Feynman, at several points in his life, the book QED, The Strange Theory of Light and Matter, one of the last, the lectures at Caltech to non-mathematicians about physics another, took it upon himself to explain science to a wider audience.  He did this to dispel a remarkable predisposition toward fantasy and illusion even in the science press, but particularly in popular media.  Why?  Feynman knew that science was a recent and fragile structure in human life, and that to retain popular support for funding big and small projects it was necessary to file a readable report once in a while.  With considerably less fanfare -- Feynman, like Linus Pauling, was a gifted promoter -- Gell-Mann proposed to do something similar with a book called The Quark and the Jaguar.

In The Nature of Physical Law, that series of lectures to non-mathematicians, Feynman remarked at one point that he doubted that we would ever find the connection between, say, a quantum event and foreign policy other than the obvious, that without quantum events there would be no foreign policy.   Gell-Mann, who had spirited disagreements with Feynman throughout the latter's life, disagrees again.  How he presents his case, however, may be daunting.  Why?  He presumes that the reader will take the trouble to explore meanings of words used differently in science or new, spend time on difficult concepts, perhaps even take notes, all in an effort to build an internal summary of the ideas in the book that will enable the reader to touch upon the meaning of two essentials of contemporary physics, the simplicity of quantum events and the complexity of larger systems, and how one generates the other.  It's not quite the same as reading an aging physicist's ruminations on the magical properties of starlight.

The case is difficult and slow-going, though the prose is crystal-clear.  Gell-Mann, like Feynman, tries to skirt the mathematical proofs, using only a few equations, more as signs of a final achievement than as comprehensible mathematical sentences, and a few illustrations, most of them Feynman diagrams.  This is a good approach.  Unless you're directly engaged in theoretical work, the depth of the proofs is not so important.  And, for most of us, even those who have gone well beyond the minimal requirements in public and higher education, such mathematical proofs would stretch both ability and time far past personal limits.  Does he suggest the connection Feynman said was missing?

Feynman's scientific career, though very long by most standards, came to an end during his final illness before he passed away in 1988.  As such, he missed what Gell-Mann describes as the next great advancements in the physical sciences, both the elaborations of what is called superstring theory, which permits partial unification of the strong, weak and gravitational forces in one mathematical structure, and the integration of the sciences which is taking place at such places as Los Alamos and the Santa Fe Institute, and in many like institutions throughout the world.  On the latter, for instance, it turns out that the same structures can be used to predict the behavior of systems as diverse as economies, stars and living beings.  Whether or not a casual science reader will get to this is hard to guess.  The clarity of the writing, evidence of the extraordinary care Gell-Mann took in preparing this book, suggest that readers are in luck.  And his sharp critique of the various mythologies that have sprung out of popular and academic misinterpretations of science, particularly regarding quantum, chaos and complexity theories, is worth paying attention to.  Uncertainty principle or not, we do not make the world and the universe we live in; we adapt to and transform it as we are able, usually unaware of the full consequences.

The overall effect is a readable history which, if you work through the problems, teaches you a great deal about mathematics, from the basics of types of numbers to geometry and trig, and on to algebra, linear and matrix, probability, and the calculus.   It would be unfair to put this down as "breezy."  The examples and problems Hogben provides are demanding and, though it's worthwhile to skim the book for the history of mathematical development first, learning the ideas, discovering the processes, and doing the problems is presumed to be the reason why someone buys the book.  The reviewer has worn out two copies.  Mathematics is not only for scientists.  Indeed, the current hot markets in the world are in many instances driven by mathematical applications no business person would have heard of fifty years ago.

Such an eminently practical approach to mathematics, but which nevertheless allows sight of the abstract beauty that mathematical mystics have noted for thousands of years, is startling for two reasons:  1) that it isn't the standard approach to educating students in math, and 2) that except from David Berlinsky (A Tour of the Calculus), it may be the only text "with legs" that's written this way about the subject.    Hogben's constant drubbing of the mystical side, from the Pythagorean boys society to his present, which in the book is about 1930, will dishearten lovers of the French school, described as entirely about the abstractions (and which the reviewer remembers well in the strange way math was taught in high school and college).  It will also irritate those pure researchers who have followed logic to mathematical structures which have, sometimes generations later, proved valuable in the physical sciences.  Hogben doesn't entirely trash such approaches.  He spends a great deal of time covering the gematria of Greek mathematics, which evolved into powerful tools sometimes as late as 2000 years after they amused upper crust schoolboys in Athens.

Even if you don't do the problems, it's an enlightening and fascinating book.  One hopes for many more editions, even though the author will never edit another.
 

                                Arthur Mortensen

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