# Dividing by Zero is defined now!

• I'm so very, very afraid.

Nullity.

I'm somewhat surprised it wasn't represented with WTF instead of Captital Phi.

• # On one hand....

Ok, so nullity is zero divided by zero. Under normal circumstances, any number divided by zero is considered infinite. In fact, the guy *redefines* infinity as 1/0, and "negative infinity" as -1/0. Either of those numbers can be extracted from another number divided by zero. For example, 344/0 = 344*(1/0). It's normalized, that's good. But how do you extract a 0/0 from a result? That's the only issue I see. Is it necessary to define 0/0? What is 0/0, anyway, infinitely nothing? I can see how this might be good for math, in much the same way imaginary numbers were good for math -- i.e. to solve problems only mathematicians care about.

Where I object is the guy bitching about how computers can't handle divide by zero. Computers handle divide by zero just fine, thankyouverymuch. They recognize it and set a bit that says "you divided by zero. Shame!" Anybody who programs knows how to handle a divide by zero. Sure, they don't know what to *do* with it, aside from object to it, but they can handle it. The idea that this will somehow revolutionize computing is bunk. It might revolutionize math and allow you to find answers to unsolvable equations by fudging the numbers the same way imaginary numbers do, but it won't revolutionize the computer industry. The computer industry has been handling the problem gracefully for decades.

Basically, it looks like the guy wants to modify IEEE floating point to include 1/0, 0/0, and -1/0. Fine. But don't sensationalize it as revolutionizing computing, solving global warming and raising the dead.

• Heh. Nullity. I guess 'infinity' is just too sensible!

• [quote user="Whiskey Tango Foxtrot? Over."]Basically, it looks like the guy wants to modify IEEE floating point to include 1/0, 0/0, and -1/0. Fine. But don't sensationalize it as revolutionizing computing, solving global warming and raising the dead. [/quote]

It already does define them as +INF, NaN and -INF

[quote user="Whiskey Tango Foxtrot? Over."]Basically, it looks like the guy wants to modify IEEE floating point to include 1/0, 0/0, and -1/0. Fine. But don't sensationalize it as revolutionizing computing, solving global warming and raising the dead. [/quote]

It already does define them as +INF, NaN and -INF

[/quote]

Yes, I know. But from his posts in the article discussion (I assume it really is him) he implies he wants to encode those three values along with the result of the operation. I.e. 2445.003/0.0 = 2445.003 * (1/0). IEEE can't do that: +/-INF is an all-1 exponent and an all-zero mantissa; no other values are encoded.

• [quote user="Whiskey Tango Foxtrot? Over."]

I can see how this might be good for math, in much the same way imaginary numbers were good for math -- i.e. to solve problems only mathematicians care about.

[/quote]

Ever heard of electrical impedance or the Fourier Transform? Both hit imaginary numbers pretty hard, and are rather important to information theory and modern telecommunications.

• aghhh....

I can't get rid of my divide by zero Nullity errors!

• [quote user="sparked"][quote user="Whiskey Tango Foxtrot? Over."]

I can see how this might be good for math, in much the same way imaginary numbers were good for math -- i.e. to solve problems only mathematicians care about.

[/quote]

Ever heard of electrical impedance or the Fourier Transform? Both hit imaginary numbers pretty hard, and are rather important to information theory and modern telecommunications.

[/quote]

I was going to say the same thing, "Them's fightin' words. I didn't study j for years for nothin'."

Artificial Intelligence research has been held back by constraints, particularly in computability theory.  Some things are not "computable" because everybody accepts limitations like the Turing machine and division by zero.  Nobody else had the absolute brilliance to discover that by defining all the undefined things, one could create a superior mathematics in which all things are computable.  Likewise, the Perspex Machine can compute results in finite time that would require infinite time and space for a Turing machine.  It is not possible for a robot built from a Turing machine to have free will.  Therefore it is possible for a robot built from a Perspex Machine to have free will.

Dr. Anderson explains it much better than I can:

Arthur: Why did you invent the number nullity explained in Exact Numerical Computation of the General Linear Transformations?

Dean: Because of the homunculus problem.

Arthur: Let me see. It was once thought that the eyes focus the world on the pineal body in the brain. So it was thought that the pineal body sees the world. But the real question is how the pineal body could do that without involving a little person, or homunculus, sitting in the pineal body that does the seeing. In fact, what is wanted is a neurophysiological explanation of the visual pathways such as, or better than, we have today. Whenever explaining a human faculty we want a mechanistic explanation, not one that pushes words around and leaves the problem right back in the human. But what has that to do with arithmetic?

Dean: Integer arithmetic is fine, but rational arithmetic, and more advanced arithmetics all involve division by zero. Division by zero is not defined so whenever it arises a human mathematician has to get involved and try to sort it out. Division by zero turns up in an awful lot of mathematics and, so too, do related geometrical properties. Points that are co-punctal or co-linear or co-planar are banned from all sorts of geometrical operations. When they turn up a human mathematician has to sort them out on a case by case basis. And so it goes on. Almost all of mathematics is infected by the homunculus problem - corner cases are not defined so a human mathematician has to get involved to try to sort them out. This is one of the things that makes computer algebra so hard. All of the corner cases have to be defined so that a computer can solve algebraic problems without assistance from a human.

Arthur: So how does the number nullity help?

Dean: I defined a canonical form for numbers divided by zero and let the rules of arithmetic hold regardless of division by zero. This produced a new arithmetic that contains the arithmetics that people commonly use, but one in which division by zero is well defined. Thus, I removed the homunculus problem from this part of mathematics. That was all that I needed to do in order to define the perspex machine in a way that is guaranteed to be able to operate without human intervention. The number nullity, together with the number infinity, makes the perspex machine a suitable physical substrate for a robot's mind and body. That is what I wanted to achieve. Back then I did not know how to implement a mind using conventional mathematics - so I changed mathematics. Now I could do it in conventional mathematics, but I do not want to. Nullity makes all sorts of calculations easier.

There, isn't it all clear and obvious now?  Next we'll have a new theory on perpetual motion machines.

• [quote user="Whiskey Tango Foxtrot? Over."]Under normal circumstances, any number divided by zero is considered infinite. In fact, the guy redefines infinity as 1/0, and "negative infinity" as -1/0. Either of those numbers can be extracted from another number divided by zero. For example, 344/0 = 344*(1/0). It's normalized, that's good. But how do you extract a 0/0 from a result? That's the only issue I see. Is it necessary to define 0/0? What is 0/0, anyway, infinitely nothing? I can see how this might be good for math, in much the same way imaginary numbers were good for math -- i.e. to solve problems only mathematicians care about.[/quote]

Um, no. Any number divided by zero is not considered infinite, it is considered undefined. Consider: the limit of 1/x as x approaches zero from the positive side is, indeed, infinity. But the limit of 1/x as x approaches zero from the negative side is negative infinity. It's not just a discontinuity, it's (arguably) the largest possible discontinuity.

You just plain can't define 0/0 at all, and still have the rules of arithmetic work. Suppose we allow it. Start with identity: 0/0 = 0/0. But (1 - 1) = 0, so we can substitute 0/(1 - 1) = 0/0. Multiply through by (1 - 1) and you get 0 = (0 * (1 - 1))/0. But then again, (1 - 1) = 0, so 0 = (0 * 0)/0. But 0 times anything is 0, so we once again get 0 = 0/0. Since we assumed at the beginning that this was not true, it means that this assumption leads to a contradiction.

And if you try the same thing for 1/0, you get similar results: start with identity: 1/0 = 1/0. We know (1 - 1) = 0, so 1/(1 - 1) = 1/0. Since (1 + 1)/(1 + 1) is just 1, that means we can multiply the left side by it, and get: (1 * (1 + 1))/((1 - 1) * (1 + 1)) = 1/0. Carrying out the multiplication on the left side, we get (1 * 2)/0 = 1/0. Which means 2 * 1/0 = 1/0. Divide out the 1/0, and -- presto! -- 2 = 1. Once again, this assumption leads to a contradiction.

These things have been known, now, for over a century. How someone whose job is to teach mathematics could be ignorant of them is... well, on this website I hesitate to say "beyond comprehension" but clearly there's a teacher interviewer in England who is on a par with the developer interviewers we read about here.

The only way you could try to make the definitions work would be by making arbitrary rules like "in the presence of 1/0, you can no longer multiply". But that makes the whole construction useless, since the whole point is to allow you to carry on doing arithmetic.

Mathematicians -- the real ones, not amateurs like this nitwit -- hear from people like this all the time. There was a guy back in the 1930s or thereabouts named Carl Theodore Heisel who "proved" that pi was exactly 256/81, which is 3.16049... It's not entirely unlike the way that physicists hear from people who have "proved" that God created the universe by looking at numerology in the Bible.

• [quote user="The Vicar"]

The only way you could try to make the definitions work would be by making arbitrary rules like "in the presence of 1/0, you can no longer multiply". But that makes the whole construction useless, since the whole point is to allow you to carry on doing arithmetic.

[/quote]

Seems they picked "x * nullity = nullity" instead.  The wikipedia page on the subject shines a little more light on the subject, although I was hoping there'd be a "Criticisms" section.

http://en.wikipedia.org/wiki/Nullity_%28transreal%29

• [quote user="merreborn"][quote user="The Vicar"]

The only way you could try to make the definitions work would be by making arbitrary rules like "in the presence of 1/0, you can no longer multiply". But that makes the whole construction useless, since the whole point is to allow you to carry on doing arithmetic.

[/quote]

Seems they picked "x * nullity = nullity" instead.  The wikipedia page on the subject shines a little more light on the subject, although I was hoping there'd be a "Criticisms" section.

http://en.wikipedia.org/wiki/Nullity_%28transreal%29

[/quote]

Like I said: it makes the whole operation useless. It's a fundamental rule of arithmetic that you can make substitutions of equal terms, and as soon as you allow that, the whole construction falls down. This is the mathematical equivalent of building a wrapper library for the Windows API and then claiming that, since you can program without making calls to the Windows API, your new library must be cross-platform.

As for Wikipedia not covering criticisms, well, that's hardly surprising. The new attempts to make Wikipedia "respectable" by requiring sources makes this sort of situation inevitable. I doubt that many reputable mathematicians are going to waste their time proving -- again, after all these years -- that defining division by zero leads to contradictions, so there are no sources to quote. Wikipedia is becoming trash faster and faster these days...

• I love taking quotes from articles out-of-context :

<font face="Verdana, Arial, Helvetica, sans-serif" size="2">"..it seems the Year 10 children at Highdown now know their nullity.."</font>

Hope it feels good to know you suck kids.

• [quote user="The Vicar"]

I doubt that many reputable mathematicians are going to waste their time proving -- again, after all these years -- that defining division by zero leads to contradictions, so there are no sources to quote. Wikipedia is becoming trash faster and faster these days...

[/quote]

Now that he's made BBC news, someone bound to take the time to point out how absurdly stupid this whole thing is.  Especially since he's teaching it to kids, and making rather bold claims.

• [quote user="The Vicar"][quote user="merreborn"][quote user="The Vicar"]

The only way you could try to make the definitions work would be by making arbitrary rules like "in the presence of 1/0, you can no longer multiply". But that makes the whole construction useless, since the whole point is to allow you to carry on doing arithmetic.

[/quote]

Seems they picked "x * nullity = nullity" instead.  The wikipedia page on the subject shines a little more light on the subject, although I was hoping there'd be a "Criticisms" section.

http://en.wikipedia.org/wiki/Nullity_%28transreal%29

[/quote]

As for Wikipedia not covering criticisms, well, that's hardly surprising. The new attempts to make Wikipedia "respectable" by requiring sources makes this sort of situation inevitable. I doubt that many reputable mathematicians are going to waste their time proving -- again, after all these years -- that defining division by zero leads to contradictions, so there are no sources to quote. Wikipedia is becoming trash faster and faster these days...
[/quote]

Maybe you should look again.  The article is listed for deletion.

• [quote user="Carnildo"][quote user="The Vicar"][quote user="merreborn"][quote user="The Vicar"]

The only way you could try to make the definitions work would be by making arbitrary rules like "in the presence of 1/0, you can no longer multiply". But that makes the whole construction useless, since the whole point is to allow you to carry on doing arithmetic.

[/quote]

Seems they picked "x * nullity = nullity" instead.  The wikipedia page on the subject shines a little more light on the subject, although I was hoping there'd be a "Criticisms" section.

http://en.wikipedia.org/wiki/Nullity_%28transreal%29

[/quote]

As for Wikipedia not covering criticisms, well, that's hardly surprising. The new attempts to make Wikipedia "respectable" by requiring sources makes this sort of situation inevitable. I doubt that many reputable mathematicians are going to waste their time proving -- again, after all these years -- that defining division by zero leads to contradictions, so there are no sources to quote. Wikipedia is becoming trash faster and faster these days...
[/quote]

Maybe you should look again.  The article is listed for deletion.

[/quote]

Nice.  The page was created two hours ago, and listed for deletion minutes ago.  I suppose we've just witnessed one of those periods of extreme turbulence that wikipedia experiences.

• This whole /0 thing is even dumber then people who believe that .9999~ is not equal to 1.

www.winemakingjournal.com

• [quote user="sparked"][quote user="Whiskey Tango Foxtrot? Over."]

I can see how this might be good for math, in much the same way imaginary numbers were good for math -- i.e. to solve problems only mathematicians care about.

[/quote]

Ever heard of electrical impedance or the Fourier Transform? Both hit imaginary numbers pretty hard, and are rather important to information theory and modern telecommunications.

[/quote]

No, I was a CS major, not an EE major; I never encountered a practical use for sqrt(-1). I admit my ignorance on the subject.

Still, if i has helped in many ways, perhaps 0/0 can help in the same ways, by abstracting out the undefined and allowing calculations to continue. If you use NaN, everything returned is NaN.

• umm

what about nullity / nullity ?

I guess we are back to the same problem

•

• [quote user="Kev777"]

what about nullity / nullity ?

[/quote]

wouldn't that equal 1.

so (5/0) / (10/0) = 1

garbage.

• [quote user="themagni"]

I was going to say the same thing, "Them's fightin' words. I didn't study j for years for nothin'."

[/quote]

The real WTF is that everyone knows it's called "i".

flees

• [quote user="Benanov"]

I'm so very, very afraid.

Nullity.

I'm somewhat surprised it wasn't represented with WTF instead of Captital Phi.

[/quote]

I'm
somewhat surprised it wasn't represented with NaN instead of capital
Phi, since that's the existing term that everyone has been using for
years untiul this charlatan came along and made up a new name for it
and pretended he invented the whole idea.

• [quote user="iwpg"]

The real WTF is that everyone knows it's called "i".

flees

[/quote]

They're not imaginary numbers, they're [b]jmaginary[/b] numbers.

• [quote user="dllexport"]They're not imaginary numbers, they're [b]jmaginary[/b] numbers.[/quote]

• [quote user="newfweiler"]

Artificial Intelligence research has been held back by constraints, particularly in computability theory.  Some things are not "computable" because everybody accepts limitations like the Turing machine and division by zero.  Nobody else had the absolute brilliance to discover that by defining all the undefined things, one could create a superior mathematics in which all things are computable.  Likewise, the Perspex Machine can compute results in finite time that would require infinite time and space for a Turing machine.  It is not possible for a robot built from a Turing machine to have free will.  Therefore it is possible for a robot built from a Perspex Machine to have free will.

[/quote]

While the "transreal aka nullity" stuff is a (presumably) consistent, though (most likely) worthless algebraic structure, the idea that any machine that can be simulated on a PC can be more powerfull than a Turing machine simply proves complete ignorance.

• [quote user="The Vicar"]

Um, no. Any number divided by zero is not considered infinite, it is considered undefined. Consider: the limit of 1/x as x approaches zero from the positive side is, indeed, infinity. But the limit of 1/x as x approaches zero from the negative side is negative infinity. It's not just a discontinuity, it's (arguably) the largest possible discontinuity.

You just plain can't define 0/0 at all, and still have the rules of arithmetic work. Suppose we allow it. Start with identity: 0/0 = 0/0. But (1 - 1) = 0, so we can substitute 0/(1 - 1) = 0/0. Multiply through by (1 - 1) and you get 0 = (0 * (1 - 1))/0. But then again, (1 - 1) = 0, so 0 = (0 * 0)/0. But 0 times anything is 0, so we once again get 0 = 0/0. Since we assumed at the beginning that this was not true, it means that this assumption leads to a contradiction.
[/quote]

Using the transreal axiums, (0/(1-1))*(1-1) is not 0, but 0/0. They avoid the conclusion of your proof by making 0/0 (aka nullity) a tarpit, comparable to NULL in SQL.

And if you try the same thing for 1/0, you get similar results: start with identity: 1/0 = 1/0. We know (1 - 1) = 0, so 1/(1 - 1) = 1/0. Since (1 + 1)/(1 + 1) is just 1, that means we can multiply the left side by it, and get: (1 * (1 + 1))/((1 - 1) * (1 + 1)) = 1/0. Carrying out the multiplication on the left side, we get (1 * 2)/0 = 1/0. Which means 2 * 1/0 = 1/0. Divide out the 1/0, and -- presto! -- 2 = 1. Once again, this assumption leads to a contradiction.
1/0 has no multiplicative inverse in "transreal arithmetic", so you cannot divide out the 1/0.

The only way you could try to make the definitions work would be by making arbitrary rules like "in the presence of 1/0, you can no longer multiply". But that makes the whole construction useless, since the whole point is to allow you to carry on doing arithmetic.

No, there is another way: Just abandon the requirement to define a field. Or even a ring, btw. The result might be a relatively worthless algebraic structure, but at least it's consistent and gets you into the newspapers. Imagine how many mathematicans work on really hard problems and are never heard of.

• > They're not imaginary numbers, they're jmaginary numbers

Don't forget the k-maginary - suitable for those that don't like GNOME ;-p

• [quote user="ammoQ"][quote user="The Vicar"]

Um, no. Any number divided by zero is not considered infinite, it is considered undefined. Consider: the limit of 1/x as x approaches zero from the positive side is, indeed, infinity. But the limit of 1/x as x approaches zero from the negative side is negative infinity. It's not just a discontinuity, it's (arguably) the largest possible discontinuity.

You just plain can't define 0/0 at all, and still have the rules of arithmetic work. Suppose we allow it. Start with identity: 0/0 = 0/0. But (1 - 1) = 0, so we can substitute 0/(1 - 1) = 0/0. Multiply through by (1 - 1) and you get 0 = (0 * (1 - 1))/0. But then again, (1 - 1) = 0, so 0 = (0 * 0)/0. But 0 times anything is 0, so we once again get 0 = 0/0. Since we assumed at the beginning that this was not true, it means that this assumption leads to a contradiction.
[/quote]

Using the transreal axiums, (0/(1-1))*(1-1) is not 0, but 0/0. They avoid the conclusion of your proof by making 0/0 (aka nullity) a tarpit, comparable to NULL in SQL.

[/quote]

That breaks arithmetical axioms, so if you don't allow it, you aren't allowing standard arithmetic. But, again, the point of allowing division by zero is to allow arithmetic to continue uninterrupted when division by zero occurs, so why bother?

[quote user="ammoQ"]
And if you try the same thing for 1/0, you get similar results: start with identity: 1/0 = 1/0. We know (1 - 1) = 0, so 1/(1 - 1) = 1/0. Since (1 + 1)/(1 + 1) is just 1, that means we can multiply the left side by it, and get: (1 * (1 + 1))/((1 - 1) * (1 + 1)) = 1/0. Carrying out the multiplication on the left side, we get (1 * 2)/0 = 1/0. Which means 2 * 1/0 = 1/0. Divide out the 1/0, and -- presto! -- 2 = 1. Once again, this assumption leads to a contradiction.
1/0 has no multiplicative inverse in "transreal arithmetic", so you cannot divide out the 1/0.
[/quote]
So subtract from both sides, and get 1/0 = 0. Big whoop.

No, wait, I bet it has no additive inverse, either. So, in order to avoid causing math errors from division by zero, this idiot has created a new batch of entities which requires special rules which are even more complex (which means less intuitive and more difficult to implement) than just "don't divide by zero". Obviously a major human accomplishment.

If math were medicine, this guy just invented a special new pill which, if you take it, one Planck Time unit before you die, puts you into a coma-like state in which you do not eat, breathe, communicate, or think, and in which you eventually decompose. Your legal status becomes hopelessly confused, and your doctor has to spend twenty times as much effort to deal with it as he would have to sign a death certificate. But it sure beats dying!

[quote user="ammoQ"]

The only way you could try to make the definitions work would be by making arbitrary rules like "in the presence of 1/0, you can no longer multiply". But that makes the whole construction useless, since the whole point is to allow you to carry on doing arithmetic.

No, there is another way: Just abandon the requirement to define a field. Or even a ring, btw. The result might be a relatively worthless algebraic structure, but at least it's consistent and gets you into the newspapers. Imagine how many mathematicans work on really hard problems and are never heard of.
[/quote]

Um, you're thinking backwards. It isn't that inverses and identities exist because the real numbers are a ring, it's that the real numbers are a ring because inverses and identities exist. If you abandon inverses and identity elements, you basically render your system worthless for purposes of arithmetic as it is used in 99.999% of applications. (Including theoretical mathematical ones.)

Any way you slice it, "nullity" is just a fauly wrapper for division by zero -- it adds a new terminology and limitations to avoid the old limitations, but the new limitations aren't any easier to implement (or productive of new insights) than the old.

• [quote user="The Vicar"]

[quote user="ammoQ"]
And if you try the same thing for 1/0, you get similar results: start with identity: 1/0 = 1/0. We know (1 - 1) = 0, so 1/(1 - 1) = 1/0. Since (1 + 1)/(1 + 1) is just 1, that means we can multiply the left side by it, and get: (1 * (1 + 1))/((1 - 1) * (1 + 1)) = 1/0. Carrying out the multiplication on the left side, we get (1 * 2)/0 = 1/0. Which means 2 * 1/0 = 1/0. Divide out the 1/0, and -- presto! -- 2 = 1. Once again, this assumption leads to a contradiction.
1/0 has no multiplicative inverse in "transreal arithmetic", so you cannot divide out the 1/0.
[/quote]
So subtract from both sides, and get 1/0 = 0. Big whoop.

No, wait, I bet it has no additive inverse, either. So, in order to avoid causing math errors from division by zero, this idiot has created a new batch of entities which requires special rules which are even more complex (which means less intuitive and more difficult to implement) than just "don't divide by zero". Obviously a major human accomplishment.

[/quote]

No additive inverse, no multiplicitive inverse.  And as I recall, in order for something to be considered an identity element, there has to be only one of it, so no additive or multiplicative identities, either.  At least he managed to retain the associative, commutative, and distributive properties -- I think.

• I think we've been able to establish so far that, whether or not this is valid, it's of little use.  I liken it to the "inventor's" example of the calamity that would ensue if an autopilot system were to crash because of a divide-by-zero.  Now, if instead we received a null, how would this carry through the equations/program used within the autopilot?  Then the settings for the thrust, alerones, elevators, and everything else would become null.  What do you do with a null?  Nothing!  It's just as indeterminate as the divide-by-zero you were originally stuck with.  So if the resulting values are of no use, then what was the point of continuing with the calculation?  You gain nothing by continuing past the null condition.

In CS, we have a term for this: an exception.  When something happens that stops the show, you back out and try to recover from it.  How does a null change this?  Instead of immediately backing out (as the divby0 interrupt on any processor would do), you have to constantly check the "nullity" of your return value and manually back out.  Congradulations, you've reinvented the square wheel.

What really scares me though, is that he's teaching this to his high school class.  This isn't yet an accepted mathematical representation, let alone part of the curricullum that he is required to follow.  So why is this being taught to high school students who will blindly accept his useless nonsense?

I also like how he insists that this is original.  Check the discussion on the wikipedia page; this is hardly a new concept.  Surely we all have heard of NaN before.  His set of rules varies only slightly from the IEEE standards regarding NaN.

• [quote user="Carnildo"]

No additive inverse, no multiplicitive inverse.  And as I recall, in order for something to be considered an identity element, there has to be only one of it, so no additive or multiplicative identities, either.  At least he managed to retain the associative, commutative, and distributive properties -- I think.

[/quote]

Identities exist, since 1*a=a, 0+a=a even for a in {INF, -INF, NaN}

• [quote user="The Vicar"]

So subtract from both sides, and get 1/0 = 0. Big whoop.

No, wait, I bet it has no additive inverse, either.
[/quote]
How did you know On your way to eternal fame and glory, you have to make some sacrifices.
INF - INF = NaN in transreal arithmetic.
So, in order to avoid causing math errors from division by zero, this idiot has created a new batch of entities which requires special rules which are even more complex (which means less intuitive and more difficult to implement) than just "don't divide by zero". Obviously a major human accomplishment.
Yeah, his axioms are full of special rules, e.g.

a-a=0: a != INF, -INF, NaN

I think this makes solving equations a bit more "interesting".

If math were medicine, this guy just invented a special new pill which, if you take it, one Planck Time unit before you die, puts you into a coma-like state in which you do not eat, breathe, communicate, or think, and in which you eventually decompose. Your legal status becomes hopelessly confused, and your doctor has to spend twenty times as much effort to deal with it as he would have to sign a death certificate. But it sure beats dying!
This "new" algebraic structure allows uninterrupted straight-forward calculations, but makes everything else (equation solving, proving etc.) much more difficult.
[quote user="ammoQ"]No, there is another way: Just abandon the requirement to define a field. Or even a ring, btw. The result might be a relatively worthless algebraic structure, but at least it's consistent and gets you into the newspapers. Imagine how many mathematicans work on really hard problems and are never heard of.
Um, you're thinking backwards. It isn't that inverses and identities exist because the real numbers are a ring, it's that the real numbers are a ring because inverses and identities exist.
[/quote]
Oh really?
If you abandon inverses and identity elements, you basically render your system worthless for purposes of arithmetic as it is used in 99.999% of applications. (Including theoretical mathematical ones.)
Worthless for theoretical applications, yes. For normal calculations (as done by a computer), it's probably not much difference, considering all the limitations of the implementation of numerical datatypes (int, double, etc.) in computers.

• This has also been covered on Bad Science.

• Fascinating.  Right up front in his paper he states "<font face="TimesNewRomanPSMT" size="2">We note that the axioms have been shown to be consistent by machine proof." and "<font face="TimesNewRomanPSMT" size="2">Transreal arithmetic is a total arithmetic that contains real arithmetic", which I interpret as containing natural numbers as well.  He also states that 'total' arithmetic contains no exceptions, so presumably every theorem which can be stated in transreal arithmetic is either true or false (nothing undefined).</font></font>

I wonder what Godel would say (http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorem)?

<font face="TimesNewRomanPSMT" size="2"><font face="TimesNewRomanPSMT" size="2"></font></font>

• <font face="Helv" size="2">

My favourite line so far

"That is, whilst the real logarithm of a negative number is undefined, transreal logarithm of a negative number is defined to be nullity. In both cases the logarithm can be extended to give complex solutions, though we do not describe the transcomplex numbers here."

So he's saying that transcomplex algebra gives a different solution from transreal algebra! Nice consistency!

</font>

• Hang on, I've got it now.

I have no sense of direction but it's embarassing to say that I'm lost.  Because of my sense of direction, I'll never find my way back to the roads I know.  Therefore I'm going to define a new place called nullity.  Instead of getting lost, I'll just say that I've gone to nullity.  I could define all the places I visit as different nullities, but what's the point.  I don't recognise them anyway.  I'll just define nullity to be the same place, wherever I am. That way, if someone case to come to find me, they only have one place to look.

That's so much better than my previous idea of defining being lost as NaN.  Then every place that I got lost in was different, so it took ages for someone to find me.

• [quote user="Alchymist"]

<font face="TimesNewRomanPSMT" size="2"><font face="TimesNewRomanPSMT" size="2">He also states that 'total' arithmetic contains no exceptions, so presumably every theorem which can be stated in transreal arithmetic is either true or false (nothing undefined).</font></font>

<font face="TimesNewRomanPSMT" size="2"><font face="TimesNewRomanPSMT" size="2"></font></font>

[/quote]

You are asking for too much. Theorems proving in transreal arithmetics? Probably not a funny task.

Anyway, the totalness he claims only means that every term, especially terms like 23/(4-2*2), has a defined result in transreal arithmetics.

• [quote user="Alchymist"]

Hang on, I've got it now.

I have no sense of direction but it's embarassing to say that I'm lost.  Because of my sense of direction, I'll never find my way back to the roads I know.  Therefore I'm going to define a new place called nullity.  Instead of getting lost, I'll just say that I've gone to nullity.  I could define all the places I visit as different nullities, but what's the point.  I don't recognise them anyway.  I'll just define nullity to be the same place, wherever I am. That way, if someone case to come to find me, they only have one place to look.

That's so much better than my previous idea of defining being lost as NaN.  Then every place that I got lost in was different, so it took ages for someone to find me.

[/quote]

Since this place is the black hole, you are easy to find but impossible to retrieve. That's the funny thing in transreal:

if you have a term like

(2*(a+b)*(c-d))-(e/(3-f)-g/h) and you know that one of the variables has the value Nullity (aka NaN), or one subterm evaluates to Nullity, the whole term inevitably evaluates to Nullity. Saves a lot of work. Granted, a normal computer might have saved the same amount of work by throwing an exception, but calling it "Nullity" makes it look like a sophisticated optimization.

• My favorite line, from the bookofparagon website:

Back then I did not know how to implement a mind using conventional mathematics - so I changed mathematics. Now I could do it in conventional mathematics, but I do not want to.

Back then I did not know how to implement a perpetual motion machine using the conventional laws of physics -- so I changed the laws of physics.  Now I could do it in the conventional laws of physics, but I do not want to.  I really could.  But I don't wanna.  You can't make me.  But I can and you can't.  Nyaa nyaa.

• [quote user="ammoQ"][quote user="newfweiler"]

Artificial Intelligence research has been held back by constraints, particularly in computability theory.  Some things are not "computable" because everybody accepts limitations like the Turing machine and division by zero.  Nobody else had the absolute brilliance to discover that by defining all the undefined things, one could create a superior mathematics in which all things are computable.  Likewise, the Perspex Machine can compute results in finite time that would require infinite time and space for a Turing machine.  It is not possible for a robot built from a Turing machine to have free will.  Therefore it is possible for a robot built from a Perspex Machine to have free will.

[/quote]

While the "transreal aka nullity" stuff is a (presumably) consistent, though (most likely) worthless algebraic structure, the idea that any machine that can be simulated on a PC can be more powerfull than a Turing machine simply proves complete ignorance.

[/quote]

Gee!  You think?

How do you like my new rule of inference?  Move over Aristotle!

Major premise:  A Turing machine cannot have free will.

Minor premise:  A Perspex machine is not a Turing machine.

Conclusion:  Therefore a Perspex machine can have free will.

This will revolutionize logic!  We must teach it to schoolchildren!  [Glances at today's newspaper.]  Awww, too late.  George Bush and Tony Blair are already using it.

• [quote user="ammoQ"][quote user="Alchymist"]

Hang on, I've got it now.

I have no sense of direction but it's embarassing to say that I'm lost.  Because of my sense of direction, I'll never find my way back to the roads I know.  Therefore I'm going to define a new place called nullity.  Instead of getting lost, I'll just say that I've gone to nullity.  I could define all the places I visit as different nullities, but what's the point.  I don't recognise them anyway.  I'll just define nullity to be the same place, wherever I am. That way, if someone case to come to find me, they only have one place to look.

That's so much better than my previous idea of defining being lost as NaN.  Then every place that I got lost in was different, so it took ages for someone to find me.

[/quote]

Since this place is the black hole, you are easy to find but impossible to retrieve. That's the funny thing in transreal:

[/quote]

Much better analogy.  I was wondering how to add the concept of 'once you are here you aren't getting out again'.

Note to those who have suggested that transreal arithmetic is just as useful as complex arithmetic.  There's your difference.  Give me any complex number & I can get back to the real line, so my auto-pilot (or electrical circuit) can still give a sensible answer.

• @iwpg said:

[quote user="themagni"]

I was going to say the same thing, "Them's fightin' words. I didn't study j for years for nothin'."

The real WTF is that everyone knows it's called "i".

flees

[/quote]

I believe that in EE, 'j' is used for sqrt(-1) due to the fact that 'i' is used for current.

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