![]() | The following Wikipedia contributor may be personally or professionally connected to the subject of this article. Relevant policies and guidelines may include
conflict of interest,
autobiography, and
neutral point of view.
|
![]() | This article was nominated for
deletion. Please review the prior discussions if you are considering re-nomination:
|
This is the
talk page for discussing improvements to the
James A. D. W. Anderson article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
This article must adhere to the biographies of living persons (BLP) policy, even if it is not a biography, because it contains material about living persons. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately from the article and its talk page, especially if potentially libellous. If such material is repeatedly inserted, or if you have other concerns, please report the issue to this noticeboard.If you are a subject of this article, or acting on behalf of one, and you need help, please see this help page. |
![]() | This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||||||||||
|
![]() | This article links to one or more target anchors that no longer exist.
Please help fix the broken anchors. You can remove this template after fixing the problems. |
Reporting errors |
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)
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)
Please rename this article!
article renamed to James Anderson (computer scientist) fintler 16:32, 11 December 2006 (UTC)
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)
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)
(talk) 19:31, 11 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.
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)
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)
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:
http://www.bookofparagon.com/Pages/Downloads.htm
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)
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)
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)
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]
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= then:
= 1/0 = 1/(-1 * 0) = -1 * (1/0) = -1 * = -
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 != -
any takers? -- Frank duff 17:43, 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. , he gives this example in his own paper:
If the following identity is true:
Then we have
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)
This cannot be anything else than a hoax. There are so many problems with the article ( makes no sense, nor does having two different and 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)
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)
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)
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).
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)
How does "transreal arithmetic" with and differ from standard IEEE floating point math?
In standard floating point math, the same axioms hold, if we simply replace with NaN:
So, how is transreal arithmetic anything other than a restatement of IEEE floating-point arithmetic??? Moxfyre 19:29, 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)
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)
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)
Then 2*nullity = nullity doesn't it? Poktirity 22:38, 7 December 2006 (UTC)
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 ( talk • contribs) 23:34, 7 December 2006 (UTC).
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)
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)
I was looking through Perspex Machine IX: Transreal Analysis and I found this, [E 8], on page 5:
But, google says otherwise: ln(-1)=Pi*i.
I think this just reiterates the fact that is just am overglorified symbol for error. - Exomnium 02:03, 9 December 2006 (UTC)
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)
• 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)
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)
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)
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)
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 ( talk • contribs) 18:54, 15 January 2007 (UTC).
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)
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)
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)
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)
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 ( talk • contribs) 22:59, March 31, 2007 (UTC)
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)
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)
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)
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)
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 should suffice as an example. Does everyone see the difference? 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 is formally considered as shorthand for the operation to multiply by the multiplicative inverse . Since it is an axiom of the theory of fields that (read for all elements of a field, it is true that multiplied by zero equals zero), and since the axiom for mulitplicative inverses state of satisfies , 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 . In fact, there is one unique field where 0 has an inverse: the trivial field with one element. In this case we have . But, recall, that we also have , so this field is rather uninteresting. (Not inconsistent though, the use of 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)
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)
I believe the 2nd AfD established notability. Is there reason to believe consensus has changed? — Arthur Rubin (talk) 15:56, 23 January 2011 (UTC)
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)
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)
I am the subject of this wiki. The wiki says that I have published two papers on division by zero but there are more publications than this, two of which won best-paper prizes. See: http://www.bookofparagon.com/Pages/Papers.htm
There is a Google+ group on transmathematics here: https://plus.google.com/communities/103261551046378190173 — Preceding unsigned comment added by 109.147.66.221 ( talk) 19:59, 6 June 2018 (UTC)
I am the subject of this page.
I established the Transmathematica journal https://www.transmathematica.org in 2017. You might want to put this in the external links section of the page.
I retired from Reading University with effect from New Year's Eve, 2019.
86.178.110.164 ( talk) 16:00, 8 January 2020 (UTC)
I am the subject of this article. I removed a libellous comment and replaced it with a true statement of fact. This is your second warning - DO NOT LIBEL ME AGAIN.
I have corrected false statements about IEEE floating-point arithmetic. These are matters of fact, which you can check by consulting the IEEE standards or the Wikipedia pages about them. Specifically, there is a multitude of NaNs none of which is equal to any NaN. Evaluating NaN = NaN is either FALSE or ERROR, depending on context switches.
The article makes several false statements about transreal arithmetic. These are matters of fact, which can be checked in peer reviewed papers. Is an editor happy for me, the subject of this article, to correct these matters of fact?
SERIOUSLY - DO NOT LIBEL ME AGAIN
James A.D.W. Anderson ( talk) 11:37, 28 October 2022 (UTC)
I am the subject of this article so am not permitted to make biographical edits. I therefore propose that someone else should consider making the following additions to improve the article.
Add a note that I am the Editor in Chief of the Transmathematica journal - https://transmathematica.org - so that readers can easily find papers on total systems, including transreal arithmetic, written by multiple authors.
Add a link to my Google Scholar page - https://scholar.google.co.uk/citations?user=IPoHbOoAAAAJ&hl=en - so that readers can quickly find my publications and other papers that cite them. This will allow readers to more easily assess the range, value, and increasing acceptance of transreal arithmetic and of the other transnumber systems (currently transcomplex and transquaternion numbers).
Note that I was a Fellow of the British Computer Society and remain a lifetime member of the AAAI.
James A.D.W. Anderson ( talk) 11:42, 29 October 2022 (UTC)
I am the subject of this page. The description of my academic and employment history is missing significant detail. Here is the detail from a source that is relatively easy to check.
Book:
J.A.D.W. Anderson, “Pop-11 Comes of Age: the advancement of an AI programming language,” Ellis Horwood, 1989.
There is a biography inside the back cover which reads as follows.
“Graduating with a B.Sc. (Hons) in Experimental Psychology from the University of Sussex in 1980, James Anderson held the post of research assistant in Engineering and Applied Sciences at Sussex University (1982-1983); and from 1983 to 1987 he held the same position in the Department of Electrical and Electronic Engineering at Plymouth Polytechnic. Previous to his current position as Lecturer in the Department of Computer Science at Reading University, Mr Anderson held a research fellowship in that department (1987 to March 1989). He is Chairman of the Poplog and Pop Languages User Group.”
![]() | The following Wikipedia contributor may be personally or professionally connected to the subject of this article. Relevant policies and guidelines may include
conflict of interest,
autobiography, and
neutral point of view.
|
![]() | This article was nominated for
deletion. Please review the prior discussions if you are considering re-nomination:
|
This is the
talk page for discussing improvements to the
James A. D. W. Anderson article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
This article must adhere to the biographies of living persons (BLP) policy, even if it is not a biography, because it contains material about living persons. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately from the article and its talk page, especially if potentially libellous. If such material is repeatedly inserted, or if you have other concerns, please report the issue to this noticeboard.If you are a subject of this article, or acting on behalf of one, and you need help, please see this help page. |
![]() | This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||||||||||
|
![]() | This article links to one or more target anchors that no longer exist.
Please help fix the broken anchors. You can remove this template after fixing the problems. |
Reporting errors |
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)
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)
Please rename this article!
article renamed to James Anderson (computer scientist) fintler 16:32, 11 December 2006 (UTC)
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)
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)
(talk) 19:31, 11 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.
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)
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)
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:
http://www.bookofparagon.com/Pages/Downloads.htm
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)
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)
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)
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]
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= then:
= 1/0 = 1/(-1 * 0) = -1 * (1/0) = -1 * = -
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 != -
any takers? -- Frank duff 17:43, 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. , he gives this example in his own paper:
If the following identity is true:
Then we have
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)
This cannot be anything else than a hoax. There are so many problems with the article ( makes no sense, nor does having two different and 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)
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)
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)
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).
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)
How does "transreal arithmetic" with and differ from standard IEEE floating point math?
In standard floating point math, the same axioms hold, if we simply replace with NaN:
So, how is transreal arithmetic anything other than a restatement of IEEE floating-point arithmetic??? Moxfyre 19:29, 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)
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)
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)
Then 2*nullity = nullity doesn't it? Poktirity 22:38, 7 December 2006 (UTC)
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 ( talk • contribs) 23:34, 7 December 2006 (UTC).
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)
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)
I was looking through Perspex Machine IX: Transreal Analysis and I found this, [E 8], on page 5:
But, google says otherwise: ln(-1)=Pi*i.
I think this just reiterates the fact that is just am overglorified symbol for error. - Exomnium 02:03, 9 December 2006 (UTC)
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)
• 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)
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)
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)
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)
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 ( talk • contribs) 18:54, 15 January 2007 (UTC).
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)
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)
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)
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)
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 ( talk • contribs) 22:59, March 31, 2007 (UTC)
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)
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)
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)
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)
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 should suffice as an example. Does everyone see the difference? 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 is formally considered as shorthand for the operation to multiply by the multiplicative inverse . Since it is an axiom of the theory of fields that (read for all elements of a field, it is true that multiplied by zero equals zero), and since the axiom for mulitplicative inverses state of satisfies , 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 . In fact, there is one unique field where 0 has an inverse: the trivial field with one element. In this case we have . But, recall, that we also have , so this field is rather uninteresting. (Not inconsistent though, the use of 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)
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)
I believe the 2nd AfD established notability. Is there reason to believe consensus has changed? — Arthur Rubin (talk) 15:56, 23 January 2011 (UTC)
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)
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)
I am the subject of this wiki. The wiki says that I have published two papers on division by zero but there are more publications than this, two of which won best-paper prizes. See: http://www.bookofparagon.com/Pages/Papers.htm
There is a Google+ group on transmathematics here: https://plus.google.com/communities/103261551046378190173 — Preceding unsigned comment added by 109.147.66.221 ( talk) 19:59, 6 June 2018 (UTC)
I am the subject of this page.
I established the Transmathematica journal https://www.transmathematica.org in 2017. You might want to put this in the external links section of the page.
I retired from Reading University with effect from New Year's Eve, 2019.
86.178.110.164 ( talk) 16:00, 8 January 2020 (UTC)
I am the subject of this article. I removed a libellous comment and replaced it with a true statement of fact. This is your second warning - DO NOT LIBEL ME AGAIN.
I have corrected false statements about IEEE floating-point arithmetic. These are matters of fact, which you can check by consulting the IEEE standards or the Wikipedia pages about them. Specifically, there is a multitude of NaNs none of which is equal to any NaN. Evaluating NaN = NaN is either FALSE or ERROR, depending on context switches.
The article makes several false statements about transreal arithmetic. These are matters of fact, which can be checked in peer reviewed papers. Is an editor happy for me, the subject of this article, to correct these matters of fact?
SERIOUSLY - DO NOT LIBEL ME AGAIN
James A.D.W. Anderson ( talk) 11:37, 28 October 2022 (UTC)
I am the subject of this article so am not permitted to make biographical edits. I therefore propose that someone else should consider making the following additions to improve the article.
Add a note that I am the Editor in Chief of the Transmathematica journal - https://transmathematica.org - so that readers can easily find papers on total systems, including transreal arithmetic, written by multiple authors.
Add a link to my Google Scholar page - https://scholar.google.co.uk/citations?user=IPoHbOoAAAAJ&hl=en - so that readers can quickly find my publications and other papers that cite them. This will allow readers to more easily assess the range, value, and increasing acceptance of transreal arithmetic and of the other transnumber systems (currently transcomplex and transquaternion numbers).
Note that I was a Fellow of the British Computer Society and remain a lifetime member of the AAAI.
James A.D.W. Anderson ( talk) 11:42, 29 October 2022 (UTC)
I am the subject of this page. The description of my academic and employment history is missing significant detail. Here is the detail from a source that is relatively easy to check.
Book:
J.A.D.W. Anderson, “Pop-11 Comes of Age: the advancement of an AI programming language,” Ellis Horwood, 1989.
There is a biography inside the back cover which reads as follows.
“Graduating with a B.Sc. (Hons) in Experimental Psychology from the University of Sussex in 1980, James Anderson held the post of research assistant in Engineering and Applied Sciences at Sussex University (1982-1983); and from 1983 to 1987 he held the same position in the Department of Electrical and Electronic Engineering at Plymouth Polytechnic. Previous to his current position as Lecturer in the Department of Computer Science at Reading University, Mr Anderson held a research fellowship in that department (1987 to March 1989). He is Chairman of the Poplog and Pop Languages User Group.”