The BBC *Horizon* series will show a programme about “Infinity” some time in the first half of 2010. Last Thursday I spent a long hard day with the team making the programme. Here are a few impressions of the filming. If I get any visual material I will post some here.

I arrived at the Mathematics building at about 7:45; the crew were just arriving in a car and a van. Karen had managed to work some magic and arrange a permit from security to allow them to leave the vehicles right outside the building, which saved a lot of trouble.

It was a beautiful morning, with the sun lighting up the 60s concrete of the building. The cameraman and soundman, who were obviously very much in love with their job, enthused over the sight, and immediately got out their kit and started filming.

We went into the building. Some readers of this will know what the inside of my office looks like; I made a half-hearted attempt to get them to use a different room, but they were so struck with the sun shining in the window and onto the whiteboard in my office that they wouldn’t look at anything else. Later on, they took atmospheric shots of me over a pile of papers or books.

While the light lasted, I had to go over the fact that there are as many even numbers as whole numbers several times while they filmed it from different angles, with different lighting. So the pattern of the day was set.

At 10:30 we had to break to drive to the Grange City Hotel near Tower Hill to film there. This was a sequence to illustrate Hilbert’s hotel, an idea dreamed up by Hilbert to illustrate the point I had just been making:

Suppose that an infinite hotel is full, and someone walks in and asks for a room. No problem: the manager asks each guest to move into the next room, freeing up the first room for the new arrival.

What happens if infinitely many people arrive? Still no problem. Simply move the guest in room number *n* into room 2*n*. Then all the odd-numbered rooms are freed up to accommoodate the newcomers.

The conceit was to be that the infinitely many new guests would be infinitely many clones of me. So I had to be filmed checking in many times; this will be edited to make it look like infinitely many arrivals. After this (the girl doing the dummy check-in was quite an accomplished actor by the end of it), we went upstairs to a rather grand room with a stunning view over the Tower and Bridge, backlit by the sun. I had to sit on the bed and explain the idea to the camera (and have closeups of testing the firmness of the bed). Finally, we had the use of a corridor, so I had to be filmed coming in and out of doors and tracked walking down corridors.

Then back to Queen Mary. The light had completely changed; it was overcast and night was falling, since we had taken longer than budgeted in the hotel. We had to get some pictures of me reading a book, some conversation about various things related to infinity (questions which all participants will be asked), and finally explain Cantor’s discovery that there are more real numbers than natural numbers (several times from different angles again.)

We had been going for twelve hours with hardly a break by the time we were finished; the security man was about to lock the gates. A hard day’s work!

I had to learn a few things about filming. I am not at all sure how well I learned them; and you won’t be able to judge from the finished programme because hopefully all the glitches will be edited out.

- First, and rather obviously, I must not look at the camera, even if what I am looking at is in the same direction.
- Anything I say must be self-contained. If I am asked a question, I must turn it round into a statement and say it before going on to comment or amplify. Moreover, I must wait until the question is finished before beginning my reply, to leave room for the editorial scissors.
- It must be self-contained in another way. Even if we have just been talking about something, I shouldn’t refer to it; that might be later in the finished programme, or even edited out completely.
- In particular, I should always say “infinity”, not “it”, even if I have just been talking about infinity.
- I mustn’t mention the word “set” since that will be introduced quite late in the programme. (I didn’t always succeed with this.)
- I must also give up my teacher’s habit of explaining the same thing in several different ways. They may be filming this bit for the third or fourth time, but I must keep the explanation the same, so that they can cut between takes if necessary.

What didn’t we cover? There are a couple of things that I think are important about infinity, one of which we attempted:

- Infinity is simpler than a large finite number, because it is less structured. The line we took on this is that, for example, the question “Is the number of elementary particles in the observable universe odd or even?” is meaningful (modulo some physical assumptions), but much too hard to answer, whereas the question “Is infinity odd or even?” doen’t have any meaning, since this concept does not apply to infinity.
- Undecidable questions in set theory such as the Continuum Hypothesis seem to lead to different kinds of mathematics, using the same logic but different assumptions about the foundations. I think it is possible that these questions will be answered in an unexpected way. We might change our view of the foundations of mathematics, and build it up in a completely different way, which will actually resolve these questions.

We didn’t talk much about the layperson’s view of infinity either. At one point the director led me briefly onto the topic of mysticism, as direct perception of infinity; I didn’t really say much worth remembering. But there is a view of infinity which for me is captured well by Bob Dylan in *Visions of Johanna*:

Inside the museums, infinity goes up on trial.

Voices echo, “This is what salvation must be like after a while.”

The infinity of space and time pushes us into insignificance, but is also infinitely tedious if we have to experience it.

Finally, I have to say: a very different day, but what an enjoyable day working with intelligent professionals who love their job and pick up so quickly the point I am trying to make (and even suggest better ways of putting it across).

For example, I thought that it might be worth explaining a bit about decimals in the segment about Cantor’s proof, and even thought it would be possible to do this while writing down the list of numbers. But I was told not to worry: people who don’t understand decimals won’t be watching this programme!

The programme will be shown at 9pm on Wednesday 10 February 2010 on BBC2. The web page is http://www.bbc.co.uk/programmes/b00qszch and the programme will be available for replay here for a while. No idea yet how much of my day’s filming will survive.

Pingback: Infinite horizons « Peter Cameron’s Blog

Interesting episode! Interesting beard too!

As a layperson, ‘infinity’ to me is a concept: From what I understand the universe is finite in size and expanding (ad infinitum or eventually looping back into a big bounce??). But is the ’empty space’ outside the known universe infinite? The mind boggles…

If an end-point cannot ever be measured now or in the future, or theorised, then the subject matter can be considered boundaryless or infinite (assuming it doesn’t loop back on itself which is what someone on Horizon suggested about numbers to infinity). If the universe is toroidal in shape, then it is not infinite in size.

Anyways, saying that ‘there are as many even numbers as whole numbers’ seems to imply that the sequence of numbers has a limit – even if the statement is theoretically true. Similarly you could deduce that there are as many zeros (extracted out of 10, 20, 30 etc) or tenths as whole numbers.

If I remember correctly, in the hotel scene with regard to infinity (i) minus infinity equals one or two or whatever [ i-i=1 ]… when you (or your clones) checked-in, I’d say you became a part of that ‘infinity group’ of hotel guests (since the group size is limitless). So if everybody else checks-out except for you, I wouldn’t assume that infinity has checked-out, but rather that infinity minus one (you) has checked-out, therefore: i-(i-1)=1 and so i-i=0 on a technicality.

Just my own rantings and ravings.

I have had a lot of email already about the programme; sometime I will try to summarise some of the points.

On Raul’s comment:

First, I have to say that I am in no way committed to the comments of various cosmologists in the second half of the programme about infinite universes. I think it is likely that the observable universe is finite, in the strong sense that there is only a finite amount of information we can get about it.

As to infinity minus infinity: what I said was that this question has no answer. You cannot say that, if all the guests except one leaves, then this is infinity minus (infinity minus one), since that suggests that the result of an arithmetic operation depends on the history of the numbers being subtracted. Once there are infinitely many guests in the hotel, that number is “infinity” (actually mathematicians have more precise ways of saying this since there are many different infinities). What if all the people who checked in an odd number of days ago left (and there were infinitely many of them)? What if all redheads, or all females, or everyone who watched the Horizon programme, left? There is no consistent way to answer the question “What is infinity minus infinity?” any more than to answer the question “What is one divided by zero?”

In light of the above remark “Some readers of this will know what the inside of my office looks like; I made a half-hearted attempt to get them to use a different room (…)”, as an ex-student of Prof. Cameron I have my own personal story on this point to share. Given that the foregoing quote reads somewhat like an admission of the fact(?) that Prof. Cameron’s office makes for an authentic representation of the stereotypical eccentric Mathematician’s study, I am hoping the following will be taken in good humour and cause no offence at all.

Three or so years ago when I was a student at QMUL and needed clarification on some algebra question, I visited Prof. Cameron during his office hour accompanied by a fellow student. It was the first time we had seen the inside of his office and, if I’m honest, I was certainly a little taken back. You must understand it was unlike any other office in the department; from floor to ceiling stacks of books and papers were piled high, loose papers were scattered about and one had to wade knee-deep through great mathematical monographs to even reach the chair. One might go so far as saying the directors of A Beautiful Mind would have done better to emulate Prof. Cameron’s room for their reproduction of John Nash’s mathematical den at the peak of Nash’s mathematical genius. Sure, this is quite an exaggeration for literary purposes but many, I feel, will be able to identify allthesame. Undeniably, something about it strikes you as quite appropriate; you have entered a place where new and exciting mathematics is generated and where no other environment could work even half as well.

I put my question to the professor and he kindly began to explain and jot a few things down to aid the process. Then, a rather hearty bang on the door and, without any invitation at all, another student enters. He must have been in the year or two above because I did not recognise him. He was assertive and direct in the way he spoke–to my mind, plain rude. His complaint was regarding the marking of coursework. In particular, his friend had received marks for some question but he hadn’t, even though they submitted the same answer. I’m not suggesting for a moment they copied each other. After all, as To Infinity and Beyond points out, living in an infinite realm (as we do?) it is inevitable that such repetitions must happen and indeed must happen infinitely many times (recall the piece on the monkey typing out Shakespeare).

Prof. Cameron responded that he would be happy to consider the matter if the student were to provide him with both scripts but warned him that, while he might gain the additional marks desired, there was also a chance his friend might lose them. The student did not seem to like the idea and this point was debated. Anyway, this is all by the way; their dialogue continued for a little while longer and the student left, leaving us to finally return to my algebra question.

I, however, could not come to terms with the fact this guy had barged in, that he had no qualms whatsoever about interrupting our conversation and that he did not utter a single word by way of excusing himself or, later, a single word by way of thanks. Prof. Cameron seemed completely unfazed by this behavior which makes me think that perhaps I am being unfairly critical but this is how I remember it. I might add that throughout this whole episode the female student who had originally accompanied me had remained completely silent.

As soon as we left the office and still very much bothered by the behaviour of this intruding student, I turned to her and said “That was just unbelievable; outrageous!”. She turned to me and said “I know. I have never seen such mess in my life!” At that point I realised we had been troubled by very different things and laughed.

On a more relevant note to this blog topic, it was a pleasure seeing Prof. Cameron in the BBC Horizon documentary. The hotel checking-in scenes go to show that apart from being a fine mathematician, PJC makes a fine actor too! Of course, it was an added bonus to get a glimmer of the famous office!

Hey!

I’ve been trying to explain the concept of there being more than one type of infinity to my colleagues for months (they’re all computer scientists). I’ll point them to the link of this show.

It’s quite funny how they stylised it all out with Steven Berkoff doing the sinister narration. The only other thing I’ve ever seen him do was this spoken word intro to a top 20 single from 1997, which he did using the same creepy voice of doom.

Elliot