Talking of things that don’t exist … I have been reading a fascinating book by Paul Nahin called “An Imaginary Tale : the story of the square root of minus one“. Somebody somewhere didn’t want me to read this book. I bought it in December for a trip back to the UK. I lost it, but it turned up in the kitchen of the hotel I was staying in. The waitress who returned it stared at all the maths and said “Is it in Chinese ?”. Really. Then, a week later, just as I was reaching chapter four, I turned a page … and it was blank. Every alternate page was blank for the next sixty four. I felt like I was inside a Borges story. It all accentuated the feeling of unearthing arcane knowledge. Back in California I returned the book to a puzzled bookstore, and eventually got a new copy back. It then sat on my shelf for a few months until I re-discovered it. I am happy to report it was worth the wait. Lots of fun.
It took hundreds of years for i to be accepted. The first key step was seeing imaginary numbers appear as an intermediate step in the solutions of cubics, where the final solution is perfectly real – so the appearance of imaginary numbers is not simply the sign of a non-physical situation. The second key step was the invention of a way to visualise complex numbers a+ib as points in a 2D plane – the Argand diagram, actually first invented by Wessel. This made complex numbers feel real, but also, especially together with the complex exponential form, made a whole bunch of calculations much quicker. Since that time, complex numbers have been an indispensable part of the weaponry of mathematicians, physicists, and engineers, and people love using them to make things somehow seem simpler. A beautiful example from the early twentieth century is the use of “imaginary time” in relativity.
In ordinary space, the interval between two points, ds2=dx2+dy2+dz2 is conserved if you transform to a different co-ordinate system. In spacetime, the quantity that is conserved is the distinctly less obvious expression ds2=-c2dt2+dx2+dy2+dz2. But you can recover the nice spacelike expression if you replace time with imaginary time, t’=ict.
But even in the twentieth century some people were uncomfortable with this sort of thing. Nahin has a beautiful quote from a physicist criticising Einstein and Minkowski’s use of imaginary numbers this way :
The square root of minus one has a legitimate application in pure mathematics, where it forms part of various ingenious devices for handling otherwise intractable situations. It has also a limited value in mathematical physics … as an essential cog in a mathematical device. In these legitimate cases, having done its work it retires gracefully from the scene… The criterion for distinguishing sense from nonsense has been lost; our minds are ready to tolerate anything if it comes from a man of repute and is accompanied by an array of symbols in Clarendon type.
This distinction between reality and mathemical convenience is a worrying one. The neat thing about i is that even though it doesn’t exist, you can manipulate it using the ordinary rules of arithmetic and get the right answer. Hamilton was unfcomfortable with this, and painfully reproduced the advantages of complex numbers in a more acceptable way by defining algebraic couples (a,b) and defining the product of two couples as (a,b)(c,d)=(ac-bd,bc-ad). Many years before, mathematicians were even uncomfortable with the idea of a negative number; in a similar manner you can of course carefully define operational rules so such things never appear.. but, hey, relax, it works !
So is mathematics invented or discovered ? Of course the same issue arises for physical theory. Do our concepts and theories describe a true reality that we have unearthed – or they simply calculating devices, that enable us to predict measured quantities ? Our nervousness about this problem depends critically on distance from sensory experience. You have to be pretty much of a pedant to deny the existence of magnetic fields. Wave those two magnets near each other and you can feel it. Electrons and protons are weird but pretty safe. You can’t feel the effect on your muscles, but you can see the needle deflect each time an electron at a time hits the cathode in your lab. Quarks hover around the border. Their presence seems clear in that plot you read in the consortium paper, from data collected from a huge machine over many years and carefully filtered. You know that was all rigorously done, but you can’t help feeling maybe if you were smart enough there could be a different set of concepts and calculations that would produce pretty much the same curve. Then finally we reach string theory, where some folk are messianic, and others are openly sceptical.
At the end of the day, most scientists are pragmatic. We only worry about the metaphysics when the facts aren’t in. One good hard prediction is all we need …