News from the Cheltenham Science Festival - Day 3 - Infinity

'Infinity' was presented by Jim Al Khalili (Professor of Physics and Public Engagement, University of Surrey and BBC broadcaster) and Richard Pettigrew (Reader of Philosophy, University of Bristol).

I didn't really know what to expect with this talk.  Given that physics and mathematics are not subjects I discuss daily, exploring infinity was always going to be a challenge for me.  Al Khalili and Pettigrew, however, managed to present this very inaccessible topic in a way that enabled me to come away feeling I had certainly learned something.

Jim Al Khalili started by giving an example of the paradoxical nature of infinity; try to divide 1 by 0 and your calculator might state 'not a number'.  But this is not strictly true, because infinity is indeed a quantity.

He then gave a historical summary of how philosophers, physicists and mathematicians have try to understand the paradox of infinity, starting with Aristotle,  who postulated that 'nature abhors a vacuum' suggesting that there was no such thing as empty space, but space was filled.  Al Khalili then described two paradoxes that have tried to explain infinity, first,  Zeno's paradox of Achilles and the Tortoise.  Achilles and the Tortoise were running a race (much like the hare and the tortoise), only Achilles gave the tortoise a headstart.  Zeno suggested that even though Achilles was faster than the tortoise he could never win the race because there were an infinite number of times when as Achilles moved to the point where the tortoise was, the tortoise would then be in a position further ahead.  We now know that you can have an infinite number of stages within a series with the final answer still being finite (so Achilles can overtake the tortoise!).

Another paradox Al Khalili discussed was Olber's paradox (1758-1840).  This was based on the question 'if the universe is infinite, why does it get dark at night?'  If the universe were infinite, then every line of sight should ultimately result in reaching a star, so therefore, the sky should be complete light from all the stars.  Various theories have been put forward to address this paradox, including postulations the universe is not infinite (Keppler) or that the light from distant stars fades (Halley).  These arguments have been refuted, but the problem of the dark night was finally resolved by Edgar Allan Poe, who stated that the sky was dark despite the fact that the universe might be infinite, because the universe had a beginning, so what we see is limited by the beginning of time.  So if you go outside at night and see it is dark, that is proof that the universe had a beginning. That is a pretty incredible concept. As for the end of the universe, the truth is, we still do not know if the universe is infinite.

Richard Pettigrew then gave a mathematician's perspective on the subject of infinity.  He started by explaining that in mathematics infinity also presents as a paradox.  He outlined different individuals who through time have tried to calculate infinity or try to grasp how infinity operates as a number.  Through his explanations of the work of mathematicians Georg Cantor and David Hilbert, Pettigrew demonstrated the bane of that playground squabble where one child tries to win a numbers argument by saying "infinity +1!" The paradox of Hilbert's hotel identified that infinity +1 = infinity, infinity + infinity also = infinity, and infinity x infinity also = infinity.  He showed, however, that infinity was not a number that you could do anything to and still get infinity, because if you add up all the measuring numbers  and try to add them to infinity there is a point where infinity cannot accommodate all the numbers.

Al Khalili and Pettigrew's talk sparked a lively question and answer session with questions ranging from the purely mathematical to profound questions about the origins and nature of the universe, and of time.  I would say that this talk has stretched the levels of my understanding the most.  I'm glad I went along though, as it's great to challenge your brain to think in a completely different way sometimes.  Another, unexpected benefit, is I will have a riposte to my son's attempts to get one up on me using "infinity + 1!".