it makes sense if you read the rest of the book but I gave up about 5 seconds after I read the chapter titles (e.g. Chapter 8 Free Will, Chapter 9 Intelligence, Chapter 10 Feeling, Chapter 11 Time etc. etc.)

## cribbins said,

December 9, 2006 at 6:49 pm

Bootboy – As far as I can tell — and I haven’t taken the time to read all of his papers and online posts — the guy appears to be doing valid pure mathematics research while making grandiose claims for its potential usage, both philosophical and computational. This is not a unique phenomenom in mathematics. The only real curiosity is his apparent fascination for dividing by zero; it’s a strange element to boast about, as producing new number systems which contain the real numbers as a proper subset is not exactly the sort of thing one cites as ‘big and clever’ these days. We’ve been there and done that, several times over.

## Leonor said,

December 9, 2006 at 10:18 pm

What did he solve, I don’t get it. Didn’t he just make up a symbol for it?

## Ephiny said,

December 10, 2006 at 12:40 pm

That’s the impression you would get from the article, but read his actual paper (link above) and he’s actually done rather more than that. Whether he’s done anything sensible or useful is another question, of course, but it’s a little more than just saying ‘Let’s call it ‘phi”.

## Ephiny said,

December 10, 2006 at 1:02 pm

“Also, I dont think itâ€™s fair to equate the classroom of kids the group for peer review. I think his point is that dividing by 0 with using nullity is so simple kids can do it.”

I”m not sure about this…yes it is quite simple, but I don’t see how the children can understand it properly without having covered things like set theory and a bit of real analysis (axioms for the real numbers) and as far as I know these subjects aren’t really covered until undergraduate level.

And maybe it’s just me, but I think I would have just been completely confused by this if I’d had to learn it in Year 10 (age 14?).

## dbhb said,

December 10, 2006 at 1:30 pm

I think the issue of whether he should be ‘teaching this to school kids’ is a problematic one.

While I was a PhD student (of genetics) I became a ‘Science and Technology Ambassador’ with the SETpoint scheme and did quite a few talks at schools. The most common one was about alien life. It was meant to try and get kids interested in a whole bunch of stuff, from simulations of star formation, to very basic biochemistry and biopoiesis, to simple concepts like divergent and convergent evolution, and yes even the pseudoscientific Drake equation.

Was the talk scientific in the strictest sense? Was I qualified to talk about each and every aspect? Probably not, on reflection. Was it worth doing? Did it get the kids interested in learning more? Well I don’t know that either, but I hope so.

I don’t know the circumstances surrounding this guy’s talk, but given that he’s a university academic it was probably a similar sort of deal, not something he does every day. Let’s not crucify the guy for trying to get kids interested in maths without being a bit surer of our facts.

## Huang said,

December 10, 2006 at 5:00 pm

Dr James Anderson makes a bold attempt to understand the trivial. He fails, but I say bravo for the effort. All he did was generate nonexistences, but at least he tried.

The reason I say bravo is because everybody knows that there are some wierd things in math and physics which remain unexplained. Something fundamental is missing. People want to understand, and they have every right to.

This is what Dr. Anderson missed.

There are 3 distinct existential types.

1) That which exists.

2) That which does not exist.

3) That, for which existence is indeterminate.

Trivials fall under category (3). Trivials are generated as a natural consequence of uniqueness and existence.

A trivial cannot be generated by dividing by zero. Division by zero will result in things which do not exist.

For example:

The number 1 which is equal to 2 is an example of something which “does not exist”.

The “round square” does not exist.

However, the number 1 which is identical to the number 1, and yet is distinct from the number 1, this number is TRIVIAL .

For example:

It is impossible to determine if an object is really itself, or if it is a trivial clone of itself. As long as uniqueness is not actually violated, you cannot distinguish between the actual artifact and the clone. The distinction is trivial. Actually violating uniqueness results in a type(2) nonexistence. But, if uniqueness has not been violated there is no way to determine if you are looking at the original object or an identical twin.

Trivials are NOT strictly nonexistent. They are, however, strictly speaking “Indeterminately Existent”.

Ayn Rand is spinning in her grave, but that’s the truth. There are 3 distinct existential types or forms. Existent, nonexistent, and trivial.

Ayn Rand’s premise “Existence Exists” seems incomplete, because there are some things for which existence is “indeterminate”.

Dr. Huang Xien Chen

## Lazzer said,

December 10, 2006 at 7:36 pm

Delicious science, I have laughed a lot. Right, there is nothing new. I cannot imagine how he became professor. I have read his paper “PerspexMachineIX” and I think it is complete nonsense. He pretends to have something newer and better than hundreds of years old theories. It is not that this could not be, but both the proof and the effect are zero.

He mixes up some mathematical analysis with computation theory. Computation theory deals with complexities of functions. The common term ‘feasibility’ is used to describe the neccessary effort to calculate the result of a function. For example the sqrt(2) is an irrational number thus cannot be represented in discrete systems. Descrete systems have a limited number of states. The feasability lies in computing an arbitrary precise result which approximates the irrational state. The Heron approximation of the squareroot takes a view steps to get a useful result.

The nullity cannot enhance any calculation, neither in speed nor in precision. The nullity is a symbol as every other symbol, you can choose greek, latin, hebrew, kyrillic, , ancient or extraterristic ones. Your computation power does not change, whatever symbols you try to use. A symbol just represents a state. And a state machine is a machine that transforms one state into another state. The machine does not care what symbols you use for representation.

And the problem with this phi is, that it never can solve any calculus. Consider it as a state, table 1 and table 2 of the “PerspexMachineIX” as transition matrices. You will discover, that state phi is terminal. The state phi is terminal because you can get into state phi but you can never get out of that state. You just stuck in that state. Symbolically there exists no reduction rule for phi. Once you have the phi in an equation you’ll never get rid of it. This means you never can get a conventional equation out of it.

It is proven that phi cannot solve any additional equation and the perspex machine is touring equivalent. Remember, combine two scientific words and claim to have some historical problem solved and you get in the tv.

## Robert Carnegie said,

December 10, 2006 at 11:56 pm

The point of a Turing machine is that any stepwise computing machine is mathematically equivalent to a Turing machine. The X machine cannot compute anything that a Turing machine can’t compute. You may need to perform a mechanical translation on a program for the X machine to produce a program for the Turing machine, but the Turing machine can do the translation. Alan Turing PROVED that and then he KILLED himself, so it’s disrespectful to walk all over his grave.

As for dividing by zero and inventing a new number to do it… division isn’t a natural mathematical operation. Division is a convenient label for an inverse function of multiplication; it relies on a property of rational or algebraic or real numbers, that for a number x there is another number y so that x multiplied by y equals 1, and that to “divide” w by x you multiply w by y, to find an answer z = wy and x times z = w.

This does not work when x is zero, because whatever y is, x times y is going to be zero. If you invent a number that, multiplied by zero, isn’t zero, it breaks the whole system.

I learned this at university you know… twenty years ago though. It actually probably doesn’t even show now.

As others have pointed out already, computer programs such as spreadsheets, databases, and interpreted or compiled languages usually have an exceptional case for dividing by zero. A spreadsheet will store an error code to indicate any of several conditions which are usually regarded as faults. You probably appreciate that cells in a spreadsheet program contain formulas that depend on the values of other cells. When a formula depends on a cell that contains an error code, the formula result is also an error code, generally the same one. So the fault message is transmitted all through your formulas and you are prompted to investigate where it came from. I think it’s possible to write a special formula that disposes of error codes.

A modern database has at least one special result, NULL, which is not a value but usually represents “Don’t Know”, although that is up to the programmer. It is not a proper representation of division by zero, but an attempt to divide by zero probably will produce a NULL, or else a program-stopping exception. Again, most operations and formulas that receive NULL as one of the inputs will have NULL as output, although there are exceptions.

And other programs will treat division by zero as an error which typically breaks off the sequence of execution and either terminates the program at once, or executes an error-handling routine that “cleans up the mess” in some sense. After all, if you’re typing in Microsoft Word, you don’t want to have the program disappear and to take the document you were typing away as well, just because you typed a formula with division by zero.

But a divide-by-zero symbol has no proper place in school mathematics.

Perhaps the story is very badly misreported and the school lesson described was to make exactly that point – to put this firmly in the category of “mistake” for students of varying ability.

More than twenty years ago, I saw a teacher “prove” that any two angles of a triangle are equal. I think this was done by sleight of hand in the diagram equivalent to changing the “sign” of a term in an equation from “plus” to “minus”. I forget now if the lesson was “Trust no one” or “I hate teaching lessons after the exams”.

And many of us have seen a teacher “prove” that 1 = 0 or 1 = 2 The simple ways to do this usually include, well well, dividing by zero. But you don’t call it that, you write an equation that includes maybe (x – 2y) on each side of it and you score out (x – 2y) to leave 1 = 2. But, you’ve guessed, x = 2y, so x – 2y = 0, and dividing the formula by zero is cheating. Again, the message of the lesson…

On the other hand, there also is a long history of people who don’t understand mathematics, or don’t understand science, of inventing a new mathematics or a perpetual motion machine or, in a famous spectacular case, a legal bill to declare a new value of pi. Fact and fancy vary mostly on exactly how far this measure got in one American legislative chamber, and with what prospect of actually being enacted.

## Ephiny said,

December 11, 2006 at 12:51 pm

“As for dividing by zero and inventing a new number to do itâ€¦ division isnâ€™t a natural mathematical operation. Division is a convenient label for an inverse function of multiplication; it relies on a property of rational or algebraic or real numbers, that for a number x there is another number y so that x multiplied by y equals 1, and that to â€œdivideâ€ w by x you multiply w by y, to find an answer z = wy and x times z = w.

This does not work when x is zero, because whatever y is, x times y is going to be zero. If you invent a number that, multiplied by zero, isnâ€™t zero, it breaks the whole system.”

This is true for the real numbers, as you say, however the point of the paper was that it defines a new set of ‘transreal’ numbers where the operation we call division is NOT defined as the inverse of multiplication but as multiplication by the reciprocal, where the reciprocal of zero is defined. So within this system, it is possible to divide by zero. Of course you still can’t divide by zero in the real numbers, but he never claimed that you could!

Again, it’s up for debate whether this was a sensible thing to do or will have any useful applications in computing. But it is NOT just the same thing as assigning a new symbol to an undefined value. I don’t even want to be in the position of defending this guy, I just wish people would read the paper (instead of the BBC article) before commenting, and that we could have a discussion about the actual issue.

## conejo said,

December 11, 2006 at 2:07 pm

Ephiny – bravo.

## rueroy said,

December 12, 2006 at 5:39 am

I think that the major bad science being spouted by Anderson (aside from his needless confusion of lots of high school children with something that will either never be of any use or they’ll have to unlearn the moment they enter a calculus class) was saying was the implication that his arithmetic would save your plane from crashing or your pacemaker from failing. Once our autopilot has determined that we’re heading at a bearing of Nullity degrees, what position is it supposed to set the rudder to fix the problem?

Incidentally, I don’t know what programming language Twm is using, but I know that in Java and C (and I expect in every other language), on any operating system, a division of a floating point number by zero will not result in the current thread or process being terminated. The result is +/- infinity or NaN (depending on whether the numerator was positive, negative or zero. Integers are a different matter, but then the +0.00001 trick doesn’t work there, so that doesn’t seem to be what he or she was talking about. I expect that he or she is getting a little confused with dereferencing a null (or zero) pointer.

## MJ Simpson said,

December 12, 2006 at 12:05 pm

I have invented a new letter, Ibob, which sits outside the alphabet and allows me to spell lots of new words. Who do I speak to at the BBC?

## cribbins said,

December 12, 2006 at 12:35 pm

Robert Carnegie said, “division isnâ€™t a natural mathematical operation.”

You’ve never encountered the concept of profit-sharing, have you?

## tg said,

December 12, 2006 at 7:08 pm

He has replied to the nay-sayers. Unfortunately it requires realplayer and working speakers on the computer, so I don’t know what it says. But for those with both, and 18 minutes (!) to spare, here you are:

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

## Ben Goldacre said,

December 14, 2006 at 3:02 pm

heh, some mate/colleague of the journalist in question has replied to the naysayers to, and he is, to quote the great homeopath Dr David Spence, “eggy”.

i’ve reproduce the joy here for posterity:

www.dayorama.com/archives/002155.html

In other news we’ll pop back to the maths story briefly for some stats. Today I discovered that, in its first full day online, the article’s two videos were viewed approximately 51,000 times. To give you some idea of scale, the third-placed piece of audio or video in our list – behind the two maths videos – was accessed 31 times. Our servers shifted over 100 gigabytes of maths video to the world that day.

I’m also pleased to see that the comments to my follow-up, while still largely disagreeing with Dr Anderson, have been of an altogether far higher class of literacy and numeracy. Clearly the folk with enough interest in the issue to come back for the follow-up are the ones with something to contribute – and if that contribution is to disagree strongly then that’s fine by me (I’m conscious of the opinion, in certain places, that we’ve somehow already made our minds up that Dr Anderson is 100 per cent correct). Given our local university’s refusal to put anybody up to challenge Dr Anderson, it’s vital that people have written in.

It’s disappointing how hypocritical some people can be when taking time out of their busy schedules to criticise others. I came across the blog of a Guardian science writer (Ben Goldacre) earlier in the week, a man who took it upon himself to write the following:

What is odd is a reporter, editor, producer, newsroom, team, cameraman, soundman, TV channel, web editor, web copy writer, and so on, all thinking itâ€™s a good idea to cover a brilliant new scientific breakthrough whilst clearly knowing nothing about the context. Maths isnâ€™t that hard, you could even make a call to a mathematician about it.

Now if you’re going to accuse somebody of being inaccurate and knowing nothing about a subject, it helps to get any references to that person or organisation right. Of our Guardian science writer’s list of people who could have stopped this apparently heinous crime against science (for which read: local “and finally” story) being published, very few actually exist:

* There was no cameraman or soundman – our reporter is a video journalist, he does all of that himself.

* I wrote the copy for the web but there was no “web editor” involved (how many people does he think work for us? I’m running the site entirely on my own this week!)

* Of the list of editor, producer, newsroom, team and TV channel, I can grant that an editor and producer are involved in getting the report onto TV (but importantly, not involved in getting it onto the web). But what’s this “newsroom” and “team”? How is a “team” different from a “newsroom”? What is the TV channel supposed to do about it? The TV channel is the end product – it’s a thing, not a person! BBC1 can’t just stop transmitting if it realises it’s broadcasting a maths report it doesn’t fully agree with, it’s a bloody television set!

The point here is that it’s all well and good picking us up for not knowing our mathematics inside out – but, if you’re a science writer writing about journalists getting things wrong (in your view), you can’t get lots of things about the journalism in question wrong. It devalues your entire argument. You could even make a call to a journalist about it.

## Robert Carnegie said,

December 15, 2006 at 2:32 am

Ephiny, #67:

(i) Division is the inverse of multiplication.

(ii) The reciprocal is the multiplicative inverse.

(iii) There is no reciprocal of zero.

(iv) Anything multiplied by zero comes out zero. That is what zero is -for-.

(v) If Andersonation is not division in “real numbers”, but an alternative operation in an alternative number domain, then andersonation by zero is not division by zero. If flibble is the andersonative inverse of zero, so that zero andersonates with flibble to produce 1, this tells us nothing about division.

(vi) When I got to university the first mathematics lecture I attended appeared to be nonsense. It then turned out that it actually was nonsense because the real lecturer turned up five minutes into the show and the joker gleefully fled. In this case a walkout seems potentially more dignified.

## conejo said,

December 15, 2006 at 9:54 am

There is a rationale for the invention of ‘new numbers’ which is derived from the inability to solve equations which can be formulated but not solved using an existing set. It goes something like this (forgive the repetition – I want to establish a pattern):

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.

## conejo said,

December 15, 2006 at 9:58 am

Bugger, couldn’t post the “less than” and “greater than” symbols together because they look like HTML tags! the first bit of code should read:

if X (not equal to) 0 then

print(1/X)

else

print(â€Nanâ€)

## phayes said,

December 15, 2006 at 10:40 am

So what is the solution to 0x + 1 = 0? 😉

## conejo said,

December 15, 2006 at 11:22 am

Dunno, but you could try:

www.bookofparagon.com/Mathematics/PerspexMachineVIII.pdf

😉

Quote (#75):

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

## Robert Carnegie said,

December 16, 2006 at 10:50 pm

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

## richard_p_auckland said,

December 20, 2006 at 2:56 am

The thing with computer programming that’s different from other fields of endeavour is that you are typically dealing with a bounded problem. So it’s often valid to make an assumption that’s correct in the context of the problem but wildly wrong generally.

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!]

## T_or_F said,

December 28, 2006 at 11:55 pm

Analysis:

-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.

## timos said,

January 14, 2007 at 6:27 am

Back in highschool we were taught that, say I had the simple formula x = 4/y. when y = 0 x is undefined. Does undefined equal infinity? From my (probably misinformed) understanding undefined means no possible value within the number spectrum (as is the case in my example formula), while infinity is a value so large it cannot be represented, however it still exists compared to undefined. Hmmm.. now I think over that that dosen’t really make sense… Anyone care to offer an opinion on infinity vs. undefined?

## conejo said,

January 15, 2007 at 9:00 am

Post #78: Robert says:

“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.

## Andrew_R said,

February 7, 2007 at 3:12 am

In response to #74, conerjo:

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 * âˆž = nullity (0*âˆž = 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

## conejo said,

February 7, 2007 at 8:43 am

Andrew,

” … 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.

## Andrew_R said,

February 8, 2007 at 1:00 am

conejo,

“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.

## Andrew_R said,

February 8, 2007 at 1:08 am

From Wikipedia (en.wikipedia.org/wiki/Limit_of_a_function):

“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 â‰¤ x â‰¤ 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 +âˆž when approached from the right, and -âˆž 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.”

## conejo said,

February 8, 2007 at 9:41 am

Andrew:

“I think it foolishly arrogant for Dr. Anderson to proclaim it as â€œsolving a 1200 year problemâ€ and to describe it even indirectly as a â€œparadigm shift.â€

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.

## urbane legend said,

February 20, 2007 at 1:10 am

I’m no mathematician: let’s get that straight. But shouldn’t a number divided by zero = the original number? isn’t 1 cake divided among zero people 1 cake?

## JL said,

March 10, 2007 at 6:50 am

I am a mathematician, and his nulity concept is pure horse shit! This is some kind of joke; he can’t be serious! He defined 1/0 as infinity, and then -1/0 as -infinity. The limit as the denominator approaches 0 is infinity huge, but 1/0 is simply undefined! This is an error on his part. He then pulled the nullity concept out of his ass an said, let me define nullity as 0/0…

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!

## Nonformation said,

July 12, 2009 at 8:35 am

Let us consider the definition of space, where a point has the dimensions (0,0). We are told that there are an infinite amount of points between any two points, (a,b) in real space. Thus, x times infinity equals a real number, where x=0. If, therefore, x/0 is nullity, then the universe is null space and does not exist. This is preposterous, because “time is” and “we are,” meaning that existence exists as opposed to existence does not exist. Or does existence not exist? In which case he does not exist, and neither do perspex machines.

Does a singularity exist in 0/0=nullity?

What is more, you cannot treat infinity like a quantity, for it is not. Unless you think that you can define infinity, which is like saying you can define that which by definition is beyond definition. Consider the age old paradox of definition of God. To define god is to limit that which cannot be limited, which is blasphemy (thus the commandment not to make idols, supposedly). Not to say that infinity is God, but the two have often been equated.

It sounds like this guy is trying to redefine zero in terms of nullity, which is not a new concept at all. What is more, this guy is trying to say what does and does not exist is mathematical terms, which clearly state that nothing exists! So, how can nothing actually exist?

The answer is simple: Nothing does not exist (0/0 is undefined), and if it does, then the universe does not. Now, think about this real hard: Do you exist? Easy.

## Nonformation said,

July 12, 2009 at 8:39 am

****what does and does not exist IN mathematical terms…

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ed hardy kidsed hardy kids## Linklothos said,

June 30, 2010 at 4:40 am

Urbane Legend said: “But shouldn’t a number divided by zero = the original number? isn’t 1 cake divided among zero people 1 cake?”

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

## Peter Specht said,

November 26, 2014 at 2:40 am

I have a solution to this problem.

Take, a = 2a = b.

a/b = 2a/b = 1

1 = 2a/b = 1

2a/b = 1

b = a

2a/a = 1

a = 0

2a = 0

0/0 = 1

Zero divided by zero is equal to one.

When we divide one by a number that is less than one, the decimal exponent of that number is equal one to the one plus the number of zeros put after the decimal point, and that

shows that dividing one by a numbers less and less than one

becomes higher as the divisor approaches zero.

1/.1 = 10

1/.01 = 100

Therefore,

1/0 = infinity

## Peter Specht said,

November 26, 2014 at 2:43 am

Multiples of infinity are relevant.

For instance, the length of distance forever to the right of your right hand is, infinity m.

The distance forever to the left of your left hand is also, infinity m.

Added, this is, 2infinity m.

## Peter Specht said,

November 26, 2014 at 2:50 am

It’s not correct to refer to zero divided by zero as nullity. According to my practice of mathematics, zero divided by zero nominally equals one (like when you’re hunting a deer and trying to focus on the target). Nullity implies zero. A number divided by itself is nominally equal to one no matter what that number is in my opinion.