Slashdot Mirror

Errors are something you better get handled, but if you're uninspired, you can create your own math system like J. A. D. W. Anderson did on his paper "PERSPEX MACHINE VIII: AXIOMS OF TRANSREAL ARITHMETIC". source: http://www.bookofparagon.com/M...

How is this "nullity" different from "Not a Number"?

http://www.bookofparagon.com/Mathematics/SPIE.2002 .Exact.pdf (PDF warning) article seems to directly talk about nullity. (havent read it though) I am quite open to the idea that there is some way of defining something that can deal with x/0 properly. (though i think it will be slightly more complicated (and entirely different) as the complex numbers)

because he doesn't define how the + and * operators map up with it (nullity + a = ?)...

Actually, he does: Axiom 4: nullity + a = nullity, and Axiom 15: nullity * a = nullity. He also defines additive and multiplicative inverses to all new "numbers" he introduces. Look at his paper, it's at

http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf

I went through the paper and during a brief examination, I wasn't able to find a contradiction. I am not saying there isn't one, but he does claim to have proved consistency of his axiomatic system with standard axioms of arithmetic, even though he used an automatic theorem prover of some sort, if I recall correctly.

i think it is wrong, given his axioms (as defined here: http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf).

(inf) = 1/0 [A20]

= 1/(-1 * 0) [T77]

= -1 * (1/0) [A13]

= -1 * (inf) [A20]

= -(inf) [A24]

which contradicts his axiomatic supposition of (inf) and -(inf) as unique entities [T41]

A fundamental part of his explanation pivots on the following being true: 1/0 = infinity -1/0 = -infinity

So, according to that, the following would hold: if 1/0 = infinity then infinity * 0 = 1 which does not work, for obvious reasons. This I told my teacher in 6th grade.

If you throw away one of the axioms of any system of arithmetic then it shouldn't come as a surprise if your bastardised version of that system is inconsistent and/or incomplete.

In his new system of arithmetic, axiom 16 is:
infinity * 0 = nullity

Check out: http://www.bookofparagon.com/ It has what seems to be a fuller explanation of this "idea": http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf My 2c: I am no math-genius, but unless he comes up with some new way to actually use this "nullity", I do not for the love of God understand what he has done that merits even 5 secs of fame... And here's the new and interesting bit, our friend James Anderson of Reading has apparently also solved the problem of how mind relates to matter: It is something about a perspex, which is "a simple physical thing that is both a mind and a body", "a particular kind of matrix", "a physical shape, a physical motion, an artificial neuron, and an instruction for a machine that is more powerful than the Turing machine". It is also: "an instruction for a perspex machine that is more powerful than any theoretically possible digital computer". And of course, "...[a] perspex machine operates in a 4D space of perspexes called perspex space." And it "can describe any aspect of the universe we live in, and can be built from any part of our universe." Now I have actually read philosophy but that gets me nowhere with this guy. He seems to have swallowed some Leibniz and is trying to mix it with material realism, and further believes that eg. two discrete software programs can change continuously into another - while retaining the ability to use the same stored data (I am going to make miniscule changes to my OS after writing this, with a magnet, and see if it won't still be able to run my programs). What does help me understand him, however, is having tidied up in wikipedia and this seems a case of "things made up in school" and... well, nut-job-theries.

Check out: http://www.bookofparagon.com/ It has what seems to be a fuller explanation of this "idea": http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf My 2c: I am no math-genius, but unless he comes up with some new way to actually use this "nullity", I do not for the love of God understand what he has done that merits even 5 secs of fame... And here's the new and interesting bit, our friend James Anderson of Reading has apparently also solved the problem of how mind relates to matter: It is something about a perspex, which is "a simple physical thing that is both a mind and a body", "a particular kind of matrix", "a physical shape, a physical motion, an artificial neuron, and an instruction for a machine that is more powerful than the Turing machine". It is also: "an instruction for a perspex machine that is more powerful than any theoretically possible digital computer". And of course, "...[a] perspex machine operates in a 4D space of perspexes called perspex space." And it "can describe any aspect of the universe we live in, and can be built from any part of our universe." Now I have actually read philosophy but that gets me nowhere with this guy. He seems to have swallowed some Leibniz and is trying to mix it with material realism, and further believes that eg. two discrete software programs can change continuously into another - while retaining the ability to use the same stored data (I am going to make miniscule changes to my OS after writing this, with a magnet, and see if it won't still be able to run my programs). What does help me understand him, however, is having tidied up in wikipedia and this seems a case of "things made up in school" and... well, nut-job-theries.

Take a look at what the man has actually written instead of some terrible news article before bandying about the "idiot" moniker too freely:

http://www.bookofparagon.com/

I'd lump him in the same delightfully fun kook category as Rudy Rucker and Wolfram (and he might just have something)...

Here's a link to his paper:

http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf

I haven't bothered reading it, but in the first couple of lines of the paper he says:

"Transreal arithmetic is a total arithmetic that contains real arithmetic, but which has no arithmetical exceptions. The absence of exceptions makes it a more secure basis for computation than standard arithmetics,"

At the moment high security or safety critical developments spend a lot of money to ensure there are no arithmetical exceptions, so a scheme that ensures they can't occur sounds attractive.

Of course I haven't read the paper, so no idea if what he says makes sense or is useable, but the objectives look good.

OK, I have to say that maybe I jumped to conclusions too fast here. He has submitted to detailed papers on this and I am currently trying to digest them.

For anyone interested: http://www.bookofparagon.com/News/News_00012.htm

Anderson is somewhat of an avid writer.

On page 140 (type in page 154 in Acrobat Reader) in his book you can read a letter he has written to the future fleet of elite androids, powered by his super-Turing machine (which he calls a "perspex").

An excerpt:

I am not a god. I am not your creator. I am an ape of the species homo sapiens. I proposed to create you. I tried, within the span of my life, to create the simplest of you. You know, better than I, what you have become.

[...]

If you harm my kind, take comfort from this: I forgive you. I hope that your kind will have the wisdom and power to atone for the evil you have done.

I say this report is Bullshit.

I say you should've read up on the subject first. (But then again this is Slashdot, after all.) There are some papers available. So, at least the children weren't his first audience but merely the strange byproduct you get when you contact the media.

His stuff is still a bit weird though -- if his stuff were really such a groundbreaking mathematical discovery, it wouldn't have been published in a journal of the International Society for Optical Engineering...

Maybe, though, you can derive more consistent rules for using the IEEE 754 NaN and Inf numbers using his findings, but I'd think he still has a long way to go to prove his findings useful enough for that.

AC is right, the paradox of the grandparent does not hold, since
0*inf=nullity
and
a*nullity=nullity
these are the axioms A16 and A15 in his http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf

This silly number is completely uninteresting compared to the other amazing work that can be found on some website someone linked to. Like the Perspex Machine!

"The compiler generally lays out perspexes along a straight, spacetime line that changes only in time, t. However, the compiler implements C conditionals by jumping additionally in the spatial dimensions x, y, or z. C loops exploit a backward jump in time to iterate the loop. This exposes an infelicity in the specification of the perspex machine. Whilst the perspex machine can support such arbitrary temporal jumps it cannot copy them directly. It must use additional geometrical transformations and the access column to prevent an arbitrary temporal jump component being reduced to a normalised jump component of 0, 1, nullity, or infinity. This is remedied by the introduction of the universal perspex machine."

This silly number is completely uninteresting compared to the other amazing work that can be found on some website someone linked to. Like the Perspex Machine!

"The compiler generally lays out perspexes along a straight, spacetime line that changes only in time, t. However, the compiler implements C conditionals by jumping additionally in the spatial dimensions x, y, or z. C loops exploit a backward jump in time to iterate the loop. This exposes an infelicity in the specification of the perspex machine. Whilst the perspex machine can support such arbitrary temporal jumps it cannot copy them directly. It must use additional geometrical transformations and the access column to prevent an arbitrary temporal jump component being reduced to a normalised jump component of 0, 1, nullity, or infinity. This is remedied by the introduction of the universal perspex machine."

ruh ruh. Crank alert.

Heres his book


The Book of Paragon is a web site that offers one solution to the centuries old philosophical conundrum of how minds relate to bodies. This site shows that the perspective simplex, or perspex, is a simple physical thing that is both a mind and a body.
....or philosopher alert. Havent worked it out yet :)

Wow. Looking over the guy's axioms, as soon as you introduce "nullity" the result of all of your computations is nullity:

- the sum of anything and nullity is nullity (his axiom A4)
- the product of nullity and anything is nullity (his axiom A15)
- the reprical of nullity is nullity (his axiom A22)

So, his arithmetic is normal arithmetic, but as soon as you hit nullity anywhere, it's a black hole you can never get out of. All he's essentially done is take the "error state" and add it into the system as an object. You still can't compute anything you couldn't compute before. So yes, he has truly discovered NaN.

Surprised no one's come up with these yet. Here's his website http://www.bookofparagon.com/ Here's his paper [PDF Warning] http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf And a bit disheartened at the low math IQ around these parts. Though we were all nerds here.

Surprised no one's come up with these yet. Here's his website http://www.bookofparagon.com/ Here's his paper [PDF Warning] http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf And a bit disheartened at the low math IQ around these parts. Though we were all nerds here.

Surprised no one's come up with these yet. Here's his website http://www.bookofparagon.com/ Here's his paper [PDF Warning] http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf And a bit disheartened at the low math IQ around these parts. Though we were all nerds here.

Actually, in his own paper here:

http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf

he agrees with closely what you just said. It says nullity*14 = nullity, he's created a new mathematic Field. It still might be a load of hot air but this paper is at least more rigorous than the video's 0/0 = nullity craziness.

If you don't know what a Field is its kind of like a different universe for numbers with different rules than the one's we're taught in school. Another example of a Field would be discrete mathmatics which is used for encryption.

The way you have it matches with the axioms in the original source: http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf

Anderson has a website with his theory explained in a series of PDFs. My take is that first he has to show that his idea is internally consistent. But there are uncountable (literally...) numbers of consistent, but useless theories. I don't see how it can have any practical significance -- just asserting 1/0 = nullity doesn't solve any problems that I can see. Certainly I don't see how it's relevant to computing, that can't handle real numbers like pi (without approximation), let alone infinity which does have a long established mathematical pedigree and use in analysis.

Seriously though this is the sort of thing that you don't want to sneeze at, it can sound both inane and brilliant. Anderson is not such a crackpot, I found a presentation of his on optical computing and an introduction to its underlying theory called perspex algebra ( "Representing geometrical knowledge."). He seems to be a geometer stating his perspective in the first line of that presentation: "Aims: To unify projective geometry and the Turing machine".

He's a geek hero! Who knows if his nullity will end up just NaN with a British twang or the next best thing to sliced bread and i?

I was unable to hear the realaudio casts but from Book of Paragon, The Perspex Machine (Anderson mentions transreal arithmetic) and Exact Numerical Computation of the Rational General Linear Transformations (a mathematical treatise with applications to computer vision and robotics) just glancing I'd have to say the guy seems to be a real mathematician, geek and philosopher-king. I don't know if he's up there with Newton but he at least deserves an honorable mention for his wonderfully witty (and to me as yet inscrutable) naming of the Walnut Cake Theorem (see page 10 of Perspex.pdf). It seems that he was motivated to create nullity in order to make reliable advanced computers that would not barf when asked questions about the universe, and to him "Not-a-Number" is vomit. I'd say read some of his stuff before assigning him to the 9th Hell. Would like to hear what any mathematicians or other people with brain cells over the age of 12 have to think about it. It's okay if he reinvented something but it appears he is trying to make a machine that can handle infinities and other tough numerical concepts with ease, and that's worth something. Oh, that and his quantum computer looks neat.

Seriously though this is the sort of thing that you don't want to sneeze at, it can sound both inane and brilliant. Anderson is not such a crackpot, I found a presentation of his on optical computing and an introduction to its underlying theory called perspex algebra ( "Representing geometrical knowledge."). He seems to be a geometer stating his perspective in the first line of that presentation: "Aims: To unify projective geometry and the Turing machine".

He's a geek hero! Who knows if his nullity will end up just NaN with a British twang or the next best thing to sliced bread and i?

I was unable to hear the realaudio casts but from Book of Paragon, The Perspex Machine (Anderson mentions transreal arithmetic) and Exact Numerical Computation of the Rational General Linear Transformations (a mathematical treatise with applications to computer vision and robotics) just glancing I'd have to say the guy seems to be a real mathematician, geek and philosopher-king. I don't know if he's up there with Newton but he at least deserves an honorable mention for his wonderfully witty (and to me as yet inscrutable) naming of the Walnut Cake Theorem (see page 10 of Perspex.pdf). It seems that he was motivated to create nullity in order to make reliable advanced computers that would not barf when asked questions about the universe, and to him "Not-a-Number" is vomit. I'd say read some of his stuff before assigning him to the 9th Hell. Would like to hear what any mathematicians or other people with brain cells over the age of 12 have to think about it. It's okay if he reinvented something but it appears he is trying to make a machine that can handle infinities and other tough numerical concepts with ease, and that's worth something. Oh, that and his quantum computer looks neat.

Seriously though this is the sort of thing that you don't want to sneeze at, it can sound both inane and brilliant. Anderson is not such a crackpot, I found a presentation of his on optical computing and an introduction to its underlying theory called perspex algebra ( "Representing geometrical knowledge."). He seems to be a geometer stating his perspective in the first line of that presentation: "Aims: To unify projective geometry and the Turing machine".

He's a geek hero! Who knows if his nullity will end up just NaN with a British twang or the next best thing to sliced bread and i?

I was unable to hear the realaudio casts but from Book of Paragon, The Perspex Machine (Anderson mentions transreal arithmetic) and Exact Numerical Computation of the Rational General Linear Transformations (a mathematical treatise with applications to computer vision and robotics) just glancing I'd have to say the guy seems to be a real mathematician, geek and philosopher-king. I don't know if he's up there with Newton but he at least deserves an honorable mention for his wonderfully witty (and to me as yet inscrutable) naming of the Walnut Cake Theorem (see page 10 of Perspex.pdf). It seems that he was motivated to create nullity in order to make reliable advanced computers that would not barf when asked questions about the universe, and to him "Not-a-Number" is vomit. I'd say read some of his stuff before assigning him to the 9th Hell. Would like to hear what any mathematicians or other people with brain cells over the age of 12 have to think about it. It's okay if he reinvented something but it appears he is trying to make a machine that can handle infinities and other tough numerical concepts with ease, and that's worth something. Oh, that and his quantum computer looks neat.

Submitter couldn't be bothered to do the research, but there is a paper written by this guy about the concept.

http://www.bookofparagon.com/News/News_00012.htm It's way too late for me to read these right now. Anybody know how this might be related to Conway's surreal numbers?

The first paper he describes as:

describes how to divide by zero consistently in a non-trivial way. This shows that division by zero is no longer an error. Amongst other things, the paper explains why the standard model of arithmetic is not valid.

The second paper he says:

explains how to extend calculus so that it works with transreal numbers. This paper disposes of various counter "proofs" that attempt to show that division by zero is impossible. The paper ends with a very simple equation demonstrating the possibility of division by zero and challenges the reader to accept it.

The first paper he describes as:

describes how to divide by zero consistently in a non-trivial way. This shows that division by zero is no longer an error. Amongst other things, the paper explains why the standard model of arithmetic is not valid.

The second paper he says:

explains how to extend calculus so that it works with transreal numbers. This paper disposes of various counter "proofs" that attempt to show that division by zero is impossible. The paper ends with a very simple equation demonstrating the possibility of division by zero and challenges the reader to accept it.

The first paper he describes as:

describes how to divide by zero consistently in a non-trivial way. This shows that division by zero is no longer an error. Amongst other things, the paper explains why the standard model of arithmetic is not valid.

The second paper he says:

explains how to extend calculus so that it works with transreal numbers. This paper disposes of various counter "proofs" that attempt to show that division by zero is impossible. The paper ends with a very simple equation demonstrating the possibility of division by zero and challenges the reader to accept it.