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
this just in: Anderson discovers Universal Woo Theology Machine, hits Alan Turing's corpse over the head with it, claims victory. fans of Turing later beat him up outside pub.
I should make up a machine that can do all that whilst making the perfect cup of coffee, it will beat his machine into the ground. Never mind how impossible my machine might happen to be.
The user should rest assured no harm will come from this research for a long time. Perhaps for millenia, probably for eternity.
Perspex is a registered trademark for polymethyl methacrylate. Does he even own a dictionary?
In Anderson's defense I do have to add that the animations of the (rational) perspex machines are kind of entertaining.
Again, I'm left to wonder if Anderson and his ilk understand that their 'theories' are worthless, merely wishing to pass as much greater minds than they really are, or is it possible for an obviously educated person to hold such absurd delusions of grandeur.
As I said in my post: it's an interesting model of computation. I didn't mean that sarcastically. The idea of expression computational instructions as projective transformations, and modeling a computation by things moving around in projecting space is really interesting. The animations of perspex computations are, actually, more than just entertaining - they're genuinely interesting, as a different way of visualizing a computational process.
The problem is that Anderson seems to genuinely *not* understand what he's done - in the sense that he's never bothered to think through his claims and what they really mean. He's decided that he's a revolutionary, and just won't consider anything that doesn't jibe with that conclusion.
From my armchair, this guy sounds like he has bipolar disorder/manic-depressive disorder. Unfortunately I had a friend who suffered from the same. He also had grandiloquent theories like this. He wrote long tracts as Word documents explicating his theory of "artificial intelligence" and detailing how his theory explained everything from God to the universe to human behavior, etc. This book of paragon thing all reads weirdly the same. I think my friend's theory also involved some sort of AI machine (not turing machines). Whenever he was questioned about his theories, he would chuckle because his theory at once made everything clear to him, and yet no one else could understand it.
The tragedy is that he was also a really smart person.
alright you guys, i get the point.
At least, unlike many creaters of woo, he's actually DONE something interesting and potentially useful. If he'd stop getting all mystical, look at some of the other research in this area, and act like a computer scientist, maybe he'd be onto something.
Reminds me of a mathematician I came across recently, who seems to have real issues with the idea of infinity, irrationals, and in general, things that are "too complicated". I confess I don't know enough to really rebut him in this paper http://web.maths.unsw.edu.au/~norman/views2.htm but I find his section "Why Real Numbers are a Joke" quite amusing.
um, CRM114, speaking of dictionaries, try looking up the meaning of "trademark" and exactly how they are applied. "Perspex" as a trademark for a type of plastic does not preclude someone else from using it for a design of a computer.
Susan B.; that page is amazing.
MarkCC: I'd love to see a Bad Math post about the page Susan B. posted. Although it might be difficult since he's not really doing any math, it seems... just showing us some math straight out of a textbook and saying "this is religion!"
Yeah, that page to which Susan B provided a link would make a great springboard for a blogpost.
Ah, the perspex machine is so simple, you can see right through it.
Hey, that's what Orac must have been.
I read the section "Does mathematics require axioms?" in the Wildberger text linked by Susan B. It's awful. Prof. Wildberger doesn't appear to understand that mathematics is a language that can be used to describe reality, as we perceive it, not some sort of property of reality itself. Of course, the idea is to try to pick axioms that make math useful, but from the perspective of the language itself, the axioms are arbitrary. The whole point of this is to have a strong logical foundation for mathematics, without needing to rely upon unreliable human intuition.
As for the "Why real numbers are a joke", I'm myself intrigued by the idea of having a model restricted to computable numbers for mathematics. Would such a model have serious obstacles in dealing with any practical bits of mathematics?
Does someone know more about Wildberger's claim that "computable real numbers are not countable , and are complete?"
Anything that starts with "The walnut cake theorem guarantees..." is guaranteed to be a real chestnut.
Still regarding Wildberger, I do agree on a lot of what he wrote about teaching mathematics. It's really a shame that kids aren't even taught what numbers really are.
Flaky: I can't get past the awfulness to see anything good.
"Think clearly about the subject for a few days, and you will see that the computable real numbers are not countable , and are complete."
It's always a good sign when someone's proof consists of "think clearly for a few days" and it will be obvious! I don't even care about what he's trying to say. He just comes off as such a quack.
Another amusement was Anderson's Walnut Cake theorem where he attempts to model "paradigm shifts" by the non-monotonic character of resolution-limited approximations. In real life this is but one of many reasons why we avoid using measurements or maps at the limits of resolution as basis for decisions, but use safeguard limits. Similar methods can be used in simulations.
Prof. Wildberger doesn't appear to understand that mathematics is a language that can be used to describe reality, as we perceive it, not some sort of property of reality itself.
Speaking of algorithms, couldn't Wildberger mainly be a constructivist? And some of them might rely on the above unsupported assumption. But I'm not sure I understand constructivists need for proving constructible objects, as idealizations (such as infinities and axiom of choice) looks like fine and working tools to me.
From my armchair, this guy sounds like he has bipolar disorder/manic-depressive disorder. Unfortunately I had a friend who suffered from the same. He also had grandiloquent theories like this.
I have a neighbor that the neighborhood divide in two camps about, either that or he is a drug user. (Or both.) It is hard to tell, and he won't. :-)
Another group of grandiloquents are the pathological liars.
There was this, not so bright fellow, at a former work place that made the damnedest claim to anyone who listened. (Which is their regress problem of course - once it is obvious they are regular liars, people stop listen, and they have to bloviate ever more to perceive that someone listen.)
The best rumor was that he might have claimed that he had a nuclear reactor in his basement. It wouldn't surprise me if he did, even considering the usual mistakes in perception and rumor mongering.
Computational models that deal with real numbers are nothing new. See, for example, Blum, Cucker, Shub, and Smale, _Complexity and Real Computation_, Springer, 1998.
TorbjÃ¶rn is of course right Prof Wildberger is a constructivist and apart from the somewhat over the top style of his polemic and some of the rhetoric that he uses he says nothing that has not been said before by other constructivists. There is nothing mad about constuctivism and some of the thoughts of constructivists on mathematics and logic are well worth persuing but I not sure if Prof Wildberger is the right person to turn to for explanations. That which he write is certainly not stuff for a "Bad Maths Post".
The computable real numbers are countable; this is a fairly easy result. By definition, a number is computable iff there is a program which will compute it. The set of programs of any given length is finite; the set of all programs is a countable union of finite sets and is hence finite.
They are (algebraically) complete, because given any algebraic equation with computable coefficients, one could use Newton methods to obtain a solution. These methods can be programmed; hence the result is computable.
Now I, personally, disagree with the constructivists, but they do have a compelling and (apparently) internally self-consistent philosophy of mathematics.
Oops, meant "... and is hence countable."
@TorbjÃ¶rn: Yes, Wilderberger probably is a constructionist, but even on a second reading that offending section about axioms doesn't come across as any less offensive. That section, of course, isn't about mathematics, but philosophy of mathematics. In my (rather arrogant) opinion mathematics is a language, a game, and any choices made therein are arbitrary in terms of the language itself. To claim otherwise sounds awfully lot like that religiosity that Wilderberger objects to.
@Craig: Computable numbers map to a subset of natural numbers, but to actually enumerate them is not possible, as the computing programs are not countable, due to the undecidability of their terminability. For the same reason, determining equivalence among the numbers is not possible. Are they really countable?
It doesn't matter that the computing programs terminability is undecidable, nor that one may have duplicates. Every positive integer is computable, and every computable number is the output of some program. The set of programs is countable, and hence can be placed into a 1-to-1 correspondence with the positive integers. By the above, the computable numbers can be placed into a 1-to-1 correspondence with some (non-computable) subset of the positive integers, and the positive integers is a subset of the computable numbers, therefore both sets are infinite and have the same cardinality.
So the computable numbers are countable.
(Now, if you want to argue that I've used the axiom of choice in my proof, you're welcome to do so :-)
@Craig: Yes, the computable numbers map 1-to-1 with integers, which is usually given to mean that they are countable, but isn't that a rather strange use of the word 'countable', since you actually cannot even begin to count them?
I have no idea, if this is what Wilderberger is hinting at though.
The claim that computable numbers were complete is rather strange as well. If they do form a complete metric space, since they include all rationals, the space would contain all Cauchy sequences of rational numbers, therefore being the set of reals, which in turn would then be map 1-to-1 with integers.
Flaky: You *can* count the computable numbers, though.
If you dovetail (for those who don't know the term, it's just doing one step of the first, then another step of the first and a step of the second, then another step of the first and second and then a step of the third, etc ad infinitum) computations of *every* program (in enumerated order, easily done if you devise the right kind of encoding), and each time a program in the dovetailed computation halts, you take the output, compare it to all previous outputs, and if it's new, you count it.
You're counting them in a well-defined order that way. It doesn't matter, of course, that some of the computations won't halt - you're ordering by when in the dovetailing they do halt (which is computable for those that do!) which simply means that those that do NOT halt aren't in the ordering at all - and that's fine.
(Idly, this puts the computable numbers into a 1-1 correspondence with a computable subset of the positive integers, and is therefore an improvement over what Craig said. NYAH!)
I see. Personally, I don't begrudge mathematicians their philosophy. As a physicist I can start with seeing math as a tool, a set of models, and have to extract observations and theories before I can start philosophize from my context. So there are different perspectives.
(Not that some, like Tegmark, not ends up with some math inspired metaphysics. Btw, surely double negatives must be allowed on a math blog? :-P)
With that background I can also make whichever choice suits me. My main problem with constructivist math then becomes that idealizations seems easier and more general to use and understand. Even if they may at times yield weak results or even counterintuitive ones beyond such limits that they don't correspond to physical objects. (Eg. the Banach-Tarski paradox.)
(My minor problem is that mainstream philosophies, or even Chaitin's picture of math as inherently random, is easier to get too. :-)
Does using NULLITY as HALT give it any useful properties? I'm wondering why he did it, and my best guess so far is that it makes the equations nicer.
It's kind of ironic, though. When he came out with nullity, he made a bogus claim about planes falling out of the sky and pacemakers crashing on divide by zero -- and how nullity was a way to fix that.
I'm afraid that won't do it. The programs computing numbers generate rational approximations of the (computable) reals that they represent, given a rational parameter that is the desired upper bound of the error of the approximation. You could use dovetailing for some value of the parameter, but the programs representing computable numbers are supposed to terminate for all values.
"As for the "Why real numbers are a joke", I'm myself intrigued by the idea of having a model restricted to computable numbers for mathematics. Would such a model have serious obstacles in dealing with any practical bits of mathematics?"
I'm not a mathematician, but isn't that kind of the position of Leopold Kronecker circa the 1800s and doesn't it pretty much rule out calculus for one thing?
Flaky: It seems you define computable numbers in a way I find counterintuitive. I view a number as computable iff there is a program that terminates with it as output. By that definition, my argument holds - and shows that the set of "computable reals" is roughly equivalent to the rationals.
(I say roughly, because it's possible to propose an encoding that can represent finite equations - and thus is capable of representing irrationals such as sqrt(2). But such an encoding will still be placable into 1-1 correspondance with the integers, and is thus a smaller degree of infinity than the reals.)
Doug: I've heard of something like that as well. As I'm not much of a mathematician either, I cannot comment on that. Except to note that it is peculiar that we shouldn't be able to model mathematics around what's computable, when at any moment of time, all the mathematical works produced, all the formulas, equations, statements, are finite in size. This is true also of all the potential, but unrealized mathematical products, as there's only so many letters and symbols that anyone could write in a given amount of time, leading to a huge, but finite number of permutations of mathematical symbols. If these are all products of manipulation of symbols under strict rules (I know that this is a bit of an idealization, but I believe a reasonable one), isn't there an implicit countable model?
Michael Ralston: You should check out Wikipedia and Mathworld for the definition of Computable Number.
@Michael: Flaky's definition of computable reals is the standard one. For instance, see Wikipedia's article on computable reals.
According to this standard definition, pi is a computable number, because it's possible to produce a program which computes an arbitrarily close approximation to pi.
@Flaky: Your intuition is good. To the extent that mathematics can be said to be based on ZFC set theory (or any other first-order theory), if it has a model at all (that is, if it is consistent), it has a countable model. (This is the (downward) LÃ¶wenheim-Skolem theorem.) The proof depends on the axiom of choice.
"It is claimed..."
It always tweaks me when cranks abuse the impersonal passive this way. Whenever somebody uses that construction, it often turns out that they are the only one doing the claiming.
I'm not a mathematician, but isn't that kind of the position of Leopold Kronecker circa the 1800s and doesn't it pretty much rule out calculus for one thing?
Constructive analysis is perfectly possible, a small number of theorems in classical analysis are not constructive otherwise there are no problems.
Except to note that it is peculiar that we shouldn't be able to model mathematics around what's computable, when at any moment of time, all the mathematical works produced, all the formulas, equations, statements, are finite in size.
Yes, that is exactly one of those idiosyncrasies of constructivist philosophy I was thinking of. It isn't exactly like non-constructivists are guessing.
Constructivists must see physics as the Dark side - since we can, at least temporarily, use ad hocs to model data. :-o
TorbjÃ¶rn Larsson: I'm by no means against using what ever works, but I think that there is value in having as simple foundations as reasonably possible, even if that leads to a bit of extra work in producing useful theorems.
Oh, I agree. We trade of a simple, hopefully correct, axiom set (science: theory) with sometimes complex theorems (science: models).
My specific problem is when we have alternatives for proving theorems (science: model complexity).
Curious - Dr. Wildberger was actually my linear algebra instructor in my first year at Toronto. It was a rigorous course, and he did an admirable job, as I recall.
He seems to be one of those rare people who straddles the line between sane and cranky. For example, he wrote a book on trigonometry that sets things up slightly differently, avoiding some irrationals, which seems to be substantively correct as far as I can tell. But then he makes all these claims about his method being "revolutionary" and changing the face of all science etc etc.
His views on foundations sound like the intuitionists, a more radical subsect of the constructivists. ("Brian: Excuse me. Are you the Judean People's Front? Reg: Fuck off! We're the People's Front of Judea!") Misguided, I think, but certainly not crazy, no more than Hermann Weyl on a cranky day.
I hate to disagree with you, but I side with the ..
nontraditional exploratory theorist
in my professional opinion, he has indeed achieved nullity with this project.