Nullity – the Nonsense Number

Tons of folks have been writing to me this morning about [the BBC story about an idiot math
teacher who claims to have solved the problem of dividing by zero][bbc-story]. This is an absolutely *infuriating* story, which does an excellent job of demonstrating what total innumerate idiots reporters are.

[bbc-story]: http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/divide_zero_feature.shtml

What this guy has done is invent a new number, which he calls “nullity”. This number is not on the number line, can’t be compared to other numbers by less than or greater than, etc. In other words, he’s given a *name* to the basic mathematical concept of “undefined”, and proclaimed that this is somehow a solution to a deep and important problem.

The thing is, **there is no problem**. We understand what division by zero *means*. You *can’t do it*. There is no number that meaningfully expresses the concept of what it means to divide by zero.

You *can* assign a name to the concept of “not a number”, which is what this bozo has done;
but *that’s not new*. The standard floating point representation used by computer hardware
manufacturers ([IEEE 754][ieee-float]) specifically defines a set of values for operations
that can’t return a meaningful number: [NaN][] (**N**ot **a** **N**umber). NaN works as you
should expect a non-number to work: it can’t be compared to anything, and no arithmetic
operation works on it – because comparisons and arithmetic *only work on numbers*.

What he’s done is to take projective geometry – which (as I mentioned in the Steiner post a few weeks back) gives you some useful ways of using infinity; added the concept of a NaN value called nullity, and redefined the multiplication and division operators so that they’re defined to be able to produce nullity.

What *good* is it? Well, the crank behind it claims two things:

1. That currently, dividing by zero on a computer causes problems because division by
zero is undefined. But if computers adopted nullity, then division by zero errors could be gotten rid of, because they’ll produce nullity. Except of course that modern floating point hardware already *does* have a NaN value, and it *doesn’t help* with the kind of problem he’s talking about! Because the result *is not a number*; whether you call it undefined, or you define it as a value that’s *outside* the set of real numbers, it doesn’t help – because the
calculation you’re performing *can’t produce a meaningful result*. He says if your pacemaker’s software divides by zero, you’ll die, because it will stop working; how will returning nullity instead of signalling a divide by zero error make it work?
2. That it provides a well-defined value for 00, which he claims is a 1200 year old problem. Except that again, it’s not a problem. It’s a *meaningless expression*. If you’re
raising 0 to the 0th power in a calculation, then something’s wrong with that calculation. Modifying basic math so that the answer is defined as NaN doesn’t help that.

Basically, he’s defined a non-solution to a non-problem. And by teaching it to his students,
he’s doing them a great disservice. They’re going to leave his class believing that he’s a
great genius who’s solved a supposed fundamental problem of math, and believing in this silly
nullity thing as a valid mathematical concept.

It’s not like there isn’t already enough stuff in basic math for kids to learn; there’s no excuse for taking advantage of a passive audience to shove this nonsense down their throats as an exercise in self-aggrandizement.

To make matters worse, this idiot is a computer science professor! No one who’s studied CS should be able to get away with believing that re-inventing the concept of NaN is something
noteworthy or profound; and no one who’s studied CS should think that defining meaningless values can somehow magically make invalid computations produce meaningful results. I’m ashamed for my field.

Further, the reporters who are breathlessly repeating his nonsense claims need to be sent back to school. If you can’t understand the subject that you’re reporting on well enough to recognize this as nonsense, then you shouldn’t be reporting on it.

[IEEE-float]: http://en.wikipedia.org/wiki/IEEE_754
[NaN]: http://en.wikipedia.org/wiki/Not_a_Number

1. #1 Phil
December 7, 2006

Heh… I was one of the “ton” who emailed you; sorry for adding yet another to your inbox, this must have been posted just before I hit “send.”

I don’t consider myself a math wizard by any means, but I was still shaking my head while reading the article, trying to figure out how in the world this guy could be serious…

2. #2 Eighty
December 7, 2006

It seems he’s done the same thing as this guy: http://www.math.su.se/~jesper/research/wheels/

It’s not nonsense, and the new value (he introduced two actually: infinity and nullity) does not mean undefined. However, he should not be teaching it to school children, because it is a different algebra (for instance, neither of 0x=0, x/x=1, x-x=0 apply) and is not very useful. He should also not be touting that he’s solved an age-old problem.

3. #3 p
December 7, 2006

I was surprised when I followed up on that article this morning, this guy is the same guy who came up with the, erm, unique, “Perspex” model of general AI. I have an extremely high tolerance for weirdos and their weirdo ideas, but could not make sense of Perspex and I don’t have too high hopes for nullity.

4. #4 Eric
December 7, 2006

This is only tangentially related to the topic, but a lot of the arguments against this absurd idea I’ve seen about the place were themselves absurd.
What this lead me to thinking was, it would be enlightening (if you have the time) for a great many people to introduce the hyperreals to a lay audience, which actually manage to define operations on what many people have been taught are meaningless – infinitudes and infinitessimals.
Give it a go, it’ll be fun for all ðŸ˜›

5. #5 Jurgen
December 7, 2006

What a guy!

He also invented the perspex machine that is more powerful than the Turing Machine. I can’t believe such a crackpot can be teaching at a university!?!?!

6. #6 p
December 7, 2006

To be fair, hyperturing computation is a legitimate (and fascinating) field of theory. (If you think impossible machines are not worthwhile to study, consider that there is no genuine Turing machine in physical existence, only stochastic finite state machines.) I’m just dubious about his particular attempt to link this with “intelligence” in general.

7. #7 Ron Avitzur
December 7, 2006

What’s wrong with 0^0? In Concrete Mathematics (Graham, Knuth, and Patashnik):

Some textbooks leave the quantity 0^0 undefined, because the functions x^0 and 0^x have different limiting values when x decreases to 0. But this is a mistake. We must define x^0Â =Â 1, for all x, if the binomial theorem is to be valid when xÂ =Â 0, yÂ =Â 0, and/or xÂ =Â -y. The theorem is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant.

8. #8 Blake Stacey
December 7, 2006

I once tried to invent an arithmetic in which dividing by zero gave a reasonable result. I knew it was a meaningless operation as given, but following the inspiration of complex numbers, I thought that maybe I could make something interesting out of it. So, I cranked away and wrote lots of pencil scribbles, and got inconsistencies everywhere, yielding in the end nothing useful.

I was in seventh grade, OK?

In high school, I tried to devise a formalism in which mass could be a vector quantity (we got really bored during AP Physics class), and I nearly invented tensors. Nearly.

9. #9 Elia Diodati
December 7, 2006

If nullity = 1/0, then should nullity/nullity = 1? Because then you can cancel the zeroes… ðŸ˜€

10. #10 Mark C. Chu-Carroll
December 7, 2006

Elia:

Based on his handwaving explanation of why nullity “solves” the 00 problem, I suspect that yes, nullity/nullity=1.

11. #11 Bernhard Bauer
December 7, 2006

If you look at the problem of 0^0 from a discrete point of view, you can define a^b as the number of mappings from a set with b elements to a set with a elements. In this case, 0^0 is 1, as there is only one mapping from the empty set to the empty set: the empty mapping.

Of course, the value you get when defining 0^0 as the limit of a function depends on the function…

12. #12 Blake Stacey
December 7, 2006

Baez makes the same point Bauer did above in Week 240 of This Week’s Finds in Mathematical Physics.

13. #13 Mikael Johansson
December 7, 2006

And there is a research report (“half a phd thesis” – sort of halfway between M.Sc. and PhD) from my alma mater, written by a friend of mine from that time, on ‘Algebraic Wheels’. In 2001.

It’s a formalisation of what happens if we introduce a new element: “bottom”, denoted by _|_ (note the similarity to the Haskell notation for undefined stuff), to deal with 0^0 and 0/0 and similar expressions. This _|_ was to have the property that once you got there, you wouldn’t get out – just as infinity gets sticky wrt addition, _|_ gets sticky with regard to everything.

All this leads up to a consistent algebraic theory, where no operations are left undefined. Of course, the way it’s presented in this research report isn’t nearly as funky as the way “nullity” is presented. But still.

14. #14 Jens Ayton
December 7, 2006

At the top of the second-last page of his paper “Perspex Machine VIII: Axioms of Transreal Arithmetic” he explains what’s wrong with NaN: “We cannot accept an arithmetic in which a number is not equal to itself (NaN =Ì¸ NaN)” – which is easily resolved, as NaN is not a number (the name is something of a hint). There’s more on actually applying this amazing new theory (cough, cough) here.

15. #15 Jeff Petkau
December 7, 2006

I think you’re slightly wrong on the IEEE spec–while it does have a NaN value, it also has +infinity and -infinity. 1/0 is +infinity, and -1/0 is -infinity. (0/0 and infinity/infinity are both NaN, of course).

Like 0^0, 1/0 doesn’t have to be undefined. You can always define any mathematical operator to mean whatever you like; the trick has coming up with a definition that’s useful and powerful. As someone already pointed out, 0^0=1 is a much more useful definition in practice than 0^0=0 or 0^0=undefined.

(Also the “bottom” value from the Algebraic Wheels reference sounds an awful lot like IEEE NaN’s, in that it’s “sticky” and turns everything it touches into another NaN.)

16. #16 Edmund Schluessel
December 7, 2006

A thought occurs (I may be entirely wrong about this): Consider two polynomials P(z) and Q(z) where z is complex. By the fundamental theorem of algebra, the equation P=Q must have a number of solutions on C equal to the greater of the order of P or the order of Q.

Now make the projective transform u = 1/z. P(u) and Q(u) must both go to some kind of infinity in the limit of u -> 0. But in Anderson’s system, these are replaced by nullities, which are all equivalent regardless of whether the limit is +infinity or -infinity; so suddenly these polynomials have an additional root at u = 0, regardless of their order or their signage.

Does this mean that nullity is inconsistent with the fundamental theorem of algebra?

17. #17 Nasorenga
December 7, 2006

This blog entry could have been a useful commentary if it hadn’t been for all the invectives and ad-hominem attacks. As it stands, it’s a piece of garbage.

18. #18 Nuri Yeralan
December 7, 2006

The Real Numbers form a complete ordered field.

Let’s just look at how well Nullity fits into the Real Field.

FIELD INCONSISTENCY
The Real Numbers form a field. In it, every number except the additive/multiplicitive identity has a unique distinct inverse element such that x*x`=E. What is nullity’s inverse? Let’s look at the multiplicitive inverse. Nullity’s multiplicitive inverse would be 0/0 (it’s reciprocal). By this result, either E=Nullity=0 or Nullity is not in the set of Real Numbers. If it is not in the set of Real Numbers, what set, group, ring, field, algebra does Nullity fit into? Would 10 year old children be able to grasp a separate set of rules distinct from the rules they’ve learned thusfar?

There’s a reason this professor is teaching primary school children… That reason is not flattering.

19. #19 grigory
December 7, 2006

When it comes to math blogs, I’d have to say John Baez carries a lot of weight… Here’s what he said about 0^0 recently:

0^0 = 1

This is something they don’t teach in school! In analysis, X^Y can approach anything between 0 and 1 when X and Y approach 0 from above. So, teachers like to say 0^0 is undefined. But X^X approaches 1 when X â†’ 0. More importantly, in set theory, A^B stands for the set of functions from B to A, and the number of elements in this set is

|A^B| = |A|^|B|

When A and B are empty, there’s just one function from B to A, namely the identity. So, for our purposes we should define 0^0 = 1.

Consider the case of functionals, which are elements of X^(X^X). If we evaluate this at X = 0 we get

0^(0^0) = 0^1 = 0

You can read what Baez wrote here:
http://math.ucr.edu/home/baez/week240.html

Could it be that there’s bad math on “good math, bad math”?
Hey, everyone makes mistakes now and then.

20. #20 Daniel James
December 7, 2006

The report is from Berkshire local news. Berkshire! Do you really expect a local news team to have a maths specialist?

Finding a newsworthy story in Berkshire probably isn’t that easy, so local journalists have to cover any piece of fluff that comes up. Your attitude to the journalist should be sympathy, not scorn.

21. #21 Graham Fyffe
December 7, 2006

I developed an algebra for infinities and infinitesimals when I was in high school. It’s not hard. It’s based on simple calculus. I later learned that there are already many books about it. It’s not new. By using infinitesimals in place of zeros, you can prevent the loss of information that occurs with zeros. For example, 0^0, x/0, log(0), all have meaning if “0” really means some variable that has the value 0. You can also work with (1/0)^0, log(1/0), etc. You are essentially computing limits of your expressions.

The reason it works is that you are basically doing l’hospitale’s rule implicitly through the algebra. HOWEVER, it does not work in every case, because l’hospitale’s rule is not valid for all expressions. It is only valid for smooth functions, if I recall, and it is easy to produce counter examples. Furthermore, it does not retain enough information when more than the first derivative of your expressions is important to find the correct answer for your problem. However, in these cases you can use higher order infinitesimals, which is equivalent to working with a Taylor expansion of your expression in order to work with limits. Again, nothing new. And again, it only works for smooth expressions.

In short, I am willing to bet that if I actually took that guy’s nullity course, and learn what he is actually doing, I could provide examples where his method still produces the same kinds of “problems” that dividing by zero, or 0^0, produce.

And as many people have already pointed out, these are not actually problems.

22. #22 Alon Levy
December 7, 2006

Well, an infinite quantity sometimes makes sense, when you think about limits. But you can’t do arithmetic operations with it; all you can do is define meromorphic functions from the extended complex plane to itself.

23. #23 Aerik Knapp-Loomis
December 7, 2006

24. #24 Joshua
December 7, 2006

Daniel,

The paper doesn’t have a maths specialist, sure, but there must be any body else at all that he could have called to check this. I mean, surely the journo could have looked up the phone number of a somewhat reputable university nearby and asked to speak to someone in their maths department. But, no, apparently true.

25. #25 Coin
December 7, 2006

I’d be curious to hear from this guy in his own words– I can’t figure out how much of the nonsense here is the BBC creating hype over nothing, and how much is the original “Dr James Anderson” creating hype over nothing.

But meanwhile… what the heck is this “Perspex engine” thing??

I look it up on wikipedia, and it appears to be some kind of attempt to define a computational model more powerful than the turing machine. This is a subject which interests me, as does the subject of perverse models of computation in general. But the wikipedia page doesn’t really explain what the perspex machine is or what it does or why it exists or in what way it is more powerful than the turing machine, and the whole thing is headed up with the banner:

It is proposed that this article be deleted, because of the following concern:

Original research. Citation from http://www.bookofparagon.com is authors own home page which publishes non-peer reviewed math articles. This article is also complete 100% nonsense from a mathematical point of view and like all articles on http://www.bookofparagon.com borders on Timecube levels of crank silliness.

What is happening here?

26. #26 Jonathan Vos Post
December 7, 2006

A George Carlin joke:

“The Nobel Prize in mathematics was awarded yesterday to a California professor who has discovered a new number. The number is ‘bleen’, which he says belongs between six and seven.”

TOTBAL (There Ought to Be a Law). (1) Any newspaper that publishes an Astrology column must also publish an Astronomy column.
(2) Any newspaper that publishes ad hoc interpretations of what the Stock market results “mean” on any give day, or un-checked stories on government-released statistics, must also publish a Mathematics column.
(3) Any newspaper that publishes a Computers & Software & Entertainment Products column must also publish a Computer Science column.
(4) Any newspaper that publishes a Health and Medicine column must also publish a Biology column.

27. #27 Michael
December 7, 2006

Yes, it’s dumb. However, it’s also a question of semantics and language, not math. Therefore it’s ridiculous from a math perspective, but not totally up shit creek if he simply proposed it as new way of referring to nothing as the lack of something, as opposed to nothing as the whole of not anything… err, wait. I can’t believe that either. Inventing synonyms is patently retarded, literally.

28. #28 Mark C. Chu-Carroll
December 7, 2006

Ron, Bernhard, et al:

Yes, you’re right, you *can* define a meaning for 00. The point still stands: it’s not really a problem. It’s a question of definition: the expression 00 can have a value defined for it by relying on
different constructions – usually either 0 or 1. This guy’s
solution, that it’s a magic NaN value outside of the number line that still functions as a number is silly, pointless, and leads to some very strange results if it’s taken seriously.

I don’t think that he understands the concept of NaN; he insists that the difference between IEEE’s NaN and his nullity is that “NaN” is *not a number*, whereas his “nullity” is a number – so in his system, nullity=nullity, but in IEEE floating point, NaN != NaN. But that’s a distinction that doesn’t make a big difference: you *can* ask “Is this value a NaN?” is IEEE float.

But it goes beyond just definitions and names; he claims that the existence of his nullary value eliminates the problems caused by divide-by-zero errors in software. How? By defining the answer to be nullary instead of NaN?

29. #29 GreedyAlgorithm
December 7, 2006

grigory: Nope, no bad math this time. You’ll find that in almost every discipline, and even more often when considering any one subject, that 0^0 only comes up as something that works out nicely as 1 or 0 but not both. In every such case it makes sense to just define 0^0 as 1 or 0 and move on. But what you’re really saying is “every time you 0^0 in this context, evaluate it as 1 (or 0) since it would end up that way if you considered it formally”. John Baez is simply saying that in a certain context you might as well let 0^0 be 1.

30. #30 Pupeno
December 7, 2006

Teachers, when they are good, make the students better, but when they are bad they make students worst.

31. #31 Flaky
December 7, 2006

I think he got the idea from homogenous coordinates:
“In the standard homogeneous models of projective geometry the homogeneous origin, which exists and is denoted by
the zero vector, is punctured from perspective space. This vector is called the point at nullity.” From his Perspex Machine (http://www.bookofparagon.com/Mathematics/SPIE.2002.Perspex.pdf)
So I suppose the idea comes from having a real x/y represented by the pair (x,y) (and all equal pairs), or something like that.

Judging this guy based on his Perspex Machine article, it seems that this guy probably isn’t a crackpot. So the question is, is this ‘nullity’ business a joke or can a professor of CS be so ignorant of the basics of math? It doesn’t seem likely that he wouldn’t have realized the similarity of his ‘nullity’ and IEEE 754 NaNs.

32. #32 Tim G
December 7, 2006

if (divisor == 0)
{
Contingency(); // Cardiac patient saved!
}
else
{
NormalStuff();
}

33. #33 Blake Stacey
December 7, 2006

@Coin:

The most likely explanation for the state of the Wikipedia article you found is that it was written by the “Perspex machine” people themselves. Someone then noticed this piece of made-up non-science and suggested that it be deleted. You can check the history of the article for more information. It was created on 31 January by a user named “Emilywinch”, who never edited any other article. Then along came “Ben thomas”, who added a picture of a matrix and wrote a screed entitled “There are no NaNs in a total arithmetic” on the IEEE 754r article’s discussion page. He never did anything else, either. Finally, somebody from the IP address 86.135.152.222 visited to contribute an “introduction” and add a link on the “perspex” page.

In short, this article was created by people more interested in boosting a fringe idea than contributing to an encyclopedia.

Does anyone know if the “perspective simplex” idea has any worth or visibility at all? The Google hits are contaminated by Wikipedia mirrors, and the ones which do not stem from the very article in question are not, ahem, promising: “This site shows that the perspective simplex, or perspex, is a simple physical thing that is both a mind and a body. . . .”

34. #34 Blake Stacey
December 7, 2006

Oh, and that IP address geolocates to London, but I couldn’t find out anything else.

35. #35 Flaky
December 7, 2006

I take back what I wrote above, on further reading the Perspex Machine article appears to be BS. Particularly, as it delves into Quantum Mechanics. It appears that the Machine might likely be Turing complete in rational space and possibly something more in real space, but the rest is rather fishy.

December 7, 2006

typical situation — msm thinks they’ve found a genius and it takes bloggers to point at that he’s a charlatan.

37. #37 Coin
December 7, 2006

I take back what I wrote above, on further reading the Perspex Machine article appears to be BS. Particularly, as it delves into Quantum Mechanics.

Uh oh. The Emperor’s New Mind strikes again? :O

38. #38 Flaky
December 7, 2006

Blake: Following the link from Anderson’s uni page (http://www.cs.reading.ac.uk/people/J.Anderson.htm) to what is offered as his personal web page takes to http://www.bookofparagon.com/,
which I believe is just the page you mentioned: “This site shows that the perspective simplex, or perspex, is a simple physical thing that is both a mind and a body.”

39. #39 Harald Hanche-Olsen
December 7, 2006

Two things …

One, I believe any life critical computer controlled hardware like a pace maker will typically include some kind of watchdog mechanism that will reset the device if any (uncaught) exception occurs.

Two, I am not an expert on IEEE floating point, but doesn’t it actually have a large number of NaN values? Meaning different bit patterns that are all interpreted as NaN?

40. #40 James Bascle
December 7, 2006

thank god someone wrote about that cook. i cant believe it got so many upvotes on reddit and it was actually o the bbc

41. #41 milty
December 7, 2006

I think this guy talks a lot of horse sense. A genius is often unrecognized in his own time. He’s solved a great mystery of maths in an ingeniousway and the rest of the mathematical world is jealous.

42. #42 gavin
December 7, 2006

2/0 = 2*Ñ„ (2 * infinity)

43. #43 Coin
December 7, 2006

Two, I am not an expert on IEEE floating point, but doesn’t it actually have a large number of NaN values? Meaning different bit patterns that are all interpreted as NaN?

I would suggest going here and skipping down to the “special values” section. Basically, IEEE floating point offers two kinds of infinity (positive and negative), plus the “denormalized” (too small to accurately represent) numbers, plus two kinds of NaN: NaNs which should signal an exception when you try to use them and “quiet” NaNs which just propagate. (There are a wide variety of different bit values which correspond to each of the two NaNs, but they’re all treated identically.)

In addition to this, it’s probably worth noting that there’s this concept of the IEEE floating point interrupts being “masked”– floating point hardware, and I’m not mistaken as the standard requires this, allows software to mark individual floating point exceptions as “masked”, which means that the hardware doesn’t trigger an exception and instead attempts to recover in some logical manner, like switching the number to a QNaN.

Basically overall the standard goes out of its way to make sure that if something goes wrong, you can find out what it was and react to it, but if you don’t want to be bothered you can just plow ahead without halting the computation and the hardware will do its best, propagating errors as NaNs where appropriate.

I think this guy talks a lot of horse sense.

I do remember this old “Superman” episode where there was this horse that could do math, and it would stomp on the ground a certain number of times to communicate the answer…

44. #44 Sigivald
December 7, 2006

Milty: They laughed at Einstein.

But they also laughed at Bozo.

Odds are a lot stronger for Bozo than Einstein – especially if, unlike Einstein, the prospective Bozo can’t provide even theoretical empirical tests of his theory (eg light refraction around stars, clock drift in orbit), and solid math that doesn’t involve simply inventing a new not-number that does whatever he wants…

45. #45 Jesus
December 7, 2006

function generatenullity(){
\$base=”O”;
\$superimposed=”I”;
\$generation=”

\$base
\$generation

“;

// \ (@_@ /

// / (@_@ \

// \ (@_@ /

// / (@_@ \
\

\$pointless=rand(0,99999);
\$dielol=rand(0,1);

if (is_nan(\$pointless/\$dielol)){
\$nullity==true;
}

}

while(true){
beatheart();
if (\$nullity==true){
break;
//zid0wned
}
}

46. #46 Jonathan Vos Post
December 7, 2006

George Gershwin:

“The odds were a hundred to one against me
The world thought the heights were too high to climb
But people from Missouri never incensed me
Oh, I wasn’t a bit concerned
For from hist’ry I had learned
How many, many times the worm had turned.

They all laughed at Christopher Columbus
When he said the world was round
They all laughed when Edison recorded sound
They all laughed at Wilbur and his brother
When they said that man could fly
They told Marconi
Wireless was a phony…”

Well, sometimes “laughter is the best medicine.”

Oh, and what probability is “nullity to one against me?”

47. #47 Jack9
December 7, 2006

Same meaningless probability as “i to one against you”, but equally important that we can ask rather than throwing our hands up and saying it’s Not a number.

48. #48 Tyler
December 7, 2006

I’m an English major, but even I remember thinking, while reading the article, “didn’t he just assign a new name to an undefined outcome?”

Thanks for writing this.

49. #49 Levi
December 7, 2006

I do remember this old “Superman” episode where there was this horse that could do math, and it would stomp on the ground a certain number of times to communicate the answer…

http://en.wikipedia.org/wiki/Clever_Hans

50. #50 Mark C. Chu-Carroll
December 7, 2006

Jack9:

No, *not* equally important to ask rather than throwing up our hands. Nullity is just a *name*. It doesn’t actually *mean* anything different than “not a number”.

Complex numbers have real meaning. Nullity does not. Arithmetic *works* on complex numbers. They have *meaning* as numbers. You can perform meaningful manipulations with them: they aren’t a meaningless dead-end.

Dividing by zero *is* a dead-end. It doesn’t *mean* anything. Giving it a name doesn’t help with that. The basic fact is that the *definition* of division in mathematics makes the concept of dividing by zero meaningless. So just like NaN, nullity is a dead-end. Once you’ve got it, you’re stuck; you can’t *add* to it, you can’t *multiply* by it, you can’t *divide* by it. Everything stops, because you’re totally blocked by the basic fact that nullity *isn’t* a number.

The bozo who proposed it insists that it *is* a number. But all he’s doing is *redefining* the term number to include his nonsense value. And, as other posters in the thread above pointed out, that means that certain things that are true for real numbers are *not* true any longer for his version of numbers.

51. #51 Davis
December 7, 2006

He’s solved a great mystery of maths in an ingeniousway and the rest of the mathematical world is jealous.

Um, no. This is not a “great mystery of maths”; if you think about what division actually means, it’s clear that dividing by zero should be problematic. What he has done contributes exactly nothing to mathematics.

52. #52 neil
December 7, 2006

I assume it was the BBC reporter who made it sound like he had invented something new and world-changing, or solved a math problem that’s been dogging humanity for years. Nullity just differs from NaN in a few minor ways that are important in his particular research. He probably also thinks it’s easier to understand the properties of NaN this way (particularly for schoolchildren). And he probably should have known not to go on TV…

53. #53 Elia Diodati
December 7, 2006

Upon second look, could you comment on whether nullity makes sense even in the context of Conway’s surreal numbers? Like it is even conceptually possible?

54. #54 clinton bowen
December 7, 2006

THANK YOU GOOD SIR.

YES, PLACING AN EXTRINSIC VALUE ON SOMETHING THAT HAD NO INTRINSIC VALUE TO BEGIN WITH, LOGICALLY, IS QUITE IMPOSSIBLE.

it creates foolish problems like “nothing is something, but something isn’t nothing.”.

55. #55 ASH
December 7, 2006

Convenience aside, can you really reach an exact point between -1 and 1, called 0?

56. #56 Blake Stacey
December 7, 2006

@Flaky:

Yes, that page was the source of the quotation I gave. I’ve learned the hard way that the ScienceBlogs spam filter drops messages into a moderation queue if they have too many URLs. Rather than have my comment sit in the queue, I figured I’d insert the most crucial URL, i.e., the Wikipedia history link.

All of which makes me wish those Seed people would implement some kind of login/ID system so that people who are willing to make a time investment and divulge a few personal details can post comments with multiple hyperlinks. I mean, I like citing my sources.

Davis said:

What he has done contributes exactly nothing to mathematics.

Well, he did contribute nullity to mathematics, even though his “contribution” was, er, null and void. . . .

57. #57 Mark C. Chu-Carroll
December 7, 2006

Elia:

Nope, nullity doesn’t make sense in surreal numbers. Surreals depend on the ability to position a number in terms of it’s greater-than/less-than relationships to other numbers that were defined before it. Nullity is *outside* of the system permitting greater-than/less-than comparisons.

58. #58 Blake Stacey
December 7, 2006

Upon second look, could you comment on whether nullity makes sense even in the context of Conway’s surreal numbers? Like it is even conceptually possible?

The short answer is no. Even in Conway’s system of surreal numbers, 0 times any number is 0. This implies that dividing by zero makes as much sense as, say, dividing by purple and calling the answer cinnamon.

The statement that 0x = 0 for all x is Theorem 21 in Knuth’s Surreal Numbers. It’s not hard to prove; you just need the definition of surreal multiplication and the fact that 0 is that number with empty left and right sets. I would work out the symbols, but for some reason the blog-comment interface no longer accepts superscript and subscript tags.

59. #59 Brandon
December 7, 2006

Wait a second, if nullity=0/0, then its multiplicative inverse is 0/0. And since its inverse it itself, then it is equivalent to the multiplicative identity! Thus nullity=1!!! Hooray for bad math!

60. #60 TheEngineer
December 7, 2006

Thank you, thank you, thank you! I thought the entire world had gone insane there for a while.

61. #61 Mark Wagner
December 7, 2006

From an abstract algebra point of view, there’s nothing particularly crankish about what this guy has done. He’s invented a new number called “nullity” that isn’t on the number line? So what? I could use the principles of abstract algebra to create an arithmetic in which there are numbers like “horse”, “lemon”, and “green”, but all operations are internally consistent.

His new arithmetic appears to be internally consistent, and it appears to give meaningful results when you divide by zero. Unfortunately, it does this by giving up all sorts of useful properties that ordinary arithmetic has, such as the concepts of additive and multiplicative inverses, and the additive and multiplicative identities (0 and 1, respectively).

In abstract algebra terms, ordinary arithmetic constitutes a “field”. His new arithmetic is merely a set with two operations and a few useful properties. A side effect of this is that all the other areas of mathematics that depend on arithmetic being a “field”, such as algebra, complex mathematics, and calculus, no longer work properly, and will need to be re-formulated from scratch.

Is it crankish? No. Is it useful? No.

62. #62 awe inspired
December 7, 2006

Another testament to the fact that incompetence + and inflated ego leads to disaster, albeit often in the form of self-inflicted punishment.

63. #63 Cale Gibbard
December 7, 2006

I just thought I’d link you to a letter which I wrote to Mr. Anderson about his system: http://cale.yi.org/index.php/Open_letter_to_James_Anderson

December 7, 2006

Only Timecube can surpass this exceptional example of extreme mathematical genius.

65. #65 Mark Wagner
December 8, 2006

Hardly. Timecube is the semi-coherent writings of a schizophrenic. This is a serious exercise in abstract mathematics — it’s just not as useful as the creator believes it is.

66. #66 TyPhyter
December 8, 2006

wow i am in tenth grade, in the U.S., and while my English isn’t the best it could be my math is well above average. i read this and was dumbfounded at the stupidity of this guy. there are not answers for those types of problems for the same reasons stated in this article, and by replies to it. the numbers that “would” come out as an answer to these problems does not exist!

December 8, 2006

Haha, this is funny because I did the same thing when I was in 11th grade. I thought it odd that mathematicians solved the otherwise unsolvable problem of taking the square root of -1 simply by defining it to be “i”, the imaginary number, yet they didn’t do the same thing with division by zero. So I went ahead and defined it to be “j”, and my math teacher encouraged me to keep working on it.

It soon became apparent, though, that when you do on to define mathematical operations with “j”, you never get anything useful. With “i” you can carry it along and eventually square “i” and get back -1. “i” also follows a lot of nice mathematical properties like following the basic transitive, associative, and commutative properties of algebra.

“j” or “Nullity” as he’s calling it turns out to be basically useless. If you divide N by zero and then multiply the result by zero later, the zeros don’t cancel out and you don’t get back N. (N/0)*0 is still undefined as any real definition would lead to total mathematical contradictions.

So basically, an 11th grader could figure out that it makes no sense mathematically to define division by zero, so this guy should have figured it out too.

68. #68 Jonathan Vos Post
December 8, 2006

Perhaps it shameful to admit, but the rantings of Timecube have a poetic non sequitur effect that is more entertaining than the straight-faced dilute gruel of nullity. My son and his computer-science student classmates agree. Truly mad ranting can do something that trained academics sometimes cannot do.

As to hypercomputing, I am still re-reading the Notices of the AMS special issue on Turing. The relationship between Turing Machines and Quantum Computing is increasingly obscure to me. Hence, although Perspex makes no sense to me (who got a M.S. in Artificial Intelligence and Cybernetics way back in 1975) I cannot get my brain around a proof that it is a preposterous as nullity.

Oh, and the song I credited to George Gershwin? Most likely, the tune was by George, but the quoted lyrics by his brother Ira Gershwin (born Israel Gershowitz). As a professional writer, I’m sorry that I got that wrong the first time. As wikipedia comments: “, Ira played a huge part in bringing about a new type of song lyric: a smart, witty, vernacular style that the common man could relate to and enjoy.”

I feel the same way about Paul Simon, Bob Dylan, and John Lennon. Dylan, by the way, said that he was a “poet of algebra, I use words the way others use numbers.”

As if preedicting Nullity, he wrote:

“Inside the museums, infinity goes up on trial. Voices echo, ‘This is what salvation must be like, after a while.'”
[Bob Dylan, “Visions of Johanna”]

He also wrote, in a song that mentions the street in Brooklyn where I grew up:

“All these people that I used to know, they’re an illusion to me now. Some are mathematicians, some are carpenters’ wives.
[Bob Dylan, “Tangled up in Blue”]

And unifying these two themes:

“On sighting mathematicians [poetry] should unhook the algebra from their minds and replace it with poetry; on sighting poets it should unhook poetry from their minds and replace it with algebra.
[Brian Patten, “Prosepoem towards a definition of itself”]. Maybe that would be clearer in terms of a fixed point theorem, or something about adjoint functors.

69. #69 Div
December 8, 2006

If this theory is true then this should be true as well,
Prove
2=3
Taking LHS
2*0=0
Taking RHS
3*0=0
RHS=LHS
so QED
2=3!!!!!

70. #70 nullity
December 8, 2006

oww c’mon, ur just jealous!

71. #71 Bob O'H
December 8, 2006

Complex numbers have real meaning.

“Life is complex: it has real and imaginary parts”

Bob

72. #72 Mike Scott
December 8, 2006

Mark Wagner is the only person in this entire long thread, including the original poster, who has any idea what he’s talking about.

Hint: any commentary on this proposal which does not include the word “field” in its technical algebraic sense is worthless.

(Nuri Yeralan knows what a field is, but he’s not read the original paper which cleary states that nullity and +/-infinity do not have additive or multiplicative inverses by definition.)

73. #73 Flaky
December 8, 2006

Cale’s letter had the link to the paper on these ‘tranreal’ numbers:
http://www.bookofparagon.com/Mathematics/PerspexMachineVIII.pdf
It strikes me that this guy knows at least the basics of the relevant theories, but doesn’t really grasp the full detail. Yet, apparently that doesn’t stop him from making grandiose claims. I think this stuff can rightfully be called Cargo Cult Math. I’m really interested in knowing what drives people to come up with this sort of stuff and publish it, when they clearly should know better.

which contains a couple of rather funky videos of the Perspex Machine running.

74. #74 Tyler DiPietro
December 8, 2006

Mark Wagner is the only person in this entire long thread, including the original poster, who has any idea what he’s talking about.

No, the original topic is about whether the author of the proposal has solved a 1200 year old “problem” of division by zero. He hasn’t. What he has done is created an abstract value (“nullity”) and assigned his own rules to it. It doesn’t solve any long-standing “problem” in arithmetic, which AFAIK is what is claimed, since division of real numbers by zero is still not defined.

75. #75 Andrew McClure
December 8, 2006

Hint: any commentary on this proposal which does not include the word “field” in its technical algebraic sense is worthless.

(Nuri Yeralan knows what a field is, but he’s not read the original paper which cleary states that nullity and +/-infinity do not have additive or multiplicative inverses by definition.)

Uh… then by definition it’s not a member of a field. The field axioms specifically require that every element has to have an additive inverse and every element except the additive identity has to have a multiplicative inverse.

No, I’ve not read the original paper either. I don’t know where to find it.

76. #76 Harald Korneliussen
December 8, 2006

Coin: “Basically, IEEE floating point offers two kinds of infinity (positive and negative), plus the “denormalized” (too small to accurately represent) numbers, plus two kinds of NaN: NaNs which should signal an exception when you try to use them and “quiet” NaNs which just propagate.”

Wow, that’s interesting. I wrote a little program to have a look at a sort of network where some quantity flowed from node to node according to certain rules. I was careless about the order in which I did my multiplications, and a floating point division became NaN. Quickly it had propagated to my whole network.

I wonder how I could convince Java to use the “other” NaN ðŸ™‚

77. #77 Davis
December 8, 2006

Hint: any commentary on this proposal which does not include the word “field” in its technical algebraic sense is worthless.

(Nuri Yeralan knows what a field is, but he’s not read the original paper which cleary states that nullity and +/-infinity do not have additive or multiplicative inverses by definition.)

Sod off. Many of the posters (myself included) are aware of the field axioms, and didn’t feel the need to invoke the word “field” (hint: some of the comments apply the field axioms). Of course, what Anderson has defined is clearly not a field.

On the other hand, this did motivate me to actually skim through the first of his two papers, where I found unsubstantiated dreck like this:

We argue that transreal arithmetic is more powerful than real arithmetic, and that it is more coherent than IEEE floating point arithmetic. Indeed, we argue that whilst real arithmetic is an interesting algebraic structure, daily practice, in electronic computation, tells us that it is not a valid model of arithmetic.

Yet the only reason he gives for its greater power is that we can now say the answer to a problem is “nullity” rather than “no solution.” And only at the small cost of no longer working over a field! And a much larger collection of axioms! There’s nothing useful here, in my opinion.

78. #78 Charlie
December 8, 2006

Oh dear. The school mentioned (Highdown School) is where my neighbour’s kids go. And where my own boy will go in about 10 years’ time.

I’ll have to stick my head around the fence and tell them to forget about nullity.

79. #79 -E
December 8, 2006

I think it’s interresting that most of you refer to 1/0 = ? as an undefined result.

If you look as this from another angle, I like that this professor actually tried to solve this problem, I’m not saying he’s correct or wrong.
But I’m saying this, if we can’t define simple math 100% then that invalidates the foundation all our advanced math and physics are based on!

Let me illustrate with the famous pie division:

1 pie / 1 person = 1 pie per person
1 pie / 0 person = 1 pie per person

The pie is still there ðŸ™‚

Another angle to another to the same great question, I suggest that we look into defining the simple math 100%, because when we have nailed that we will automaticly be visited by aliens… (Remember you heard it here first)

PS. excuse my poor english, I’m not a native speaker.

80. #80 Andrew McClure
December 8, 2006

But I’m saying this, if we can’t define simple math 100% then that invalidates the foundation all our advanced math and physics are based on!

Dr. Hilbert, is that you?

81. #81 An
December 8, 2006

You people need to get a life. Seriously, who gives a fuck?

82. #82 Pseudonym
December 8, 2006

It’s worse than wheels, he’s reinvented Knight’s Null Algebra!

KNA was invented by a bright-but-misguided 13-year-old. A CS PhD doesn’t have an excuse.

83. #83 Flaky
December 8, 2006

Pseudonym: Thanks for that! I nearly laughed my arse off picturing a wise-ass thirteen year old, who likes to call himself “The Great” and believes his book will change the thinking of Man forever. Though, I’ll say that writing a 150 page book at the age of 13 is quite an achievement in itself.

84. #84 -E
December 8, 2006

To: Andrew McClure

No I’m not Dr. Hilbert, incase you didn’t read my post entirely you would see that I refer to him in 3rd person, because I’m not him.

Anyways I totally digg the way that you tried to make a lame /. joke.

Instead of actually countering my argument with some solid facts.

You da man…

85. #85 Jud
December 8, 2006

Not a good time for computer scientists:

http://www.cs.unc.edu/~plaisted/ce/dating.html

86. #86 Mark C. Chu-Carroll
December 8, 2006

E:

There are two responses to your comment.

(1) Saying that something is *undefined* is a reasonable thing to do – in fact, something the *only* correct thing to do. Throughout math, we frequently use *partial* functions – which are functions that are *not* defined for some values. That’s fine, and even important.

(2) There was a famous effort early in the 20th century to conclusively define a perfect, complete foundation for mathematics. Gödel proved that it couldn’t be done – that’s what the incompleteness theorem is all about.

87. #87 JP
December 8, 2006

Not a good time for computer scientists:

I actually had a class with Plaisted (in 2001). He is a really nice guy who did some nice work in theorem proving which has recently had applications in my field (algebraic computing). So I was very surprised to find out about these pages a couple years ago.

From personal experience I can say that there are a surprising number of math and cs professors in North Carolina who are kooky YECs. Prof. Plaisted just one of the very few who has the bad taste of having a large set of writings online on the subject.

88. #88 Snappy!
December 8, 2006

As mentioned, I think its kind of like the complex number i, except that there may need to have a set of mathematical rules to apply to this new nullity for it have any usage at all.

instead of saying x/0 = nullity. I suggest 1/0 = N, where N is the nullity number if you will.

So that way, we represent any number divided by 0 with x/0 = xN, thereby preserving the original number involved in the division.

hmmm … is this any good … let me rethink … :p

89. #89 Billy Carson
December 8, 2006

Reminds me of another mathematician who tried to find a meaningful way to divide by zero … a fellow named Euler. He drew a distinction between numerical zero, which cannot be a divisor, and geometrical zero, which can.

Euler’s strategy was later rejected, of course, and even as you read his “Institutiones calculi differentialis” you kind of get the feeling that his heart wasn’t quite in it.

Still, Euler had a few excusues:
He was trying to solve a real problem: differentials wanted a rigorous development, and the limit had not yet been invented. Cauchy and Weierstrass wouldn’t show up until the next century, and Bernoulli’s infinitesimals were unsatisfactory (it turns out that they also were only waiting for a rigorous development, but nonstandard analysis was two centuries away).

Also … well … he was Euler. You can cut a lot of slack for the guy who invented functions, especially when his failures are more interesting than his contemporaries’ successes.

This Anderson, on the other hand …
Well, he’s no Euler. As far as I can tell, all _he_ needs is a strongly typed language.

90. #90 SparoHawk
December 8, 2006

He proposed on giviing it that name, not that he invented a solution because the solution has been there all the time: undefined, as many have said. But undefined can mean a lot of different things, he simply said: let’s call division by zero “nullity”, a term that you can simple say and all could understand.

Dunno, I am wavering on both sides to this term, but, whatever. No matter the name they give it, it will always be something that will not have an answer, I mean, division by zero.

91. #91 Blake Stacey
December 8, 2006

Sparky! said:

instead of saying x/0 = nullity. I suggest 1/0 = N, where N is the nullity number if you will.

This is what I did in seventh grade. Play with it all you want, but there’s a reason I don’t have a Fields Medal today, and it ain’t ’cause the ivory tower foax are suppressin’ the truth.

92. #92 Blake Stacey
December 8, 2006

JvP and others have brought up Time Cube. . . . Gosh, I feel sort of bad about this. . . . I helped inflict Gene Ray and his 4-corner Time Cube upon the world. See, back before he made it big, I was a physics student at MIT, and an acquaintance of mine decided to bring him there as an (ahem) “guest lecturer”. I was one of the people who went through his website to collect quotes and put them on posters. The one I remember best is “-1 x -1 = +1 is STUPID AND EVIL!”

I helped poster for the event all around campus, but then I got a guilt attack. Was it right to fill a lecture hall with people who just wanted to laugh at the man?

I stayed home during the big “debate”, which packed the largest lecture hall on campus. The moment everyone remembers is when the “-1 times -1 = +1” thing came up. Gene Ray said that equation was as stupid as saying, “A North American times a North American is a South American.”

A student asked, “So what should -1 times -1 be?”

“A North American.”

The people in this photograph are my friends, colleagues and housemates. . . . On some Hi-8 tape buried in my camera gear, I have the world’s only footage of Gene Ray being taught how to play Go. Hey, in this degenerate modern age, I should put that sort of thing on YouTube.

If you wanna talk about social networks, it’s also my fault that Gene Ray is only two degrees of separation from John Nash. That’s right, John Nash tripped over a power cord and nearly fell on me during a conference banquet this past summer. Oh yeah. Almost, but not quite, entirely unlike famous: that’s me.

93. #93 Uffe
December 8, 2006

We were all holding our breath for the next episode of MarkCC’s nice Haskell tutorial, and then comes this “idiot math teacher” cluttering up everything with his nonsense… In addition to the obvious mathematical nonsense of this, he also claims to address the real problem of NaN’s you can unexpectedly get in flight guidance computers. This is a relatively nasty problem that occurs when unexpected numbers accidentally roll through the flight guidance algorithms. That problems is not trivial, you cannot just stop flying when a calculation accidentally attempts to e.g. divide by 0. There are many examples of how bad this can go. However, in spite of his claims, also here his theory offers absolutely no help.

94. #94 DouglasG
December 8, 2006

There are two worlds colliding here. The theoretical mathematical world and the applied computational world. The computational world is more limited than the theoretical math world. For example, integers aren’t infinite for computers. Thus, you have lots more undefineds in computer programing.

In mathematics, you can define the system in any fashion. Thus, if you wish to define division by zero you can. Then, you explore to see what that gets you. That is how the complex numbers came about and the quarternions… Both have these have been largely explored.

Why hasn’t 1/0 been greatly explored? The reason is that it breaks things. Does 1/0 * 0/1 = 1 or does 1/0 * 0/1 = 0? In a mathematical group, we need a few things. We need a set of numbers with an operation, and we need associativity, we need an identity, and we need a inverse. Clearly, 1/0’s inverse would be 0, so 1/0 * 0/1 = 1 be definition of inverse. Let us look at associativity (a * b) * c = a * (b * c) This must work for all members of our set of numbers. Thus, I can pick any and it should work. Let us look at 1/0, 0, 25. (1/0 * 0) * 25 = 25 as 1/0 * 0 = 1 in our inverse definition above. 1/0 * (0 * 25) = 1/0 * 0 = 1.

Just from that simple example, defining division by zero ends up with a not very interesting result. You could potentionally work it to make it a group or even a field but it would be largely limited. Thus, mathematically uninteresting and this is why it has “been a puzzle for 1200 years”. This definition does not do anything interesting.

95. #95 Bronze Dog
December 8, 2006

[Professor Frink]

“Pi is exactly 3!”

*horrified gasps*

“I’m sorry it came to that, but I needed your attention.”

[/Professor Frink]

96. #96 Jonathan Vos Post
December 8, 2006

Blake Stacey:

That was extremely interesting!

By the way, I’ll never forget John Forbes Nash, Jr., with his first words spoken to me:

“Use the microphone!”

Technically, he was speaking (at that time) not to me, but to the speaker in a conference session (6th International Conference on Complex Systems). I was the Session Chair, and thus standing near the transparency projector where the grad student was forgetting the microphone. Nash is hard of hearing.

Fortunately, I got to speak with him extensively, later. he is still doing some very subtle and wonderful research.

Sometimes Mathematicians are attacked as crazy, and their work only later appreciated. Grassman (Grassmanians having nilpotents), Cantor (later in his life), Godel (when he showed a solution to Einstein’s field equations which allowed time travel). On the other hand, Mathematicians go crazy more often than other scientists, and at about the same level as other very high stress jobs such as policeman and dentist.

What is the Square Root of a South American?

97. #97 Matt Giuca
December 8, 2006

I dislike how he bagged out NaN in his paper and seems to have missed the whole point of NaN.

1. There is no point saying “the great achievement here is that NaN is not a number, while Nullity IS a number”. While Nullity has these axioms to support it being a number, it still isn’t useful when you stick it in a pacemaker. It still has to be handled specially. So it’s the same as NaN in that respect.

2. It seems like the only one difference between Nullity and NaN is that NaN != NaN, while Nullity == Nullity. I see this as a huge error. The wiki sums up the Nullity axioms nicely. Nul + a = Nul. Nul = Nul. Now how can that be? That means a = 0! There are a thousand ways you can bork yourself up if you think all Nullities are equal. This is precisely why Infinity != Infinity, and in IEEE, NaN != NaN.

Oh, and someone said earlier, “but you can check to see if a number is NaN, so you can check for equality”. Yes, but the point is that if I have float a, b; and both a and b are NaN (same bit representation), then according to IEEE 754, a == b should return false. Whereas his nullity would return true. Also for someone else that asked, yes there are many different bit patterns in 754 which all mean NaN, but technically all should be treated the same. (This means you can’t just use a bitwise equality check).

3. As somebody pointed out, transreal does not obey the axioms of fields, meaning a lot of things you could do with reals can no longer be relied upon. (I say this with only a wiki’s knowledge of fields).

4. Why are we all wasting our precious time bagging this guy? He clearly doesn’t deserve our attention!

98. #98 Fred Schreyer
December 8, 2006

I am not a mathematician, but it seems to me that if you divide 50 by 0 the answer is 50. In other words, you are not dividing it all. Same as 50 divided by 1. The numeral zero is the mathematical expression of the concept of none, so 50/0 is fifty divided by none. Fractions (1/2, 0.5)are just larger numbers writ small. Fifty apples, fifty people, each get one. Two people, they each get twenty-five. There is no such thing as a half of a person, so fifty apples divided by a half a person makes no sense. In short, there are no numbers in the real universe less than 1, except on paper and the minds of mathemeticians.
Or am I just stupid?
Fred Schreyer

99. #99 csrster
December 8, 2006

Mark Tyler wrote
>Is it crankish? No. Is it useful? No.

Ok, but is it crankish to think that it’s useful?

The (only) interesting thing about this blip has been to see how many of the critiques of it are more wrong than the nullity idea itself. On the BBC site there are people apoplectically insistent that 1/0 is infinity and that all numbers _must_ lie on the real number line.

For myself, I don’t doubt that you can consistently extend the reals to include 0/0, but frankly I think 0x=0 is something I would like to keep hold of for _all_ x!

100. #100 Jonathan Vos Post
December 8, 2006

The most interesting paper this week [warning: advanced Math] on multiplication of real numbers in the context of nonstandard analysis is:

http://arxiv.org/PS_cache/math/pdf/0612/0612184.pdf

math.NT/0612184 [abs, ps, pdf, other] :
Title: A Nonstandard Approach to Real Multiplication
Authors: Lawrence Taylor
Comments: 33 pages. Adapted from a chapter of the author’s PhD thesis
Subj-class: Number Theory; Logic
MSC-class: 11U10; 11R37

We explore the possibility of using Model Theoretic ideas to study certain non-Hausdorff spaces knwon as Quantum Tori with a view to their application to Manin’s theory of Real Multiplication. We study the morphisms between these spaces using Nonstandard Analysis, and describe an action of a certain Galois group on a certain classes of these spaces.

101. #101 Tim G
December 8, 2006

It’s mathturbation.

It’s self-indulgent and won’t spawn anything.

102. #102 Blake Stacey
December 8, 2006

@Fred Schreyer:

Before you can say “I have fifty apples in this pile,” you have to make the judgment call that each two things in the pile are effectively identical. But this apple is rotten, that one has a worm hole, this other one has a leaf still attached to the stem, somebody took a bite out of this bruised one here, etc. When are two apples the same? If you can’t judge that, then you certainly won’t be able to say you have fifty of them.

“Natural numbers” — 1, 2, 3 and so forth — are useful for counting things. But you need to make a judgment call before you can count, and on top of that, counting is only one kind of problem we’d like to solve! Why, then, should we give 1, 2, 3 and their friends any special priority? New problems mean new kinds of number, which we develop in such a way that the new numbers are “familiar”: we can add and subtract irrational or complex numbers, in ways which follow much the same rules as arithmetic on the integers.

103. #103 einstein
December 8, 2006

As far as I’m concerned, dividing any number except zero, by zero, produces infinity (or negative infinitity, if dividing by a negative number).

Its calculus. As you divide a number by x, as x approaches 0, the result approaches infiniity, generating larger and larger numbers.

All numbers and numerals are simply symbols that represent a concept of a quanity, like x or y for example. Infinity is simply just another concept represented by another symbol (for example, the infinity sign, an 8 turned sideways), that has its rules when anything is divided or multiplied… etc by it.

104. #104 Charley
December 9, 2006

1/0 = one divided into zero parts = leave it alone = do nothing until the zero become something not zero. When it becomes something not zero, then do something. I can live with this concept.

105. #105 Jonathan Vos Post
December 9, 2006

Sorry, Charey. Zero is not the same as nothing. The set containing one element, a zero, is not the same as the null set. The null set is not the same as the set containing one element, which element is the null set. Zero is not the same as the zero vector. Zero is not the same as the zero matrix. Zero is not the same as the empty graph.

“Nothing will come from nothing.”
[Shakespeare, King Lear]

106. #106 Bob O'H
December 9, 2006

There’s an update: you’ll all get the chance to tell him he’s a fool to his virtual face on Tuesday.

Bob

107. #107 Daniel
December 9, 2006

Joshua, slightly ironic that someone criticizing a journalist for not fact checking didn’t bother to look at what they were crticizing (it’s obviously from a local TV station, not a newspaper).

It’s an ‘and finally’ story about an interesting local character. Do you really expect it to be fact checked? The journalist has a little “I don’t have a clue what he’s talking about” bit. The commentary comes from children. Most TV viewers have the sense not to take it seriously. I suppose internet commentators are another matter.

108. #108 AJS
December 9, 2006

You can derive the existence of fractions and negative numbers just by reference to a simple number line. A tape measure, for instance. (Note that it’s even conveniently divided into tenths.) Deriving j is a little more effort, but you can prove it is meaningful. For all positive x, there exists y such that y ** 2 = x.

Note that –y also satisfies this equation, so we have a 2:1 mapping. This doesn’t utterly ruin the usefulness of the square root, provided we remember to account for it. The squaring operation can still be considered reversible, though we must accept two equally plausible solutions to any equation with a squared term in it. Of course, if we have other information available, we may be able to eliminate one solution; especially when dealing with a mathematical model of a real-life situation. For instance, you can’t have a negative length of fabric, and a projectile may have made impact the first time it reached a particular height.

It turns out that if we hypothesise that there is such a number as sqrt(-1), square roots of negative numbers obey the same rules as the numbers we already know about. We can let j = sqrt(-1), extend our number line into a two-dimensional plane and it still forms a number system.

So much for that. j works because it adds an orthogonal dimension to a number system. Or you could say that real numbers are a subset of complex numbers which are themselves a subset of multi-dimensional vectors, to which the rules of number systems can be generalised.

Multiplication by zero, however, is a many-to-one mapping. In fact, it’s an infinity-to-one mapping. There is no way to know whether 0 is the result of 1 * 0, 2 * 0, -1 * 0, π * 0, or anything else. Nor does it make any sense to try — and people have tried.

Let’s try to define o = 1 * 0. So 2 * 0 = 2o, and -1 * 0 = –o, and so forth and so on. This makes multiplication by zero reversible. Unfortunately, you don’t have to continue this thought experiment for very long to see that o doesn’t obey the rules as nicely as j did.

The whole reason why we say the result of division by zero is undefined, is that we can’t reverse the operation of multiplying by zero with anything even remotely resembling certainty (and indeed, the zero may have been obtained directly by subtraction, not by multiplying anything by zero). Now, that very irreversibility is actually a fantastically important thing in and of itself: it suggests a process that can be done to unequal things and give equal answers. As a corollary, we can say that if a product is zero, we know that at least one of the terms in that product must be zero.

109. “The Existence of a Trivial is Indeterminate”. This is very important and easy to prove. It says that you cannot prove whether an object is really itself, or if it might in fact be an identical clone of itself. That this is strictly “indeterminate”. This is also true of numbers, etc.

This is Harris’s Theorem, and is perhaps the most important theorem in all of mathematics.

I’m not sure if his “nullity” is the same as triviality, but it might be. >> I am not familiar with what this man is doing. << Triviality is not a trashcan. There is a distinction between the trivial and the strictly nonexistent. You have existence, nonexistence, and there IS a third existential type which is the trivial. It is a third type because "The Existence of a Trivial is Indeterminate", and I can prove it easily. All you need to do is play with uniqueness. Respectfully, Dr. Viktor I. Planckenstein

110. #110 Quakeulf
December 10, 2006

A B C , Its easy as
1 2 3 , as simple as
do re mi, A B C, 1 2 3
baby you and me girl

111. #111 Reinier Post
December 11, 2006

For the first time since I’ve been reading these excellent articles, I feel Mark is totally wrong.

Nullity does make mathematical sense, the explanation (about NaN not being as useful) makes sense, and it does serve a practical purpose (safety checks in code can be postponed longer than with NaN). I can’t see why Anderson would be an “idiot” or “crank”. The final straw is the suggestion that Anderson doesn’t know of NaN, while his article explicitly proposes nullity as an improvement over NaN.

BTW I found the article

from a reference in the Slashdot discussion

112. #112 Mark C. Chu-Carroll
December 11, 2006

Reinier:

Nullity only makes mathematical sense if, by mathematical sense, you mean “can be axiomatized”. Yes, you can build a set
of axioms under which nullity makes sense. But the result is by pretty much any reasonable standard an *inferior* axiomatization compared to the standard math where 1/0 is undefined. It breaks identity laws, symmetry constructions, etc. It introduces *more* problematic cases rather than less.

And the idea that it has any practical purpose is similarly silly: it does not change anything. The only difference between nullity and NaN is that nullity==nullity, but NaN!=NaN. But that’s a difference that makes no real difference: With current NaN in numeric code, when you do something resulting in an error, you wind up with NaN. Once you’ve hit NaN, you’re trapped: adding, subtracting, multiplying, dividing with NaN all result in NaN. You can keep going, but your result is going to still be NaN. What happens with nullity? Nullity is likewise a trap: adding, subtracting, multiplying, and dividing with nullity all result in nullity. The *only* difference is that in NaN code, to check for NaN, you have to call a function like “isNaN(n)” to see if you screwed up; but in nullity code, you can check “n == nullity”. Whoop-de-doo. It’s no real change: you can have one dead-end value or the other; whichever dead-end value you pick can be tested for; one system has a slightly nicer syntax for testing for the dead-end value, but that slightly nicer syntax comes at the expense taking basic arithmetic and algebraic semantics, and making a mess out of them.

113. #113 Dustin
December 11, 2006

It introduces *more* problematic cases rather than less.

That’s a vast understatement. He’s completely annihilated the algebraic structure of the reals by throwing that thing in there. Any axiomatization of his nullity can’t be consistent with the arithmetic properties we gain from the fact that the reals are a field, or else every real number must be equal to nullity. If there is an axiom system in which this would make sense, it wouldn’t resemble the reals in the slightest: not only would it not be a field, it couldn’t even be a ring. And it doesn’t seem to me that he’s even paused to give that consideration since, while he’s doing his ‘proof’ on the board, he’s implicitly using field properties left and right.

114. #114 Dustin
December 11, 2006

Ahh, pardon me. I should be more careful. His procedure would work in one ring: {0}. How exciting.

115. #115 Dustin
December 11, 2006

Oh, hell. No it wouldn’t. He said nullity was different from zero. I’m just going to stop talking now.

116. #116 Mark C. Chu-Carroll
December 11, 2006

Dustin:

You’re absolutely right; he totally screws up everything algebraic about the structure of the reals. But even without getting as deep into the math as the field structure of the real (which, unfortunately, most people don’t understand), you can see what’s wrong with it just by playing around with the basic identities that most people understand that derive from the deep properties of algebra. For example, most people understand that if you’ve got 3/4 × 4/5, you can “cancel” the 4s and get 3/5; but in his system with nullity, the basic properties that make that canceling work no longer hold.

117. #117 Mikael Johansson
December 12, 2006

Dustin: It’s right that what he defines end up not being a field. As has been said a couple of times in the comments, what he does is parallell to the theory of algebraic wheels: where a ring (or specifically a field) gets expanded to involve no illegal operations and still be consistent.

Just like you cannot really adjoin infinity to the reals and retain the field axioms for the extended reals, you cannot adjoin infinity and nullity/bottom/whatever to the the reals to get a wheel over the reals and still retain the field axioms. However, there will be a subfield working as we expect it to embedded within the wheel.

118. #118 Qubit
December 13, 2006

How come 0 is nothing but yet we have to represent it?
You can play with it and not be playing, you can win with it and not win it and we will become it… even though that should be impossible!

119. #119 Nonny
December 15, 2006

Dr James Anderson PhD
Said “I’ve created this thing – nullity.
I’ve divided by zero,
That makes me a hero”,
And his Perspex Machine said “I agree!”

120. #120 Jonathan Vos Post
December 15, 2006

Good limerick, Nonny.

Today’s Woo, from the [London] Times Liteary Supplement, is:

A cultural history of delusion
Jon Barnes

David Standish
HOLLOW EARTH
The long and curious history of imagining strange lands, fantastical creatures, advanced civilizations, and marvellous machines below the Earth’s surface
304pp. Oxford: Da Capo. Â£14.99 (US \$24.95).
0 306 81373 4

121. #121 Nonny
December 18, 2006

Is it’s equal to its own identity:
Unlike NaN, if you you take
Null from null, it will make
Exactly the value of, er, nullity…

122. #122 Gary Williams
December 18, 2006

Divide by zero DOESN’T cause things to crash though. Try it in calc in windows – Does windows crash? No.

This is just a daft idea. A fad. A piece of pre christmas fun.

123. #123 Mr. Anonymous
January 9, 2007

In his interview he says that the main improvement of nullity over NaN is that nullity is a fixed number, but what is he defining as a number? Would his definition have to be vague enough to also define NaN as a number?

124. #124 Mark C. Chu-Carroll
January 9, 2007

Mr. Anonymous:

The only difference between his “nullity” and the IEEE NaN is that in IEEE, NaN != NaN. He claims that nullity is a number because nullity==nullity.

February 26, 2007

This does appear to be pretty nonsensical, but bear in mind that there are real precedents for solving a problem by simply ‘giving a name’ to a concept – what about the number i?

126. #126 Xanthir, FCD
February 26, 2007

You are correct – there is nothing about mathematics that really makes one thing crazier than another. Negatives were considered impossible by many high-placed mathematicians well into the 19th century.

The issue, of course, is in showing that the new concept is useful. None of the reasons given for nullity do anything of the sort.

I found a good exposition on ‘nonsensical’ math in the book Negative Math. It’s a good read.

127. #127 Mark C. Chu-Carroll
February 26, 2007

Xanthir already mostly covered this, but I’ll throw in my two cents as well. But the reason that nullity is nonsense isn’t just because it assigns a name to a formerly undefined quantity. It’s because defining that quantity doesn’t add anything useful, and it destroys a lot of important properties of the field of numbers. (Field in that sentence is the mathematical term field, not field of work.)

Taking something that was formerly undefined, and defining it with a name and properties that breaks the thing it was supposed to fix is a bad, bad thing.

i is a good example of when adding something is good. Complex numbers solve a problem; and they do it without breaking the real numbers. The addition of i gives us complex numbers; it solves serious problems in algebra and analysis. So it’s useful. And it keeps the basic field structure of numbers. It doesn’t have exactly the same properties as the field of real numbers: the ordering of numbers breaks with complex numbers. But the complex numbers are a field; the real numbers are a sub-field of the complex numbers; any result that was valid in the field of real numbers remains valid in the superfield of complex numbers.

Nullity breaks the structure of the reals. There are things that are provably true about the field of real numbers that are not true about the algebra that you get by adding nullity. The real numbers are a sub-set of the real numbers+nullity, but they aren’t a sub-field – because real+nullity isn’t a field.

And that’s a deep fundamental difference. Define something that solves a problem without breaking what already works can be a good thing. Define something that doesn’t really solve a problem, and which breaks lots of things that already work – that’s bad.

128. #128 J. B. Rainsberger
March 19, 2007

From a computer programming perspective–something the original author seems to paint as important–I have difficulty distinguishing his nullity from the long-established design pattern “Null Object”. I also have difficulty distinguishing it from the null/nil object in truly OO languages where null/nil can be given sensible behavior.

129. #129 Jonathan Vos Post
May 5, 2007

“The set R union {+infty,-infty} obtained by adjoining two improper elements to the set R of real numbers is normally called the set of (affinely) extended real numbers. Although the notation for this set is not completely standardized, R^_ is commonly used. The set may also be written in interval notation as [-infty,+infty]. With an appropriate topology, R^_ is the two-point compactification (or affine closure) of R. The improper elements, the affine infinities +infty and -infty, correspond to ideal points of the number line. Note that these improper elements are not real numbers, and that this system of extended real numbers is not a field….”

http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html

130. #130 saya
May 28, 2007

interesting…i definitely agree with what you said. giving “undefined” a new name does not mean you have solved this problem. 0 has been a troubling number to mathematics and here it goes again

131. #131 gege
October 17, 2007

I think that the person who created the nullity number has a made a solution that solves no problems at all. Because, people, we understand what dividing by zero means. You can’t divide “nothing” with a “something”. So, dividing by zero is simply and understandably undefined. In the contrary, it is said in the paragraph that the “nullity number”, what the creator calls his “discovery” or shall I say “invention” does not represent any numerical quantity at all. So, making a said “value” has no sense. The creator is a professor. If he teaches this to his students he’ doing them a great disfavor. In short, I think that the creation of this “number” has no use in man’s everyday lives.

132. #132 Pete
March 10, 2008

As a doctor in mathematics myself, and having spent time in Reading, it is probably surprising how I failed to read about this (wal)nut. The thing is what he has done is trivial and while his math is correct it is of no applicable use. Read the Wikipedia entry under “transreals”. But the worst thing is that he seems to be one of those New Age crackpots. The following is from his website, a “paper” called “Visions of Mind and Body” excerpt from the sub-titled “Forgiveness”: “The walnut cake theorem guarantees that robots will be fallible. If they are sufficiently intelligent to develop spirituality we might hope that they will develop concepts of forgiveness and atonement”. Enough said? Please download visions.pdf (from this guy’s site).

I would NOT want to be in the same room with this guy. I don’t know what is more sad: BBC’s trademark of trashy journalism we’ve been used to for many years now, or Reading University’s relatively new Computer Science Department with not-exactly-great names having truly gone downhill in a desperate attempt to be more heard.

One way or another, one thing is clear: this guy has spent way too many lonely nights at the computer rooms of the Cybernetics Department of Reading University (right next to Computer Science). I hope for the sake of the University that he’s kicked out.

133. #133 Sgt. APWN
October 8, 2008

Roughly over a thousand years ago, people all over the world said zero was not a number. Can we say they were wrong?

And today, no one remembers that. Well, some calculus might help out.

There are two infinities on a number line, the positive and the negative. Although functions might put some limitations on these sets of most numbers, 1/0 is the positive half of the cartesian plane and -1/0 is the negative side.

When you put the two together, you have (1-1)/0, or 0/0. Why is this significant? Well, if you subtract a number from infinity, you still have infinity. When you subtract infinity from its self, it doesn’t cancel, but cumulates; you are left with both halves of the cartesian plane, ie NULLITY.

If that doesn’t suit you, you can think of it as what defines the cartesian plane, as in what is outside of it. The guy who made the cartesian plane, Renee Descartes, once drew two circles of different magnitudes and then said he could put an infinite amount of points in both of them, but one circle could hold more than the other.

We are just now starting to know what he meant, and how to expand on him. Spooky!

134. #134 petroski
October 20, 2008

Reading over all these enlightening opinions…

An undefinied problem… Â¿Is it a problem?… Â¿Does it need to be solved?… Eeeeh, Â¿Is it solvable?…

It is trully fun, at least; I would say ðŸ™‚

135. #135 Jonathan Vos Post
October 21, 2008

Math is fun.

Knowing a nonproblem from a problem is not trivial.

136. #136 William Wallace
February 9, 2009

The 0^0 issue just popped up. As my wife (a math major) and I were watching a DVD on exponentiation with my 7 year old son. (He’s doing algebra and beyond already). We were both shocked when they taught the students that X^0 is 1 only when X is not equal to 0.

But 0^0 is defined as 1 in math. Consider the binomial theorem, which is elegantly stated with a simple equation.

It gets ugly, though, if you have to come up with all of the different equations for special cases x=0, x!=0, y=0, y!=0, n=0, n!=0, x+y=0, and x+y!=0. And all of those special cases will generate a value that is the exact same as if you just defined (as many if not most mathematicians do) 0^0=1.

Even google defines 0^0 as 1.

In fact, the binomial theorem comes up, if my memory of my own math education is correct, long before the esoteric limit (f(t)->0, g(t)->0) of f(t)^g(t) will come up, and the binomial theorem is much more useful.

Seem strange that math educators are teaching kids that X^0=1 only when X!=0, when it won’t be that much longer before they cover the binomial theorem.

You should blog on 0^0, and the (in my view) bad math of junior high school teachers teaching students that 0^0 is not defined. They should at least teach that 0^0 is sometimes (or usually) defined as 1.0.