Not too long ago, trade books written by well-respected people in the sciences began dotting the shelves of popular bookstores. Many of them are philosophic treatises on a range of topics: sweeping overviews of the universe (Steven Hawking's *A Brief History of Time* and Roger Penrose's The Road to Reality); adjudications of scientific controversies (John Casti's *Paradigms Lost: Images of Man in the Mirror of Science*); and broad accounts of the flow of human thought (Machio Kaku's *Hyperspace*, or *Visions*).

Many of these works display amazing speculations as to the origin or fate of the universe. Using Kaku to illustrate, he writes (in *Hyperspace*, page 312) about the possibility of an evolved analog computer super-robot that, at the *big crunch*, may somehow escape to a "hyperspace" outside of our universe, and from there begin the universe again with the declaration, "Let there be light." (Much as in Isaac Asimov's story "The Last Question".) In the concluding pages of *Visions* (page 352), he states

According to this startling new picture [of infinitely many parallel universes], in the beginning there was Nothing. No space. No time. No matter or energy. But there was the quantum principle, which states that there must be uncertainty, so even Nothing became unstable, and time particles of Something began to form.

John Byl's latest book, *The Divine Challenge: on Matter, Mind, Math and Meaning*, offers a Christian alternative to the above kind of thinking, displaying an impressive grasp of philosophical and theological ideas woven together in a 298 page treatise. The proper audience for this book should be impressed not only by the breadth of topics covered, but also by the reasoned nature of Byl's arguments.

What is the divine challenge? Actually, it is a "double challenge, from God to man and from man to God, to establish who will rule" (page xiii). According to Byl, modern man, in his arrogance, has presumed that he can be the measure of all things, and, armed with a naturalistic worldview, give an adequate account of reality. Byl examines the plausibility of this thinking with special attention to four big-ticket items: matter; mind; mathematics; and meaning.

Byl frequently refers to mathematics, arguing in chapter 3, for example, that naturalism has great difficulty in explaining the existence of mathematical objects, which Byl indicates in chapter 8 — *Mysteries of Mathematics* — have a real but non-material existence located in God's thoughts. He also argues in that same chapter that naturalism cannot adequately account for mathematical intuition, nor explain why mathematics applies so widely to the physical world. Other "mysteries," Byl argues, at least point to mathematical realism; for example, the sense of discovery one encounters in exploring constructs like the Mandelbrot set.

There are two other chapters whose titles specifically deal with mathematics, of which one, chapter 7, *From Mind to Math*, argues more for the failure of naturalism to provide a satisfactory account of why we can have confidence in our reasoning ability. Byl's argument is similar to that given by C. S. Lewis in *Miracles*: genuine notions of truth cannot coherently exist if all thought arises from strictly naturalistic "blind-chance" causes. (This argument was recently expanded by Victor Reppert in his book *C. S. Lewis's Dangerous Idea*.)

Chapter 14, *A Christian View of Mathematics*, seeks to explore the relation of God to mathematics. Byl again opts for mathematical realism, but also attempts to ground a portion of mathematics — including the law of non-contradiction, the axiom of choice, and notions of a completed infinity — on attributes of God found in the Christian scriptures.

The book is well thought out, but there are some items that Byl might consider by way of improvement if the opportunity for a new edition presents itself. The first relates to the reading audience. The writing style is quite assertive, and while all of Byl's criticisms of a naturalistic worldview are good ones, many could be initially met with ready-made answers. Thus, I worry that some of my agnostic friends would dismiss his book too readily. Such a concern would be assuaged if Byl were to make two simple changes.

First, he could allow opposing views more grace by adding some preliminary remarks similar to those made by Reppert in his defense of C.S. Lewis: "It seems to me that many discussions of Lewis's arguments treat these arguments as finished products, to be accepted or rejected as they stand ... There are, of course, valid points to be made on the side opposing Lewis ... [whose argument] can either be offered as a final answer or as a spur to think the relevant issues through oneself" (Reppert, pages 12-14). Second, Byl might consider toning down the certainty with which he dismisses a variety of positions. Several times throughout the book he makes concluding statements like, "Such questions xxx is unable to answer," where xxx is the name of a person espousing a view with which he disagrees.

There are also instances of what I would call inappropriate criticisms of silence, where Byl says things like, "so and so gives no detailed argumentation for such and such." But Byl himself gives no such "detailed argumentation" in some of the theories he develops.

Byl is extremely well-read, and has an impressive bibliography. Nevertheless, there are at least two sources he omits with which he could have profitably interacted. *Conversations on Mind, Matter, and Mathematics* (Princeton, 1989), pits biologist Jean-Pierre Changeux, who argues that mathematics is merely a product of neural interactions in the human brain, against mathematician Alain Connes, who argues for an objective, independent existence of mathematical objects. There were also numerous occasions where he might have drawn from chapters in Mathematics in a Postmodern Age: A Christian Perspective, (Eerdmans, 2001).

Finally, there are some minor inaccuracies in the book. Kant is said to be an empiricist (page 38), a label that would puzzle philosophers, as no empiricist would accept Kant's notion of the synthetic a priori. A better classification is a constructivist, or maybe a rationalist-empiricist hybrid. Euclid's theorem on the infinitude of prime numbers is said to be an *indirect* proof that "starts off by assuming the number of primes is *not* infinite" (page 145). Actually, Euclid's theorem is a *direct* proof of the claim, "Prime numbers are more than any assigned multitude of prime numbers." Perhaps less significant, Gödel's theorem could be nuanced a bit more precisely, and Goldbach's conjecture is cast as "any even whole number can be written as a sum of two primes" (page 145). Byl evidently chose not to bother his readers with a subtler version of Gödel's theorem, or with the requirement that the even number in Goldbach's conjecture be greater than 2. A danger with this approach, however, is that some knowledgeable folks, who are reading skeptically and looking for things to pick at, may, because of these simplifications, tend to read the rest of the book more dismissively than they would otherwise.

All things considered, John Byl has produced a notable work. He should be commended for the time and care he put into it. I am glad I read it, for it contained many thought-provoking ideas. I would recommend it to a variety of people, but in giving them my recommendation I would want to mention some cautionary notes already discussed. Meanwhile, I will look forward to further scholarly writings from the author in the years ahead.

Russell Howell is Professor of Mathematics at Westmont College in Santa Barbara, CA. His area of interest is Complex Analysis; he also enjoys tennis, music (piano) and ocean kayaking.