L. E. J. Brouwer is best known to many mathematicians for his seminal contributions to topology. He is also the founder of mathematical intuitionism, and a key player in the debate on foundations of mathematics that raged for a brief decade in the 1920s, and then subsided. *Gnomes in the Fog* tells the story of that important and influential episode in the history of mathematics, in fascinating and delicious detail. Although some readers, like my nephew who is a graduate student in mathematics, might want to skim over some of these details, anyone with an interest in mathematics and its history and philosophy, should enjoy this book. Mathematicians (especially logicians) may find some surprises in the first chapter, on Brouwer's predecessors; philosophers and science study scholars should especially appreciate the final chapter on the cultural context of the debate.

I'll get my gripe out of the way first: Such a scholarly and important book merits more meticulous editing. There are too many places in the text, footnotes, glossary and bibliography where formulae are missing, with mysterious —s printed where mathematical symbols should be. Although the author's English is excellent, there are numerous infelicitous idiomatic usages ("physician" for "physicist," forgotted, costed, impopular, sticked), incorrect prepositions, and dropped or added letters, many of which could easily have been corrected even by a good word-processing program.

OK, now that I got that off my chest, on with the review.

Since first reading about Luitzen Egbertus Jan Brouwer (1881-1966) ("Bertus" to his friends), I have felt an affinity for this non-conformist full of contradictions. He may have been the original hippie—an iconoclast and a vegetarian who studied Christian, Buddhist and Hindu mystics. When I told my friends in graduate school whose field was Logic and Foundations of my interest in Brouwer and intuitionism, they rolled their eyes. These were friends who otherwise shared my predilection for radical viewpoints, questioning authority, and challenging unexamined assumptions. But so prevalent within their mathematical training was the dismissal of this debate as a dead end in the philosophy of mathematics that they, too, subscribed to the prevailing wisdom that it was a non-issue, a cul-de-sac not worth exploring. Even those who are sympathetic to the intuitionistic point of view have the impression that practicing mathematics as an intuitionist would be like trying to box with one hand tied behind your back. Indeed, intuitionism is commonly perceived as a poor, much less powerful cousin of mainstream mathematics, a purely negative movement that restricts the tools we have at our disposal and contributes nothing positive. My first clue that there was more to it than this was reading Dirk van Dalen's Mystic, Geometer, and Intuitionist, Vol 1: The Dawning Revolution (Oxford, 1999). Reading this book, written by his student Dennis E. Hesseling (an improved version of his doctoral dissertation), has confirmed my suspicion that intuitionism and the controversy it provoked left a positive residue behind.

Full disclosure: I am a nerd. When we went on vacation to the coast for a week, my husband took along a couple of mysteries, a collection of *Doonesbury* comics, and other "beach books." I took this volume along, and read it start to finish. My idea of a good time. Sitting with me on the deck overlooking the ocean was my friend Richard. Curious, he looked over my shoulder to see what book I was so engrossed in, and read the subtitle. My friend is a thoughtful person, but hadn't yet gotten beyond mathematics-as-memorizing-formulas. A bit puzzled, he asked me "what's the role of intuition in mathematics?" I gave my usual "mathematics as a liberal art" speech about the role of creativity, inspired guesses, leaps of faith and yes, intuition in mathematical research. How, after all, do we come up with the conjectures we try to prove? But, I continued, that's not what intuitionism is about.

As Hesseling makes clear, Brouwer was concerned with living a meaningful life, and, like the contemplative mystics he admired, looked within, relying on personal experience and inner vision as the ultimate sources of truth and meaning. In keeping with this philosophy, for him mathematics consisted of mental constructions. It is a pre-linguistic "languageless" activity. First come the ideas, obtained by looking within. Second follows the mathematical language that accompanies and describes these ideas that are "constructed intuitively," based on the "primordial intuition" of time, our experience of time as change "by itself" (p. 38). This primordial intuition is reminiscent of my spiritual mentor's instruction to meditate on permanence in change or unity in multitude.

The separation between mathematics and mathematical language is the first of the two "acts" in the development of intuitionism. The other one is the construction of sets, where the novel idea of choice sequences is introduced. Brouwer's own writing (in particular his treatment of choice sequences), as Hesseling wryly notes, "would not win the prize for didactical clarity" (p. 61). But Hesseling's writing, especially his explanations of difficult passages, would (and should) win many such prizes. One of the many treasures to be discovered in reading this book is the rich collection of original quotes in the many languages in which the debate took place, along with the author's translations.

But back to my friend Richard. Seeing as how he is in the process of building his own house, he was nodding his head vigorously in support of Brouwer's approach to the question of mathematical existence, i.e., "to exist in mathematics means to be constructed intuitively" (p. 41). He liked the idea of concrete constructions. "I never understood the use of all that abstract thinking about things that have no contact with reality," he said. Well, that's not exactly what Brouwer is saying. In fact, his mental constructions may be purely creatures of the mind with no apparent connection to material reality. "In Brouwer's view, seeing things in a mathematical way means seeing things as repetitions of causal systems in time... Since seeing things in a mathematical way turned out to be useful and gave man power, Brouwer maintains, man developed pure mathematics, thus having mathematical systems available to be projected upon reality whenever this seemed functional" (p. 37).

I gave Richard the example of non-Euclidean geometries which seemed at first to be purely mathematical inventions, but later turned out to be just the thing for framing Einstein's theories of spacetime. My husband chimed in with a brief summary of Eugene Wigner's paper on "The Unreasonable Effectiveness of Mathematics," (Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in *Communications in Pure and Applied Mathematics*, vol. 13, No. I (February 1960), reprinted in many other places). I took a moment to think about how to explain the second dominant thread (after mathematical existence) in the foundational debate—the challenge to the principle of the excluded middle, a.k.a. *tertium non datur* (Latin for "a third possibility does not exist").

I explained to Richard that in classical logic it is assumed that if a statement has meaning (makes sense), then either the statement or its negation must be true (these two cases exhaust the possibilities). But, I countered, arguing as an intuitionist, just because I don't accept something as true, doesn't mean I am asserting it is false. And proving a statement is not false, is not the same as proving it is true. In other words "from the intuitionistic point of view, 'yes' or 'no' are not the only answers that can be given to [a] question, since 'not no' presents a further possibility" (p. 236). "Is this like Heisenberg's Uncertainty Principle?" asked Richard (who's read more about physics than mathematics). At first I didn't see what he was driving at. But then I thought yes, it is. The uncertainty principle implies that there are things we cannot know—not just that we need better measuring instruments, but that uncertainty is inherent in "reality" itself. Most people, including many mathematicians, make the same mistake as Barzin and Errera (two critics who claimed to have found a contradiction in Brouwerian logic) in failing to distinguish between the habit of thinking of a proposition as either true or false, and asserting that a proposition will become true or false some time in the future. According to Hesseling, "It is exactly the latter point that we do not know, and against which Brouwer protested" (p. 261). "They do not consider the possibility that there are propositions for which one can neither prove that they are true or false" (p. 290).

Intrigued by the comparison, I went inside and pulled a book by Heisenberg off the shelf (this coastal retreat included a well-stocked library) and opened to a passage discussing the paradox known as Schrodinger's Cat.

Let us consider an atom moving in a closed box which is divided by a wall into two equal parts. The wall may have a very small hole so that the atom can go through. Then the atom can, according to classical logic, be either in the left half of the box or in the right half. There is no third possibility: "tertium non datur". In quantum theory, however, we have to admit—if we use the word "atom" and "box" at all—that there is [sic] other possibilities which are in a strange way mixtures of the two former possibilities. This is necessary for explaining the results of our experiments. [Werner Heisenberg. *Physics and Philosophy, the Revolution in Modern Science*, New York, Harper and Row, 1966, pp. 181-182.]

So, in the quantum world, there is an alternative to mutually exclusive dichotomous categories—overlapping categories where a thing is sometimes one, sometimes the other, or some of each, where the answer to a question depends on the context, like the wave-particle duality.

Hesseling provides a context for Brouwer's criticism of the law of the excluded middle (LEM) with a short history of classical logic, a kind of "Greatest Hits" (section 5.1.1). Topping the list is Aristotle (384-322 B.C.E.), who formulated two principles of logic that most of us accept without even thinking of them as "principles": the law of contradiction (a statement cannot be both true and false at the same time) and the law of the excluded middle (all statements can be classified as either true or false—no middle ground). For the next two millennia, logic became progressively more important for mathematics "up to the point that people like Frege and Russell put forward logic as a foundation for mathematics. This was a tendency Brouwer protested against" (p. 220). Although Aristotle did have some qualms about applying the principle of the excluded middle to future events, he eventually accepted it. No one balked again until Brouwer, who did not object to applying LEM to finite sets, where each element could, at least in principle, be checked to see which of 'yes' or 'no' applied, but was troubled by the unrestricted use of LEM in the case of infinite sets.

Since Aristotle, no one had had any doubts about the validity of LEM. Along came Brouwer, and all hell broke loose. "Like when Euclidean geometry was challenged by non-Euclidean variations?" Oh, to have a roomful of students like Richard! Yes, a very similar development. Hesseling points out, "[t]he effect of the discussion [on the excluded middle] was that classical logic lost its absolute status" (p. 351), just like Euclidean geometry lost its absolute status in the first half of the 19th century. And just like alternative geometries, alternatives to classical logic turned out to be "unreasonably effective" in 20th century modern physics.

"So what, finally, was all the fuss about?" Richard wondered. "I mean why did people get so exercised about this stuff?" Mathematics is about certainty, and Brouwer's critique precipitated a crisis in what gets to count as mathematical truth. The foundational debate ultimately boiled down to a "battle about the meaning of classical mathematical terms" (p. 161). In particular, what does it mean for a mathematical object "to exist" (consistency vs. constructivity) and what does it mean to negate a proposition (is not-false the same as true?). Often the protagonists on either side were talking at cross-purposes, using the same terms, but attributing very different meanings to them. This resulted in distinct paradigms of mathematical practice that seemed incommensurable, and led to disagreements that couldn't be resolved until a new generation grew up and was able to sort through the confusion and gain some clarity.

Hilbert's student Hermann Weyl, an early defender of Brouwer, later modified his exclusive allegiance to intuitionism and advocated a more inclusive view of mathematical practice. "[Weyl] maintains that Brouwer and Hilbert together demarcate a new period in modern foundational research, and that one should not only do mathematics in Brouwer's way, but also in Hilbert's symbolic way" (p. 232). In acknowledging the human yearning "to create a symbolical image of the transcendental" (p. 233), Weyl transcended the dichotomous thinking encoded in LEM. A member of the younger generation, he embodied the mystical vision that an open heart/mind can contain more than one truth.

The heat of this debate changed more than one mind. For instance, the focus of much of the philosopher Wittgenstein's later work was on "meaning in use," and Hesseling conjectures that it was witnessing the battle between Brouwer and Hilbert on foundations that stimulated Wittgenstein's shift from a normative position (prescribing what phrases such as "there is" should mean in mathematics) to a more descriptive one (describing the different uses of the expression and thus its different meanings). "[O]ne could interpret Wittgenstein's changing view in the following way. Having considered the matter again, Wittgenstein came to the conclusion that, because Hilbert and Brouwer accepted different proofs in mathematics, in fact the meaning of the mathematics they defended differed. For the meaning of a statement lies in its use" (p. 197, italics in original).

And what positive residue remains? I heartily endorse the point of view expressed by Arnold Dresden, professor of mathematics at Swarthmore College, in a lecture on intuitionism delivered during the joint meetings of the AMS (American Mathematical Society) and MAA (Mathematical Association of America) in December, 1927. "[Dresden] judges positively the fact that there are now different views on mathematical existence: in diversity, as he puts it, there may lie strength, and we can learn more about mathematics by looking at it in different ways" (p. 183). Indeed, as I reluctantly turned the last page of this book and looked out towards the horizon where the vastness of the ocean mirrored the endless sky, I realized that my own horizons had expanded and I had learned a great deal about mathematics and mathematicians. I will be reflecting on this diversity of viewpoints for a long time.

Bonnie Shulman is associate professor of mathematics at Bates College in Lewiston, ME. Coming of age in the 1960s, she has always been attracted to theories and viewpoints that challenge the status quo. Trained as a mathematical physicist, her current intellectual passions include history and philosophy of mathematics and applications of game theory to studying social conflict and cooperation. She teaches t'ai chi and also practices yoga and meditation.