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. This is an absolutely infuriating story, which does an excellent job of demonstrating what total innumerate idiots reporters are.
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) specifically defines a set of values for operations that can't return a meaningful number: NaN (Not a Number). 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:
- 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?
- 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.
Comments
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...
Posted by: Phil | December 7, 2006 10:35 AM
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.
Posted by: Eighty | December 7, 2006 10:46 AM
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.
Posted by: p | December 7, 2006 10:48 AM
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 :P
Posted by: Eric | December 7, 2006 10:55 AM
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!?!?!
Posted by: Jurgen | December 7, 2006 11:01 AM
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.
Posted by: p | December 7, 2006 11:14 AM
What's wrong with 0^0? In Concrete Mathematics (Graham, Knuth, and Patashnik):
Posted by: Ron Avitzur | December 7, 2006 11:25 AM
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.
Posted by: Blake Stacey | December 7, 2006 11:27 AM
If nullity = 1/0, then should nullity/nullity = 1? Because then you can cancel the zeroes... :D
Posted by: Elia Diodati | December 7, 2006 11:51 AM
Elia:
Based on his handwaving explanation of why nullity "solves" the 00 problem, I suspect that yes, nullity/nullity=1.
Posted by: Mark C. Chu-Carroll | December 7, 2006 11:59 AM
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...
Posted by: Bernhard Bauer | December 7, 2006 12:07 PM
Baez makes the same point Bauer did above in Week 240 of This Week's Finds in Mathematical Physics.
Posted by: Blake Stacey | December 7, 2006 12:21 PM
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.
Posted by: Mikael Johansson | December 7, 2006 12:28 PM
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 ≠ 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.
Posted by: Jens Ayton | December 7, 2006 12:35 PM
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.)
Posted by: Jeff Petkau | December 7, 2006 1:06 PM
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?
Posted by: Edmund Schluessel | December 7, 2006 1:23 PM
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.
Posted by: Nasorenga | December 7, 2006 1:41 PM
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.
Posted by: Nuri Yeralan | December 7, 2006 1:53 PM
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 → 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.
Posted by: grigory | December 7, 2006 1:58 PM
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.
Posted by: Daniel James | December 7, 2006 2:04 PM
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.
Posted by: Graham Fyffe | December 7, 2006 2:09 PM
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.
Posted by: Alon Levy | December 7, 2006 2:09 PM
Amazing. I got done reading about this nullity bullshit 2 seconds before seeing this post on the Reddit.com feed. Brilliant. I knew I was reading retardery.
Posted by: Aerik Knapp-Loomis | December 7, 2006 2:10 PM
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.
Posted by: Joshua | December 7, 2006 2:17 PM
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:
What is happening here?
Posted by: Coin | December 7, 2006 2:44 PM
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.
Posted by: Jonathan Vos Post | December 7, 2006 2:45 PM
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.
Posted by: Michael | December 7, 2006 2:52 PM
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?
Posted by: Mark C. Chu-Carroll | December 7, 2006 3:02 PM
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.
Posted by: GreedyAlgorithm | December 7, 2006 3:09 PM
Teachers, when they are good, make the students better, but when they are bad they make students worst.
Having bad teachers is dangerous.
Posted by: Pupeno | December 7, 2006 3:09 PM
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.
Posted by: Flaky | December 7, 2006 3:13 PM
if (divisor == 0)
{
Contingency(); // Cardiac patient saved!
}
else
{
NormalStuff();
}
Posted by: Tim G | December 7, 2006 3:32 PM
@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. . . ."
Posted by: Blake Stacey | December 7, 2006 3:43 PM
Oh, and that IP address geolocates to London, but I couldn't find out anything else.
Posted by: Blake Stacey | December 7, 2006 3:55 PM
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.
Posted by: Flaky | December 7, 2006 4:03 PM
typical situation -- msm thinks they've found a genius and it takes bloggers to point at that he's a charlatan.
Posted by: vlad | December 7, 2006 4:18 PM
Uh oh. The Emperor's New Mind strikes again? :O
Posted by: Coin | December 7, 2006 4:18 PM
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."
Posted by: Flaky | December 7, 2006 4:20 PM
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?
Posted by: Harald Hanche-Olsen | December 7, 2006 4:31 PM
thank god someone wrote about that cook. i cant believe it got so many upvotes on reddit and it was actually o the bbc
Posted by: James Bascle | December 7, 2006 4:43 PM
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.
Posted by: milty | December 7, 2006 4:45 PM
2/0 = 2*ф (2 * infinity)
Posted by: gavin | December 7, 2006 4:59 PM
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 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...
Posted by: Coin | December 7, 2006 5:34 PM
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...
Posted by: Sigivald | December 7, 2006 6:22 PM
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
}
}
Posted by: Jesus | December 7, 2006 6:39 PM
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?"
Posted by: Jonathan Vos Post | December 7, 2006 6:50 PM
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.
Posted by: Jack9 | December 7, 2006 7:01 PM
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.
Posted by: Tyler | December 7, 2006 7:13 PM
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
Posted by: Levi | December 7, 2006 7:15 PM
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.
Posted by: Mark C. Chu-Carroll | December 7, 2006 7:27 PM
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.
Posted by: Davis | December 7, 2006 7:35 PM
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...
Posted by: neil | December 7, 2006 7:35 PM
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?
Posted by: Elia Diodati | December 7, 2006 7:56 PM
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.".
Posted by: clinton bowen | December 7, 2006 8:02 PM
Convenience aside, can you really reach an exact point between -1 and 1, called 0?
Posted by: ASH | December 7, 2006 8:03 PM
@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:
Well, he did contribute nullity to mathematics, even though his "contribution" was, er, null and void. . . .
Posted by: Blake Stacey | December 7, 2006 8:05 PM
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.
Posted by: Mark C. Chu-Carroll | December 7, 2006 8:08 PM
Elia Diodati asked:
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.
Posted by: Blake Stacey | December 7, 2006 8:16 PM
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!
Posted by: Brandon | December 7, 2006 9:45 PM
Thank you, thank you, thank you! I thought the entire world had gone insane there for a while.
Posted by: TheEngineer | December 7, 2006 9:48 PM
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.
Posted by: Mark Wagner | December 7, 2006 10:09 PM
Another testament to the fact that incompetence + and inflated ego leads to disaster, albeit often in the form of self-inflicted punishment.
Posted by: awe inspired | December 7, 2006 10:43 PM
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
Posted by: Cale Gibbard | December 7, 2006 10:58 PM
Only Timecube can surpass this exceptional example of extreme mathematical genius.
Posted by: Brad | December 7, 2006 11:07 PM
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.
Posted by: Mark Wagner | December 8, 2006 12:18 AM
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!
Posted by: TyPhyter | December 8, 2006 12:46 AM
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.
Posted by: Conrad Poelman | December 8, 2006 12:48 AM
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.
Posted by: Jonathan Vos Post | December 8, 2006 12:56 AM
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!!!!!
Posted by: Div | December 8, 2006 1:25 AM
oww c'mon, ur just jealous!
Posted by: nullity | December 8, 2006 1:44 AM
"Life is complex: it has real and imaginary parts"
Bob
Posted by: Bob O'H | December 8, 2006 2:18 AM
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.)
Posted by: Mike Scott | December 8, 2006 3:01 AM
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.
BTW, following a link from the article takes to http://www.bookofparagon.com/Pages/Downloads.htm
which contains a couple of rather funky videos of the Perspex Machine running.
Posted by: Flaky | December 8, 2006 3:46 AM
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.
Posted by: Tyler DiPietro | December 8, 2006 3:50 AM
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.
Posted by: Andrew McClure | December 8, 2006 4:16 AM
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 :-)
Posted by: Harald Korneliussen | December 8, 2006 4:29 AM
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:
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.
Posted by: Davis | December 8, 2006 4:44 AM
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.
Posted by: Charlie | December 8, 2006 5:04 AM
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 :)
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.
Posted by: -E | December 8, 2006 5:16 AM
Posted by: Andrew McClure | December 8, 2006 5:21 AM
You people need to get a life. Seriously, who gives a fuck?
Posted by: An | December 8, 2006 5:40 AM
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.
Posted by: Pseudonym | December 8, 2006 5:50 AM
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.
Posted by: Flaky | December 8, 2006 6:33 AM
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...
Posted by: -E | December 8, 2006 7:05 AM
Not a good time for computer scientists:
http://www.cs.unc.edu/~plaisted/ce/dating.html
Posted by: Jud | December 8, 2006 7:35 AM
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.
Posted by: Mark C. Chu-Carroll | December 8, 2006 7:55 AM