God is not a mathematician; God is in the mathematics

I’ve just read a recently published book, The Universe Speaks in Numbers; How Modern Maths Reveals Nature’s Deepest Secrets by Graham Farmelo. The title says it all really, but the content takes you places you’ve probably never been, intellectually. This book, possibly more than any other book I’ve read, gives you access to the minds of the superstars in theoretical physics, both past and present, but mostly present. It’s like listening to a group of virtuoso musicians and becoming aware of how inadequate you are in the presence of such talent.

It’s written in an historical context, from Newton through Maxwell to Einstein, but rather than lingering in the so-called golden age of physics, it focuses on the last 50 years.

Farmelo covers what he calls ‘the long divorce’, a term coined by Freeman Dyson, covering much of the latter 20th Century, when pure mathematicians and theoretical physicists, not only avoided each other, but were not very accommodating of each other’s contributions.

That all changed very late in the 20th Century, and since then there has been a synergy that has benefited both disciplines. The problem is that theoretical physics has leapfrogged ahead of experimental physics, and, consequently, physicists at the leading edge of high energy physics have become more dependent on mathematical solutions to provide veracity to their ideas. ‘High energy’ means highly confined dimensions, so we are talking about the smallest constituents of matter we currently know or can imagine.

I’ve made the point in previous posts that the ultimate arbiter of truth is evidence, but Farmelo’s book has made me re-evaluate that position. Instead of testing theories with experiments that can’t be performed, theoretical physicists simply adhere to 2 founding principles: quantum mechanics and special relativity. Because those 2 specific theories have been experimentally tested in extremis, they simply ensure that any new theory, no matter at what level, or using whatever mathematical tools they have, must meet both criteria.

The book necessarily covers string theory, from its inception to its current status. But what I found most fascinating was all the discoveries that seemed to have occurred on the side so-to-speak, where mathematics appears to continually underlie the fundamentals of nature in unexpected ways.

I admit to being a sceptic of string theory, due to its 6 additional dimensions that can’t be observed and its plethora of 10⁵⁰⁰ universes. But I must also admit that the people exploring this mathematical world leave me stranded when it comes to intellectual wizardry.

Farmelo repeatedly refers to 2 talks given by Albert Einstein and Paul Dirac, respectively, where they effectively gave a call-to-arms, arguing that mathematics is the key to new theories in physics, with experimental physics providing confirmation rather than having the leading role. Whether by design or accident, this is how physics has evolved in the last 5 decades. As Farmelo expounds, not everyone has been happy with this development, yet there have been successes.

To give one example, mathematical devices called twistors (developed by Roger Penrose in the 1960s) have led to providing accurate predictions in the amplitudes of scattering gluons, which are the mediating particles for quarks in atomic nuclei. This short description belies the convoluted story, involving many theorists in the UK and the US, and the many unexpected discoveries made along the way; including a connection with a mathematical object discovered by Hermann Grassmann in 1844, called eponymously a Grassmannian. It led to another mathematical object called an 'amplituhedron'. One of the co-discoverers, Nima Arkani-Hamed (an American born Iranian, at the Princeton Institute for Advanced Study) said:

This is a concrete example of a way in which the physics we normally associate with space-time and quantum mechanics arises from something more basic.

The ‘something more basic’ is only known mathematically, as opposed to physically. I found this a most compelling tale and a history lesson in how mathematics appears to be intrinsically linked to the minutia of atomic physics.

In the same context, Arkani-Hamed says that ‘the mathematics of whole numbers in scattering-amplitude theory chimes… with the ancient Greeks' dream: to connect all nature with whole numbers.’

There is an assumption by non-physicists that the role of mathematics in understanding nature is a consequence of the fact that we need to measure everything. A common criticism is that people who emphasise the role of mathematics in their theories have ‘mistaken the map for the terrain’.

Einstein was probably the first to use mathematics alone to sculpture a theory independently of observation and experimentation, when he developed his masterpiece, the general theory of relativity. It was his mathematical prediction that gravity would bend light that clinched his theory when few people believed that relativity reflected reality.

In reference to the abovementioned metaphor about the ‘map and terrain’, there is an axiomatic inference that the map is derived from ‘surveying’ the terrain. However, it’s becoming increasingly apparent that the map is discovered before the terrain is even 'explored', which turns the metaphor on its head. It’s not a metaphor I would choose to use, but if you insist, you might have to consider the possibility that the map pre-exists the terrain.

In reference to the title, I’ll retell a joke by mathematical physicist, Robbert Dijkgraaf, from the Princeton Institute for Advanced Study:

What is the difference between a physicist and a mathematician?
A physicist studies the laws that God chose for nature to obey.
A mathematician studies the laws that God has to obey.