Readers with an interest in either the history or the philosophy of mathematics should find *Plato’s Ghost* both informative and entertaining. Those who think they are interested in neither should read Gray’s book anyway: the quotations from well-known mathematicians alone are worth the price of admission!

Jeremy Gray is professor of the history of mathematics and director of the Centre for the History of the Mathematical Sciences at the Open University. His books include: Worlds Out of Nothing (an undergraduate textbook on the recent history of geometry) and János Bolyai, Non-Euclidean Geometry and the Nature of Space (aimed at the general public). Now we have *Plato’s Ghost*, a historical monograph which describes “…the development of mathematics from 1880 to the 1920’s as a modernist transformation similar to those in art literature and music.” Modernism, in this context, refers to an “…autonomous body of ideas, having little or no outward reference placing considerable emphasis on formal aspects of the work and maintaining a complicated — indeed anxious — …relationship with the day-to-day world…” (p. 1)

For others, modernism is defined, at least in part, by a reliance on formalism — a concept explored in some depth by Douglas Hofstadter in *Gödel, Escher, and Bach.* Under this view, modernism consists in a move toward viewing mathematics as the manipulation of symbols according to an agreed upon set of rules. In other circles such systems are called Post Production systems after E.L. Post.:

A set *L* of strings is said to be *Post-generable* if there exists a finite set of strings, called axioms, and a finite set *P* of Post productions such that each string in the set can be obtained from the axiom set by some finite derivation, where each step in the derivation is sanctioned by an application of some production in *P*. (http://www.encyclopedia.com/doc/1O11-Postproductionsystem.html)

Gray’s book is also a history of the people whose ideas changed mathematics from a reality-based discipline to one which views itself as independent of reality while still deeply connected to it. This is not entirely a new idea. In 1684, Isaac Newton proclaimed the goal of his seminal *Principia Mathematica* in no uncertain terms: “it is the goal of the present work to subject nature to the laws of mathematics.” If we are to succeed at such a bold task, we need to establish with certainty that our mathematics is, in some sense, correct. If mathematics consists merely in the manipulation of symbols, this is difficult to justify. Gray’s work addresses this question in context, discussing how each branch of mathematics came to embrace modernism.

While this is beguiling enough, Gray’s work is considerably broader in scope than is indicated by its dust jacket. Even if the debate over foundations does not interest you, you will find lots to enjoy in this work. In particular, there are wonderful anecdotes about famous mathematicians sprinkled throughout the work. The strength of their convictions on matters related to the foundations of mathematics brings mathematics (and its history) to life as a subject which is practiced by human beings — human beings who often have a profound and deep interest in the questions of its completeness and its consistency. Gray’s discussion of the disagreements among those who took on these issues makes for challenging and profitable reading. Teachers of the history of mathematics will be able to add lots of interesting tales to their arsenal. A feel for the flavor of these discussions can be obtained by reading of the squabble between Sophus Lie and Felix Klein, which appears on pages 126 and 127.

Gray argues that the emergence of modernism (or formalism, or axiomatics, if you prefer) resulted in important changes in the way mathematicians viewed and practiced their craft during the time period in question. These changes created profound challenges for those who work in the foundations of mathematics — those challenges remain today. He argues that these changes took place, in part, as a result of the rise of the professoriate and the emergence of the mathematical sciences as a discipline. This created a class of professional mathematicians — people with the time to do mathematics and with outlets for the dissemination of their results.

Gray marshals an impressive array of evidence in support of his thesis — perhaps too much for those with only a casual interest in the subject. It is possible, however, to read selectively, perhaps looking closely only at matters as they relate to one’s main area of interest and expertise and browsing the rest. The book merits a complete and careful reading, but much can be gained from scanning table of contents in search of intriguing ideas to explore.

As we now know, mathematics originated as a means of organizing certain aspects of our experience. Its objects (at first just the natural numbers and the points and lines of Euclidean geometry) seemed firmly rooted in the reality which gave them birth. As such, its foundations were viewed as co-terminal with those of our conceptions of the world around us. In the words of Riemann “…a conception of the world is correct when the coherences of our ideas correspond to the coherence of things.” (pp. 92–93)

With (among others) the discovery of non-Euclidean geometry, the invention of set theory and transfinite numbers by Cantor, and a precise development of the complex number field, mathematics came to be seen as freed from reality. By the end of the period in question, many of its practitioners saw mathematics as nothing more than the manipulation of symbols according to a prescribed set of rules. The pinnacle of this approach came with the publication in 1910–1913 of Whitehead and Russell’s *Principia Mathematica,* a work whose goal was to free mathematics from its reliance on the underlying structure of reality. Whitehead and Russell described mathematics in stark terms:

Whitehead: “The sole concern of mathematics is the inferences of propositions from propositions.” (p. 285)

Russell: “…pure mathematics is the class of all propositions of the form P implies Q.” (p. 285)

A bit more concretely we have Charles Sanders Pierce:

...mathematics deals exclusively with hypothetical states of things, and asserts no matters of fact whatever. The certainty of pure mathematics … is due to the circumstance that it relates to objects which are the creation of our own minds.” (p. 242)

As Gray describes the situation, the period from 1880to the 1920’s sees mathematics becoming increasingly abstract and self-contained. It is no longer dependent on either the outside world or the sciences. This independence from physical reality arose first in arithmetic. Kronecker and Gauss both describe arithmetic as the pure product of the human mind while arguing that geometry requires knowledge of reality. With the discovery (or is it creation?) of non-Euclidean geometry, geometry as well is seen to be an abstract creation of the mind. The fact that all three versions of geometry can be modeled inside of Euclidean geometry means that they must rise or fall together. Hence, we can’t appeal to “reality” to determine which of the three the “correct” geometry is.

To return to our original definition, modernism is well and good for disciplines such as art and music. We don’t expect art or music to provide predictions about the world, only to provide us new ways of responding to it. But mathematics, despite its increasingly abstract nature, is still used to model the real world. Indeed, it is this “unreasonable effectiveness of mathematics” that led to the questioning of its foundations. If, as Laplace once suggested, the worth of a book can be judged by the amount of mathematics it contains, we had best be as certain of the mathematics as we possibly can be. Lagrange included no diagrams in his *Mécanique Analitique*, informing the reader that the equations spoke for themselves. If that is the case, argues Gray, then we are forced to evaluate the foundations upon which the equations are built. Just how do we know that mathematics is not built on sand? That difficult and still unanswered question is the unifying idea of Gray’s work.

Gray ends his discussion with a reiteration of the issues raised earlier. In particular, the question of where mathematics comes from and its epistemological status is raised for a final time. While this vastly simplifies what is a deep and thoughtful discussion, there are three distinct ways of viewing mathematics which are espoused by the various protagonists in Gray’s book.

Previous to the rise of modernism, mathematics was seen as a model for reality. We create mathematics to organize our experiences and predict future experiences. We choose amongst various possible mathematical structures pragmatically based on which one works best in a particular situation. In this view, mathematics is created, but we are not free to create just any mathematics — it must enable a coherent view of the world around us.

The modernists would argue that mathematics is created by the human mind — its objects and structures need have no relation to reality. In that sense, mathematics is much like art. In this view, mathematical proofs and musical canons display the same sort of regularity, but in neither case do they depend upon or impinge upon reality.

Finally, and herein lies a clue to the title of Gray’s book, there are those who believe that mathematics is discovered. G.H. Hardy declared: “I believe mathematical reality lies outside us, that our function is to discover to observe it, and the theorems which we prove, and which we describe grandiloquently as our ‘creations’ are simply our notes of our observations.” In this view mathematicians are abstract archeologists, digging out choice bits of mathematics from the cracks and crevices of their own brains, which somehow have access to this other reality.

And thus we return again to Plato’s notion of being and becoming. In some sense *Plato’s Ghost* consists in a discussion of which side of Plato’s divided line mathematics lies on. It’s an issue with important implications. Gray does an admirable job of tracing the changes which took place in mathematics and in our view of it during an important period in its history.

*Plato’s Ghost* is highly recommended for anyone with even a passing interest in the history and philosophy of mathematics. Indeed, Gray makes a compelling case that we should all be interested in these matters!

Richard Wilders is Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences and Professor of Mathematics at North Central College. His primary areas of interest are the history and philosophy of mathematics and of science. He has been a member of the Illinois Section of the Mathematical Association of America for 30 years and is a recipient of its Distinguished Service Award.