A not-paradox, a not-paradox, a most ingenuous not-paradox

In my post of 8 October 2016 I discussed George Calderon’s love of paradox and suggested that the ‘self-referential’ paradoxes in his plays might have been influenced by his following ‘developments in set theory in the 1900s, as he was an excellent mathematician’. In particular, I wondered if George was not taken by Russell’s Paradox (popularly known as ‘the barber paradox’), but I had to admit that my maths was not up to saying whether the Calderonian paradox is a literary form of Russell’s Paradox.

Five days later, I had a long email from my friend Bryan, who is a Cambridge graduate mathematician I have known for over forty years. He explained that ‘most mathematicians would regard contradiction as the essence of paradox; indeed would probably regard the two as synonymous’. But for mathematicians the ‘form of the paradox is not really of interest, since it is just a big “Stop!” sign saying you have a false assumption, go back and find it, correct it and proceed’. From the Calderonian literary examples I had given, my friend felt that I regarded paradox not as contradiction but mere incongruity, ‘a much weaker standard’. All this set me thinking more about what a paradox really is…

Then in February this year Bryan appeared unannounced at my front door, gave me the first mathematical explanation that I have been able to understand of why Russell’s Paradox is nonsense, and presented me with an inscribed copy of this book:

Click the cover to find this book on Amazon.

(A brief parenthesis on Bryan. He is a terrifyingly clear, logical thinker, but also a lover of the arts including, I think, Chekhov. He is a great fan of Gilbert and Sullivan, but hearing the song ‘A paradox, a paradox, a most ingenious paradox’ in The Pirates of Penzance, even as a child he said to himself: ‘There is no paradox here; it is just a description of how the calendar works.’ Not surprisingly, he knows Lewis Carroll inside out. He can be relied on to break the ice at parties by reciting ‘The Walrus and the Carpenter’. Pace C.P. Snow, he does not believe there are ‘two cultures’.)

I am exceedingly grateful to Bryan for his crystalline explanations in two long emails and over the kitchen table, and for giving me Al-Khalili’s book, which has further expanded my understanding of this subject and which I can heartily recommend to followers.

Jim Al-Khalili starts with a vital distinction:

A true paradox is a statement that leads to a circular and self-contradictory argument, or describes a situation that is logically impossible. But the word ‘paradox’ does tend to be used more broadly to include what I prefer to call ‘perceived paradoxes’. For such puzzles there is a way out. It may be that the paradox has hidden within it a trick or sleight of hand that deliberately misleads the listener or reader. Once the trick is uncovered, the contradiction or logical absurdity disappears. Another type of perceived paradox is one in which the statement and the conclusions, while initially sounding absurd or at the very least counterintuitive, turn out on more careful consideration not to be so, even if the result remains somewhat surprising.

For the purposes of this post, let’s call Al-Khalili’s ‘true logical paradox’ P1, his ‘perceived paradoxes’ relying on sleight of hand P2, and his resolvable ‘perceived paradoxes’ P3.

He further defines P1 as ‘a statement that is constructed [my italics] in such a way that there really is no way out of the loop’. It seems to me that ‘the barber paradox’ is P1 and so are the self-referential paradoxes that George has constructed in ‘The Lieutenant’s Heroine’, ‘The Little Stone House’, ‘The Fountain’, ‘Geminae’, ‘The Two Talismans’ and ‘The Lamp’, not to mention his ‘chopper paradox’ and ‘unexploded bomb paradox’ described in my original post. The classic example of a P1 is the ‘liar paradox’ (‘This statement is a lie’). All P1s, surely, are artefacts. Their makers are people endowed with powers of creative fantasy, e.g. the Greeks or George Calderon.

Without further ado, Al-Khalili tells us that his book is not about such paradoxes, it is about P2s and P3s, and especially P3s in physics, all of which, ‘or nearly all’, can be ‘resolved with a little bit of fundamental scientific knowledge’.

When Laurence Binyon said of George as a student that ‘paradox attracted him’ and ‘his dialectical skill seemed rather sterile’, it is natural to think Binyon was referring to P1s. Perhaps, however, the paradoxes that George perpetrated at Oxford were more ‘incongruities’, to use Bryan’s word? George’s love of incongruities is familiar from his writing in Russia, and is something he shared with Chekhov. But perhaps some of George’s ‘paradoxes’ were P3s.  For instance, in a long letter to the Daily News of 13 February 1899 about overcrowding in London — a great issue of the day — George states that ‘the more houses you put together in one place the less living-room will there be for the people that will come there’. This is a P3 (resolvable perceived paradox) because any modern town-planner would agree that to ease overcrowding you have to decentralise, e.g. to Eastcote where George then lived and which was ‘not in the least overcrowded’.

In his first two chapters, Al-Khalili explodes a number of P2s, especially Zeno’s, then gets down to the P3s in science that naturally interest him most as a theoretical physicist. These include such fascinating ‘paradoxes’ as Olbers’ Paradox (why does it get dark at night?), Maxwell’s Demon, The Pole in the Barn Paradox, and the Paradox of the Twins. The reason I put ‘paradox’ in quotes here is that as P3s they are of course not paradoxes at all: they are resolvable by Einsteinian relativity and quantum physics. This still, in my view, leaves the tricky category of what science calls ‘Anomalies’. For instance, contrary to expectation, the density of water suddenly starts decreasing as temperature falls below 4 degrees C. and goes on decreasing until at zero it is floating on the top as ice. The ‘anomalous expansion of water’ may be completely explicable by the change in molecular bonding of hydrogen below 4 degrees C., but what explains that change? It’s still incongruous, and very fortunate for the survival of life in Earth’s rivers and polar oceans.

The bulk of Jim Al-Khalili’s Paradox: The Nine Greatest Enigmas in Physics is about astrophysics, relativity, quantum mechanics and chaos theory. If, as a layman or arts person, you were to digest his clear, direct, ‘popular’ explanations of the P3s in these areas and combine it with a reading of John Polkinghorne’s Quantum Physics: a Very Short Introduction (OUP, 2002), I reckon you would have as good a grasp of these mind-bending subjects as you could hope for. In chapter 7, however, Al-Khalili moves broadly on from P3s and this is where I find myself taking issue with him.

Chapter 7 is entitled ‘The Grandfather Paradox: Going back to the past and killing your grandfather means you would never have been born’. The subject has not got off to a good start on page 14 of the book, where Al-Khalili says that ‘physicists have not yet ruled out the possibility, certainly in theory, of time travel’. A possibility that is certainly theoretical is a very attenuated possibility (feasibility) indeed. Al-Khalili describes the Grandfather Paradox as an ‘argument that goes round for ever in a self-contradicting circle’. It is therefore a P1 and not a P3. The P3s that Al-Khalili considered in his previous chapters had a basis in hard science. Time travel is at most an hypothesis, but one that is unverifiable. He seems to admit as much when he proposes the ‘multiverse and wormholes in space-time’ as ‘a possible solution to time-travel paradoxes’, but writes that such ideas ‘remain just outside conventional science: fun to consider but impossible to verify’. The last third of his book confusingly mixes such ‘ideas’ with P3s like Fermi’s Paradox that have an empirical basis. Actually, time travel, the multiverse and wormholes in space-time seem to me not only unverifiable, but unfalsifiable. They are therefore not science at all, but ‘thought experiments’.

‘Thought experiments’ are all the rage in science today. Anyone, of course, can devise one. Here, for instance, is the Paradox of Miles’s Cricket, which would slip easily into Al-Khalili’s penultimate four chapters:

I have a cricket that I keep in a matchbox. The cricket can travel at 1.1 times the speed of light whilst remaining subject to the Earth’s gravity. As soon as the matchbox is opened wide enough, it jumps out. To the human observer, will the cricket always appear to be just inside its box?

Whether the thought experiment is the Grandfather Paradox, Schrödinger’s Cat or String Theory, it is an act of pure creative fantasy, although String Theory is commonly said to be elegant mathematics as well. The thought experiment in modern science therefore has more in common with metaphysics and P1s — even with literary paradoxes like George’s — than with what Al-Khalili terms ‘resolvable paradoxes in physics’ (P3s). The thought experiment may be ingenious, it may be delightful, it may be intriguing, it may be disturbing, it may be maddening, but it has no scientific content.

The mathematical physicist John Polkinghorne has extended ‘thought experiments’ to theology. Thus, to take an example from The End of the World and the Ends of God (2000), for him the ‘carrier of continuity’ in our bodies is the ‘immensely complex “information-bearing pattern” in which that matter is organised’ and ‘it is this information-bearing pattern that is the soul. […] At death that human “pattern” is held in the divine memory, to be re-embodied in the “space-time matter” of the new creation’. Polkinghorne believes this. Yet it is only a thought experiment. It has no religious content. This is a fundamental disagreement between us.

A strange similarity between Al-Khalili’s book and Simon Baron-Cohen’s, which I looked at five posts back, is that the standard of writing and/or proofreading deteriorates alarmingly towards the end. I am reminded of the 79-year-old Lord Weidenfeld’s claim at a Frankfurt Book Fair that books were being rushed out before they were fit for publication and ‘extraordinary howlers’ were appearing in print largely because publishers under financial pressure were bringing out books quickly and cheaply, at the expense of editorial and design quality. I think he had a point.