Many mathematicians treat philosophers the way many people deal with dentists: if a tooth hurts, they want it fixed, but otherwise, they don’t want to hear any lectures about flossing three times a day — much less any discussion of the metaphysics of chewing.

Many mathematicians are familiar with the role of philosophy in mathematics during what Albert Einstein called the “frog-and-mouse battle” between L. E. J. Brouwer’s intuitionists and David Hilbert’s formalists. During that toothache, philosophers and logicians took a *prescriptive* role: they told mathematicians what they *should* do — or, at least, they formed cheering sections for various mathematical combatants. This particular toothache was not the only philosophical issue: other schools, like (Platonic and neo-Platonic) realists, structuralists, logicists, naturalists, empiricists, humanists, etc., were concerned with what mathematics is and what mathematical objects are: is that sponsor of numerous Sesame Street episodes, the Number Seven, a transcendent ideal in itself, an aspect of the integers (collectively a transcendent ideal), a derivative of logical structures, a derivative of contemporary mathematics, an observed phenomenon, a social convention, etc.?

In this book, Emily Grosholz goes in a direction more typical of psychologists and social scientists. She assumes a *descriptive* role and asks what doing mathematics actually entails. Much of her book involves two interrelated issues: how do mathematicians deal with interactions between mathematical objects from multiple fields? and what does this do to the meanings of the terms used to define those objects?

As for the first issue, Grosholz is interested in *ampliative* reasoning — reasoning across several fields — as opposed to *deductive* reasoning, which involves reasoning within a single framework with a single nomenclature. One of her examples is geometry: Euclid’s axiomatic geometry is the classical example of deductive reasoning within a single “homogeneous” framework, while René Descartes’s analytic geometry involved juxtaposing several frameworks into a “heterogeneous” system, which makes Descartes’s reasoning “ampliative.” And Isaac Newton’s *Principia* used the calculus in geometric clothing, and the juxtaposition was a performance in ampliative reasoning (among other things). The importation of complex analysis into number theory leads to her favorite example, Andrew Wiles’s proof of Fermat’s Last Theorem. Not being a number theorist nor an analyst, I cannot comment much on her account of Wiles’s proof, but as it involves differential equations, elliptic curves, group representation theory, modular forms, etc., it certainly is an extravagant example of ampliative reasoning.

The ampliative versus deductive dichotomy leads to the idea of a “foundation of mathematics,” a single field in which one could stand on sure ground and simulate all known mathematics and thus safely prove all theorems. Since it is all in one field, the reasoning would be deductive and homogeneous, and Grosholz oberves that the standard simulation of mathematics within Zermelo-Frankel set theory using predicate calculus is deliberately deductive and homogeneous. But as Grosholz observes, most mathematicians are not logicians or set theorists, and she observes that mathematical reasoning is often ampliative. Echoing Wittgenstein (although she does not describe mathematics as a “motley” as he did in his *Remarks on the Foundations of Mathematics*), she argues that

…deductive logic is a branch of mathematics, like set theory and category theory; we have to give up the idea that those branches can provide mathematics with foundations or that mathematics needs foundations.

After all, mathematicians express their contructions and arguments in natural language, and in the fields and nomenclatures of their choices, not necessarily in predicate calculus.

Before we go into her second issue, let’s take a brief but significant detour. Like many philosophy books, this one devotes much space to past developments, including commentary by philosophers (although, curiously, she overlooks Immanuel Kant’s distinction between *analytical* and *synthetical* reasoning, a distinction approximating her own “deductive” versus “ampliative” dichotomy). But despite some precursors, her historical approach seems relatively new: “Up until the last couple of decades, history has been a term conspicuously absent in Anglo-American philosophy of mathematics” (p. 25), while history has been in the philosophy of science for some time. But Grosholz does not stoop to note the interplay between mathematical development and the human enterprise — like the part artillery and maritime navigation played in the development of the calculus — and instead she focuses on philosophy and mathematics in themselves.

Her second issue was: how do mathematical terms refer to what they refer to? In order to do mathematics, or at least communicate mathematics, there has to be some common sense of what one is talking about when one uses mathematical terms. A *referent* is a mathematical object represented by one or more mathematical terms, like the Number Seven is referenced by the numeral “7”. The problem is that the Number Seven of the Ring of Integers is sometimes handled differently from the Number Seven in Field of Reals. It is this multiple habitation of the Number Seven that is central to Grosholz’s concern that a mathematician employing ring theory may, in employing an inhabitant of the integers, suddenly find herself in a different territory. This means that Grosholz must deal, however gingerly, with the reality (or not) and essence of these referents.

She devotes two chapters to one elusive referent, time. She leads the reader through a brief history of time, or more precisely, how various intellectuals from St. Augustine to Roger Penrose thought about time. Most notably, Gottfried Leibniz followed Descartes in viewing space and time in terms of relations between objects, while Newton saw space and time as absolutes prior to physical objects. (This seems to define the spectrum of opinion today.) After concluding that “…time seems as resistant to the idioms of science and mathematics as it is to those of philosophy…” she concludes the book in pursuit of planets, stars, galaxies, and other astronomical objects. Ever since the *Principia*, physicists — far less fastidious than logicians — have imported a motley of mathematical tools to deal with heat, kinetic energy, galactic nebulae (complete with red shifted Cepheid variables), big bangs, and dark matter.

“Mathematics inspires us because it is at once inhuman and intelligible,” and unlike the other objects we humans reason about, mathematical objects “…are so determinate [that] they render the little that that we do manage to say about them necessary” (p. 45). Psychologists, social scientists, mathematicians, and even screenwriters are describing what mathematicians do all day, so it is about time that the philosophers do, too. But Grosholz is not particularly interested in what non-philosophers have to say on the subject — not even mathematicians like Henri Poincaré, Jacques Hadamard, or J. E. Littlewood — and certainly not psychologists or social scientists. And she has little to say of mathematics exposition and education. The result is a tendency towards a review of mathematics research publication, and this may bias her presentation.

For example, the resulting heterogeneity of the research she reviews may be the result of being at the frontier. For example, when Descartes’ reasoned within an at least officially Euclidean context, his reasoning may have been ampliative. But the analytic geometry encountered by our students is less so. Our calculus texts are deliberately homogeneous (the course is hard enough without heterogeneity), and that reflects the centuries of polishing the subject has undergone. Indeed, mathematicians often talk about the “flavor” of a field, and that field seems to acquire a somewhat homogeneous flavor after enough decades, or centuries, in the mathematical crockpot.

If heterogeneity is sign of being at the frontier, this may explain the esthetic issue of “deep” results, where “deep” means “the proof involves everything including cohomology groups and the kitchen sink.” To take an example Grosholz mentions briefly, the Prime Number Theorem was originally proven by Hadamard and Charles Jean de la Vallée-Poussin using the Riemann \(\zeta\)-function — clearly an ampliative exercise. But after Atle Selberg and Paul Erdős produced “elementary” proofs, Wikipedia could stoop to remark that “These proofs effectively laid to rest the notion that the PNT was ‘deep’.”. By “deep”, we could mean “at the frontier.” Whether Fermat’s Last Theorem is still “deep” a century from now may depend on how mathematics research and exposition progresses.

The book addresses important issues on an important subject, with an interesting point of view. Unfortunately, Grosholz relies heavily on ill-defined or undefined jargon from both mathematics and philosophy, and this makes the book less accessible than it should have been. Returning to where this review started, I cannot forebear responding to her dismissal of foundations — and hence of prescriptive concerns. Many mathematicians do blithely create mathematics without foundational anxieties — except during those moments when the teeth begin to ache, and something must be done so that mathematicians can go back to creating mathematics without foundational anxieties. That is when geometers, analysts, logicians, or whatever dentists are available fix things and and send us on our merry way. Perhaps mathematicians don’t pay much attention to hygienic issues — perhaps dental hygiene is just a collective delusion — but just the delusion that we need to floss occasionally may be what keeps mathematicians relatively honest.

Greg McColm is an associate professor of mathematics & statistics at the University of South Florida - Tampa. He worked in mathematical logic but is now active in mathematical crystallography.