# Talk:James A. D. W. Anderson

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## Article Discussion

The two articles PerspexMachineIX.pdf and PerspexMachineVIII.pdf look like pseudo-science to me (only references to their own papers, and one to wikipedia) --Soyweiser 13:17, 7 December 2006 (UTC)
I just took off some stuff that belongs on the transreal article. Specifics on nullity, etc. Attempting to create a structure for this article. fintler

## Publications Discussion

This is not new. There already was a NaN ages ago. --Soyweiser 13:07, 7 December 2006 (UTC)
And what the Flying F is Transreal analysis in mathematics, as far as I can see the author of the new NaN made that up to. Don't really consider that to be proper academic research now is it? --Soyweiser 13:30, 7 December 2006 (UTC)
He came up with a name for his idea, I don't see a problem with that. It separates it from stuff that already exists. It uses NaN only in passing, it has nothing to do with it directly, take a look at the paper. It notes the existance of proofs explaining 0^0 -> 0/0 = DNE. fintler 13:39, 7 December 2006 (UTC)
Well Sadly I don't have the time right now to read another paper. So I'll believe you about the 0^0 -> 0/0 = DNE part. (There where some comments on the /. page about the transreal analysis and the paper, some strange things that could indicate work that still has to be done on the papers.) But the problem was already solved (at least partially then) and it doesn't justify all the media attention. --Soyweiser 13:50, 7 December 2006 (UTC)

## Mathematician?

Giving Anderson the title of mathematician in my opinion is inaccurate given that his academic background is in Computer Science and his "mathematical" publications are not from any established mathematical journal. 65.246.47.233 18:20, 7 December 2006 (UTC)

I agree on that one. --Soyweiser 19:16, 7 December 2006 (UTC)
I also agree. Also candidate for deletion? Lee-Jon 23:09, 7 December 2006 (UTC)
Candidate for deletion? A nope, It is a current event, and it is someone who stirred up some commotion, so in that regard I do think it should have its place in wikipedia. Soyweiser —The preceding unsigned comment was added by 195.169.215.154 (talk) 00:10, 8 December 2006 (UTC).
Fifteen minutes of fame (or infamy) doesn't qualify one for an encyclopedia article, IMHO. Lunch 00:19, 8 December 2006 (UTC)
Another vote for deletion, he doesn't have any credite published papers Masiosare 00:23, 8 December 2006 (UTC)
This deletion process is going all wrong. The guys page gets deleted, while the wierd math is staying. How are people going to see that this math ain't normal math? --Soyweiser 04:10, 11 December 2006 (UTC)

Please vote at Wikipedia:Articles for deletion/James Anderson (mathematician) whether or not you feel he deserves an article. Lunch 00:31, 8 December 2006 (UTC)

article renamed to James Anderson (computer scientist)fintler 16:32, 11 December 2006 (UTC)

## Crackpot

I've tried to explain that his notability is as a crackpot. But you may want to add more direct sources, which should be available in the wikinews article. And the text could be cleaned up considerably. JeffBurdges 12:37, 11 December 2006 (UTC)

whether he's a crackpot or a dabbling amateur I don't think "james anderson is a mathematics crackpot" as the first sentence either helps debunk him or makes for a very good Wikipedia article. Without a referenec it's very 'POV'. An introductory sentence should probably describe his job as a dr and/or lecturer of computer science, (in the UK this is equivalent to prof, because the word professor is reserved for very senior academic posts). debunking his nonsense should take the form of reasoned refutation, not name calling--Mongreilf 12:50, 11 December 2006 (UTC)
Exactly, I mean jeez, I'm sure Newton and Leibniz were both called crackpots by some crowd of people when they introduced calculus. Prove his stuff wrong instead of personally attacking him. It's more effective and helps to avoid making yourself look like a crackpot :)

Listen, you cannot prove him wrong since his theory is sound, and if there is some minor mistakes they are certainly correctable. The point is that it is very trivial stuff for a logician. It is like renaming the numbers, and publishing papers on it. It is easy to see, for e.g. an undergraduate in mathematical logic, that his theory is a conservative extention in the sense that any theorem only involving the old symbols (i.e. no nulity), can already be proven in the old theory. All new theorems involve the symbol phi, and all expressions involving phi is simply....phi or NaN or undefined or call it what ever you like....

fintler 16:32, 11 December 2006 (UTC)

• They laughed at Galileo.
• They laughed at Newton.
• They laughed at Einstein.
• They laughed at Tesla.
• They laughed at the Wright Brothers.
• They laughed at Heisenberg.
• They laughed at Bozo the Clown.
In other words, a crackpot is a crackpot until his work is accepted. However, whether he is a crackpot is not the question we should be considering. It's whether he is a notable crackpot. — Arthur Rubin |

(talk) 19:31, 11 December 2006 (UTC)

To say he is a crackpot is a POV statement, we should concern ourselves with whether he is notable, and just that. --Salix alba (talk) 19:58, 11 December 2006 (UTC)
In general, yes. But his only notability is as a crackpot. If he is not a crackpot, he's clearly not notable. — Arthur Rubin | (talk) 20:07, 11 December 2006 (UTC)
no a person who's work hasn't yet been accepted, particularly if that work is new, isn't by definition a crackpot, else a lot of research is crackpottery to begin with. sound refutation ignored is the sure sign of a crackpot (and a refutation of usefullness, originality or mathematical consistency would do here), though self-publication suggests anderson may be avoiding refutation. if he doesn't persist in the face of good criticism and takes it as a lesson learned then we can't call him a crackpot.--Mongreilf 12:15, 12 December 2006 (UTC)
oh hahaha i've added a sentence and a link on his defense of his ideas that cameout today. getting close to defining himself as a crackpot--Mongreilf 15:05, 12 December 2006 (UTC)
It is somewhat unfortunate, but the creator of a new theory is not notable for that reason until the theory is at least discussed, if not accepted, in academic circles. So, if he's not a crackpot, we have to wait for the article. — Arthur Rubin | (talk) 15:19, 12 December 2006 (UTC)

Alright, I guess I'm happy with the article as it stands, although I'd much prefer that the introduction gave some indication that he is might be a crackpot. But its a short article and we get to the sociologically invalid statment pretty quick. So I guess all thats left is to descide if all his other aticles, i.e. Perspex machine, etc. should be deleted or redireted here, no? JeffBurdges 13:39, 14 December 2006 (UTC)

Some people are VERY DENSE. The section on Transreal Computing Ltd. ought to include a note to the effect that computers capable of performing transreal arithmetic already exist, since it's the same (with some small programming changes) as IEEE floating point arithmetic, if you ignore the limited precision of the latter. This is clear if you've read the rest of the article and have any brains at all, but some people are VERY DENSE.

(Bit of a shame that the user shouting 'VERY DENSE' had not the smarts to sign his or her name quota)

## Transreal arithmetic table...

We need to distinguish equality from definitions; for example, Φ = −Φ is analogous to -NaN is defined to be NaN, even though they're not "equal". NaN is NaN, although they don't compare as equal. I'm not sure how to do this. — Arthur Rubin | (talk) 14:53, 15 December 2006 (UTC)

If you really need to show that -NaN is NaN but that -NaN ≠ NaN, then say -NaN := NaN (read as negative NaN is defined to be NaN). Welbog 16:47, 15 December 2006 (UTC)
As one Dave Korn points out in response to this BBC article, the only difference between nullity and NaN is that nullity compares as equal to itself while NaN does not; but then by the axiom, ${\displaystyle \Phi -\Phi =\Phi }$, so even though nullity is equal to itself, the difference between it and itself does not vanish -- thus the statement that it "is equal to itself" has no real meaning. -- Thomas —The preceding unsigned comment was added by Tkoeppe (talkcontribs) 18:44, 15 December 2006 (UTC).
A more sophisticated way to view this is to treat = as an equivalence relation. For IEEE numbers (NaN) this relation is not reflexive, for transreals (${\displaystyle \Phi }$) it is. There are pros and cons to each treatment. --Salix alba (talk) 19:00, 15 December 2006 (UTC)

BTW, someone might want to merge Talk:transreal number here now that the articles have been merged. Lunch 18:11, 15 December 2006 (UTC)

It is not true that division of zero by zero may yield many different results in standard arithmetic. Division by zero is not defined at all in standard arithmetic. The statement "0/0 is an indeterminate form" is a mnemonic for a theorem about limits of quotients, and it does not imply that 0/0 has a literal meaning. David Radcliffe 23:40, 22 December 2006 (UTC)

## Reply to critics giving verifiable facts

I am the subject of this Wikipedia entry. Anyone is free to contact me directly to present criticism of my work. I have responded directly to very many people in the last two weeks and, whilst I cannot respond to the many thousands of people who have expressed views on my work, I will try to respond to all reasoned criticism.

It has been claimed, as reported here, that my publications on transreal arithmetic are plagiarised. I deny this. I invite you to send me references to work that you believe has been plagiarised, indicating the passages in my work which you believe have been lifted from passages in the prior work. At the very least, it will be interesting for me to consider the substance of your claims.

I have written more than two papers on transreal numbers. In addition to the two cited in the article, I have written the following. And I have written more than these, though they are not available on the web.

http://www.bookofparagon.com/Mathematics/SPIE.2002.Exact.pdf

http://www.bookofparagon.com/Mathematics/PerspexMachineVII.pdf

It has been claimed that transreal numbers are wheels. This claim is false. The additivity and distributivity axioms of transreal numbers and of wheels are incompatible.

The IEEE standard, which I cite in my work, is different from transreal arithmetic. Transreal arithmetic makes total the operations of addition, subtraction, multiplication, division, equality, and greater-than. It is defined axiomatically on a set of numbers. The axioms are derived from only the standard algorithms of arithmetic. Transreal arithmetic does not make any appeal to calculus (analysis) to define the infinities and nullity. By contrast the IEEE standard does make such an appeal. Hence transreal arithmetic is independent of the IEEE standard and the IEEE standard is cited in my work. In these circumstances, I do not see how I can have plagiarised the IEEE standard.

The IEEE standard defines NaN /= NaN. This is dangerous. If a programmer tests equality of some datum of integer type, but the code is updated in a modular way to change the type to float, then the equality test may fail. This is more dangerous in structured types such as arrays, records, and, notably, database records. Here, a programmed test for equality of structures may fail if any datum is changed from any other type to float. This is not a theoretical risk. Wikipedia records the case of the USS Yorktown that was stranded for two hours forty minutes when a zero was typed into a database record, causing a division by zero error that cascaded into the entire network of computers on the ship. These computers were equipped with IEEE float.

I am making a very serious point here. Not handling division by zero is suicidal. Handling division by zero using IEEE float is dangerous. Handling division by zero using transreal arithmetic is safer than doing it by IEEE float. You have a choice to make. Use IEEE float or use transreal arithmetic. The choice you make is a matter of life and death. Choose.

Many words and symbols in mathematics have many uses. This is a testament to the productivity of mathematics. I do not see how it can be adduced as a criticism of my nomenclature in particular. Indeed, I specifically chose capital Phi to distinguish it from the better known uses of lower case phi.

Note that transreal arithmetic is defined only by arithmetical axioms and not by any appeal to calculus. Arithmetic is logically prior to calculus so no argument from calculus can undercut any argument from arithmetic. However, arguments from arithmetic can undercut arguments from calculus.

All discussion of calculus in relation to transreal numbers is logically irrelevant, though it is clear that the authors of the article think it is important. I take it as a sign of the paucity of your mathematical tools for handling division by zero that you have to refer to any mathematical system more complicated than arithmetic.

Consider the function f(x) = (sin x)/x. I agree with you that the limit of this function, as x approaches zero, is one. This says nothing about its value at x = 0. By transreal methods I have f(0) = (sin 0)/0 = 0/0 = nullity. Nullity lies off the number line so f(x) is discontinuous at x = 0 (and also at x = +/- infinity and x = nullity.) Thus, I can evaluate f(0) using transreal arithmetic. But you cannot evaluate f(0) using standard arithmetic, nor can you evaluate f(0) using the methods of calculus. (I can also evaluate f(+/infinity) and f(nullity)).

I maintain that it is useful to have total functions everywhere so that software is well defined everywhere. In addition to the case of f(x), immediately above, consider the case of computing eigensystems using the Jacobi method. The standard method uses the arctangent in most cases, but cases of division of a non-zero number by zero can occur, in which case the standard method uses the arccotangent. But, cases of zero divided by zero can occur, in which case the standard algorithm fails. Thus the standard algorithm has two branches and a failure state. When maintaining code it is necessary to maintain both branches, the selection code, and the error handler. By contrast the transreal implementation of this algorithm uses a transreal arctangent everywhere. It has no branches and no error states. It is easier to maintain.The transreal version computes an identical value to the standard version in every case that the standard version can compute anything at all, and it computes a solution where the standard version fails. I regard this as a useful improvement on standard arithmetic and standard programming methods. And one which generalises to very many programs indeed.

For a brief discussion of the transreal Jacobi algorithm see:

http://www.bookofparagon.com/Mathematics/SPIE.2002.Exact.pdf

An earlier paper of mine developed a transreal arithmetic with projective infinities: +infinity = -infinity.

http://www.bookofparagon.com/Mathematics/SPIE.2002.Exact.pdf

Whilst it is true that there is no international standard defining the representation of transreal numbers in binary, there are binary representations of transreal numbers. Software libraries may be down loaded from:

and more comprehensive libraries are cited in a paper on transreal compiler methods:

http://www.bookofparagon.com/Papers/PerspexMachineX.pdf

I am fully aware that many people consider me to be a crank. I deny that I am a crank. If you believe that I am a crank then I invite you to send me a substantive criticism of my work.

Now, let me make my claim very clear. I claim that I am the first person to define 0/0 as a fixed number. (In transreal arithmetic it is also the case that +/- infinity is/are fixed number(s), depending on whether the projective of affine infinities are used.) I am aware of many mathematical approaches to handling the symbol 0/0, but I maintain that none of these acknowledges that 0/0 is a fixed number, and I maintain that many of them do not acknowledge that 0/0 is a number of any kind. If you believe otherwise, then send me a reference.

When zero was introduced to mathematics, approximately 1,200 years ago, there was discussion of 0/0, but, I maintain, there was no valid, arithmetical, identification of 0/0 with any fixed number. If you believe otherwise, then send me a reference. I have provided the identification by the definition nullity = 0/0 such that nullity is a fixed number. It happens to lie off the number line.

I have described the geometry of transreal co-ordinates and their trigonometry. I have begun to describe the calculus (analysis) of transreal numbers by adopting the axiom (d e^x)/(dx) = e^x.

There is a great deal more I could say about the article, but this must suffice for now. James A.D.W. Anderson 11:35, 21 December 2006 (UTC)

You claim that you are "the first person to define 0/0 as a fixed number". What about [1] (PDF), a draft from 1997 (according to [2])?--80.136.177.142 09:48, 22 December 2006 (UTC)
You make some rather emotive claims about the need to be able to divide by zero (claiming that it's a life or death situation) and I don't think those claims improve your argument at all. Can you explain clearly why you believe that the ability to divide by zero is better than handling the exception of a divide by zero. You specifically cite the case of the USS Yorktown and its divide by zero error. It's worth noting that the Yorktown had trouble because of an unhandled divide by zero error, which caused a fault in the consumer operating system (Windows NT) used aboard the ship. If I write a program that doesn't catch divide by zero errors, in what way is that better than having a program that suddenly has nullity in calculations that occur after a divide by zero. I do not see how nullity enables a program to continue operating safely, any more than an unchecked divide by zero is safe. Jgrahamc 13:52, 22 December 2006 (UTC)
Well, in a field, division by zero cannot be defined for any dividend other than zero. If we let x != 0 and define division of x by zero, 0 * (x / 0) is zero by definition of zero but is x by definition of division, meaning x = 0, which contradicts our assumption that x != 0. Therefore, either x / 0 must be left undefined, or our system of arithmetic must not be a field. If I'm not mistaken, the transreal numbers do define x / 0 in this case, so it's not a field. However, defining only the division 0 / 0 may lead to something interesting; I'll investigate. --Ihope127 22:39, 22 December 2006 (UTC)
Actually, it seems that the definition of a field doesn't even include a division operation. It does, however, include a reciprocal operation, so 0 / 0 would be defined as 0 times the reciprocal of 0. That means defining 0 / 0 means defining the reciprocal of 0. Now, as I said above, zero times anything is zero, but anything times its reciprocal is one, so zero times its reciprocal is both zero and one, something explicitly prohibited by the definition of a field. Therefore, the transreal numbers are not a field. —The preceding unsigned comment was added by Ihope127 (talkcontribs) 22:49, 22 December 2006 (UTC).
I think the most fruitful way to think of trransreals is to go back to the earlier papers. It seem like the idea of transreals grew from considering homeogeneous coordinates, (x,y) with the equivilence relation (a x,a y) = (x,y) for non zero a. Traditionally the zero element (0,0) is excluded for the set, but you could define a set formally. In this setting it make more sense to talk of infinite values (x,0) and nullality (0,0). If your working in this set then things seem more natural.
Indeed your right, pretty much by definition you can't define 0/0 in a field. Fields are not closed under division (or recriprocal). I've a feeling problems stem from the fact that a field is an odd sort of Algebraic structure in the sense of universal algbra.
I'm also curious about the perspex machine, if you strip away the transreal and turing machine stuff do you get a novel sort of neural network? --Salix alba (talk) 00:31, 23 December 2006 (UTC)

Please remember our Wikipedia:No original research policy. Wikipedia is not the place for directly interviewing the article's subject about material that no secondary source has addressed. If you want to interview Anderson, request that Wikinews set one up. Uncle G 06:40, 31 December 2006 (UTC)

## Talk:Transreal number merged here

I am merging the content of the above talk page here, since the content is already merged. Somebody may consider removing that talk page later. AbelCheung 23:31, 5 January 2007 (UTC)

## BBC Q&A Session on Dec 12, 2006

from the same page the bbc article is at:

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.

update: [3]

Maybe some people on here will try to ask him to clear some stuff up for them. Make sure you visit the BBC site on tuesday if you're interested. fintler 23:59, 11 December 2006 (UTC)

## axioms of the transreal numbers

okay, my formal training in mathematics is limited, but can anyone see a reason that the axioms would not be attackable by the standard problem with resolving division by zero to signed infinity?

namely, if 1/0=${\displaystyle \infty }$ then:

${\displaystyle \infty }$ = 1/0 = 1/(-1 * 0) = -1 * (1/0) = -1 * ${\displaystyle \infty }$ = -${\displaystyle \infty }$

it looks to me like all of those steps are legit within his axioms (he preserves commutativity and allows multiplication of infinity by negatives to produce negative infinity). yet, this also contradicts his definition of ${\displaystyle \infty }$ != -${\displaystyle \infty }$

any takers? --Frank duff 17:43, 7 December 2006 (UTC)

I agree with you. ${\displaystyle \infty }$ as a “transreal number” is exactly the same as ${\displaystyle -\infty }$ and has nothing to do with ${\displaystyle \infty }$ as in the limit value. This article needs heavy justification. In fact ${\displaystyle 1/0}$ no longer equals ${\displaystyle (-1*1)/(-1*0)}$ with the set of axioms given in the paper. Sam Hocevar 18:13, 7 December 2006 (UTC)
really? it seems to me that given associativity [A12], commutativity [A13] and his definition of division [A17], then ${\displaystyle 1/0}$ must equal ${\displaystyle (-1*1)/(-1*0)}$. i have to admit though, it's been a long time since i did this sort of proof from axiom. --Frank duff 18:30, 7 December 2006 (UTC)
For this to work I think you need to prove that ${\displaystyle a^{-1}*b^{-1}=(a*b)^{-1}}$ 88.108.28.16 18:38, 7 December 2006 (UTC)

This looks fine to me. You have to remember that he is a computer scientist. What he has defined could be implemented as a system of computer arithmetic that would be more robust than the standard one. However I doubt very much if it is novel. 88.108.28.16 18:14, 7 December 2006 (UTC)

I just found a variation of the above counterclaim that looks more difficult to dispute. Since James defines division as reciprocal, i.e. ${\displaystyle \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}}$, he gives this example in his own paper:
${\displaystyle 0^{-1}=\left({\frac {0}{1}}\right)^{-1}={\frac {1}{0}}=\infty }$
If the following identity is true:
${\displaystyle {\frac {0}{1}}=0={\frac {0}{-1}}}$
Then we have
${\displaystyle 0^{-1}=\left({\frac {0}{-1}}\right)^{-1}={\frac {-1}{0}}=-\infty }$
This also gives the conclusion that +∞ = -∞. Actually, in James' paper (please refer to James Anderson (computer scientist), 9th item in the References section), I fail to notice him giving any formal dispute of this dilemma, other than a simple sentence: "Why represent it this way?" AbelCheung 18:55, 5 January 2007 (UTC)

Just noticed, James has covered a bit about this himself. His PerspexMachineVIII.pdf [T81] says, ${\displaystyle (a\times b)^{-1}=a^{-1}\times b^{-1}}$ for many cases of a, b, but not when one of them is zero and another is negative number. This means ${\displaystyle \left({\frac {0}{-1}}\right)^{-1}\neq {\frac {0^{-1}}{(-1)^{-1}}}}$, and implies ${\displaystyle \left({\frac {0}{-1}}\right)^{-1}\neq {\frac {-1}{0}}}$. Yet fractions like ${\displaystyle {\frac {1}{0}}-{\frac {1}{0}}={\frac {1-1}{0}}}$ can be conveniently used. Funny.
AbelCheung 02:53, 6 January 2007 (UTC)

## Hoax warning

This cannot be anything else than a hoax. There are so many problems with the article (${\displaystyle \infty -\infty =\Phi }$ makes no sense, nor does having two different ${\displaystyle \infty }$ and ${\displaystyle -\infty }$ values) that anyone remotely skilled in math can spot that it should really be removed before Wikipedia makes a fool of itself. Sam Hocevar 17:50, 7 December 2006 (UTC)

${\displaystyle \infty -\infty =\Phi }$ because ${\displaystyle {\frac {1}{0}}-{\frac {1}{0}}={\frac {1-1}{0}}={\frac {0}{0}}=\Phi }$ and he has set different specific infinities, the largest and smallest on each side of the real number line.T.Stokke 22:34, 14 March 2007 (UTC)
I tried to avoid making any affirmative claims that the system actually works, rather just describing the claims in the paper as claims. If you see any, please take them out. Wouldn't it be original research to incorporate these criticisms into the article, though? Eliot 17:53, 7 December 2006 (UTC)
Having the article at all before the paper was peer-reviewed is original research itself. Sam Hocevar 17:55, 7 December 2006 (UTC)
Since it was written about by the BBC, a supposedly credible source, that gives it an 'in,' even if (we think) they probably shouldn't have written about it. Anyway, if it comes up on AfD, I'll vote to delete it; by the time the decision is made, there will probably be no more people coming via BBC to try to create this article or a related one. In the meantime we might as well inform the people who do come, though. Eliot 17:59, 7 December 2006 (UTC)
Then the article should not dig into the “mathematics” of it until the exact goal of the authors is known. I have now read the complete paper, and it seems correct (as in non-contradictory). However to reach this result they had to declare some operations as invalid (for instance, ${\displaystyle (a^{-1})^{-1}=a}$ is no longer true for all numbers). This questions the usefulness of teaching that to kids, for instance. Sam Hocevar 18:11, 7 December 2006 (UTC)
I agree with that, but I couldn't find an NPOV to say 'whoever's idea it was to teach this to high schoolers should be fired.' Please, remove anything from the article that you think shouldn't be there. I only put the identities in because it seemed like a good way to illustrate the nullity concept. It seems bad to put those there without mentioning the identities that are sacrified by the more expansive definition of 'number,' though. Eliot 18:47, 7 December 2006 (UTC)
I've reread Anderson's paper. And sure enough, his axioms are trivially inconsisent in a way provable within his system. i've posted a rigorous mathematical refutation on slashdot (http://slashdot.org/comments.pl?sid=210408&cid=17148824). i'm sure i'm not the first person to notice this, but i didn't see a rigorous refutation anywhere else, just general dismissal. --Frank duff 18:17, 7 December 2006 (UTC)
Your refutation is hardly rigorous. See my response on Slashdot. — flamingspinach | (talk) 19:11, 7 December 2006 (UTC)
I tried to do what you did, and I would have to invoke [T81] at some point (bringing the inverse inside a product); yet the guarding clause prevents me from doing so. Your proof is flawed. --King Bee 19:24, 7 December 2006 (UTC)
It does look like [T81] is holding this system together. It now appears to me to be consistent. So that raises the question: Is it useful? Given the symmetry found in mathematics, I have a real problem with 1/0 being biased towards positive infinity. That would mean ${\displaystyle f(x)=1/x}$ isn't an even function. At least IEEE float has the symmetry of -(1/0) == (1/-0). More to the point, the author's claim that Φ is superior to NaN isn't really true. He says
When NaN is used as the argument to a function, as a database record, as a time stamp, or anything not envisaged in the standard, its semantics become problematical, and the more widely it is used, the more problematical its semantics become. Ultimately, it will fail in an incoherent morass of interpretations. By contrast nullity, Φ, is a well defined number with fixed semantics. It means that there is no unique number on the real number line, extended by the signed infinities, that satisfies the given formula. So a function with some nullity arguments may perform arbitrary processing on them, because they are just numbers. A database record with value nullity is not set to any real value. A time stamp with value nullity is not set to any real time. And so on, for all applications.
The thing is, you still can't put Φ in a database without special effort because, like NaN, Φ has no order with respect to itself or other numbers so you can't put it in a binary search tree. Furthermore, NaN is a well defined value with fixed semantics. —Ben FrantzDale 21:51, 8 December 2006 (UTC)
As much as I think buzz around the paper is much ado about nothing, I think the author can reasonably be trusted when he claims Isabel/HOL validated the system. The implications of it are quite a different matter, though. I can add Ѫ or ホ to the set of real numbers with associated axioms that make the system consistent; that does not mean it is useful for anything. Sam Hocevar 20:12, 7 December 2006 (UTC)
How could a system prove itself consistent without violating Goedel's second theorem? --King Bee 20:43, 7 December 2006 (UTC)
The system is not proving itself. Sorry if you were joking, I was not sure whether you were really asking. Sam Hocevar 02:11, 8 December 2006 (UTC)

I'm not a mathematician, but that isn't necessary to having an opinion of whether this is a valid article for Wikipedia. Wikipedia contains many controversial topics, many denounced by various people in that specific field of expertise(see Global Warming). Though this isn't quite the same as Global Warming, it still doesn't mean its controversy makes it unworthy of inclusion in an encyclopedia; rather, challenges to it should be presented in the article on it, like a Criticism section. I remember a teacher trying to teach something to similar to this in a class of mine awhile back, and though I thought they were full of it that doesn't exclude it from being a significant point of interest or discussion. Smeggysmeg 19:24, 7 December 2006 (UTC)

One can not mathematically prove the validity of global warming (yet), but there are many examples of how Anderson's theorem is inconsistent with itself. Global warming may be "controversial", but Anderson's nullity theory is plain "wrong". Why not start an article about how 1=4? Has April Fools Day come early? Flangiel 07:39, 8 December 2006 (UTC)
How is it inconsistent with itself? His arithmetic doesn't constitute a field, which means that common sense doesn't always apply, but it does appear to be internally consistent. --Carnildo 07:45, 8 December 2006 (UTC)
One example is listed on this talk page above, where Anderson's definitions can be used to show that ${\displaystyle \infty }$ = -${\displaystyle \infty }$, yet his own definition states that ${\displaystyle \infty }$ is different to -${\displaystyle \infty }$. Flangiel 05:45, 9 December 2006 (UTC)

I would just like to bring to people's attention how little mathematics Anderson actually knows, I quote from him: "It is just an arithmetical fact that 1/0 is the biggest number there is." ... I'm sorry, in what freaking universe is this true? Clearly he has been introduced to some programming language whereby this happens to work and then claimed it as an mathematical fact. Surely this alone is enough to invalidate the majority of his claims that stem from this ill-concieved claim.Sekky 22:10, 7 December 2006 (UTC)

Having come across the concept of "nullity" via a forum I frequent, I felt I had to see the wikipinion on it, and lo and behold... Eurgh. Sorry, but looking at his reasoning, it's basically "We can't well define 0/0. I will create a transreal number, called nullity. It is defined as 0/0. Hey, rest of the mathematical community! 0/0=Nullity! I just solved a 12 century old problem! Look at me, I'm great!" And this is ignoring the MASSIVE problems with it, such as ${\displaystyle \infty -\infty =\Phi }$ I don't say we should remove all reference to it, but I think that it should be marked as mathematically dubious, and it should be clearly noted that all he has done is restated a previous concept and attached his name to it. --Crane

Anyone who is capable and willing to check this out, I want to know if I've found an inconsistency or if I'm barking up the wrong tree with this one. my attempt Thanks, Welbog. —The preceding unsigned comment was added by 156.34.78.192 (talk) 01:36, 12 December 2006 (UTC).

Your proof depends on the validity of E10, and I haven't seen a proof of it. What you've found is that either E10 is invalid, or the axioms are inconsistent. If you can derive E10 from the axioms, then you've got an inconsistency. --Carnildo 06:18, 12 December 2006 (UTC)
I figured that. Either way, [E10] is invalid, though, which is the theorem he uses to prove that 00 = Φ. I really want to see the proof to [E10]. But you're sure that I haven't made an error other than my overambitious conclusion? -Welbog 156.34.78.192 11:24, 12 December 2006 (UTC)
Doesn't look like it, though to make it airtight you should find the specific axioms or theorems that support steps 1 and 2, since many of the "basic properties of reals" don't hold for the transreals. --Carnildo 19:06, 12 December 2006 (UTC)
Indeed, I didn't include real justifications for those steps because he doesn't justify those steps in his proof that 0^0 = 0/0. But this seems irrelevant now as his follow-up article on BBC states that he knows about problems with E10. --Welbog 19:13, 12 December 2006 (UTC)

## Deleted article

We had a series of articles on this topic that were deleted only a week or two ago, as hoax/unscientific nonsense. Complaints then, as now, are that: 1) Concept seems to be a kludgy reinvention of Conway's star, 2) insufficient context w.r.t. IEEE definitions of NaN. 3) Nebulous claims -- e.g. "might help with projective geometry". I'm disapponted to see this on slashdot, and the recreation of the aricle. linas 19:22, 7 December 2006 (UTC)

Yes, the projective geometry claim is totally ludicrous. I’m removing it. Sam Hocevar 20:07, 7 December 2006 (UTC)

## How is this any different from IEEE floating point?

How does "transreal arithmetic" with ${\displaystyle \pm \infty }$ and ${\displaystyle \Phi }$ differ from standard IEEE floating point math?

In standard floating point math, the same axioms hold, if we simply replace ${\displaystyle \Phi }$ with NaN:

• ${\displaystyle 1\div 0=\infty }$
• ${\displaystyle 0\div 0=NaN}$
• ${\displaystyle \infty \times 0=NaN}$
• ${\displaystyle \infty -\infty =NaN}$
• ${\displaystyle NaN+a=NaN}$
• ${\displaystyle NaN\times a=NaN}$

So, how is transreal arithmetic anything other than a restatement of IEEE floating-point arithmetic??? Moxfyre 19:29, 7 December 2006 (UTC)

That wasn't a comprehensive list of the axioms but just a few illustrative identities. But, for example, in IEEE, NaN != NaN but in transreal, ${\displaystyle \Phi =\Phi }$. Eliot 19:46, 7 December 2006 (UTC)

Which makes perfect sense, right? Why should NaN != Nan? I mean, if we can't possibly know the value of a result, all unknown values might as well be the same. Right? Oh wait.... Jamesg 20:37, 7 December 2006 (UTC)

Hmmm... okay. Transreal arithmetic doesn't seem to introduce any more internal consistency than IEEE floating point, in my opinion. (I've looked at the original sources now.) I don't get how a floating-point library based on Transreal arithmetic would be any more robust or useful. It would just have slightly different quirks in the corner cases. Moxfyre 20:01, 7 December 2006 (UTC)
Look at the paper, the conclusion has an explanation of what he's hoping to achieve with this. It seems that he found NaN to be a limitation in some other research and decided to try to create a way around it. Eliot 20:19, 7 December 2006 (UTC)
It seems to me that he made a slight change to the IEEE standard and is trying to introduce it into mainstream math. I personally prefer NaN over nullity because NaN means that the result is undefined. This is a very useful concept. By switching to nullity, any two previously undefined values are now equal to each other. This is contrary to limit theory. 71.36.13.85 20:02, 7 December 2006 (UTC)
Exactly. If we want to throw all of analysis out the window, then transreal arithmetic is fine. Otherwise, we're stuck with the real numbers. --King Bee 20:09, 7 December 2006 (UTC)
His point was to make a complete arithmetic, you take two numbers a and b, use the mathematical operators +,-,*,/ and get the number c as an answer. no matter what you do, as long as you use the mathematical operators and two numbers you get a number as an answer. in IEEE floating point you dont always get a number as an answer. —The preceding unsigned comment was added by T.Stokke (talkcontribs) 22:42, 14 March 2007 (UTC).
Sometimes in IEEE floating point, you get NaN, which acts the same as ${\displaystyle \Phi }$. — Arthur Rubin | (talk) 22:46, 14 March 2007 (UTC)
The difference is NaN is defined by computers as a completely unique kind of variable in that it doesn't necessarily equal itself. Phi, on the other hand, would be a completely typical variable that does. This is definitely less messy, especially from a programming standpoint. Most likely, the effect of this on computing will be minor and long-term, but regardless of its trivality, the result is something arguably more stable than IEEE float. It does introduce new problems, but there are overall fewer problems in general. 76.93.65.34 (talk) 10:20, 19 November 2008 (UTC)

## A description of a mathematical formalism

As a description of a mathematical formalism, I don't see why there is anything wrong with this article. Whether or not that formalism is useful is really not significant -- quite a few are not (e.g. Sedenions) but are still worth mentioning. However, this article needs to state in more unambiguous terms that this is a specific formalism promoted by a specific matematician, and isn't widely used. --Hpa 21:32, 7 December 2006 (UTC)

## (0-0)/0 = ???

First of all sorry for not using that fancy math-mode but: 0/0 = (0-0)/0 = 0/0 - 0/0 = nullity - nullity = 0 iff nullity = nullity.

Can someonte tell me were the above goes wrong? Poktirity 22:14, 7 December 2006 (UTC)

I had a look at the axioms an according to A9 nullity = - nullity.

Then 2*nullity = nullity doesn't it?Poktirity 22:38, 7 December 2006 (UTC)

That’s correct. Sam Hocevar 02:13, 8 December 2006 (UTC)
Lets not forget that he has defined a - a = 0 :a≠∞,-∞,Φ (: means if)

The "theory" can be explained as an introduction of a new "value" that are assigned to undefined computations made on the extended real line, and does just the same as the normal NaN computation does, exept NaN!=NaN, but nullity==nullity.

This is just a clean? workaround a try-catch statement. Are there any real applications with this theory? —The preceding unsigned comment was added by Paxinum (talkcontribs) 23:34, 7 December 2006 (UTC).

## Newsworthy

All of the talk about whether or not this is a valid mathematical concept seems to miss this fact that as of right now, this is a discussed concept. NPOV requires, IMO, not that we decide upon the merits of this article's inclusion based on our individual opinions on the accuracy of the concept, but on the noteworthiness of the topic. By this measure the article clearly belongs here. If near-universal disagreement with their respective arguments doesn't keep us from having articles on the Flat Earth Society and holocaust denial, then there doesn't seem any reason to me it should keep this out either.

That it is just a proposed as opposed to accepted explanation/theory is good and probably necessary to include in the article, as is text about disagreements and possible inconsistencies in the concept, provided they conform to Wikipedia policies (NPOV, no original research, etc.) But to delete this page because some disagree with the concept included in it misses the point. The purpose of Wikipedia is to catalogue what is known, and right now what is known is that a professor has proposed this concept, and it has gained at least enough acceptance that he is teaching it to his students. IMO, that makes it worthy of inclusion. Fractalchez 01:04, 8 December 2006 (UTC)

If we still had Category:Pseudomathematics, and this article were in it, I'd have less objection. If it's to be in Category:Mathematics or a "traditional" subcategory, it's wrong. It's not a discussed mathematical concept, and, as there's no published debunking (as no publisher thinks it worthy of debunking), we cannot include the clearly true statement that it's not new or interesting. Hence we should probably include nothing at all. — Arthur Rubin | (talk) 01:11, 8 December 2006 (UTC)

I say just start the article as "is a concept proposed by blah blah blah, and is not yet confirmed nor rebuked by any larger scientific community" and be done with it... It IS a mathematical concept, though not yet or maybe never-to-be a universally accepted one. So what? That's what we know and that much is true. 83.24.211.76 02:27, 8 December 2006 (UTC)

Wikipedia is an encyclopedia. It is not a site of first publication for original research. --Carnildo 03:11, 8 December 2006 (UTC)
I am aware of that. Encyclopedia contains established facts. It is an established fact that a certain professor proposed a peculiar off-the-scale number. Just as it's an established fact that some people created the Flying Spaghetti Monster for some purposes, even though the Monster doesn't actually exist. And later, if/when it's officially disproven, one might change the page into an established proof why nullity is wrong - starting the page as "Nullity is a concept introduced by... disproved A.D.2007 by ..." 83.24.211.76 11:30, 8 December 2006 (UTC)
I would like to propose that the professor in question is actually from another dimension. That proposal is now an established fact. I'll be expecting my page shortly.
The proposed guidelines for science were the closest I could find to apply here.
An item in the field of science is probably notable enough to merit an article on Wikipedia if it meets at least one of the following criteria:
1. It is part of the corpus of generally accepted scientific knowledge.
2. It is considered a possible explanation by a part of the scientific community independent of its creator.
3. It is advocated by at least one researcher who is prominent in the relevant field.
4. It is represented by a number of peer-reviewed papers, and is the work of several, not just one researcher
5. It is supported or examined by major scientific institutions, such as by funding, sponsoring seminars, or invited presentations.
6. It is previously thought of as correct or plausible, or is otherwise of historical interest.
7. It is advocated by a prominent persons or for political or religious reasons, or is a tenet of a notable religion or political philosophy, or is part of a notable cultural tradition or folklore.
8. It is well known due to extensive press coverage, or due to being found within a notable work of fiction.
9. It is believed to be true by a significant part of the general population, even if rejected by scientific authorities.
10. It is notable because there is strong criticism from the scientific community.
The closest this gets is probably #10...but I think most of the scientific community will ignore it rather than criticize it.
Yes, it's a fact that this guy made up a new number. It's not notable just because the BBC did a story about it.
In my opinion, he should be fired for teaching bad math to impressionable youth... --Onorem 11:50, 8 December 2006 (UTC)
Philosophical Question, indirectly related to the subject - impressionable or otherwise, would you consider it better to teach good math to disinterested kids or bad math to interested kids? I ask because the article also noted that his class was calling this 'cool' and 'exciting', expressions that students don't often associate with Math. Assuming that his theory is incorrect, is there any value whatsoever in removing him from the one position where his output is successful?
I appologize for the derail, but I've seen that statement numerous times on this page and it strikes me as being terribly nieve.

On the subject of notability: One thing I find troubling about Wikipedia, as an online resource, is its extreme difficulty in actually accounting for the internet when seriously looking at something's notability. The BBC webpage that contained this article garnered more than enough popular attention both from lay persons and mathematicians, all of which is still archived online and availble to search through google or BBC's own site. Why such things are rarely if ever taken into consideration - or specifically and overtly ignored as I've seen in some cases - continues to baffle me. 76.93.65.34 (talk) 10:48, 19 November 2008 (UTC)

## Transreal numbers are inconsistent with Complex numbers

I was looking through Perspex Machine IX: Transreal Analysis and I found this, [E 8], on page 5:

${\displaystyle \ln {-x}=\Phi :-x<\Phi }$

I think this just reiterates the fact that ${\displaystyle \Phi }$ is just am overglorified symbol for error. -Exomnium 02:03, 9 December 2006 (UTC)

You're mixing incompatible mathematical systems. Traditional complex arithmetic depends on the real numbers being a field over addition and multiplication. The transreals do not form a field, so you need to re-formulate complex math to deal with them. --Carnildo 04:29, 9 December 2006 (UTC)
Looking at it further, you're also misreading what he wrote. What the statement means is "If the results of taking the natural log of a number are constrained to the transreals, then the result of taking the natural log of a negative number is ${\displaystyle \Phi }$". It's just like taking the natural log of a negative number in the reals: if your results are constrained to the real numbers, then the result is undefined. --Carnildo 04:36, 9 December 2006 (UTC)
They're not incompatible, at least according to him. From Axioms of Transreal Arithmetic, "Having obtained the transreal numbers the complex, quaternion, and octonion numbers can be extended to total systems of arithmetic by replacing the components of these numbers with transreal numbers." And reformulating complex math is destructive of the purpose of transreal arithmetic, "they are supposed to produce the same result as standard arithmetic for all calculations where standard arithmetic defines a result." Where did you get that quote from? If he really said that, it's tantamount to him admitting that nullity is an over glorified undefined. -Exomnium 00:52, 10 December 2006 (UTC)

## redirect to NaN vs. redirect to James Anderson

I didn't make the change, but I support it. Other editors have effectively merged the transreal number into the James Anderson article anyway. Lunch 18:07, 15 December 2006 (UTC)

## Second reply to critics giving some verifiable facts and some opinion

• A machine proof of the consistency of transreal arithmetic has been released. [4]

• Real arithmetic is partial and allows considerable freedom to chose continuity constraints in real analysis. Transreal arithmetic is total and allows correspondingly less freedom to choose continuity constraints in transreal analysis. However, I know of no case where this reduction in freedom prevents transreal analysis from obtaining a solution. Furthermore, I conjecture that transreal arithmetic obstructs all and only the cases of continuity that are problematical in real analysis. I suggest that it is useful to have this obstruction because it forces machine proofs of transreal or real analysis to deal explicitly with these cases which might otherwise be missed erroneously. So far as I know, any desired continuity constraint that is valid in real analysis can be imposed on transreal analysis by defining an auxiliary function. Thus, there is no loss in expressive power by adopting transreal analysis, but there is a gain in the security of machine proof. I regard this as useful.

• I am continuing to assess claims to prior invention. Initially, I check the equivalence of the systems. So far, no prior system has proved to be equivalent to transreal arithmetic. I then check the mathematical scope of the other system to understand how it relates to the problem of developing a total arithmetic that is consistent with standard topology, geometry, and analysis. I then seek to understand the motivation for the other system. Where the author(s) are contactable, I engage in a correspondence to get these issues clear. Finally, I draw up conclusions on how the systems relate. This takes some time. If I do find a case of prior invention I will acknowledge that fact. (I will also note close cases, there is only one so far, to save others the trouble of conducting these comparisons, and to highlight similar work. For the avoidance of doubt, I would not have considered this one case to be similar had there not been repeated claims in favour of it.) James A.D.W. Anderson 15:04, 9 January 2007 (UTC)

• Dr. Anderson, I know I've asked before, and you've been gracious enough to reply, but has anyone independently published research on transreal numbers? Your answer to me before seemed to be that no, no one has.
• I am not aware of any scientific paper on transreal numbers of which I am not an author or co-author. James A.D.W. Anderson 18:36, 15 January 2007 (UTC)

To the other editors of the article, should the section on transreal numbers be deleted? IMHO, it falls entirely into the category of "original research". The only citations in that section of the current article only cite Dr. Anderson's work. Further, the notability of transreal numbers is adequately mentioned in the other parts of the article about Dr. Anderson; I don't think it merits an independent exposition here. Lunch 20:40, 9 January 2007 (UTC)

• A fair question ... but Dr. Anderson was not the person who originated the discussion and articles in Wikipedia (as far as I can tell). So those other contributers are in some sense an example of 'original research'. And no doubt others are, too. So I would suggest that it would be appropriate to give this a few months, or a year or so, before any 'move to delete'. In any case, if deleted, someone else will recreate the section anew, without the multiple contributions from others; probably not progress. quota 20:49, 9 January 2007 (UTC)
• The only citations in that section of the current article only cite Dr. Anderson's work. — You are conflating citations and the cross-links that link them to the text. The citations are in the "References" section of the article. The cross-links are merely a convenience. And as mentioned in the original edit summary, and again in a second edit summary, the source is the Wikinews article, as cited. Uncle G 15:21, 15 January 2007 (UTC)
• I did not initiate, nor induce anyone to initiate, any article under discussion here. James A.D.W. Anderson 18:36, 15 January 2007 (UTC)

Uncle G, can you look past the terminology and address the substance of what I'm saying? The Wikinews article that you keep mentioning does not address the technical merits of transreal numbers. Wikinews is not a computer science research journal. As the section "Transreal numbers" currently stands, it is a summary of Dr. Anderson's papers. It appears to be a fine summary, but -- again -- it is a summary of Dr. Anderson's work alone. Lunch 21:14, 16 January 2007 (UTC)

## Transreal Arithmetic Section

This section should be removed, or at least summarized in a few sentences. We don't need to summarize all of Dr. Anderson's works into his article here. It's too much detail for an article about Dr. Anderson. An article for Transreal Arithmetic was removed for being not notable enough, and not having any secondary sources for this seems proof enough. If anyone wants to summarize this section, feel free to do so, otherwise I will likely delete all of it except for the first two paragraphs. Vir4030 17:16, 15 January 2007 (UTC)

• Two falsehoods. transreal number was not deleted. It was redirected here, where one can find discussion of transreal numbers, in context. And there are secondary sources. They are cited in the "References" section in the article. Please do not remove sourced encyclopaedia content. Uncle G 20:22, 15 January 2007 (UTC)
Wikinews articles are not reliable sources since they are self-published wiki articles. I am restoring the cleanup and source tags to the article. This article needs cleanup. Vir4030 15:21, 16 January 2007 (UTC)
Oops - perhaps I should have put my comments in the above section here instead. At the expense of being repetitive, I think I'll say it again though: this section has no references other than to Dr. Anderson's papers. The Wikinews article is not a "reliable" technical source of information on transreal numbers. Lunch 21:30, 16 January 2007 (UTC)

## Third Reply to Critics Giving Verifiable Facts

Transreals and wheels are different things. Transreal numbers are defined on the set of real numbers, augmented with three strictly transreal numbers: negative infinity, positive infinity, and nullity. Wheels are defined variously on an integral domain and/or a commutative ring, augmented with at least two objects: infinity, and bottom. These sets are always different because transreals have minus infinity is less than infinity, whereas wheels have minus infinity equals infinity. This leads to a difference in the operations of addition and subtraction, and the property of distributivity, amongst other differences.

The transreal numbers preserve all of the properties of real numbers and extend some of these properties to the strictly transreal numbers. Wheels do not preserve all of the properties of real numbers. For example, the transreals preserve ordering of the reals so that the sentence “0 < 1” is true, whereas wheels do not preserve ordering so this sentence is undefined. As a second example, the sentence “0/0 = nullity” is a true sentence in transreals describing a property of the real number zero. The corresponding sentence in wheels, “0/0 = bottom,” does not describe a property of the real number zero, it describes a property of the zero element of an integral domain and/or a commutative ring, as the case may be.

IEEE float is dangerous because the specification that NaN is not equal to itself breaks a cultural stereotype. This is illustrated in the fragment of pseudocode, “statement_1; if x = y then statement_2 else statement_3 endif.” Suppose that statement_1 calculates x and y as identical quotients. If the code is executed in integer arithmetic and involves a division by zero in statement_1 then statement_1 raises an exception, otherwise statement_2 is executed. In no case is statement_3 executed. Now, if the code is executed in floating point arithmetic and statement_1 involves a division by zero then statement_1 may or may not raise an exception, depending on how flags are set in the processor. If no exception is raised in this case then statement_3 is executed; but if there is no division by zero in statement_1 then statement_2 is executed. Thus, the behaviour of conditional tests is radically different when a modular change to code is made that converts integer to floating point arithmetic. This is just one example. There are many possible examples of how breaking this cultural stereotype results in erroneous computer code, and it is hard for programmers to find such errors because their cultural stereotypes make it difficult to conceive of such cases.

The transreals, wheels, and floating point arithmetic are all methods of obtaining total functions. If a computer program uses only total functions then it will execute in every case, but if it uses partial functions, such as the functions of real arithmetic, then it may fail in some cases.

(The article asks for citations relating to my biography. I will supply third-party citations that list the required information where I know of these, but it will take some time to collate this information. All of the biographical information can be verified under the Freedom of Information Act by asking a question, in writing, of the Universities in question.) —The preceding unsigned comment was added by James A.D.W. Anderson (talkcontribs) 18:54, 15 January 2007 (UTC).

• You have the wrong idea of what Verifiability is on Wikipedia, Dr Anderson, and the wrong idea of what encyclopaedia article talk pages are for. Wikipedia is not a soapbox. Neither encyclopaedia articles nor talk pages are the places for you to harangue supposed "critics" or to promote additions to your theories about how "IEEE float is dangerous", as you are doing here. The place for that is your own web site, which we know that you are capable of publishing on. And verifiability, as far as Wikipedia is concerned, is the ability for content to be checked by readers against a secondary source that has been researched, fact checked, submitted for peer review, and published outside of Wikipedia. If you wish to give "verifiable facts", then please cite sources whence such facts may be obtained. What you are doing is not giving verifiable facts.

Incidentally: If the only way that information can be verified by readers is by writing off to Reading University requesting unpublished information, then that information is original research, which is forbidden here. Fortunately, we already have a source for what biographical information is given in this article. It is the Wikinews article cited in the "References" section. Uncle G 20:45, 15 January 2007 (UTC)

You seem to misunderstand the function of a talk page. It most specifically is where Dr. A. should discuss what appear to be inaccuracies in the article. Also, you call Dr. Anderson's posts here 'haranguing' despite that he's nicer than most other posters. You're trolling. 24.82.203.201 (talk) 09:47, 26 January 2008 (UTC)
• Sorry, I am new to Wikipedia. It is taking me a little time to get to grips with all of the policies and working practices. I did not mean to harangue anyone, only to answer questions posted here. Would you like me to supply references to printed catalogues of university staff so that the article can cite evidence that I was employed where the article states? Here are the bibliographic details of my thesis: Anderson, J.A.D.W. (1992) Canonical Description of the Perspective Projections, Ph.D. Thesis, Department of Computer Science, Reading University, England, RG6 6AY. James A.D.W. Anderson 17:57, 16 January 2007 (UTC)

Thank you for your replies, Dr. Anderson. Though Uncle G may be indelicate, I think what he's trying to say is that as an encyclopedia -- ideally, at least -- the information here is not autobiographical. It's also not meant to be a primary source of information; in principle, Wikipedia only presents information that others have already digested and "reported" on. (Mind you, though, there are zillions of independent "editors" here that each have their own notion of what does and doesn't belong. YMMV.) Lunch 21:19, 16 January 2007 (UTC)

Actually, it is much more important for information to be neutral, not autobiographical. Furthermore, how would these catalogues of university staff be primary or "original" research, anyway? mike4ty4 05:42, 18 June 2007 (UTC)

## wikinews as a source on transreal numbers

Uncle G, I've said it before above, but I'm not sure you've read my comments so I'll say it again. The Wikinews article that you keep mentioning does not address the technical merits of transreal numbers. Wikinews is not a computer science or mathematics research journal. As the section "Transreal numbers" currently stands, it is a summary of Dr. Anderson's papers. It appears to be a fine summary, but -- again -- it is a summary of Dr. Anderson's work alone. The Wikinews article is not a reliable source on this matter. I think other editors agree with me here. Lunch 20:51, 25 January 2007 (UTC)

• (That is, I reverted you, too, but it seems Arthur Rubin beat me to the punch by seconds. Funny, no warning of an edit conflict...) Lunch 20:56, 25 January 2007 (UTC)
• The article covers the territory covered by the Wikinews article upon which it was based. The section "Transreal numbers" (actually entitled "Transreal arithmetic", you will find) is not a summary of Dr. Anderson's papers. It wasn't based upon that. For the sixth time: Please read the summary of the edit. Uncle G 21:10, 25 January 2007 (UTC)
• I read the edit summary. Others did too.

I think the "self-published" template serves as a fair warning to readers that the references (explictly) cited in the section on transreals all come from Dr. Anderson.

You have reverted four times today. If you do it again, I'm going to ask that you be blocked from editing the article. Lunch 21:18, 25 January 2007 (UTC)

• You clearly hadn't. And you do not understand what is and is not a citation. The links next to sentences are not citations. They are just cross-links. The things in the "References" section, that the cross-links link to, are the citations, and one of those citations is the Wikinews article that I have pointed out eight times now, including in the original summary of the edit, formed the basis for the content. It does not "come from Dr. Anderson" as you assert. Asking for more sources with tags makes no sense when a citation for the source used is right there in the article (and was there even before the content was, as a matter of fact). Asserting that the source is Anderson when one can read the citation right in front of you and see that it isn't also makes no sense. Uncle G 21:29, 25 January 2007 (UTC)
To me the tags seem partially pointless. We are pretty certain that we have all the sources for the article so there is no use in a tag which asks for more sources. Self-published is not an acurate description as they have been reviewed (lightly) and published by a third party. On one of the other JA related articles I briefly created a custom template anlong the lines of: This work has appeared in published journals, however they are not fully peer-review so do not meet normal wikipedia criteria for reliable sources. Something along this line may be a suitable compremise. --Salix alba (talk) 23:41, 25 January 2007 (UTC)
Thank you, Salix, for that thoughful comment. That seems like a reasonable compromise and signals that, hey, there's really only one source on transreals. Lunch 02:06, 28 January 2007 (UTC)

## Misunderstandings and article accuracy

I see that this talk page is full of misunderstandings about transreal numbers, and the article is almost 49% about Transreal numbers, 49% IEEE floating point and 2% about James himself. Honestly there should be created 1 article about James Anderson himself, and 1 article about transreal arithmetics. —The preceding unsigned comment was added by T.Stokke (talkcontribs) 22:59, March 31, 2007 (UTC)

James Anderson's theories are covered in the article, and I cannot distinguish between transreal numbers and IEEE floating point in any (trans)real sense. — Arthur Rubin | (talk) 03:58, 1 April 2007 (UTC)

## The limits in the box on the right...

The second two limits in the box on the right are wrong.

you cannot evaluate x/0 as x -> 0+ 163.1.148.48 15:45, 2 May 2007 (UTC)

Changed x/0 to 1/x, for the moment. That might be the intent. — Arthur Rubin | (talk) 16:36, 2 May 2007 (UTC)

## BBC

I do not yet have a source for this, but the BBC report was not only shown on South Today, as I remember seeing this, and I live in an area where South Today is not broadcast —Preceding unsigned comment added by 86.161.138.219 (talk) 10:02, 10 December 2007 (UTC)

## Transreal numbers are not useful

I mean, did he publish a single article about the EXACT application of this? He mentions that a pacemaker might fail if it reached an exception while having 0 dividing 0. As one of the "Further Readings" state, with transreal numbers, what can you actually do if you get an answer of nullity? If he does state actual applications for this, it's VERY IMPORTANT that someone add them to the article. --68.161.190.195 (talk) 19:07, 13 December 2007 (UTC)

We don't need to note whether or not they're useful, only that he claims they're useful, and that claim is reported. Personally, I agree that none of his concepts (at least as reported here and in the references here) are useful, but I'm afraid being "useful" is not a requirement for the subject of a Wikipedia article. — Arthur Rubin | (talk) 19:27, 13 December 2007 (UTC)
However, if you're going to have an article about a concept which has notability, potential usefulness is vital in the article, provided it exists. --68.161.190.195 (talk) 09:40, 14 December 2007 (UTC)
Actually, I think that should be: provided there are sources that document it. Otherwise it is original research which is not game for Wikipedia regardless of whether or not it "exists". mike4ty4 (talk) 08:50, 17 December 2007 (UTC)
Wikipedia is not the place to make claims about things, only to journalistically and neutrally report on claims that have already been made -- no original research is one of the core principles of Wikipedia. Furthermore, although I know this ain't relevant, I just have to say, the software for pacemakers better darned be fault-tolerant and be able to catch and handle the exception if generated in a way that will allow it to continue to run! If I was running the company making the thing, I'd fire the programmer if I found out they hadn't implemented a proper exception handler! mike4ty4 (talk) 08:50, 17 December 2007 (UTC)
"If he does state actual applications for this..." Why are you people nitpicking over each of my posts in succession as if I knew nothing of how Wikipedia works, rather than JUST ADMITTING THAT YOU DON'T KNOW? --68.161.190.195 (talk) 06:52, 18 December 2007 (UTC)
I decide to have a look at what Anderson has been up of late and found a blog Transreal Computing - The Elevator Pitch. Apparently they are developing a parallel computer which use transreal arithmetic in the processing units. If this actually gets built, I see no reason why this system could not be built in a chip, then it could be said that Transreal numbers are useful. --Salix alba (talk) 11:30, 17 December 2007 (UTC)
It could be said that they are used. Whether or not they are useful--whether or not they would have been better off using IEEE floating point--would still be up in the air.--Prosfilaes (talk) 14:13, 23 January 2008 (UTC)

## Stupid MF

It's really criminal and inhumane that they let this nullity-guy 'teach' to mostly acritical, defenseless children. If he's such a tough guy, why doesn't he submit to Nature? They a have a 'dequackination' review board always striving for more blood. The parents of these pupils should sue. --Quackinator —Preceding unsigned comment added by 89.152.242.155 (talk) 07:02, 23 January 2008 (UTC)

I tend to agree, but this has little to do with editing the article, and probably should be deleted. — Arthur Rubin | (talk) 14:04, 23 January 2008 (UTC)

## The theory is void (as in empty)

Dear Mister Anderson,

Your introduction of "nuillity" is old news to the logic community, we usually use the symbol called \bottom in standard LaTeX to denote the undefined, and I know of no papers which treat the stuff your theory is made of, since it is a very easy excersise for undergraduates to verify by induction on the build-up of terms, that any term containing an occurrence of the constant for nullity, will be provably equivalent to nullity.

In contrast the introduction of complex numbers proved invaluable to many fields of mathematics. I am not an expert on the subject (complex analysis), but the simple identity ${\displaystyle i^{2}=-1}$ should suffice as an example. Does everyone see the difference? ${\displaystyle i}$ actually makes new computations possible. One ventures out into the complex plane—and imporantly—returns to the real line. In Anderson's model however, once you go to nullity, there is no coming back. Calling the undefined a number is actually a historically erroneous use of the term number, which usually is reserved for solutions of equations in a field (and their extenstions). One cannot, and this is an easily provable fact, find a multiplicative inverse for 0 in a field.

In fact, in more theoretically and abstractly flavoured mathematics, division as such do not even enjoy an independent status as an operation on numbers. To divide by ${\displaystyle n}$ is formally considered as shorthand for the operation to multiply by the multiplicative inverse ${\displaystyle n^{-1}}$. Since it is an axiom of the theory of fields that ${\displaystyle \forall x\left(0\cdot x=0\right)}$ (read for all elements ${\displaystyle x}$ of a field, it is true that ${\displaystyle x}$ multiplied by zero equals zero), and since the axiom for mulitplicative inverses state ${\displaystyle x^{-1}}$ of ${\displaystyle x}$ satisfies ${\displaystyle \forall x\left(x\neq 0\to \exists x^{-1}x\cdot x^{-1}=1\right)}$, it is evident that the only field where division by zero is possible, is the field with exactly one element 0, i.e. 0=1. Note that this last axiom does not state that there is no inverse for 0, only that by defintion, in a field, every nonzero element has an inverse. That 0 does not have an inverse, is a logical consequence of other axioms, and the assumption that ${\displaystyle 0\neq 1}$. In fact, there is one unique field where 0 has an inverse: the trivial field with one element. In this case we have ${\displaystyle {\frac {0}{0}}=0}$. But, recall, that we also have ${\displaystyle 0=1}$, so this field is rather uninteresting. (Not inconsistent though, the use of ${\displaystyle 0=1}$ as the canonical inconsistency is only sound on the implicit assumption that the theory in question implies nontriviality).

Best Mathias —Preceding unsigned comment added by 80.212.86.168 (talk) 22:45, 8 February 2008 (UTC)

## Perspex?

According to his website "perspex" stands for "perspective simplex", whatever that means. Perhaps that should be included in the article, so people don't get a mental picture of some sort of magic Wonkavator made of acrylic glass. Salvar (talk) 12:06, 1 June 2009 (UTC)

## Notablility

I believe the 2nd AfD established notability. Is there reason to believe consensus has changed? — Arthur Rubin (talk) 15:56, 23 January 2011 (UTC)

## A disproof

I'm going by the axioms stated in this paper of his. http://www.bookofparagon.com/Mathematics/PerspexMachineVIII.pdf By A15, nullity * a = nullity for any a. By A16, infinity * 0 = nullity. Letting a = nullity, A15 tells us that nullity * nullity = nullity. This means that nullity * nullity = infinity * 0. Dividing both sides by 0 and applying A17 as well as A12, we get nullity * nullity * 0^(-1) = infinity. By A15, this simplifies to nullity * 0^(-1) = infinity, which simplifies by another application of A15 to nullity = infinity. This contradicts his axioms. Q.E.D. — Preceding unsigned comment added by 99.241.98.119 (talk) 02:11, 14 July 2011 (UTC)

When you divide both sides by 0, you're assuming that 0/0 = 1, when it does not. Correcting your mistake, we have (after dividing by zero) Φ*Φ*0-1 = ∞*Φ, which (by A15) simplifies to Φ = Φ, which is a tautology. — Preceding unsigned comment added by Ianmathwiz7 (talkcontribs) 03:16, 10 September 2011 (UTC)

## Transreal arithmetic in a history of mathematics book

Transreal arithmetic is discussed in a recent history of mathematics book. See http://www.bookofparagon.com/News/News_00039.htm I had no contact with the author prior to publication. James Anderson - the subject of this page. — Preceding unsigned comment added by 134.225.31.99 (talk) 11:42, 1 October 2013 (UTC)

FYI - Here is a video interview I gave to BCS The Chartered Institute for IT. Part 1 deals with transarithmetic and computing. Part 2 deals with women in computing and education. http://www.bcs.org/content/conWebDoc/51645 - James Anderson — Preceding unsigned comment added by 86.143.120.170 (talk) 11:38, 19 November 2013 (UTC)