Universe all summed up

Our voyage of understanding has just left the garage, says David Hughes.

Deep in the heart of this book lies the relationship between the strictures, temptations and revelations of modern mathematical theory and the physical behaviour of the universe. Note the order. It is clear that Sir Roger Penrose is a real mathematician and here mathematics certainly comes first and the universe second. Quite rightly, this book dismisses those arty pseudo-intellectuals who suggest, smugly, that anything more than adding two to two is both beyond them and immaterial. The message is clear. No appreciation of the extraordinary power and accomplishment of modern science can be achieved without at least some acquaintance with mathematical techniques and attitudes.

Even though The Road to Reality is aimed at the "general reader", it becomes apparent that GCSE, A level and even a couple of years of university subsidiary-level mathematics is the minimum requirement. Even then, the reader must be prepared to be stretched. The Penrose view is similar to that of Martin Johnson, my former Birmingham University astronomy tutor. Only via mathematics do you appreciate the "beauty and the convincement of tracing an argument in symbols and actual quantities", of "exploring the sole way in which... critical minds can escape the endless ambiguity of mere verbal persuasion, and therefore the sole claim of the exact sciences to be knowledge" ( Astronomy of Stellar Energy and Decay , 1950).

When setting out on The Road to Reality one passes 382 pages of mathematics before the physical world rears its head. This protracted mathematical section took me back to the hard benches of my university days. There, the physics student was presented with lecture after lecture on mathematics by mathematicians, without any enlightenment as to why. Questions such as why am I being told this? Where is it all leading? Is it ever going to be useful? were ever present in one's mind. The whole attitude appeared to be mathematics for mathematics' sake, and not mathematics as the handmaiden of science. Physicists know that they need mathematics, but most of them do not wish to be overindulgent. "Just enough to get by" seems to be good manners.

Penrose has produced a superb review of the progress so far in the physical sciences, and cogently stresses that these extraordinary advances are due to a symbiotic partnership. On the one hand, there is careful and accurate observation of physical parameters coupled with probing experimentation, on the other theoretical reasoning, aided by mathematical arguments, ranging from the routine to the inspirational.

We are told how the insight of the ancient Greek philosophers into the geometry of Euclidean space led eventually to the Newtonian mechanical and gravitational understanding of moving objects on Earth and in the Solar System. The 19th century saw the blossoming of electromagnetic theory and thermodynamics. The 20th century brought us not only special relativity and then Einstein's extraordinary and precisely verified general theory of relativity but also quantum mechanics, which in turn led to quantum field theory. Relativity, with its insistence on the curvature of space and the inter-relationship of mass and energy, became the foundation stone of cosmology, nuclear science and astrophysics. Quantum mechanics explained many of the 19th-century's puzzling phenomena. Spectral lines, the stability of the atom, the nature of chemical bonds, the strengths and colours of materials, ferromagnetism, the transition between the solid, liquid and gaseous phases and the variability of a body's colour with its temperature suddenly became clear.

Quantum mechanics went on to form the theoretical foundations of superfluidity, superconductivity, liquid crystals, lasers and Bose-Einstein condensates. It also led to successful standard models in the small world of particle physics. We have moved far from the relatively simple situation where the material world consisted of photons, electrons, protons, positrons, neutrons and neutrinos. We now encounter elaborate mathematical schemes underpinning a realm in which muons, pions, kaons, lambda, sigma and omega-minus particles crowd together with quarks, gluons, W and Z bosons and a horde of particles whose existence is so fleeting they are never directly observed and are usually referred to as mere resonances.

We are confronted with a modern mathematical physics that enjoys working at the boundaries. It seems unfair that these are so distant. The edge of the observable universe is 10,000 times further away than the most distant object visible to the naked eye (the Andromeda Galaxy). This means that the recent shift towards a positive cosmological constant or, putting it more poetically, the existence of a "dark energy" that seems to be forcing the universe to expand faster and faster, is less easy to comprehend. The Big Bang singularity from which everything supposedly started took place about 14,000,000,000 years ago. And this makes the singularity's special state, absurdly low entropy and very precise spatial isotropy and homogeneity difficult to investigate and justify. The lack of time-symmetry in inflationary cosmology is also a problem. And the positive cosmological constant means that the cosy, closed and finite world of bang followed by crunch, followed by bang followed by crunch, and so on, has been replaced by a singular universe with no known precursor, that is simply expanding inexorably to a more tenuous and colder future.

In trying to explain the elementary particles Penrose evokes strings and twisters. Here, at the tiniest levels, the macroscopic world of continuity breaks down into discontinuity and discreteness: we are forced to worry about the possibility that space might disintegrate into a multiplicity of dimensions and confronted with the conundrum that even though the absolute Planck units of time and length are so very small, the Planck mass is 10-5 g, about the mass of a small midge. There is also the underlying problem as to whether strings and twisters can influence space-time geometry.

Much is made by Penrose of the need for a Grand Unified Theory, in which gravity can be brought into the fold of the other much stronger forces, and the quantum mechanics of the very small can be wedded with the general relativity of the very large. When confronted with the travelling child's question, "Are we nearly there yet?", Penrose clearly thinks that we are hardly out of the garage. Even though mathematical physics has made immense strides, there is no reason why 21st-century physics should be devoted to tidying up a few loose ends. He envisages some major rethinks, especially in the realm of quantum mechanics. At present, in the interplay between mathematical ideas and physical behaviour we seem to be at an unfortunate stage in which experimental guidance is absent. The mathematical/physical balance has been thrown out of kilter. Karl Popper's philosophical criterion that scientifically admissible theories have to be observationally refutable is not being satisfied. The big particle accelerators are just not big enough to provide the evidence.

The Road to Reality covers a huge range of topics. Hilbert space jostles with quantum entanglement and the paradox of Schrodinger's cat vies with the Copenhagen interpretation of quantum mechanical ontology. The uncertainty principle and the wave-particle duality are still a source of worry; it remains difficult to explain what an electron is. The anthropic principle, which insists that the form of the universe must be restricted to ensure the existence of sentient forms of life, is still debated, while some dismiss the principle as a "cop-out". The quantum approach to the gravitational field remains enigmatic and controversial. There is a strong belief that complex number theory is an indispensable, even magical, ingredient of theoretical physics. Cosmologists are encouraged to continue struggling with the horizon problem, the smoothness problem and the flatness problem.

Penrose insists that the second law of thermodynamics lies at the heart of Big-Bang cosmology throughout its stages. And there is the underlying, supposedly self-evident belief that the explanation of nature has to appear beautiful and elegant to its human observers.

This book repays multifold the effort one has to invest in it. It is hardly easy, but what is that is worth while? Penrose is a skilful communicator, and a great ambassador for his subject. He makes you want to understand, but he demands that you pay attention. For me, a sentence on page 401 rather sums up the author's attitude: "We shall start with a blank slate - or, rather, with a featureless real 4-manifold." If you are happy with visualising "blank slates", and do not want to take on the burden of understanding "a featureless real 4-manifold", then The Road to Reality is not really for you.

As a mere physicist, I enjoyed it hugely. I revelled in its picturesque descriptive passages, appreciated the informative illustrations and was challenged by the mathematical ideas and symbolism. Penrose has got to me.

He has made me slightly ashamed that I have spent my life as an astrophysicist and have not dedicated myself wholly to the pursuit of mathematics. But let us return to the subtitle of the book, A Complete Guide to the Laws of the Universe . Scientists - does that word cover mathematicians? - must surely endeavour to keep the horse before the cart.

As a scientist, you first make great efforts to observe the universe, then you try to understand its laws. The experimentalist and the observer should be in the vanguard. The explanation comes afterwards. Science becomes terribly sterile when the mathematicians try to barge into the lead.

David Hughes is professor of astronomy, Sheffield University.