Imaginary Worlds

April 13, 2009

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 …


Everything you know is wrong – or is it ?

December 1, 2008

There is a paradox at the heart of science. The root of science is scepticism; doubt authority and do the experiment. However the result of all that scepticism and experiment over four hundred years is the most secure body of knowledge ever produced. If you go around doubting Newton’s Laws every day you’re a nutter.

Around the edges of the clearing, where we still face the darkness of the forest, its a different story. We could be hacking uselessly into dense growth, when the path to the meadow is hidden just a few yards away to the left. Its fashionable now to be sceptical about String Theory  – see Lee Smolin’s book The Trouble With Physics – but inflationary cosmology worries others, and some folks are still nervous about the logical basis of quantum mechanics. Three years ago in Warsaw I saw David Gross and and Roger Penrose  give alternating public talks. This was very entertaining. Penrose’s title was “Faith, Fashion, and Fantasy in Modern Physics” – thats Quantum Mechanics, Inflation, and String Theory. You could almost hear David Gross’ teeth grinding. More recently, Robert Simpson (the Orbiting Frog) wrote a lovely post called Five Scientific Ideas that could just be Bullsh*t . Do give it a read.

Of course some people are just career long rebels. Recently on astro-ph Geoff Burbidge set out his dish of sour grapes yet again. Mostly this was quietly ignored, but over at Cosmic Variance Sean Carroll picked it apart. But its not just Burbidge you know. Accomplished, distinguished and terribly serious young cosmologist Douglas Scott, along with his possibly even more distinguished colleague Dr Frolop, has written a series of three papers questioning many of the key assumptions of modern cosmology : see here, here, and here.

So here’s some fun. Below I paraphrase a few statements from Burbidge and Scott, not saying which is which. A big no-prize for who-ever can give the best one-sentence rebuttal or confirmation of each statement. John Peacock has to wait until everybody else has finished.

(1) Dark Energy is just Hoyle and Narlikar’s C-field, so the Steady State Universe can come back.

(2) H_0 * t_0 is consistent with 1.0 to within 3%;  there is no reason this should be the case, so it tells us we are missing something obvious.

(3) We believe in a hot Big Bang because there isn’t time to make the Helium in stars; but if the Universe is cyclic, with a bounce before the Big Crunch, some stars could be much much older, and the Helium can be easily made.

(4) We believe that black holes powering radio galaxies are 10% efficient; however, man made particle accelerators like SLAC are a thousand times less efficient than this, so we are almost certainly kidding ourselves.

(5) The biggest galaxies have the oldest stars; younger galaxies are smaller; therefore galaxies are coming apart.

(6) If likewise clusters of galaxies are breaking up, the virial theorem doesn’t apply, and there is no need for dark matter.

(7) The energy density in the CMB is almost exactly the energy density associated with converting Hydrogen to the observed density of Helium; therefore the latter is the cause of the former.


Treating String Theory with the respect it deserves

June 18, 2007

Well I spent the last few days musing about writing a post on computational biology .. how pretentious is that … but instead here is a picture of a cute cat testing string theory. Thought I should get there before the Cosmic Variance Team.

LolCats is in the Word Press Top Ten EVERY DAY. I keep trying not to like it but I can’t help it.

This one comes from a blogger called TechnoBuddhist. What does the Bad Astronomer make of that ?