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• British computer scientist's new "nullity" idea provokes reaction from mathematicians

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: [1]

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) reply

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) reply

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) reply

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) reply

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) reply

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) reply

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) reply

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) reply

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) reply

Having the article at all before the paper was peer-reviewed is original research itself. Sam Hocevar 17:55, 7 December 2006 (UTC) reply

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) reply

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) reply

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) reply

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) reply

Your refutation is hardly rigorous. See my response on Slashdot. — flamingspinach | (talk) 19:11, 7 December 2006 (UTC) reply

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) reply

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) reply

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) reply

How could a system prove itself consistent without violating Goedel's second theorem? -- King Bee 20:43, 7 December 2006 (UTC) reply

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) reply

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) reply

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) reply

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) reply

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) reply

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) reply

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). reply

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) reply

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) reply

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) reply

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) reply

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) reply

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

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) reply

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) reply

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) reply

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) reply

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) reply

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) reply

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) reply

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) reply

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) reply

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) reply

That’s correct. Sam Hocevar 02:13, 8 December 2006 (UTC) reply

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). reply

The theory has defined several other "things", seeing as it accepts limits as accurate calculations. T.Stokke 14:47, 11 March 2007 (UTC) reply

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) reply

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) reply

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) reply

Wikipedia is an encyclopedia. It is not a site of first publication for original research. -- Carnildo 03:11, 8 December 2006 (UTC) reply

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) reply

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) reply

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

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

But, google says otherwise: ln(-1)=Pi*i.

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

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) reply

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) reply

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) reply

These are transreal numbers, not transcomplex. I don't see how you can say they are inconsistent, seeing as ln(-1) is not defined in standard real numbers. And it does not say that ${\displaystyle \ln {-x}=\Phi :-x<\Phi }$ it says ${\displaystyle \ln {-x}=\Phi :-x<0}$, since nullity (${\displaystyle \Phi }$) lies off the number line, you can't have a number smaller than it. T.Stokke 14:45, 11 March 2007 (UTC) reply

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) reply