I’ve just read John D. Barrow’s Pi in the Sky, published in 1992, and hard to get, as it turns out. I got a copy through Amazon UK, who had one in stock, and it’s old and battered but completely intact and legible, which is the main thing.
Those of you who regularly read my blog (not many of you, I suspect) will know that I’ve read lots of Barrow’s books, possibly The Book of Universes is the best, which I reviewed in May 2011.
Pi in the Sky is a very good title because it alludes to the Platonist philosophy of mathematics that seems to dominate both mathematics and physics as it’s practiced, in contrast to how many of its practitioners would present it. Barrow points out, both in his introduction and his concluding remarks (after 250+ pages), that Platonism has religious and mystical connotations that are completely at odds with both mathematics and science as disciplines.
He points out that there is a divide between mathematicians and physicists and economists and sociologists in the way they approach and view mathematics. For the economist and sociologist, mathematics is a tool that humans invented and developed, which can be applied to a range of practical applications like weather forecasting, economic modelling and analysis of human behaviours.
On the other hand, pure mathematicians and physicists see an ever-increasing complex landscape that has not only taken on an existence of its own but is becoming the only means available to understanding the most secret and fundamental features of the universe, especially at the extremities of its scale and birth.
This is an ambitious book, with barely an equation in sight, yet it covers the entire history of mathematics from how various cultures have represented counting (both in the present and the ancient past) to esoteric discussions on Godel’s theorem, Cantor’s transfinite sets and philosophical schools on ‘Formalism’, ‘Constructivism’, ‘Intuitionism’ and ‘Inventism’. Naturally, it covers the entire history of Platonism from Pythagoras to Roger Penrose. It’s impossible for me to go into any detail on any of these facets, but it needs to be pointed out that Barrow discusses all these issues in uncompromising detail and seems to pursue all philosophical rabbits down their various warrens until he’s exhausted them.
He makes a number of interesting points, but for the sake of brevity I will highlight only a couple of them that I found compelling:
‘Once an abstract notion of number is present in the mind, and the essence of mathematics is seen to be not the numbers themselves but the collection of relationships that exists between them, then one has entered a new world.’
This is a point I’ve made myself, though I have to say that Barrow has a grasp of this subject that leaves me well behind in his wake, so I’m not claiming any superior, or even comparable, knowledge to him. It’s the relationships between numbers that allows algebra to flourish and open up doors we would never have otherwise discovered. It is the interplay between ingenious human invention and the discovery of these relationships that creates the eternal philosophical debate (since Plato and Aristotle, according to Barrow): is mathematics invented or discovered?
One cannot discuss this aspect of mathematics without looking at the role it has played in our comprehension of the natural world: a subject we call physics. Nature’s laws seem to obey mathematical rules, and many would argue that this is simply because we need to quantify nature in order to study it, and once we quantify something mathematics is automatically applied. This quantification includes, not just matter, but less obvious quantifiable entities, like heat, gravity, electromagnetism and entropy. However, as Barrow points out, the deeper we look at nature the more dependent we become on mathematics to comprehend it, to the point that there is no other means at our disposal. Mathematics lies at the heart of our most important physical theories, especially the ones that defy our common sense view of the world, like quantum mechanics and relativity theory.
The point is that these so-called ‘laws’ are all about ‘relationships’ between physical entities that find analogous mathematical ‘relationships’ that have been discovered ‘abstractly’, independently of the physics. There may not be a Platonic realm with mathematical objects like triangles and the like but the very peculiar relationships which constitute the art we call mathematics have sometimes found concordant relationships in what we call the ‘laws of nature’. It is hard for the physicist not to believe that these ‘mathematical’ relationships exist independently of our minds and possibly the universe itself, especially since this mathematical ‘Platonic’ universe seems to contain relationships that our universe (the one we inhabit) does not.
In 2010, or thereabouts, I read Marcus du Sautoy’s excellent book, Finding Moonshine, which is really all about dimensions. The most fantastical part of this book was the so-called ‘Atlas’, which was a project largely run by John Conway with a great deal of help from others (in the 1970s), which compiled all 26 ‘sporadic groups’ that I won’t attempt to explain or define. Part of the compilation included a mathematical object called the ‘Monster’ which existed in 196,883 dimensions. Then a friend and colleague of Conway’s, John Mackay, discovered a most unusual and intriguing connection between ‘The Monster’ and another mathematical entity called a ‘modular function’ in number theory, even though it first appeared as an apparent ‘coincidence’ - as no reason could be conceived - but a sequence in the modular function could be matched to the sequence of ‘dimensions’ in which the Monster could exist.
I’m only telling snippets of this story – read du Sautoy’s book for the full account – but it exemplifies how completely unforeseen and unlikely connections can be found in disparate fields of mathematics. The more we explore the world of mathematics, the more it surprises us with relationships we didn’t foresee; it’s hard to ignore the likelihood that these relationships exist independently of our discovering them.
Because the only mathematics we know is a product of the human mind, it can be, and often is, argued that without human intelligence it wouldn’t exist. But no one presents that argument concerning other areas of human knowledge like the laws of physics, where experimentation can validate or refute them. However, no one denies that mathematics contains ‘truths’ that are even more unassailable than the physics we observe. And herein lies the rub: these ‘truths’ would still be true even without our knowledge of them.
This brings me to the second insight Barrow made that caught my attention:
He points out that our mathematical theories describing the first three minutes of the Universe predict specific ratios of the earliest ‘heavier’ elements: deuterium, 2 isotopes of helium and lithium, which are 1/1000, 1/1000, 22 and 1/100,000,000 respectively; with the remaining (roughly 78% ) being hydrogen. And this has been confirmed by astronomical observations. He then makes the following salient point:
‘It confirms that the mathematical notions that we employ here and now apply to the state of the Universe during the first three minutes of its expansion history at which time there existed no mathematicians… This offers strong support for the belief that the mathematical properties that are necessary to arrive at a detailed understanding of events during those first few minutes of the early Universe exist independently of the presence of minds to appreciate them.’
As Barrow points out more than once, not all conscious entities have a knowledge of mathematics – in fact, it’s a specialist esoteric discipline that only the most highly developed societies can develop, let alone disseminate. Nevertheless, mathematics has provided a connection between the human mind and the machinations of the Universe that even the Pythagoreans could not have envisaged. I’ve said this before and Marcus du Sautoy has said something similar: it’s like a code that only a suitably developed intelligent species can decipher; a code that hides the secret to the Universe’s origins and its evolvement. No religion I know of can make a similar claim.