"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful."

—Henri Poincare

If I contend mathematics is the purest form of conceptual art, created without regard to the material world and realized only within the biochemical process that generates conceptualization, you would likely consider me daft. If I further contend physics (and science in general) is philosophy guided by aesthetic principles to answer questions arising from (possibly extended) perceptual experience within constraints often misleadingly termed the "scientific method," chances are my ideas would be at odds with yours. Not necessarily, but with probability increasing in proportion to your training in an art, social science (which is not science by the definition above) or engineering.

The reason I bring this up is because an old acquaintance of mine, a thoughtful man and a lawyer, recommended I read *The Time of Our Singing* by Richard Powers. He thought it a work of genius, a masterpiece. I read the book and have since recommended it to many readers; nonetheless, a central flaw haunts me. That misgiving prompts this essay.

Let me make clear that this analysis is not meant as a review of the novel. I am not concerned to discuss the authors prose style nor his character development except as it relates to this flaw, though I will temper the discussion lest it seem negative. The goal is to elucidate a common misapprehension using Power's book as vehicle.

There are two aspects to the novel that reveal a viewpoint of science held by those with a background in humanities, a word I use with some trepidation since I consider science and mathematics to be humanities. It seems interesting that this viewpoint is also common to engineers (a group in which I include physicians), even though humanities graduates tend to confound engineering with science.

Power's story centers on a family with a European Jewish immigrant father, David Strom, and a black mother from Philadelphia, Delia Daley. David is a physicist expert in relativity at Columbia University and amateur musician of some skill and knowledge. Delia's hope of developing her gift by studying voice has been dashed by racism. The initial chance meeting of the couple is the 1939 protest concert of the black soprano Marian Anderson at the Lincoln Memorial.

They have two sons, both gifted musicians, one of whom is the narrator, his brother a musical prodigy who becomes a famous classical singer. There is much about music and the hideous history of racism in the US that makes this story powerful. The characterization of the prodigy is vivid; he overpowers others with his genius and his force of will. Their story alone is enough to make this an amazing book.

I cannot shake the theme built around Rodrigo's *Concierto de Aranjuez* which seems overplayed until suddenly, at a party, the narrator experiences an epiphany listening to Miles Davis version of this work. More amazing, I recognized Miles's *Sketches of Spain* from the description of the music well before the narrator finds out and tells the reader what is playing. This example only to provide a feel for the author's skillful narration.

The flaw of the novel is with the physics and the mathematics. I have read that Powers studied some physics as an undergraduate before turning to literature; that he displayed an interest in the subject from an early age. But anyone who has completed a bachelor's degree in a subject like mathematics or physics in the US and then gone on to higher levels knows that US undergraduate studies barely scratch the surface of these disciplines. Moreover, students at the undergraduate level seldom interact with a working mathematician or physicist, their teachers most often graduate students, junior faculty, or teachers at four year colleges who are there because they are not working as researchers. So to assume that an undergraduate science dropout is somehow expert in physics is absurd, much like assuming someone who has taken bonehead English to be an expert in literature. The fact that he worked as a computer programmer for a few years is irrelevant; the vast majority of computer programmers I worked with in my career were mathematical illiterates. I would guess his knowledge of physics is the sort of shallow misunderstanding one gains from popularizations, his experience of physicists and mathematicians second-hand from reading.

From the beginning David is not a real person. He is the cliché bungling little man of science with nothing but music and physics and his memories of Germany, clueless regarding the "real world," a stereotype no working mathematician or physicist I know fits.

Moreover, anyone familiar with mathematics and physics and their history in modern times would find problems with Power's characterization of how the two fields work. In reality, there is almost no interplay. Mathematicians create in a near vacuum removed from physical influences. Physicists use mathematics and invent mathematics in ways that mathematicians consider loose, if not downright immoral and worse, meaningless. Physicists often rediscover old mathematics. The immoral use of mathematics by physicists leads to mopping up by mathematicians which in turn inspires physicists to charge mathematicians with being freeloaders. (A famous case in point is the mathematical theory of distributions of Laurent Schwartz.)

Yet in the 1950s David is already discussing knot theory, an area of topology in mathematics that no one in physics found of much interest until sometime in the eighties. David is said to be a masterful teacher and yet is unable to explain simple ideas in relativity to his father-in-law, coming up with the sort of half-truths one finds in popular accounts. He publishes no papers and yet is kept around Columbia because he wanders the halls with a coffee cup in hand reading a musical score until someone stuck on a problem asks him for help. He listens to the colleague describe some "obstinate equation," tells him he must stop trying to visualize the problem instead of listening to it, and then solves the problem by some mysterious process.

This is pure nonsense, akin to telling a poet he needs to pull his thoughts out of the dictionary or grammar book and get them into the music of words. Physicists and mathematicians seldom if ever wrestle "obstinate equations." Relativity lives on abstract geometric objects called manifolds that are locally four-dimensional Euclidean as a sphere is locally two-dimensional Euclidean. Globally it can be almost anything. And the measuring devices on these manifolds are vector spaces tangent to points and bundled into a single space, each with an indefinite metric tensor and a very abstract something called a connection that, well, connects these unrelated tangent spaces so a comparative calculus can be done. No one visualizes this machinery. That is not how this work proceeds.

Worse still, he would need to ask his colleagues working on the bomb to stop visualizing the infinite dimensional Hilbert spaces that are the basis for quantum mechanics. If David had been a mathematician working in Cantor's universe of infinite sets of bigger and bigger infinities, some so big their existence is not guaranteed by the standard axioms of set theory, so big they are inaccessible from below, his advise to stop visualizing would be seen as ludicrous. (I suggest that anyone believing one listens to "obstinate equations" in science or mathematics become familiar with the work of Andrew Wiles in solving the old problem called Fermat's Last Theorem using very heavy machinery from the last century.)

I understand that few readers will make these objections. That is what worries me, that Powers reinforces for these readers the mythological one-dimensional viewpoint of how physics works or mathematics works, along with the one-dimensional nerds who create this work. And nothing could be farther from the truth.