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Does the fact that good explanations are limited in scope and power in the brilliant mathematical world of Hilbert's Infinity Hotel have any significance for Deutsch's optimistic paradigm shifting claim that 'explanations' have a fundamental place in the universe to create infinite knowledge?

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Deutsch's latest book

According to Karl Popper, one of our greatest philosophers, and David Deutsch, a major pioneer in quantum computation, good explanations with 'reach' are those that are hard to vary and can be refuted.

Professor David Deutsch's (2011) book: The Beginning of Infinity: Explanations that transform the world, has been a major influence on my recent work in the field of dysology. Deutsch's clear writing helped me to see more clearly how the philosophy of science allows us to know what makes an explanation good or bad. I used his book as a framework to criticise the current notion of so called 'crime science' that is being propagated within the Jill Dando Institute for Security and Crime Science at the University College London.

Deutsch's explanation of Karl Popper's philosophy made clear for me several things that I already knew before reading it. What Deutsch did was to enable me to understand those ideas better, and to appreciate the magnitude of their importance in helping us tell good explanations from bad. To repeat the point already made, essentially, good explanations are at the most fundamental level those that are hard to vary and easy to refute. This endows good explanations with a quality that stops their originators and adherents from otherwise always being wise after 'events' don't turn out as they first predicted, and elevates them above things such as simple truisms, which add nothing to our knowledge progression, as explanations for causality.

Deutsch is an expert universal explainer of existing knowledge and adds to what we know with more than a few new and important explanations of his own. For example, he has reversed my opinion regarding the hypothesis (that has become orthodoxy) that underpins Jared Diamond's award winning book Guns Germs and Steel. If you are seriously interested in knowing what makes a good or bad explanation for anything then I suggest you buy Deutsch's book today.

So far so good then because in many areas Deutsch meets the kind of test that could be generalised to any field of knowledge, not just mathematics, that was set by the great mathematician David Hilbert who said at the second International Congress of Mathematicians in Paris in the summer of 1900:

"An old French mathematician said: ‘A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.’ This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.” (Hilbert 1900).

In his book, Professor Deutsch (2011) seeks to make mathematics, quantum computation, quantum mechanics, astrophysics, biology and many other areas of science accessible to all and in doing so writes about a famous mathematics thought experiment conducted inside a famous imaginary place called Hilbert's Grand Infinity Hotel.

HIbert's Grand Infinity Hotel

The Grand Infinity Hotel was built inside the German mathematician David Hilbert’s brilliant’s mind and was first opened to the public in Paris in 1900 (Hilbert 1900; Olein 2001) at the second International Congress of Mathematicians as the first of 20 mathematical problems .

Hilbert later used the analogy of Infinity Hotel as a place with an infinite number of rooms and an infinite number of guests in those rooms. And even when it is full, the hotel can always accommodate a countably infinite number of new guests should they turn up.

Hilbert built his Grand Infinity Hotel, by way of a thought experiment, to demonstrate simply just how weird and counterintuitive infinity is. Infinity Hotel is still standing as an idea in the cyberspace of our culture, on the Internet on YouTube and websites, within our books and papers and the minds of those who know of it. Infinity Hotel is where mathematicians, philosophers and scientists sometimes go to conduct thought experiments

Once you’ve visited Hilbert’s Infinity Hotel it’s one of those places that will keep you coming back. But be warned, for this hotel is not so safe and infinitely accommodating as it might seem. Some visitors can never come back. Professor Deutsch has visited it many times. He even invented the hotel’s incredible rubbish removal system and regales us with a harrowing tale of what happened on one such visit when a little puppy climbed into a trash bag. No one noticed and the dog was passed along from room to the next higher numbered room and so on up the infinite corridor along with all the other trash bags in the hotel.

Deutsch’s trash removal system works like this (Deutsch 2011 p. 172-173):

‘Infinity Hotel has a unique, self-sufficient waste disposal system. Everyday, the management first rearrange the guests in a way that ensures all the rooms are occupied. Then they make the following announcement. ‘Within the next minute will all guests please bag their trash and give it to the guest in the next higher-numbered room. Should you receive a bag during that minute then pass it on within the following half minute. Should you receive a bag during that half minute, pass it on within the following quarter minute and so on’. To comply the guests have to work fast – but none of them has to work infinitely fast, or handle infinitely many bags. Each of them performs a finite number of actions, as per the hotel rules. So, two minutes after they begin, none of the guests has any trash left.’

Within two minutes all the trash, including the dog, disappeared from the universe. There was no way of getting the little dog back either. Because that dog was more gone than the bone in poor Mother Hubbard’s cupboard, which might for all we know have simply been stolen and stashed. The little dog really did vanish weirdly into what is known in physics as a singularity. The dog was actually annihilated - gone forever.

What happened to that hypothetical dog got me thinking. Could the same thing happen to a good idea with infinite reach? If it could, then surely the main idea in Deutsch's book that good explanations have infinite reach, and his explanation as to why that is so, could fail to have infinite reach because of what infinity can, theoretically, do to physical objects. I wanted to test Deutsch's claim by way of an experiment. And yet, as I am a complete duffer in quantum theory, the possibility of me doing such a thing seemed remote. Nevertheless, I was motivated and inspired by the words of the physicist and Nobel laureate Richard Feynman (1964):

“In general, we look for a new law by the following process. First, we guess it. No, don’t laugh, that’s really true. Then we compute the consequences of the guess to see if this law that we guess is right, we see what it would imply and then we compare the computation results to nature, or we say compare to experiment or experience; compare it directly with observation to see if it works. If it disagrees with experiment, it’s wrong. In that simple statement is the key to science. It doesn’t make any difference how beautiful your guess is. It doesn’t make any difference how smart you are; who made the guess, or what his name is, if it disagrees with experiment it’s wrong.”

And so it was, inspired by the great Feynman and with some trepidation, that I, on the third of June 2011, entered the foyer of Hilbert’s Infinity Hotel, to embark on my own imaginative and somewhat fanciful thought experiment, with an aim to attempt to determine whether Deutsch was right or wrong.

My Thought Experiment in Infinity Hotel

Thought Experiment Begins

I came to Infinity Hotel, not to stay as a guest, but to visit an acquaintance in the field of dysology to discuss her top secret work on information warfare. Apparently she’d developed a complex explanation for infinity. It was too complicated for anyone to explain or comprehend on the telephone and could only be understood by reading it. For reasons of national security I cannot reveal her name, and neither could she. I knew her only by her code name which was Cardinal One.

Cardinal One was an expert on memes. For the past few months she had been working in a secret Government facility in Malvern, England.

For those unfamiliar with the concept, memes are ideas that survive in human culture in some of the same basic kind of ways that genes do in living matter by selfishly passing themselves on. Religion, for example, is a meme (Dawkins 2006).

Cardinal One was working with a crack team of scientists on a top secret project to discover the necessary components of braced myths. The sole aim of this project, apparently, was to create a munitions level super meme.

In the foyer, I approached the desk clerk and told him that I was here to meet the occupant of Room number one. I asked if he would call her room for me on the hotel telephone. He did so but there was no reply. After several failed attempt to get an answer we took the elevator to the first floor. The corridor stretched into infinity.

We knocked on the door. But there was no reply. The clerk opened the door with his pass-key. There we found Cardinal One on the floor. There was no pulse she was stone cold dead. Beside her lay one of the hotel writing pads and a pencil. I picked the pad up and read the following words:

“Feeling deadly unwell. All my Meme Team are dead. Just wrote down the super meme and passed it on along with the rubbish to next room according to the hotel instructions and asked them to do likewise.”

We knocked on the door of Room Two. There was no answer so the clerk opened it and we went in. The guest, a portly man in his 70’s, lay on the floor. He too was dead.

Further down the corridor all the guests in the rooms we looked in were dead.

Quick.” I said to the clerk “Make an announcement on the Hotel’s public address system that everyone is to stop passing the meme to the room with the next highest number but instead to pass it back down the corridor. That way we can stop it spreading and stop it or whatever else might possibly be killing all your guests. If we don’t act fast we might lose it forever.”

I’m sorry Dr Sutton. It won’t work” The Clerk said. “The Meme’s been annihilated just as Professor David Deutsch explains on page 173 of his new book, The Beginning of Infinity: “Nowhere where all the trash has gone is called, in physics, a singularity. In other words, Dr Sutton, the meme has become unboundedly large, while remaining everywhere finite."

“Say what? But how can that be?” I asked. “That lost meme is supposed to be a good explanation for infinity and Professor Deutsch himself tells us in his book on infinity that good explanations have a fundamental place and are not unlimited in their scope and power. Good explanations have unlimited potential to create an infinity of knowledge. So can infinity kill infinite knowledge?”

Infinity doesn’t kill the guests who pass on the rubbish using Deutsch’s rubbish system so presumably it never killed the last guest who passed on the dog.” The clerk suggested knowingly.

Then something else must have killed them all, perhaps a deadly virus?

And that means that if the meme was passed on by word of mouth only it would have survived, had there been no virus. But since the meme was recorded on a physical object, namely the piece of paper it was written on, that was annihilated. So good explanations, even good explanations for what infinity does to physical objects by way of singularities can be annihilated if they are in a kindle or in a book. That’s troubling if they are so complex they can only be understood if written down or recorded on physical objects.”

The clerk smiled and shrugged. “Maybe if the piece of paper had been passed on at a lower rate than the trash…? But then what if a deadly virus was passed on into infinity at the same rate as the rubbish? Maybe it would not happen that way in the physical universe? I don't know. I don’t know, because I don't understand Deutsch's rubbish explanation. By which, I mean that I don’t understand the written explanation in his book for why that dog disappeared in the trash bag. Sorry Dr Sutton. You’ll have to ask the Professor himself. . I am, after all simply an ordinary man on the street working in this imaginary hotel.

Thought Experiment Ends

Hilbert's Grand Infinity Hotel is a weird fictional place and is obviously used as a mathematical analogy as well as a mathematical thought experiment to explain the weirdness of infinity, but unless the analogy can tell us more about these obvious 'what happens if and why' questions about Deutsch's dog then as Deutsch himself teaches us in this book:

"Arguments by analogy are fallacies. Almost any analogy between any two things contains some grain of truth, but one cannot tell what that is until one has an independent explanation for what is analogous to what and why." (p.371)

Although it makes sense mathematically to have an imaginary dog disappear in a physically impossible imaginary hotel with an infinity of rooms and guests along an infinitely long corridor, I am not at all sure that it makes sense that an idea with the potential for infinite reach, that explains the principle behind the missing idea going missing in infinity, would disappear in the process of proving itself in that very thought experiment simply because it was so complex it had to be recorded on a physical object such as a piece of paper. But then the whole point of Infinity Hotel is that what goes on in the mathematical world of infinity is counter-intuitive and just plain weird.

To me it seems that if we stay stuck within the weird parameters of Hilbert's Grand Infinity Hotel thought experiment then we will we find ourselves inside a cyberspace that is a mathematical proof of itself as an infinite serial culture killer. And in this nightmare world, if not our own physical world, David Deutsch’s thesis that good explanations have infinite universal reach does not hold true because Infinity Hotel becomes the embodiment of the movie Groundhog Day where everything we learn, including good explanations, can be infinitely vanished.

Given that the central idea of Deutsch’s book is that good explanations have universal reach, I am surprised that he uses a puppy dog in his analogy to demonstrate the effect of infinity on physical objects. Why not a copy of his book, which contains that very paradigm challenging thesis?

Because it was Deutsch who started this Infinity Hotel analogy let’s stick with it a while longer. If the only copy of his book anywhere in the universe (real or imagined) was in Infinity Hotel and Deutsch dropped dead would his central thesis fail to have infinite reach if infinity annihilated it into a singularity like it did the dog?

Weirdly, where Deutsch does use his own book to represent what happens to physical objects in infinity (pp. 176-177) he merely uses it to tells the tale of what would happen if infinite real numbers of his earlier book ‘The Fabric of Reality’ was to be wrapped up so that it was on the outside indistinguishable from infinite wrapped copies of his latest book and then distributed randomly to the guests in Infinity Hotel. The story ends with Deutsch telling us that nobody could predict the probability that they would receive one or the other because the rules of probability have no meaning with regard to comparing infinite sets of countable numbers. Whereas in a real finite hotel the chances would be even. Deutsch then uses this analogy to consider, from the point of view of physics, what is likely or not to occur in an uncountable infinity of universes.

At this point you might well be wondering where on Earth I am headed with this and what possible ‘real’ purpose can this attempt at intellectualising by a self-confessed quantum duffer serve for the real physical world. Well, I am heading beyond Earth with it. Because I am thinking about the application of these ideas for mankind’s future adventures in space travel and the possibilities that science offers for human immortality.

Firstly, given the huge leaps in technology since the 1950’s we should expect it to be possible one day to be able to back up our brains onto computers in the same way that today we can all take it for granted that we can simply back up everything that is on our laptop computers.

Imagine that. You could then travel through the Internet as yourself – perhaps represented by a holographic avatar – and arrive as a sentient hologram in cyberspace or in the real space of someone’s living room via a holographic projector and laptop computer. And if you could do that on Earth then why not through space to other universes?

Well for one thing the distances might be so great that your brain data would have to bounce (as current Internet data does) from some kind of satellite to satellite – or more likely space station to space station. What if tens of thousands of years from now we are doing this and develop the technology to build self-replicating space stations that create infinite copies of space stations stretching off into infinity? Does knowing what Infinity Hotel can do theoretically to the only surviving copy of Deutsch’s book warn us of what infinity might do to the only surviving copy of his brain, particularly when it is perhaps carrying one of his brand new and as yet untranscribed good explanations?

Perhaps Deutsch is right after all and good ideas do have infinite reach. Because taking his Infinity Hotel puppy dog analogy and applying to it future human transportation possibilities might well save his future relatives, or their pet dogs, from annihilation and their good explanations from failing to have infinite reach. To be honest I don’t know whether it would or not. Because I’m just a criminologist who is even more completely out of his field and depth with the physical sciences and quantum mechanics than the average man in the street.


If something is completely forgotten and not written down in the physical word then socially and physiologically it is 'as if' it never existed at all (Ford 2011). If something is forgotten yet written down and passed along with increasing speed in the mathematical world of infinity then it really does not exist at all. And so as an ordinary man on the street, I am left wondering whether the fact that good explanations that are not written down in the physical world and good written explanations in the mathematical world of infinity do not have a fundamental place and are not unlimited in their scope and power has any significance for Deutsch's optimistic paradigm shifting claim that 'explanations' have a fundamental place in the universe.

Of course, one answer to my conundrum is that anyone could destroy a good explanation before it became a meme. We might wonder, for example, how many good explanations were wiped out by the Nazi holocaust before they could be transcribed.

Finally, we could provide an answer to the questions raised in this short essay by confirming that Deutsch's core thesis is supported by new knowledge, contained within the essay itself, that knowledge is not infinite. We could conclude that this represents infinite knowledge in itself and therefore does not refute his thesis that good knowledge has infinite reach. Or does such an easy affirmation indicate that Deutsch's thesis is in fact a bad explanation for the 'reach' of explanations and good ideas because it is easy to vary and impossible to refute?

If only Deutsch had used his book as an analogy for how infinity creates a singularity, rather than the weird example of a puppy dog in a rubbish sack, then he would have needed to address this 'threat to theme' and I'd have my answer.

8th June 2011 (all rights reserved)

Note: The above essay takes forward and develops with further personal confusion my earlier confused thoughts on Professor David Deutsch's (2011) excellent and highly recommended book The Beginning of Infinity: Explanations that transform the world. The above essay includes a creatve 'thought experiment' that is based on my confused ideas in a longer essay: (Sutton, M. 2011) HILBERT'S GRAND INFINITY HOTEL IS A PLACE WHERE MATHEMATICS IS A SERIAL CULTURE KILLER BECAUSE GOOD EXPLANATIONS CEASE TO BE.


Thinker Media incUsed only with express written permission

The book that changes everything about the known history of the discovery of natural selection

Mike Sutton is the author Nullius in Verba: Darwin's greatest secret the book that famously used big data analysis to prove that Darwin and Wallace committed the world's greatest science fraud.


Dawkins, D. (2006) The God Delusion. Bantam Press

Deutsch, D. (2011) The Beginning of Infinity: Explanations that transform the world. London. Penguin Books.

Feynman, R.P (1964), Cornell University:

Ford, J. (2011) It Doesn’t Exist If It Isn’t Written Down. June 6th. Professor and:

Hilbert, D. (1900) Mathematical Problems. Lecture delivered before the International Congress of Mathematicians at Paris in 1900.

Olein, R. (2001) Hilbert’s Problems. Mission College. Final Paper - Math G S01

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Dr Mike Sutton is the author of 'Nullius in Verba: Darwin's greatest secret'.

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