If you remember, a while back, I wrote about a British computer scientist named James Anderson, who

claimed to have solved the “problem” of “0/0” by creating a new number that he called nullity. The

creation of nullity was actually part of a larger project of his – he claims to have designed a computing

machine called the Perspex machine which is strictly *more powerful* that the Turing machine. If

this was true, it would mean that the Church-Turing thesis is false, overturning a huge part of the theory

of computer science.

Of course, just overturning the theory of computer science isn’t grandiose enough. He also claims that this solves the mind body problem, explains how free will works, and provides a potential

grand unified theory of physics. From Anderson’s own introduction on his website:

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.The perspex can be understood in many ways. Mathematically, the perspex is a particular kind of matrix; concretely, it is simultaneously a physical shape, a physical motion, an artificial neuron, and an instruction for a machine that is more powerful than the Turing machine. In other words, a perspex is an instruction for a perspex machine that is more powerful than any theoretically possible digital computer.

The perspex machine operates in a 4D space of perspexes called perspex space. This space is related to the 4D spacetime we live in. It is claimed that the perspex machine can describe any aspect of the universe we live in, and can be built from any part of our universe. In other words, the universe can be understood as a perspex machine. And, on the materialistic assumption, our bodies and minds are physical things so they, too, can be understood as perspex machines.

This site contains mathematical formulas for the perspex machine and for properties such as feeling, consciousness, and free will. These things are described in scientific papers, books, and software that you can download and run. The site also contains news items that explain the perspex machine in a non-technical way, and it has links to old research on the perspex machine.

He also claims that the Perspex machine can prove the existence of free will, God, and original sin.

One thing you’ve got to give to Anderson – the guy’s got ambition.

Anyway – back to the computer science part of it – the Perspex machine itself. I’ve been meaning to take a look at this. It wasn’t a high priority, because given the guy’s history and the grandness of the claim, I didn’t expect it to be correct – but I did want to wait until I had the time to give it a fair reading. Well, I finally found some time, and gave it a reading. And it’s pathetic. Truly pathetic. Quite

possibly even *more* foolish than the whole nullity nonsense. But we’ll get to that.

What is a perspex machine?

Basically, it’s a computational device based or perspective geometry. Perspective geometry is Anderson’s variant of projective geometry, which adds nullity as a discontinuity at the origin. (Projective geometry is a non-euclidean geometry where parallel lines meet at infinity.)

In a perspex machine, you’re performing computations through projective transformations

is perspective space. The computational state of a machine consists of a set of locations in projective space. Each computational step of the machine is a projective transformation of the machine state. NULLITY is essentially the HALT instruction: any projection to nullity halts the computation.

It’s a moderately interesting model of computation. It’s pretty trivial to show that a perspex machine with one point state can simulate a Minsky register machine – which in turn means that it can simulate

a Turing machine. That much is simple, and obvious. So why is it *more* powerful than a Turing machine? Let me quote Anderson’s explanation:

Projective geometry is usually carried out in a real or complex space. If a theoretical perspex machine

were defined to operate in such a space it would be able to do more than a Turing machine. For example, it

would be able to decide if any given irrational number is equal to zero. Whereas a Turing machine faced

with the decimal expansion of a small number with indefinitely many leading zeros after the decimal point

and before the significant figures is not be able to decide this case. It is interesting to note that

such a simple condition as implementing a perspex machine in a real space would falsify the Church-Turing

thesis. However, all contemporary, practical perspex machines are defined in a rational space.

So, theoretically, a Perspex machine can project its state to real numbers – not just rationals,

but reals. Since the state of a Turing machine can’t represent an irrational number, but a real-valued

Perspex machine *can*, the Perspex machine is more powerful than the Turing machine.

Of course, we can’t really *build* a real-valued Perspex machine, because the values of

the irrational numbers can’t be represented with finite state by an real machine. But hey, practical reality is no concern of Anderson. The Perspex machine uses real numbers, by golly, and therefore, it’s super-Turing.

This is truly pathetic. I can devise all sorts of computational devices that do all sorts of

things that don’t exist in the real world. This is no big surprise – there are literally decades

of study of the theoretical property of machines that can’t be built. (They’re studied as part

of an effort to understand the nature of computation.) For example, there’s a class of problems studied by computer scientists that are based on using a Turing machine with a *halting oracle*. A halting oracle is a theoretical component which can solve the halting problem: in a halting-oracle augmented Turing machine, I *can* solve problems that can’t be solved on a real computing device.

All Anderson has done is introduced a new variant of this kind of theoretical but impossible machine. That’s it. He’s introduced an interesting model of computation – programming by projective geometry. He’s created a fancy name and foolish semantics for a “halt” instruction in this projective computation model. And he’s shown that this model, like *every* other computation model that’s ever been devised,

can be augmented by impossible extensions that allow it, in theory, to perform super-Turing computation. But in the real world, it can’t, because any real incarnation of the machine is limited to Turing-level

computation, because at any point in time, it can only have finite state – and there are no

finite state representations of arbitrary irrational numbers. Eliminate the arbitrary irrationals,

and reduce to some countable set of numbers in the projective space, and you’re Turing equivalent.

And that’s it. That’s his glorious result.

What he does with that introduces whole new levels of inanity. I’ll give you one taste: Anderson’s

explanation of original sin:

The walnut cake theorem guarantees that any robot we make, that does not have

access to a time machine, will be prone to error. No matter how well it is made, no

matter how good it tries to be, it will almost certainly do some evil. This is the

nature of original sin in a perspex robot.A perspex robot will suffer harm. It will run the constant risk of senility, mad-

ness, and death. These are all consequences of free will. If we give robots free will,

we subject them to these harms. We should only do this if we are convinced that the

harm is outweighed by good. I believe that free will is the ultimate good, so I am

prepared to construct robots with free will as described in this book. I will, how-

ever, take as much care as I can to ensure that no evil is done by this research. Part

of that care is to publish this book alerting others to the risks of the scientific

research, before doing the experiments aimed at creating a robot with free will.The reader should rest assured that progress in science is so painfully slow that

there will be no substantial danger from this research for a very long time – perhaps

millennia.