It should go without saying that to reach the end of David Foster Wallace's "Everything and More: A Compact History of Infinity," you will first have to reach the book's halfway point. Before arriving there, however, you will first have to get a quarter of the way through - but not before reading the first one-eighth, the first one-sixteenth, and so on. Eventually, you will realize that to make any progress in the book whatsoever, you will have to read an infinite number of textual subdivisions in a finite period of time - which is, of course, impossible.
Fortunately, by the time you reach that halfway point, Wallace will have given you the intellectual tools to overcome this hurdle. But it's likely that you'll be having too much fun to care - provided, of course, that you're not the sort of spoilsport who refuses to believe that a good time can be had in higher math.
Wallace approaches his subject matter with a surprising degree of humor, genuine enthusiasm, and technical depth. He teaches English at Pomona College and he's best known for his enormous and enormously hip novel "Infinite Jest" (1996). There's a faction among his adoring fans that may feel betrayed by this turn, and for once there will be more copies of DFW floating around MIT than Harvard.
"Everything and More" is ostensibly a history of the concept of infinity. It begins with infinity's debut in the paradoxes of Zeno (see the first paragraph of this review) and the "incommensurable magnitudes" that so troubled the Pythagorean Brotherhood. The book culminates with Georg Cantor's creation of a set theory of transfinite numbers in the 19th century.
It would be more accurate, however, to say that what we have here is a history of infinity's creeping intrusion into the Western mind, despite that mind's best efforts to stuff infinite quantities back into the toothpaste tube. The influence of Aristotle's decree that infinities can only be potential, never actual, held sway over first the Greeks and later the Roman Catholics. The latter assigned actual infinities to the province of the Divine, well outside the acceptable realm of inquiry for earthly mathematicians.
For two millenniums, the study of infinite quantities - both the "big infinities" that scroll off the ends of the number line, and the "small infinities" of points that reside in any interval on that line - was sidelined by piety and fear of the great abstract unknown.
With the arrival of Leibniz's and Newton's calculus in the 17th century came an unexpected development. Despite the high level of abstraction associated with infinite quantities, they turned out to be quite useful for solving a whole array of real-world, concrete math problems. As sheer utility broke the stranglehold of the apeirophobes, the race began for a rigorous theory of infinity. Karl Weierstrass, Richard Dedekind, and especially Cantor - the three men most responsible for working out that theory - are the unmistakable heroes of this tale.
It is tempting to blame any confusion here on Wallace's famously complex style of presentation. While trying to parse Cantor's diagonal proof establishing that some infinities are larger than others, you'll also be attempting to parse Wallace's abbreviations, interpolations, footnotes, in-jokes, "emergency glossaries," and invitations to refer to earlier points in the text (most of which you won't be able to find, due to Wallace's reader-hostile convention for delineating sections). The second time I found myself unable to determine whether a superscripted numeral was an exponent or a footnote reference, I began to judge postmodernity by its fruits.
It's no apologia, however, to suggest that forcing readers to be actively engaged with the mechanics of presentation also gets their thoughts greased for math concepts several times abstracted from experiential reality.
Many of us can remember enjoying a professor who seemed to occupy his own rarified air of cognition, whose stream-of-consciousness ramblings left you confused, frustrated, and bewilderingly well-informed. One would hope for more pop technical "booklets" from Wallace in the future, were infinity not such a tough act to follow.
• Darren Abrecht is on the Monitor's staff.