The Infinite Monkey Cage – How to Build a Universe. Robin Ince
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Название: The Infinite Monkey Cage – How to Build a Universe
Автор: Robin Ince
Издательство: HarperCollins
Жанр: Юмор: прочее
isbn: 9780008254964
isbn:
From our first episode, we started to get emails and letters with listeners’ conjecture on what exactly was meant by an infinite monkey cage. This was on top of the complaint we received from an angry animal rights activist who wrote that ‘yet again, the BBC is celebrating animal cruelty and vivisection. Who spends their time imagining monkeys crammed in a cage?’ We wrote back to explain that an infinite monkey cage is roomy. We heard nothing more.
By week three, we were receiving letters complaining that the show was another of those arrogant shows that thinks you can prove anything with evidence and that the very title was based on a lie.
‘The idea that an infinite number of monkeys would write the works of Shakespeare is rubbish, as a recent experiment has proved.’
This was exciting news. A maverick scientist had gathered together an infinite number of monkeys? Surely we would have heard. It’s hard to be surreptitious with an infinite number of monkeys. It’s the noise and the smell.
Sadly, the experiment was somewhat smaller. One typewriter and six monkeys at Paignton Zoo. After a month, they had broken the typewriter and done a poo in it, and not so much as a disembowelling scene from Titus Andronicus was found. We attempted to explain that six monkeys really wasn’t enough, it was too far from an infinite number of monkeys – but to no avail. The correspondent was certain that at the very least it would be an accumulating system:
10 monkeys = a leaflet on banana safety.
100 monkeys = an article for GQ on aftershave. 1000 monkeys = a simian version of 50 Shades of Grey (Langur).
Just as with the complainant who was annoyed that we had no ghosts on the panel for our show, eventually the fury faded into an email memory, to be archived and trashed when in need of a new memory.
INFINITY
‘Infinity always gives me the urge to scratch my head. Perhaps it’s a rash. I worry about infinity,
it is much bigger than my brain.’
Professor Carlos Frenk
Series 10, Episode 5 (5 August 2014)
First, let us convince ourselves that the concept of infinity makes sense. Consider adding an infinite number of numbers together:
This is called a geometric series. At first sight, you might guess that adding an infinite series of numbers together one after the other, forever, should lead to an infinitely large number, but this isn’t necessarily the case. For this particular series, the result of adding them all up is 1. You can see this by using a little simple algebra.
Let’s call the sum of this series S:
Now consider a different series; the original one, but with each term divided by 2:
Now subtract S/2 from S. Every term in the series disappears except the first term in S:
Adding an infinite series of numbers together is something that we can do, at least in this example, and get a finite answer.
The type of infinity we’re thinking about here is an infinite set of fractions; ½, ¼, ⅛, and so on. How many of these fractions are there? An infinite number, and we assumed this in our proof because for every fraction in the infinite series S, other than ½, there was a corresponding fraction in the infinite series S/2 to cancel it out. But this raises an interesting question. We subtracted an infinite number of fractions from an infinite number of fractions and we had one term left: ½. Does this mean that there was one more fraction in S than in S/2? The answer is no; the two infinites are precisely the same. The first mathematician to think about what we mean when we speak of an infinite set of numbers was the German mathematician Georg Cantor, at the end of the nineteenth century.
Consider, for example, the set of all integers; 1,2,3,4 … We could imagine making a table of the integers by writing them all down in a vertical column from 1 to infinity. We could then write each of the terms in our set S alongside in a neighbouring vertical column. Each fraction – ½, ¼, ⅛ – would be paired up with an integer, all the way down the list. We could do the same for our set S/2; the column would begin with ¼ rather than ½, but it would carry on all the way to infinity. This one-to-one correspondence between all three sets of numbers is the reason why Cantor claimed that the three sets have the same ‘infinity’ of numbers contained within them. Mathematicians would say that the sets have the same cardinality.
There is certainly something odd about these infinite sets, because they don’t behave as we might expect. Notice, for example, that the set S/2 is a subset of S, because S contains every entry in S/2, but it also contains ½. And yet S and S/2 are the same size! This counter-intuitive nature of infinite sets led to one of the great Infinite Monkey Cage arguments that took place between Brian and comedy producer John Lloyd in the form of Hilbert’s Grand Hotel paradox.
John Lloyd: Infinity plus one is just intellectual brain bending and I cannot see the use of it. Infinity is a word. That belongs to the wordy people like me and Robin. The point is you cannot place a numerical value to infinity and therefore you cannot add a plus one to it or a minus one.
Brian: There are either an infinite number of numbers or there aren’t. I don’t see what the problem is?
John: There aren’t an infinite number of numbers, because you can always have more than infinity and so infinity is a meaningless concept.
Series 9, Episode 4 (9 December 2013)
Hilbert’s Grand Hotel has infinitely many rooms, and they are all occupied. What happens if a further guest turns up unannounced? The guest in Room 1 can be moved into Room 2, the guest in Room 2 can be moved into Room 3, and so on, freeing up Room 1 for the new guest. There is always room in Hilbert’s Grand Hotel, even when it is full. We haven’t increased the size of the hotel, and yet we’ve accommodated another guest. Using the language above, we can say that the cardinality of the set of rooms in Hilbert’s Grand Hotel is the same as the cardinality of the set of guests. Notice that this implies that we can slot an infinite number of extra guests into Hilbert’s Grand Hotel, even when it is full. To see this, note that we could have moved the guest in Room 1 to Room 2, the guest in Room 2 to Room 4, the guest in Room 3 to Room 6, and so on. This frees up ALL the odd-numbered rooms, and since there are an infinite number of odd numbers, the Grand Hotel can now accommodate СКАЧАТЬ