book reviews largely intractable, we are setting up a much who are all at the International Institute for maticians in Paris. In it he posed 23 problems, more pressing problem for the future. Applied Systems Analysis in Vienna. A first- not all original to him, “from the discussion of A probably still more important factor is rate exercise in multi-disciplinary analysis, which”, as he put it, “an advancement of sci- that the book repeatedly speaks of a 2050 the book examines a host of associated issues ence may be expected”. His lecture turned out world in which huge numbers of people will such as fertility shifts, population ageing, to be extraordinarily influential — today’s still be in developing countries. Is this realis- agriculture, energy consumption fostering millennial conferences have all alluded to it, tic? By 2025, China may well be the world’s greenhouse-gas emissions, health patterns and could hardly have done otherwise. biggest economy, at least in terms of pur- and environmental security, together with One of the many virtues of Jeremy Gray’s chasing power parity. Already, at least 900 the many policy options they entail, especial- book about the Hilbert problems is that it million people in so-called developing coun- ly institutional adaptations. It amounts to a convincingly explains how one relatively tries qualify as ‘new consumers’ with semi- timely exploration of the major challenges young man, giving one lecture — which, we affluent lifestyles. The perceived divisions of ahead. I are told, did not make much of an impres- the world are becoming ever more blurred. Norman Myers is at Green College, University of sion on its immediate audience — could But despite these caveats, overall the book Oxford, and at Upper Meadow, Old Road, Oxford have had the effect he did. One reason is that is an authoritative assessment of what we can OX3 8SZ, UK. Hilbert’s problems were very well chosen. He expect for population totals at global, intended them to be difficult, but not so dif- regional and national levels over the next ficult as to be unapproachable, and, with one half-century. It concludes that, for the most or two exceptions, his judgements were cor- part, this will be a period when rapid popula- rect. Another reason is that Hilbert had tion growth will slow to a virtual halt. The A scientist to worked in several areas — indeed, he is often discussion of analytical methodologies is held to be the last truly universal mathemati- especially helpful to the non-demographer. count on cian — so he was able to choose problems The book might, however, have done more The Hilbert Challenge: A that spanned most of mathematics. A third to stress that these are only demographic Perspective on Twentieth-Century reason is simply that the idea of presenting projections, and take no account of other Mathematics the world with a list of carefully selected factors, such as environmental constraints. by Jeremy J. Gray unsolved problems was a brilliant one. Had Ethiopia has 65 million people today, many Oxford University Press: 2000. 328 pp. Hilbert taken a more obvious path, such as of them struggling to survive. Is it not likely £20, $34.95 outlining important programmes for that the country’s mortality rate could rise, W. Timothy Gowers research, he would not have captured the rather than decline as implied by most pro- imagination of twentieth-century math- jections? Is it reasonable to expect that, with- It will come as news to many that the year ematicians in anything like the same way. in two generations, Ethiopia’s population 2000 was World Mathematical Year. Why An important point that Gray brings out will grow to its projected 169 million? Per- was this great event not more widely recog- very well is that many of Hilbert’s problems haps it will, but the book should have high- nized? Partly, of course, because mathemat- were linked by underlying philosophical and lighted potentially constraining factors. ics is a minority interest, but also because metamathematical themes. In other words, The second book, Population and Climate mathematicians themselves did not agree on they were chosen not just for their intrinsic Change, does indeed do just that. It considers a single official way to celebrate. Several con- interest, but also for what they might reveal environmental factors in terms of climate ferences claimed to be the main occasion on about the nature of mathematical argument change, whether caused naturally or anthro- which leading mathematicians would take itself. For example, it is a well-known fact of pogenically, and shows how climate change stock of their subject and predict its future, three-dimensional geometry that if two can induce demographic change, and vice and there is more than one book with a tetrahedra have the same heights and the versa. The fourteenth and fifteenth centuries similar purpose. same triangle as their base, then they have featured unusually heavy rainfall and a high A century ago, mathematical futurology the same volume. But the only known proofs incidence of disease in Europe. Crop blights was dominated, indeed almost invented, by of this fact use calculus, or at least a limiting led to increased sickness in humans, with the one person. At the age of 38, the German argument of some kind. By contrast, the result that the average longevity in England mathematician David Hilbert gave a lecture at equivalent two-dimensional theorem con- dropped within a century from 48 years to 38 the 1900 International Congress of Mathe- cerning areas of triangles can be proved by years. The Black Death may well have origi- nated in China or Central Asia during an exceptionally wet phase around 1332. New in paperback Today we are no longer so susceptible to most forms of established or conventional Vestal Fire: An Environmental History, really wants to understand development.” John climate change, thanks to technological Told Through Fire, of Europe and Maynard Smith, Nature 398, 302–303 (1999) advances. But suppose that global warming Europe’s Encounter with the World causes the Gulf Stream to fail, as has been by Stephen J. Pyne Galileo’s Daughter proposed in certain scenarios. This would University of Washington Press, £24 by Dava Sobel profoundly disrupt agriculture, among Penguin, $14 other activities, in much of Western Europe. The Art of Genes: “This is a harrowing tale of two victims: And the possible increase in the number and How Organisms Make Themselves Galileo Galilei, forced to deny the evidence of intensity of hurricanes along Florida’s coasts by Enrico Coen his telescopes, and his daughter Virginia, forced would come on top of a fivefold increase Oxford University Press, $16.95 at the age of 13 to enter the living sepulchre in the state’s population since 1950. Some “In The Art of Genes, Coen tries to explain to of the cloistered convent. Dava Sobel’s latest 80% of today’s populace lives within 35 kilo- readers with no special knowledge of biology or well-timed bestseller provides a woman’s angle metres of the coasts. development what is happening in the world of on history’s greatest conflict between science These and many other links between genetics … I would have loved this book at 16, and religion.” Brenda Maddox, Nature 403, climatology and demographics are dealt with and so should anyone — aged 16 to 60 — who 702–703 (2000) in an illuminating way by the three authors,

632 © 2001 Macmillan Magazines Ltd NATURE | VOL 410 | 5 APRIL 2001 | www.nature.com book reviews cutting one triangle into a few pieces and rearranging them to form the other one. Hilbert’s third problem asked whether such a Going one proof could be devised for tetrahedra. But his real interest was in the metamathematical better than nature? question of whether the use of calculus was Enriching the Earth: Fritz Haber, necessary. This was the first of his problems Carl Bosch, and the AUSTRIAN ARCHIVES/CORBIS AUSTRIAN to be solved. In 1902, Max Dehn showed that Transformation of World Food calculus was needed. by Vaclav Smil Two of Hilbert’s problems have, famously, MIT Press: 2001. 339 pp. $34.95, £23.95 had metamathematical solutions. His first John Emsley problem was to prove or disprove Cantor’s continuum hypothesis, which is the state- The greatest catastrophe that the human race ment that there is no infinite set larger than could face this century is not global warming the set of positive integers but smaller than but a global conversion to ‘organic’ farming — the set of real numbers. Thanks to Kurt Gödel an estimated 2 billion people would (in 1938) and Paul Cohen (in 1963), it is now perish. That is the underlying message of this known that this statement can be neither remarkable book, which charts the discovery proved nor disproved. Hilbert’s tenth prob- of nitrogen fixation — the conversion of unus- lem asks for a systematic method for deciding able atmospheric nitrogen to useful ammonia which Diophantine equations have solu- — and its impact on the world’s food supply. Fritz Haber: discovered a way of converting tions. (Diophantine equations are polynomi- If crops are rotated and the soil is fertil- nitrogen in the atmosphere into ammonia ... al equations whose solutions are required to ized with compost, animal manure and be integers.) Building on the work of many sewage, thereby returning as much fixed support continued agriculture if properly mathematicians, Yuri Matiyasevich proved nitrogen as possible to the soil, it is just managed, but it imposes a maximum on the in 1970 that there was no such method. possible for a hectare of land to feed 10 peo- density of the human population. Results such as these have had a profound ple — provided they accept a mainly vegetar- All this changed on 3 July 1909, when two effect on the philosophy of mathematics. ian diet. Although such farming is almost German chemists, Fritz Haber and Carl The author of any mathematical book sustainable, it falls far short of the productiv- Bosch, proved that it was possible to convert aimed at the general reader has to decide ity of land that is fertilized with ‘artificial’ atmospheric nitrogen into ammonia on an what background knowledge to assume, and nitrogen; this can easily support 40 people, industrial scale. Today there are Haber– Gray, like many others, is not consistent in and on a varied diet. Of course, ‘organic’ Bosch chemical plants around the world, his demands. This can be seen from a quick farming should be encouraged in order to producing 150 million tonnes of ammonia a inspection of his ‘boxes’, those receptacles recycle compost and dung. But it can never year, most of which goes into making fertiliz- much loved of popular science publishers, compete with the bountiful supply of agro- er. The nitrogen input into farmed land from which contain illustrations and (necessarily chemical nitrogen, which now meets about these fertilizers now exceeds the natural inadequate) explanations of some of the 40% of the world’s dietary needs. input. Even low-income countries can afford technical points in the text. That said, it Nitrogen is abundant in the atmosphere, Haber–Bosch factories, and these should would be misleading to describe this book as but in a form that is difficult to extract; only a begin to turn around food production there, popular science. Two indications of its seri- few microbes and plants have the capacity to just as they did in high-income economies. ous intent are that its title does not make silly do this. Yet, thanks to their efforts over aeons In the final chapter of Enriching the Earth, use of the words ‘history’ or ‘biography’, and of time, a whole planetary ecology can now Vaclav Smil of the University of Manitoba that we learn next to nothing about Hilbert’s be sustained. This organic nitrogen will even admits that he originally intended to write a personal life. (For example, I still do not biography of Haber and Bosch, but he quickly know whether he ever married.) As for the realized that an account of the effects of their intended readership, at least some exposure research would be far more interesting, and CORBIS to university-level mathematics is essential concentrated on this. He was right to do so. to appreciate the book properly. Smil begins by looking at the fact that all My one complaint (apart from a few living things need nitrogen in order to make minor quibbles) is that Gray’s prose contains amino acids, the building-blocks for the pro- far too many clumsily constructed sentences teins on which life depends. He explains how that I had to read twice. Here is one example nitrogen is fixed naturally, and how tradi- from a long list: apparently, Hermann tional farming takes this from the soil, but Minkowski thought it “unlikely that any with only partial success at returning waste polynomial in several variables which was material to fertilize future crops. The first never negative was expressible as a sum of successful nitrogen fertilizers came from the squares”. Surely, several of them are, one Chilean guano deposits in the nineteenth wonders, before realizing that the word century, a clearly limited supply. “any” is supposed to be understood, unnatu- The central theme of Enriching the Earth rally, as “every”. This sort of writing lessens tells of Haber’s struggle to make hydrogen the pleasure of reading the book, which gas (H2) react directly with nitrogen gas (N2) nevertheless remains illuminating and high- to form ammonia (NH3), and of Bosch’s faith ly recommended. I that the process could be made to work com- W. Timothy Gowers is in the Department of mercially. Bosch then convinced the German Pure Mathematics and Mathematical Statistics, chemical company BASF to invest in it. Thus ... And Carl Bosch: had faith that the process Centre for Mathematical Sciences, Wilberforce was an industry born. But it was not immedi- could be made to work commercially. Road, Cambridge CB3 OWB, UK. ately seen as the answer to the world’s food

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