The Horizon programme “To Infinity and Beyond” was repeated on BBC4 recently. I didn’t know in advance, but this showing brought another crop of emails, to most of which I have tried to reply.
Alif Mirza pointed me to chapter and verse in the holy Quran where, he says, the Big Bang is discussed. I haven’t replied in detail yet (apologies, Alif), since I have been a bit preoccupied lately and haven’t had time to read the verses. In any case, I don’t claim any special knowledge about the cosmology part of the programme, though I have tried to react to particular questions I get.
Pete Searle asked about Hilbert’s hotel: “To my mind, a hotel with an infinite number of rooms and an infinite number of guests is full.” I tried to explain how we view this: if only the even-numbered rooms are occupied, there are an infinite number of guests but still plenty of empty rooms for more guests who could arrive.
Counting is, basically, just establishing a correspondence between two sets. Nowadays one of these sets is usually a set of numbers, and the other the unknown set we want to count. There is archaeological evidence from Iraq that an innumerate shepherd would match up his flock with a set of pebbles or clay balls, and the numerate owner of the flock (or his accountant) could then count the balls and know how many sheep he had. The argument I presented in the programme shows that we can match up the set of even numbers with the set of all counting numbers; so these two infinite sets contain the same number of elements, even though one is a proper subset of the other.
Paul Brookes was puzzled about whether Infinity × Zero = One, on the grounds that
|1 × 1||=||1,|
|10 × 0.1||=||1,|
|100 × 0.01||=||1,|
|1000 × 0.001||=||1,|
and so on. Mathematicians would not agree with the deduction, since by slightly varying the calculations we can make Infinity × Zero come out to be anything we like, including infinity or zero, or not have a defined value at all. So we would say that it is not possible to give meaning to the calculation.
However, his question had a metaphysical basis, which I don’t think I quite properly came to grips with.
Paul also delighted me by pointing me to the record of his coast-to-coast walk on his blog.
Harcharan Singh was delighted with the mathematical part of the programme, and hoped that I would do more public exposition. Well, I would love to!
Finally, a correspondent I won’t name (since the conversation is ongoing) tried to persuade me that he has refuted Cantor’s diagonal argument and shown that the sets of natural numbers and real numbers have the same number of elements. Needless to say, he hasn’t convinced me yet!
Mathematics is, par excellence, the subject in which we don’t have to take anyone’s word for anything unless it is backed by a convincing argument. But the word “convincing” is important there!