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Dr James Anderson demonstrates his theory
Dr Anderson demonstrating his theory

'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"'.

Listen in full to Dr Anderson responding to comments and criticism with BBC Berkshire's Ollie Williams:
audio Listen: Dr James Anderson (18 minutes) >
Audio and Video links on this page require Realplayer
Read the original report and watch the video here:
1200-year-old problem 'easy' >
Read more about Dr Anderson's theory on his personal website:
Dr Anderson's website >
The BBC is not responsible for the content of external websites

Presented below are Dr Anderson's answers to a selection of questions asked by many people following that article.

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?

Whiteboard showing the symbol for nullity (bottom)
Dr Anderson's symbol for nullity (bottom)

"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."

Read more about Dr Anderson's theory on his personal website:
Dr Anderson's website >
The BBC is not responsible for the content of external websites
last updated: 12/12/06
 
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costis
yes ...it doesn't..but only if you are using transreal numbers...i think that transreal calculus is under construction for the time being:P

shivakumar and saravana kumar
It is ambiguous,if is true the differentiation becomes false.It does not create a continuous curve in graps

Bruce
As the number line approaches zero .0(bar))and -.0(bar the result of self division remains 1. The answer is 1 or you can't get there from here.

Quackinator
Hey doc: So you say that nullity is a number; could you then please tell me an example of a larger number than nullity, or a smaller number than nullity? Kindly note, current accepted 'exceptions' to real numbers, namely, positive and negative infinity, have either majorants or minorants, that is, they properly fit with the strict ordering of the reals. Where does nullity stand? Is it greater than 1? Less than minus 10? What?

Raul
Something to keep in mind: mathematics is about self-consistent axioms, and not all self-consistent systems of axioms need to be consistent with other self-consitent axioms.Personally, for practical work with computers, I prefer using 0 as the result for 0 divided by 0. Of course, this is just one of many possible answers. Nonetheless, it is also a valid answer.

Jonathan
Hi. I'm a mathematician. This was pointed to me by a student.Dr. Anderson's theory does seem to be correct, but it is not what I would call new or useful. The "number" nullity is different from NaN. NaN is the computer's response to any undefined expression (including both infinite expressions and undefined expressions). Dr. Anderson proposes to distinguish indeterminate expressions, such as 0/0 or (infinity - infinity), from infinite expressions, such as 1/0. This is an entirely legitimate thing to do, but not a new idea.This is not very useful for calculations, since "nullity" acts as a sink for arithmetic. It you perform any arithmetical operation with nullity, you will get nullity.

timmy tonga
hahahahaha nice one ross

Smart MST Kid
i created this theory in class and no one believes me. except i defined the variable to be j cuz my name is joe. rock on!

William Volterman
This guy is a quack... he says 0/0 is a number... but he is misrepresenting the statement. In colloqial language 'number' means 'real number' and clearly it would not be a 'real number' Nullity is as much a 'number' as infinity. Fine I can define operations on trancendental numbers... but they are not 'numbers' in the sense that they are not 'real numbers'

miss hawkins
i think my maths teacher should read this, mrs kennedy and ms bickley at kingsley college in redditch hehehehe

matt
how is nullity useful if it is equal to everything? Good computer programmers ensure that /0 doesn't happen. Bad computer programmers don't, and during testing add an "if denominator is 0, do something else". Either way, it is absolutely no use to programmers; thus to computers. To quote you: "It is a paradigm shift in the way you think." that is all it is. Well I'm going to define "Frue" to be a boolean which is both true and false. No, it's not undefined, I'm defining it right now; no matter that it never occurs; sure it's helpful, all these unspecified computer programs will use it! Anyone want to be hit over the head with a computer that uses this value? By the way, computers work at the basic level by comparing values to zero, and jumping to a different instruction, or not, depending on the answer. So no speed-up there; nullity would compile away... no more help than NaN in programming.

Jon
I can see where it would help the computer world. In today's technology computers use the if x then y rule. This theroy allowing the 0/0 to be a number then the computer could be expanded by memory usage or even speed. For me to understand this theroy I had to look at a 3 tier plain with the middle tier being the logical numbers used in everyday use. with 0/0=nul (while still indif, but on a different plain) means a new set of numbers that could be used. The main change to the known theroies would be adding the the nul sign. (we will use k for this) If 0/0=k then -0/0=-k giving 2 different possibilities. with the common number line adding a number above and below to the already known number to the left and right copuld possibly allow computers to run programs more accurate. Thus 0/0=k and -0/0=-k then k-k=0 and puts the computer back into the number line we know. This is then just completing the the gap in the number line that is known as undefined.

Andrew
Ralph has it with his limit as x approches zero of x * (1/x). Analysis tells us that a limit is unique, but assuming 0*infinity = nullity (that's one of Dr. Anderson's assertions) we get two values for a limit. One could consider the set of numbers containing nullity, positive infinity, and negative infinity. That's certainly possible. But it would have very different properties than the real numbers (as above, nonunique limits), and so can't be used in all cases the same way that real numbers can be. This also makes his statement (from the interview above) that "[t]he trans-real arithmetic is total, and complete, and contains real arithmetic as a sub-set" completely false, because in at least one way the arithmetic behaves differently.

AndyB
Hmmm. I think this is about encoding a divide by zero error in a single bit in the numerical buffer. The "advantage" of this is if I do 5.a/a I *should* get 5. But when a = 0 I get an error from which I cannot recover. With this I get 5."-0-" so my code can recover (i.e. by ignoring the "-0-"). Presumably to do this we have to set aside an additional bit. So, 5 is equal to the binary 101. If we say all integers used in my new nullity-compatible programming language will be 8 bits long this is 00000101. If I want to encode negative numbers I have to increase this to nine bits with the first 1/0 being 1 for negatives and 0 for positives. Silimarly I can add another digit for imaginary numbers and another for nullities, so my "8 bit" number now becomes an "11 bit" number with 3 marker bits: +/-, i, and -0-. That's the computer side of it I guess. As for the maths, well, I suppose in some ways this is similar to the idea of root -1. In the end we need to see this being of use. If it is, nullity will find a place in the history books. If not, it will quite naturally find it's own way into my computer's recycle bin. Where upon it will be mercillessly zero-ed and the disk space used for something else.

Mathematical Aardwolf
I think that I need to pass on a remark made by the philosopher Mary (G.E.M.) Anscombe. She said: "There is a harmless idea that everything which exists has a name. Far less harmless is the idea that everything which has a name, exists." Whatever Doctor Anderson has done, it cannot be consistent. Doesn't he know the difference between a ring and a group?

Brian
"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." Somebody tell this man about Godel's Incompleteness Theorem.

Maria S.
Wouldn't 0/0 be 0? If you are trying to split up 0 into an indefinite amount, wouldn't it stay 0? I think I'm wrong, but whatever. New mathematical theorems and concepts always seem so revolutionary, but some of them don't live long. Who knows what'll happen to this one.

Ganesh
What is the value of the fixed number so called nullity?

Bharani
I wonder !! This is a new frontier in mathematics. Where does the nullity lies in the number line..

ross
why not just make 9 the loudest? but these go to eleven.

Saif
Interesting. What can be nullity*nullity? If he is saying 0/0 = nullity, then 0/0 * 0/0 are two rational terms, and can cancel each other. We should throw away our calculus and integration books :).

Max
You say nullity is a fixed number, yet what would you define a number as?

Anonymous
Why don't we just define nullity as all real numbers?

Shadowkiller
So wait, he has had his MATHIMATICAL argument reviewed in a COMPUTER paper? Does computing define math these days? Well I do guess that 2+2=5 for extremely large values of the number 2. How about instead of going back to the guy making seemingly wild claims and asking him to deny remarks against him you go to a mathimatics expert and have the two duke it out. I wonder if the mathimatics expert would even show up.

Ralph
Assume 0*infinity = nullity. Then: 1. Consider the limit as x approaches zero of x * 1/x. 2. x * 1/x = x/x; the limit approaches 1. 3. However, by the properties of limits, we could expand (the limit as x approaches zero of x * 1/x) to (the limit as x approaches zero of x) * (the limit as x approaches zero of 1/x). 4. Using this expanded version, we arrive at 0 * infinity, normally considered an indeterminate form, a sign that we need to evaluate it differently to reach an answer; luckily for us, we have discovered nullity, so--using the transreal system in which infinity and nullity are numbers--by definition 0 * infinity = nullity. 5. Thus the limit evaluates, in fact, to both 1 and nullity, depending upon the representation used. 6. Thus either the properties of limits (which screws up calculus) or the properties of algebra (which screws up math in general) do not hold. Conclusion: nullity is ridiculous and completely undermines limits, and therefore calculus, by doing away with indeterminate forms--which is what happens if you call infinity a number and define "numbers" such as 0/0 and infinity * 0.

Annon
People keep using "infinity" as though it were a number. It's not, it's a concept, and there are different "types / sizes" of infinity, so saying "0 x infinity" is a little like saying apple x sheep, they're 2 very different things and shouldn't be confused.

Ralph
How can you say nullity = nullity, when nullity/nullity /= 1 and nullity - nullity /= 0? What can it mean to say a = b when none of the implications of a = b (even just the implications for nonzero a and b) hold? Isn't your only recourse to say "Just as a = b does not imply that a/b = 1 for a=0, a = b does not imply anything whatsoever for a = nullity", and doesn't that make nullity = nullity utterly meaningless and therefore fly in the face of your entire argument about nullity being different from "undefined"?

David
To Raymond: I know that my argument with 0^1 is not entirely correct under the axioms. HOWEVER, you'll note that Dr Anderson used exactly this argument when teaching the Year 10 students about his 'theory'. He did exactly what I did in demonstrating his work to the kids. Now, you and I know know that that's where my argument below doesn't work, and that I haven't shown his axioms inconsistent. But if he himself is using such arguments does that fill you with confidence that he knows what he's doing?

Me
nulity= what? Is there really a number that you can say it's the only one that, when multiplied by zero, will equal zero? Aren't all the numbers able to do it?

Aaron
Mathematically, this is useless. It doesn't take too much thought to figure that. If you want to think about it, then, very basically: if a theorem doesn't derive directly from ZFC, or from a (consistent) superset thereof, it's trash. While you can probably extend the definition of "division" to include an error handling symbol within this paradigm, that's not division anymore - it's division-prime, and exists in "arithmatic-prime." Usually I'm not such a platonist, but this is idiotic. If you want to know about actual infintesimals and infinites in a consistent setting, take a course in non-standard analysis. Actually, do that anyway - it's good stuff.

IG
I believe that the problem with 'nulity' is that if you have a number, 0/0, then you also have this proof, where x is anything: 0/0=0/0 (1*0)/0=0/0 (1/0)*(0/0)=0/0 (1/0)*((0/0)/(0/0))=((0/0)/(0/0)) (1/0)*1=1 1/0=1 0*(1/0)=1*0 1=0 x*1=x*0 x=0 the only way out of this is that the initial statement, '0/0=0/0', isn't true. This stament is true for any number, so 0/0, 'nulity' is NOT a number.

Keith
@ Drivel Spotter: 1^x=1 for x not equal to plus/minus infinity (*grudgingly* nullity, too). @ David: Fortunately for Dr. Anderson, he slips out of your grasp by failing to define axioms dealing with powers of numbers. However, unfortunately for him, your example requires him to reformulate the usual power axioms if he wants to maintain that zero does not equal nullity. For example, David's objection points out that 0^a does not equal 0^(2a-a) under the ordinary axioms of power manipulation and associativity. Prima facie, this means that a does not equal 2a-a; therefore a /= a (this objection can be met by defining logarithms to deal with such cases). He can sidestep this by: a) leaving powers undefined, b) reworking power axioms when zero is the base involved, or c) reworking power axioms in general. Option a) is clearly the least appealing for the transreal numbers to be applicable to analysis. Option b) isn't particularly appealing, either, because avoiding David's objection requires elevating zero to a status where commonsense power arithmetic no longer applies; for example, setting 0^a=nullity for all a (but then we lose everything: let's say a^2=n, a /= 0; then take b=0, and we get (a-b)^2=nullity; therefore n=nullity for all n. So our entire "field (not really a field)" is reduced to 0, plus/minus infinity, and nullity. Yuck.) Option c) seems the only real viable option (there's probably a slippery way to get option b to go through), and of course it's the most work. So if Dr. Anderson is to convince us that nullity and its panoply of axioms are at all useful, he's got a hard road ahead of him. History will be the judge ultimately, but as his work stands right now, history has already decided against him.

paradigm shift
"paradigm shift" is ungrammatical

Brian
In fact this system has a very straightforward mathematical explanation, and the fact that Dr. Anderson has to resort to machines or tedious considerations to prove his results is a sad commentary on his mathematical abilities. It would take any decent professional less than an hour to sort out what is going on. Here is the situation, which will show that he has really discovered an interesting kind of "weird example" for elementary topology homework. The real projective line can be constructed in several ways, one of which (the "algebraic geometer's approach") is to take the punctured plane (remove the origin) and pass to the quotient by nonzero scaling. The transreals is the following curious variant in which we do two things differently: first, we only pass to the quotient by *positive* scaling (thereby retaining a sense of direction, which is to say two senses of infinity, represented by the points (1,0) and (-1,0)), and second (and more dramatically) we actually don't remove the origin! In other words, the transreals are the quotient of the usual plane modulo scaling by positive numbers. The usual real number line is embedded by sending a to the class of the point (a,1), and we call the classes of the points (1,0) and (-1,0) positive infinity and negative infinity respectively. Finally, the class of (0,0) is called nullity. Give it the quotient topology. Now why is this (slightly) interesting? The topology is rather non-Hausdorff (due to nullity), but one can check that the above embedding of the real line gives it its usual topology, likewise for the "extended real line" of real analysis, and (here is the nifty part for topology homework for a student) the usual arithmetic operations of addition and multiplication and negation and reciprocation on the real line have *unique* extensions to continuous binary operations on the transreals. Likewise for his definition of the "sign" (or "sgn" function). Uniqueness is a tad bit surprising since the topology is not Hausdorff. These uniquely characterized operations coincide with Anderson's ad hoc definitions, thereby making rather more natural what he presents like a rabbit out of a hat, and if we *define* a/b to be a times the "reciprocal" of b then we have a conceptual point of view from which all of his results become very easy to verify in one's head in a short amount of time (if one has a little experience with elementary topology). In particular, there is no need to make some absurdly long list of axioms as he does. But of course it is not a field, and has virtually no chance of being of any real mathematical interest beyond topology homework as a "weird example". Its "virtue" over the extended real line is the surprise that the operations can be defined for all pairs without exceptions (in a manner uniquely determined by continuity and the classical case). This hardly merits the grandiose claims being made by Anderson.

Barking_Mad
Quote: "It is clear that Dr Anderson knows very little maths. The only 'real' maths he tried to quote in his interview was wrong, namely that 'cos^2(x)+sin^2(x)=1^x'. This is basic 'A' level maths and it's 1, not 1^x !! Enough said." Did they teach you what 1^x is when x = 0?

Dan
From my brief reading of the interview and Perspex Machine VIII, I've gathered that what Anderson did was define a new number that inherits all the properties of zero and overrides the additive property of it so that instead of a + 0 = a, a + 0 = 0. Also, he defined this new version of zero (nullity) to be the result of some operations on postive or negative infinity. Any programmer with about a spare half-hour could set up a math library with functions for all basic operations that other programmers could use instead of any built-in arithmetic functionality. On the division function in the library, there would just need to be a check if something is being divided by zero. If so, then return some defined constant, something which can in turn accepted by all of the other arithmetic functions. Given that any computations in a program involving nullity will equate to nullity, in effect, then the program may as well stop doing any further arithmetic involving it. Alternatively (and what is currently done in practice), an exception could occur that stops the current computations when the program attempts to divide by zero. The effect is really same. Thus, Anderson's work can safely be disregarded by the computer programming community.

Bill Gates
when i use my Microsoft Calculator v.5.1 and input 0^0 i get 1

Raymond
David: question for you. Where did you find that a^(x+y) = a^x a^y? I haven't found that axiom anywhere. Can you prove it holds for zero unconditionally? Don't think you've been clever and beaten this unless the tiniest steps you can write have an axiom or theorem number.

Rasmus
You said it is possible to access new equations with this nullity. What equations? And isn't 0/0 = 1? Isn't a/0 = infinity?

Bob
x/x may be equal to 1 for all (real) x bar 0, but 0/x is equal to 0 for all real x bar 0 also, so 0/0 cannot sensibly be defined as either. Transreal arithmetic does get round this problem, though using the axioms that Dr. Anderson has defined 1+nullity=nullity, 2*nullity=nullity.. i.e. any calculation with nullity equals nullity. So how can nullity be used in any meaningful calculation?

A different proof
The following is NOT an attempted proof that Dr A's the axioms are inconsistent. He's claimed that they are (which is impossible, by Godel), however all attempts I've seen so far have failed to prove it inconsistent. (Actually, I have seen one: David's 0 = 0^1 = 0^(2-1) = 0^2 * 0^-1 = 0 * infinity = nullity proof. But that's more a problem on the conditions for [E10] than anything else, since it relies on using ^, which is not actually part of the axioms.) Instead I present a proof that his work adds nothing more to maths than we already have (and indeed, actually takes stuff away!). The _only_ equations that cannot be solved in a field (other than pathological ones, such as solve x = 1 = 2) are ones that can be boiled down to 0 * a = b, for varying b. (Remeber, division does not exist in a field, only multiplication by an inverse, thus saying that 1/0 and 0/0 are defined means nothing to a field, since / itself is undefined!). For a field, when b = 0, we have that any a will solve the problem. Otherwise, there is no solution. There is no unique solution, wihich is what division is meant to achieve. However, Dr A's axioms _do_ define division, and also division by zero, so let's try some equation solving in Dr A's "new maths": Problems: 1) 0 * a = 0 2) 0 * a = 1 3) 0 * a = infinity 3b) 0 * a = -infinity 4) 0 * a = nullity Can any of the above be solved uniqely, thus providing a use for his work? Solutions: 1) a = 0 / 0 = nullity? Not on your life! Any real a will do, but nullity does NOT solve this. If a = nullity then 0 * a = nullity. We've still got exactly the same set of non-unique solutions to this equation that we always had. 2) a = 1 / 0 = infinity? No, wrong again! 0 * infinity = nullity, so a = infinity doesn't solve 0 * a = 1. In fact, it still has no solutions. 3) a = 1 / (-)infinity = 0? Of course not! a = 0

Bob
Ha ha ha!!! Anderson references wikipedia in his papers! How can anyone take him seriously after that??!!!

Rehal
I just have a simple question to Dr James Anderson. Is "Nullity", 0*(infinity)?? As you have expressed in your theory that "Nullity" is a term that doesn't lie in a real number and is a transreal number. So,0/0 can be expressed as 0*(infinity)? so can infinity be a transreal number that brings up the subject of nullity? NO!! And if that is true how couldn't it be right from 0*(infinity) So, that's your so called number as nullity!! Well, I appreciate what you did but wasn't clear enough for anyone, meaning anyone to comprehend! Well, I am a High School Junior residing in Virginia! So, I dont think i know much about your theory, but its least comprehensive and is not proved right!! Thats what i have figured out after going through your explanation. =Rehal

Kirk Twist
so yeah, complicated proofs aye? hows this for simple a/b = a * (1/b) therefore 0/0 = 0 * (1/0) that CLEARLY equals 0 (The truth is that you can never define 0/0 because it could equal ANY number, the equation simply does not contain the information needed to formulate an answer. Its like trying to say that 0 * x = 0 and then say that x "is a special number i just thought of" instead of simply saying there is no way to tell what it is)

Don McMorrow
Have any mathematicians done any work to see if the concept of nullity leads to other findings? Does it help solve problems heretofore unsolved or lead to simpler proofs for problems that have been solved?

David
If you read page 7 of his second 'Paper' you'll see the tell-tale sentence which sums up his 'theory' and the mathematical rigour of it when he says, (of trying to evaluate trigonometric functions) "We accept nullity unless there is good reason not to." Formal logic is not a domain for 'good reasons', whatever one of those is. You need things to be uniquely defined, not just choosing the values you fancy on a case by case basis. No wonder so many mathematicians are despairing at his claims.

David
So, axiom 12 of his first paper tells us that a*(b*c) = (a*b)*c for all transreal numbers a,b and c. (Multiplicative associativity) So now if write 0=0^1 = 0^(2-1) just like he did when finding 0^0. Only this time we get: 0^1 = 0*0*0^(-1) and this is the same as 0*0^(-1) by axiom 12. But this is Nullity apparently. So we've deduced that Nullity is zero.

Drivel Spotter
It is clear that Dr Anderson knows very little maths. The only 'real' maths he tried to quote in his interview was wrong, namely that 'cos^2(x)+sin^2(x)=1^x'. This is basic 'A' level maths and it's 1, not 1^x !! Enough said.

S.H.
"an argument why 0/0 is not equal to 1." If you do NOT have any kind of any apples how many apples do you have?

Mark Wagner
The reason you can't use real analysis to disprove Dr. Anderson's work is that isn't the right tool for the job. Once you get into graduate-level theoretical maths, you learn techniques such as abstract algebra, which *is* the right tool for the job. Abstract algebra says that transreal arithmetic is internally consistent. It also says that transreal arithmetic isn't too terribly useful, because in the process of making the system closed under division, he's given up many of the properties that make ordinary arithmetic useful for reasoning about math and science.

Gonzalez
I absolutely agree with Larry. Dr. Anderson has not the slightest idea what is going on in mathematics. Almost every math student discovers a similar extension (what if I define division by zero? what if I add infinity to the real numbers?), then they realize that they lose many useful properties of the real system. In practice we never want to divide by zero, so the whole game is pointless, and extremely banal. Extension of real numbers to a larger field were well understood even 150 years ago. The main problem here is claiming that he did anything new, claiming that he solved any existing problem, and confusing those poor kids.

Dave Korn
Under the weight of scrutiny pointing up the total lack of meaningful content to his supposed new idea, Dr. Anderson has now been forced to resort to playing specious games with semantics. Here he claims that the difference between nullity and NaN is that NaN is Not A Number, but nullity IS a number. Yet when you read his theory, you see that what he is quite clearly describing is not a number at all: it doesn't have any value you can write down, it does not have the properties or behaviour of any of the other numbers, you can't add with it or subtract with it and every single equation in your entire scheme of mathematics has to be riddled with ifs and buts and special cases to avoid it. The very point that he uses to claim nullity is different from NaN - the claim that it is equal to itself - is where the nonsense comes in. He says nullity is equal to nullity; but then he says that you can't subtract it from itself and get zero. So, it is equal to itself, but you cannot express the difference between it and itself as nothing? Then how can it be equal to itself? The very concept is an immediate contradiction in terms. So, he says that the difference in his scheme is that nullity is a number; but his own axioms make clear that it is not in any way like a number. To insist that, despite being almost but not quite entirely unlike a number, nullity somehow still has some undefinable quality that makes it a number is theology, not mathematics, sheer meaningless metaphysical gibberish. No, nullity is not a number. Nullity is a black hole, a tar pit, a void; it's a way of throwing your hands up and saying "I don't know". Anything it touches turns to nullity and is lost for ever. In conventional mathematics, researchers in some fields have come up with ways for handling mathematical singularities and infinities. Some of the field equations of quantum electrodynamics involve handling infinities, but cleverly arranging for these infinities to cancel against each other. Nullity is like one of these techniques, a means of dealing with and handling singularities, but it is far less powerful, because nullity loses information; you can know two infinities in an equation are equal infinities and cancel them out, but you can't do that with nullities. Nullity discards information, it is the absence of any information about what you do or do not have in front of you - and that's Not A Number.

Raymond
I am, in fact, a decent mathematician. As a decent scientist I am open to the possibility of new ideas and don't begrudgingly hold to the old ways. At first I thought Dr Anderson was a complete dolt. (my apologies). Now I think that there might be merit in his work. I don't mean that in a positive light - I mean show me a use for this system that actually simplifies a problem. Indeterminate forms are not a problem - we like them just the way they are. This system, even if it is brilliant and revolutionary will not depose the old system because it is at best a product of 2 topological semigroups, maybe compact. (Compactness is a good thing) The set of real numbers is not compact. The real numbers are a metrisable topological product of 2 groups. This system might have a use in counter examples. We're always looking for weird sets that go against intuition. The question everyone else should be asking is "Does this solve a problem or does it make things easier?" My guess is no. I am not beyond error. Direct me to a specific use that isn't giving a value to an indeterminate form or giving extra values to trig functions that show how the theory looks to itself.

AP
I have 2 follow-up questions for Dr. Anderson: 1) Could you briefly encapsulate how it is possible that you managed to PROVE that your set of axioms are consistent; as far as I know this is forbidden under Godel's Second Theorem (unless your system is inconsistent of course, in which case it is possible to prove anything). 2)Why did you decide to attempt to teach this to kids before it was peer-reviewed in an established mathematical journal, and when there has so far been no good reason demonstrated for the use of this transreal arithmetic? As a brain-stretching exercise there are countless areas of mathematics suitable for kids to think about which are similarly abstract but have applications in higher maths or the real world, and which have been completely checked numerous times over by mathematicians.

Larry Teabag
To be honest, this is a lot of fuss and bother about nothing. So, Dr Anderson has defined a new system comprising the real numbers plus three extra points: plus infinity, minus infinity, and nullity, and he's shown that his system satisfies various basic algebraic constraints. Well excuse me for not being blown away - mathematicians play these sort of games the whole time. What I'd to hear from Mr Anderson is an admission that in passing to his enlarged system, one loses a great deal: algebraically, his system is not a "field", or even a "group" under addition; topologically his system isn't a "metric space", and one probably loses a lot more besides. Assuming that it's right, this research should not be controversial per se. The reason it's attracting so much negative criticism is the extraordinary gulf between the overinflated and arrogant claims being made (the suggestion that he's solved an ancient problem, or that this research constitutes a "paradigm shift"), and the reality, which any decent mathematician will recognise as being of marginal interest at best, or banal and trivial at worst.

TO: Raymond and JMS
While his axioms may be consistent with his new pseudomathematics he has defined in his papers, they do not correlate with the current mathematics that is universally accepted. Rather than solving the problem within the current restraints of mathematics, he's redefined the rules of mathematics to serve his purposes to show his proof is valid. That's where I have a problem. Is he so arrogant to think all the great mathematicians before him where incorrect in their formulations of the rules of mathematics? Furthermore, he even states in his papers that real analysis cannot be used to disprove his pseudotheory. Thus, he is not allowing us to show with the current universally-accepted mathematics that his pseudotheory is in fact flawed. Therefore, basically, we're forced to accept his pseudotheory as gospel simply because he claims it is correct. And you wonder why many mathematicians and fellow scholars have such a problem with his pseudotheories and his pseudomathematics.

Welbog
I don't understand why people care about how Dr. Anderson intends to encode nullity in binary. It's not at all important or of consequence. Not currently having an encoding for nullity in binary doesn't make his axioms inconsistent. Unless the people asking this want Dr. Anderson to say, "I'll just encode it as NaN currently is," so that they can reply, "AHA! So Nullity is just NaN after all," which is, of course, absolutely wrong. What I really care about is the guard clause for equation 10, since that clause causes a rather important inconsistency in his system. Can someone help me find or state the updated guard clause?

plh
@Steve "...So it is not a wheel." No it's not - I did not say that it was!

Raymond
Have a look on his website, guys. All these criticisms have been answered convincingly in two of his papers. Stop appealing to common sense because it doesn't work. You cannot intuit Nullity because you need to follow the axioms provided. For Vid, Nullity eats everything. Once you see it you can skip to the end. Nullity does not compare to limits. They have nothing in common. There is nothing wrong with ASSERTING that 0^{-1} = infinity and so on as long as you define all its behaviour consistently. Limits are not involved. The floor is really open to Dr Anderson to show us something useful that you can do with Nullity that you can't do without it. That's what enables change.

JMS
Firstly, I wish people would stop posting messages on here as to why Anderson is wrong, when they have a misunderstanding of maths themselves. 0^0 IS NOT EQUAL TO ONE. Nor is x/0 equal to 0 or x. That out of the way, I'd like to direct people towards Anderson's website, which includes papers in which he sets out his axioms of "transreal" maths: www.bookofparagon.com . Once you understand that he is not dealing with the usual real numbers then you may comment, as I'm about to... (1) It seems that the axioms are just a work-around to allow aritmetic to be performed on nullity. They are consistent but any equation involving Nullity gives an answer of Nullity. Anderson fails in his interview to explain how Nullity would actually be used in a computer program to provide understandable information. (2) He says that the 10 year olds he taught this to followed his method. Is he seriously suggesting they understood that you must first axiomatise the new definitions of Nullity, infinity and -infinity before performing the arithmetic? This is what mislead a lot of people, and it is quite fair to say that he was irresponsible in telling children that they could go away and divide by zero to their hearts' content. (3) This is NOT NEW MATHS! As others have pointed out, Cantor dealt extensively with the problem of infinities, and there are areas of mathematics in which some of his definitions are utilised. But his transreal maths is not useful to mathematicians since it is not developed with any specific mathematical field or problem in mind. (4) This is not a problem for mathematicians! And if one of our equations didn't work, shoving Nullity into it, even if we were working in the transreal axioms, wouldn't solve the problem. (5) He still fails to explain exactly how the computer program would cope with Nullity. Fair enough, it knows it's a number, but how is this then used? Why is Nullity better than exception handling? He needs to use real examples if he wants to change people's minds. (6) He has read a dangerously small amount of literature on this subject. Just look at his references: a wikipedia article is among them. Need I say more?

4d5e6f
Anderson's responses were vague and inconclusive. He avoids going into any details with his answers, like not stating what the number is in binary or giving any real example equations we can use this "radical new concept" in.

Vid
OK. What is 10 * nullity? Is it still nullity? What about 10 / nullity? Is that also still nullity? And 10 ^ nullity... is that 1? Or 10? or nullity? What I'm saying is, if this is just a new number, how does it work in simple arithmetic? Is it still a special case number, like 0, 1, Undefined, lim(x->infinity ){1/x}... and if it is, how is it different?

Jared Haines
If nullity is a concept on par with the twin infinities, it cannot be a fixed number, because the two infinities are not fixed numbers: they are constantly growing in magnitude beyond what you can conceive. His computations working on computers is also about half worthless; I seem to remember hearing stories about a Fortran machine that said -0 did not equal 0. TI-89s, some of the strongest personal use calculators in the world, screw up on some derivatives. We know computer calculations are only as strong as the programmer's skills. I'm sure I could find a computer that says his calculations don't work and retaliate by hitting him over the head. Math is not a subject of theories like science. Math is a subject of axioms, postulates, conjectures, theorems, etc.

Reid
Well Dr. Anderson said in this interview that there is no problem with what he presented to the public in the last spot. Well, this isn't exactly true, though it doesn't change the outcome. The problem is his proof that 0^0 = nullity. His proof is as follows: 0^0 = 0^(1 - 1) = (0/1)^1 * (0/1)^-1 = (0/1)*(1/0) = 0/0 = nullity The problem here is that he manipulates 1/0 as a fraction which it cannot be manipulated as; he assigns it a special value. Here's the proof that this cannot be so: By definition: 1/0 = infinity We now cross multiply and get: 1 = 0*infiniity By an axiom of the transreals 0*infinity = nullity. Thus 1 = nullity. Clearly this cannot be the case as nullity lies off the number line. A proper proof that 0^0 = nullity in the transreals is as follows: 0^0 = 0^(1-1) = 0*(0/1)^-1 = 0*(1/0) = 0*infinity = nullity

kn
There is this problem: ab = cd, then a/c = b/d. 1(0) = x(0), then 1/0 = x/0, then 1/0 = 0/0 = -1/0. This also makes sense because 1/x is not positive, nor negative (Look both sides of the function 1/x when aproaching to x = 0), thus it is neather infinity, nor -infinity. As far as I can tell, Nullity stands for Undefined and thats it

Jeff
1/0 is NOT infinite. The only way you can even attempt to think that is by using limits (calculus). The limit of 1/x as x approaches 0 is not infinite, it does not exist. Why does it not exist? Well, a limit must be approachable from both sides. The limit of 1/x as x approaches zero from the right is in fact infinite, but the limit of 1/x as x approaches zero from the left is negative infinite. Therefore the limit does not exist. Therefore even by incorrectly using a limit to define division by zero, you cannot derive that 1/0 is infinite, therefore your assumption fails. Also, division by zero can be prevented in computer code with a simple statement that checks the variable before it is divided, and does not divide it if the variable is zero.

Andy
Carrie: No. In your system, the moon IS cheese is correct, it follows from the axiom(s) of your system. This is even though the same statement is false in the real world. It is easy to prove consistency in a trivial system. Dr. Anderson is claiming consistency for the extended real number line + nullity system. The proof of consistency for such a system is much harder. This proof would need to be checked or peer reviewed by real mathematicians. Then we can proceed to ask if such a system is useful ie. can we prove theorems in this system that we cannot in 'normal' arithmetic.

Dicky
It is stated that: "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." So why is the work you reference a non-peer-reviewed conference paper? If this is a step-change in mathematical thinking, shouldn't we be reading about it in 'Nature' or 'Science'?

Mark Wagner
Norman, the reason that 0/0 is considered to be undefined in the general case is not that there is no good answer, but that there are too many good answers. There are good cases to be made for 0/0 to be 0, 1, +infinity, or -infinity. For example, x/x is 1 for any x, 0/x is 0 for any x, and x/0 is +infinity for any positive x, -infinity for any negative x. In specific cases, particularly in calculus, you can reason that 0/0 is any other value as well (see l'Hopital's rule).

Ed R
What is going to happen if the maths journals don't feel this new theory is correct? Is Dr. Anderson just going to agree to disagree?

spencer
Well, until I see his work, I can't really critisize it. I would like to see an example. It was hard for many people to accept and/or consider the imaginary unit: sqrt(-1). There is no real number whose square is -1. By the same token, there is no real number divisible by zero. Perhaps 1/0 is some other type of "imaginary unit". I don't know. If Dr. Anderson is going to convince me that his ideas are worth while, then I'm going to need to see some real rigour.

Steve
As I understand it, in a wheel there is only a single infinity, not plus and minus infinities. Furthermore //x = x in a wheel whereas Anderson's arithmetic has //-inf = inf. So it is not a wheel.

BlackNexxus
Norman Logan If any number (other than zero) divided by itself, gives the answer 1 and even algebraic terms such as x/x equal 1, then I would love to hear an argument why 0/0 is not equal to 1. u have no apple which u divide amid no people. where form should the apple come? Furthermore there would be every result possible: 0/0 = 1 |*2 0/0 = 2 |*5,2 0/0 = 10.4 |*(-1) 0/0 = -10.4 so u would can say that 1 = 2 because 1 = 0/0 = 0/0 = 2 btw sry for my bad english ^^

plh
"It is entirely normal for the work to be controversial ... before it is accepted by mathematicians." The "controversy" is partly an artificial construction of Dr. Anderson's own making (knocking down straw man arguments based on wikipedia articles written about division by zero in the *field* of reals etc. achieves nothing other than the irritation of those who are well aware that Dr. Anderson's "transreals" are not a field), partly due to the hyperbolic and (AFAICT) unsupported claims made for the utility of this "transreal" structure *in physics* - not just comp. sci. - and partly due to Dr. Anderson's failure to acknowledge that his "transreal arithmetic" may not in fact be the astounding and revolutionary new mathematics he seems to think it is. "I am saying that the number 0/0 is a number... That is different to what goes on currently in computing, and in mathematics." That is just not true: 0/0 has certainly appeared before, even in the work of some of his own (unacknowledged) comp. sci. peers, and by simply identifying ∞ and -∞ and trivially modifying a couple of his axioms, what I get sure looks like a wheel to me.

Andy
I pity the PhD candidate that takes up Dr. Anderson's offer to implement fluid dynamics on a simulation of his perspex machine. CFD is hard enough on real computers. To implement CFD on a non existent machine which is itself based on shaky maths must equal 3 to 4 years of wasted effort. Hint: Once nullity appears in any step of any operation in this perspex machine, the rest of the result is always nullity. One resolution to this problem: Always check for nullity and recover. But how is this different from checking for division by zero and recovering? Yes, this is a total waste of time, effort and grant money.

carrie
As far as I know a system can be logically consistent but still false? Example: 1. "If the Earth is round the moon is a cheese", 2. "The Earth is round". Conclusion: "The moon is a cheese". This is a logically consistent system but it is false (the moon is not a cheese) because one of the premisses are wrong (number 1)

RL
A great deal of care should be taken before teaching this to children. Assuming that it is both correct and useful it makes simple calculations much more difficult. Currently we teach them to ignore infinity (it's not a real number) and be careful of zero. Now we want to teach them to be careful of zero, +/- infinity and nullity, all for different reasons. Also, most equalities will now have a solution of nullity which must be properly discarded to get a solution. Example: x = x has nullity as a solution; x - x = 0 doesn't. If we ignore the nullity some of the time we have to teach them when it's ok to ignore it... when? When in high school can't you ignore it? When you get indeterminate or undefined expressions.

Sindhudweep N. Sarkar
Nullity is the dimension of the null space of a matrix. Please choose a new word for your idea.

Raymond
A brief review suggests that the system presented is correct. Whether that makes it useful is another matter entirely. Can you give an example of code (such as in pacemakers) where Nullity is more useful than NaN? Is Nullity still intended to set off alarmbells when encountered like indeterminate forms? Can you give an interpretation of differentiation (and continuity) with regard to nullity? Can you give an explanation of why f(x) = Nullity is more useful than calling it indeterminate? For Paul, 0^0 is indeterminate because it can also be useful using 0^x = 0. The complex number construct, i, can lead us to alternating current very simply. Is there some physical interpretation of nullity or does it simply allow more streamlined formulas at the expense of having 1^x = 1 and other similar inconveniences?

My biggest quarrels ...
To me, Mr. Anderson's "nullity" still seems to be a placeholder for 0/0, which is indeterminable. Unlike i (imaginary numbers) which has practical applications in the field of engineering and other applications in the field of mathematics (Mandelbrot and fractals), his "nullity" has no more practicality than the IEEE's NaN standard in the world of computing. How is the statement: if (x/y == nullity) then ..., any more simplistic and easier to use and understand than the current approach: if (x/y == NaN) then .... The answer of "nullity" to a mathematical equation provides just as much information as displaying "Error" or "NaN". Once you arrive at a "nullity" answer during any part of the calculations, your answer will always be "nullity" and thus, it has the equivalent practicality of an error trap in the world of computing. I should add that error traps have already been well defined by the likes of Turing and other well known mathematics and computing geniuses. I can understand his desire to create a method to compare indeterminable or undefined solutions, as described in his paper. However, mathematicians defined the use of undefined and indeterminable for these sorts of problems this way for a reason. How can you say, without a shadow of a doubt, that 1/0 = 2/0, or 1/0 < 2/0, or 1/0 > 2/0, or etc., (in accordance with Mr. Anderson's) if you don't know the precise solution to either statement. In all cases, a fallacious statement may be arrived at, since we do not have a precise or defined answer for either of the unknowns. Furthermore, I should also mention that the assumption that 1/0 = infinity is incorrect. As defined in calculus, THE LIMIT of 1/x as x approaches 0 from the positive side approaches infinity. Without the use of the proper notation, there is once again a fallacious statement made since THE LIMIT of 1/x as x approaches 0 from the negative side approaches negative infinity. In addition, infinity is the state of being greater than any finite (real or natural) number, however large. Thus, it is mathematically impossible to equate the solution to 1/0 and infinity. Furthermore, like Darren mentionned below, Cantor has done work related to mathematical infinity which I would suggest that Mr. Anderson study. While Mr. Anderson claims he can code "nullity" in binary for computers, how does that provide any additional usefulness than the existing binary encoding of NaN as set out by the IEEE's standard? This whole "nullity" issue seems to have no more practically than the IEEE's NaN standard. He is merely applying a computer science concept to mathematics, which survives little purpose in mathematics. It seems like an attempt to rename the existing NaN standard in computing which is universally recognized by electronics manufacturers.

Andy
I'd like to see Dr. Anderson's second published result to equation 10 to see there is another gaping hole in his analysis. Did he say his original result has been properly peer reviewed?

mongreilf
i'd like to know details of the claimed peer review. peer review isn't strictly that, as the peers are actually experts in the field that the author writes in, rather than the one the author belongs to. in addition conferences are not (generally) considered a form of peer review

Darren
Georg Cantor created all the necessary axioms for dealing with mathematical infinity over one hundred years ago (1870s) and showed that there are many different sizes (cardinalities) of infinite numbers. Simply defining 0/0 as one particular thing (nullity) cannot possibly handle the richness that Cantor proved was involved in dealing with trans-finite numbers. This approach seems far too simplistic to be correct.

Norman Logan
If any number (other than zero) divided by itself, gives the answer 1 and even algebraic terms such as x/x equal 1, then I would love to hear an argument why 0/0 is not equal to 1.

Paul Green
Your questions and Dr. Anderson's responses fail to address what seems to me to be a central issue: Dr. Anderson has claimed that deriving the formula 0^0=nullity is a demonstration of the significance of his ideas. Mathematicians are extremely unlikely to regard this result as an advance over the standard and useful identification of 0^0 with 1.

Pennywise the clown
The theorem T81 [(a*b)^-1 = (a^-1)*(b^-1) if a=/=0 and b=/=x, x<0] needs to be an axiom, I think, because the axioms don't seem to justify the guarding clauses. Plus, before I just had to check if I was dividing by zero, now I have to check if I'm dividing by -|x|*0, if I want to use T81. What's the big gain? It apparently just complicates things.

Welbog
Where can I find the proof of equation 10?

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