Richard C. Brown, *Are Science and Mathematics Socially Constructed?: A Mathematician Encounters Postmodern Interpretations of Science*, World Scientific, 2009.

In this book, Brown recounts the rise of what is now known as the “postmodern interpretations of science” (PIS) or “sociology of scientific knowledge” (SSK) movement. In addition to pioneers Karl Popper and Thomas Kuhn (the latter of whom Brown personally knew), the author describes the contributions of Berkeley philosopher Paul Feyerabend; Harry Collins and Trevor Pinch at Bath University; Steve Woolgar at Brunel; Michel Callon and Bruno Latour in Paris; a group of scholars at the University of Edinburgh; and numerous others.

Brown emphasizes that many of these writers start with a premise that is basically sound (and with which the present reviewer fully agrees): both science in general and mathematics in particular are unavoidably human enterprises, and are subject to all the varieties of human weakness. There are numerous instances of major errors in mathematical proofs (for example, in the original proof of Fermat’s Last Theorem by Andrew Wiles). Some scientific “discoveries”, such as “N”-rays and “cold fusion”, proved short-lived and vacuous (although some still hold hope that some form of cold fusion will prove real). Other scientific discoveries were founded or bolstered by experimental evidence that was later found to be flawed (such as the original observations of the bending of light around the sun, in tests of general relativity). In some instances, it appears that new theories initially prevailed as much because of the persuasiveness and personality of a leading researcher as the scientific merits of the claims. In this sense, mathematics and science are indeed “socially constructed”.

Some SSK writers, such as Karl Popper (who emphasized the value of highly falsifiable theories) and Thomas Kuhn (who analyzed the phenomenon of paradigm shift) have been quite influential and accepted, and their teachings (with some notable exceptions) have become part of the scientific enterprise. But many of the more recent SSK writers go further. They argue that claims of mathematics and science are strongly determined by the ideology and economic class of the actors, and in many, if not most, cases not because of compelling logic or experimental evidence. Thus nature cannot be interpreted independently of the theories and conceptual paradigms that scientists bring to their tasks. As a consequence, the vaunted objectivity of science, and the claimed progress of science through the years, are illusions, and science is merely an ideology on a par with various religions, the myths of native cultures, astrology, and even the “creation science” and “intelligent design” theories advanced by modern-day evangelicals.

Many of these writers emphasize the downside of technology and argue that mathematics and science are tools of western corporate capitalism and have been employed in the repression of women and minorities. SSK scholar Sandra Harding once described Newton’s *Principia* as a “rape manual”. Many have a rather low opinion of the scientific enterprise. SSK scholar Andrew Ross contemptuously dedicated one of his books to “all the science teachers I did not have; it could only have been written without them”.

Latour and Woolgar argue that laboratory procedures, experimental results and scientific theories should be regarded as rituals, no different from the rituals of some tribal shaman; they may be reported and analyzed, but certainly should not be believed by the “anthropologist”. They also argue, for instance, that the Thyrotropin Releasing Factor (TRF) hormone was “constructed” by Guillemin’s laboratory in 1977; that Robert Koch “constructed” the tuberculosis bacillus in 1882 — it had no real existence prior to that date; and, similarly, Pasteur’s microorganisms did not exist in a strict sense before Pasteur “discovered” them.

Until the 1990s, few mathematicians or scientists were even aware of the SSK movement, even though articles of the SSK genre were widely published in some leading humanities and social studies journals. During that decade, some scientists (Gross, Levitt and others) launched a counter-offensive. They highlighted errors of scientific fact and common sense, noted passages of meaningless technical jargon, and accused the SSK community of greatly exaggerating various scientific controversies. The resulting “science wars” came to a head in 1996 when NYU physicist Alan Sokal wrote the parody “Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity”. Sokal’s article included approving quotes from the writings of SSK scholars, profuse usage of erudite scientific jargon and flawed references to scientific theories, all wrapped in leftist political rhetoric. One sample paragraph is the following:

“In this way the infinite-dimensional invariance group erodes the distinction between the observer and observed; the pi of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity; and the putative observer becomes fatally de-centered, disconnected from any epistemic link to a space-time point that can no longer be defined by geometry alone.”

Note, in addition to the gratuitous technical jargon, Sokal’s assertion that pi and G are not constants! In spite of these flaws (deliberately inserted by Sokal so that any knowledgeable scientist could spot them), the article was accepted and published in *Social Text*, a leading postmodern science studies journal (in fact in a special issue devoted to the “science wars”). When shortly afterward he exposed the hoax, the episode drew worldwide attention, including front-page coverage in the *New York Times*. Sokal emphasized that he composed the hoax out of a sincere attempt to steer the SSK community, many of whose ideals he shares, away from nonsense and irrelevance: “Theorizing about the social construction of realityâ€š won’t help us find an effective treatment for AIDS or devise strategies for preventing global warming. Nor can we combat false ideas in history, sociology, economics, and politics if we reject the notions of truth and falsity.”

In the present book under review, mathematician Richard C. Brown briefly summarizes the history of the “science wars”, although he does not attempt to rehash these debates, referring the reader to books by Gross and Levitt, Sokal and Bricmont, and others. Instead, Brown’s objective, as stated in the Preface, is to examine in some detail the philosophical and political genesis of the SSK movement, and then discuss how these debates relate to the field of research mathematics. His account is often deeply personal, for example when he gives a first-hand account of the political clashes (which resulted in at least one death) at the Mathematical Research Center at the University of Wisconsin-Madison during the early 1970s.

Brown’s most interesting and most useful material is his discussion of how these issues relate to mathematics. This material begins in Chapter 10, which he appropriately titles, “The Deconstruction of Mathematics”, followed by chapters on “Epistemic Issues” and “The Fallibility of Conventionalism and Fallibilism”. Here he points out some significant errors of reasoning in some of the SSK literature, such as when philosopher Paul Ernest denies that mathematical proof “has the absolute and extra-human basis of certainty presumed by absolutism”. Brown points out, for instance, that Ernest evidently does not clearly distinguish between mathematics as a formal game based on axioms and mathematics as a tool that permits approximate applications in the real world. In any event, Brown points out that few, if any, professional mathematicians truly hold an “absolutist” view in Ernest’s sense — the results of mathematics are not necessarily tied to the real world, and the mathematical literature has numerous mistakes (Brown acknowledges that two of his own papers were later found to have significant errors). Brown further points out that the proof of a mathematical proposition from certain axioms can be “absolutely” valid, whether or not the axioms are consistent. In summary, he observes: “The fearmongering, therefore, concerning mathematics of supporters of SSK seems an exaggeration, and considering the real contradictions within their system, calling mathematics ‘fallible’ on the basis of Godel’s theorems is especially ill-conceived”. [pg. 234]. Brown concludes,

“Whatever the philosophical problems about the status of mathematical ‘truth’, settled areas of mathematics — the pureed kind found in undergraduate or graduate level textbooks as distinct from the frontier — appear more certain than the claims of almost any other human discipline, including the hard sciences such as physics or chemistry.” [pg. 235]….

“Like Ol’ Man River, mathematics just keeps rolling along and produces at an accelerating rate ‘200,000 mathematical theorems of the traditional handcrafted variety … annually’. [quoting Davis and Hersh’s book *The Mathematical Experience*, pg. 24]. Although sometimes proofs can be mistaken — sometimes spectacularly — and it is a matter of contention as to what exactly a ‘proof’ is — there is absolutely no doubt that the bulk of this output is correct (though probably uninteresting) mathematics.” [pg. 239].

In this reviewer’s view, Brown has made a valuable contribution to the philosophy of mathematics with this book. Even readers quite familiar with the “science wars” will find many new insights here into the history of these movements and their potential to further influence the scientific enterprise. The book is decidedly even-handed, offering as close to an objective view of both sides as one could hope for in a single book, even though the author, towards the end, clearly points out some of the weaknesses in the SSK reasoning. And Brown’s analysis of the interplay of these issues with mathematical research is very well done, and commands careful analysis. All of this is crafted very skillfully. It is clear that Brown has a real talent for writing to a general audience, which is unfortunately rare among practicing research mathematicians. The book is well worth the purchase price.