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A plague of infinities

Stephen Hawking is the most distinguished living physicist, who despite the catastrophe of motor neurone disease has been twice married, is a bestselling author and a media super-star.

11 September 2010

12:00 AM

11 September 2010

12:00 AM

The Grand Design Stephen Hawking & Leonard Mlodinow

Bantam, pp.199, 18.99

Stephen Hawking is the most distinguished living physicist, who despite the catastrophe of motor neurone disease has been twice married, is a bestselling author and a media super-star. He is blessed with an extraordinary intellectual energy and fearless resilience. One might also add chutzpah.

In The Grand Design he aims to give a concise and readable answer to the ‘Ultimate Question of Life, the Universe, and Everything.’ In fact he offers three such questions: ‘Why is there something rather than nothing? Why do we exist? Why this particular set of laws and not some other?’ It will come as no surprise to learn that he fails to provide a satisfactory solution to any of them. Had he succeeded, you would have already heard it on the news.

It should also come as no surprise to those who have followed the progress, or rather non-progress, of theoretical physics over the past few decades, that Hawking’s route to solving these mysteries should be chiefly concerned with GUTs and TOEs. These acronyms stand for ‘grand unified theories’ and ‘theories of everything’. They represent the Holy Grail for modern physicists, many of whom believe that a single, simple and elegant equation, such as Einstein’s E = mc², might one day be found to describe the physical behaviour of all particles and forces.


If this could be done, it would, of course, be very neat and wonderful, but the simple fact is that it can’t. Mathematics, in which physicists vest unwarranted confidence, is far too blunt a tool. It worked for Newton, Maxwell and Einstein because they found equations that accurately described the classical world. But with the discovery, in the early 20th century, of quantum mechanics, everything changed. Subatomic particles do not behave like large visible objects. One cannot measure a particle’s position and its velocity at the same time; the arrow of time cannot be observed in particle interactions; and (so Hawking believes) for a particle to travel between two points it has to take every ‘possible path’ between them simultaneously. The number of possible paths from A to B is, he claims, infinite. If this is correct, then it becomes a feature of the quantum world that all history, and all possible histories, also take place simultaneously.

How can mathematics, however sophisticated, be up to the task of dealing with this? Numbers, after all, were created by humans to describe things in the observed world. They are adjectival. How can one ascribe a number to a particle, or to its position, or its velocity if, while travelling from A to B, it is said to be in an infinite number of places simultaneously?

If numbers are not to be trusted, then it follows that mathematics is even worse. Have you ever tried asking a mathematician why a minus times a minus equals a plus? Try it. He cannot answer, except by specific reference to the man-made artificialities of algebra. Outside of these, the concept has no application and no meaning. One should be equally suspicious of mathematical infinities. In very simple terms, if you divide 10 by three in base ten you get 3.3 recurring (infinitely). Equally you could say that the answer is 31/3 with no infinite recurrence. In his famous Brief History of Time, as well as in the present book, Hawking finds himself constantly frustrated in his attempts to describe the universe because of the ‘plague of infinities’ that come into his maths at every turn.

It is because of this, one suspects, that he feels licensed to use the word ‘infinite’ with such reckless abandon, leading him to a proliferation of concepts like ‘infinite paths’, ‘infinite thinness’ ‘infinite smallness’, ‘infinite energy’, and so forth. There are so many infinities in quantum mechanics that now they’ve had to invent a mathematical process called ‘renormalisation’, which allows certain infinities to be cancelled out in such a way as to leave just a little something to work with. Strewth! If Hawking were really serious about answering the ‘ultimate questions of life’ he should have realised long ago that mathematics and geometry are not the right tools for the task. But if you cannot back your theories up with maths, what else is there? Philosophy he says is ‘dead … it has not kept up with modern developments in science.’ God, he believes, is the answer to nothing.

The problem for the theoretical physicist is not just how to find a replacement for maths but how to ditch it without losing credibility. Detailed knowledge of the extremely complicated equations that are used in quantum mechanics keeps the professional at a safe distance from the amateur crack-pot theorist. Hawking can use his recondite algebra to the same effect as the ancient wise man used his literacy to gain authority over the illiterate masses. But will future generations look back on Hawking and his confrères of the late 20th and early 21st centuries as heroic Newtons and Einsteins, or with the same lofty condescension with which we now mock the augurs of Ancient Rome, or dismiss the Mediterranean Gnostics for believing that the ultimate questions of life could only be answered by first discovering the secret name of God?

Stephen Hawking has written a short, occasionally facetious, but generally reliable and informative history of classical and quantum mechanics. That is all. That he has adverted to it as an answer to the ultimate question of life is both annoying and inaccurate, but no doubt commercially sensible. Richard Feynman, the American physicist on whose work Hawking bases much of his own theory, wrote: ‘I think I can safely say that nobody understands quantum mechanics.’ Nothing has changed since Feynman’s death in 1988, and although Hawking may have a far deeper knowledge of quantum mechanics than anyone else on this planet, he still doesn’t understand it. With a little less chutzpah he might also have realised that things of which we cannot see the bottom are not necessarily profound.


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