Throughout the history of science, a productive tension has existed between mathematicians and scientists. Such early natural philosophers as Pythagoras believed that mathematics was the key to unlocking the secrets of the natural world, and author Roger Penrose sees this Pythagorean approach as the first great breakthrough in science. However, another ancient Greek philosopher, Plato, even more enamored of mathematics than Pythagoras, denigrated the world of the senses as illusory, while exalting the world of mathematical forms and propositions as perfect, exemplary, and “unassailably true.” The contrast between the truths and beauties of this mathematical (or Platonic) world and the complexities and paradoxes of the physical world, which surprisingly often can be explained in terms of mathematical laws, is a major theme of The Road to Reality. The twenty-five-hundred-year search for these laws has resulted in significant successes, while also revealing what remains to be discovered about how physical principles may be interrelated and perhaps united.
Other scientists have written general accounts of modern physics and cosmology, but what distinguishes Penrose’s approach is his use of sophisticated mathematics that, he believes, is essential for a serious lay reader to understand the basic laws of the universe. Despite his publisher’s warnings that his highly mathematical treatment would drastically restrict sales of his book, Penrose maintained, perhaps quixotically, that he could gradually introduce novices to the complexities of the modern mathematics needed for a genuine understanding of advanced physics by a step-by-step process from simple number theory, geometry, and algebra through complex numbers and the calculus and finally to such difficult topics as topology, manifolds, groups, bundles, and infinities.
Judging by such successful earlier popularizations as The Emperor’s New Mind (1989), Penrose has the talent to make esoteric scientific ideas, such as his controversial explanation of human consciousness through quantum theory, accessible to a general readership. He is also well respected for his work in general relativity and for his contributions, in collaboration with Stephen Hawking, to black holes and big bang theory. Penrose has been knighted for his achievements in science, and he spent eight years writing The Road to Reality so that as many readers as possible would share his enthusiasm for mathematical physics. This vade mecum of all that he thinks is important in modern physics and cosmology will certainly enlighten scientists, mathematicians, and students who are pursuing technical professions, but amateurs should be cautioned that much of the material and many of the mathematical exercises may be beyond their grasp.
The title of Penrose’s book has provoked criticism and discussion. If humans are traveling on the “road to reality,” what is their goal? Is it some Platonic world of perfect forms or a “theory of everything” (TOE), the so-called Holy Grail of modern physics? Penrose himself sees mathematicians as explorers of a world of equations that has objectivity transcending “mere opinion.” Furthermore, the physical world has mysterious connections to this mathematical world, and the main thrust of his book is to explore these remarkable connections. In his book’s subtitle, Penrose presumptuously proposes that he will provide the reader with a “Complete Guide to the Laws of the Universe.” As a mathematical physicist and a reductionist, he believes that ultimately the theories of all the sciences will be explicable in terms of basic physical principles. Perhaps this is why he makes no mention of such important biological laws as Gregor Mendel’s mathematical principles of inheritance or Charles Darwin’s laws of natural and sexual selection.
In short, Penrose does not treat that many laws of scientific disciplines other than physics, thus provoking the concerns of chemists, biologists, and psychologists. On the other hand, he concedes that many important scientific laws have yet to be discovered and that some laws of this universe may, in principle, be inaccessible to human reason.
Although the thirty-four chapters of The Road to Reality are not formally divided into parts, an analysis of its contents and themes reveals a tripartite structure. Chapters 1 through 16 are devoted to an explication of the mathematical ideas that will be needed in the second part, chapters 17 through 27 show how such scientists as Isaac Newton, James Clerk Maxwell, Albert Einstein, Paul Dirac, and Richard Feynman have used these mathematical ideas to deepen the modern understanding of the universe. The final part, chapters 28 through 34, deals with speculative ideas about space, time, and matter that have a tenuous relationship with observational data and that may (or may not) be verified (or discounted) in the future.
Throughout his book, and especially in the mathematics section, Penrose uses diagrams and drawings to illustrate the equations of complex and hypercomplex numbers, the calculus, surfaces, manifolds, groups, and tensors. In this way readers who are not comfortable with equations can still gain some insight into the mathematical truths...
(The entire section is 2161 words.)