May 25, 2004

Book Review
Philip J. Davis

Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability. By Peter Pesic, MIT Press, Cambridge, Massachusetts, 2003, 213 pages, \$24.95.

If one stays in the math business long enough, one gets to know hundreds and hundreds of mathematical facts. Depending on who you are, you may know that the quaternions do not commute, that every vector space has a basis, that the Bessel functions have a closed-form generating function, that the tenth digit of pi after the decimal point is 5, that an angle inscribed in a circle is measured by half the intercepted arc, that the solution to a quintic equation can be expressed in terms of theta functions.

You may keep these facts stored away in the attic of your memory. Alternatively, you may keep them on the first floor and use them professionally, building them into something you think is significant or teaching them to others.

Once in a while, you may run into a cranky philosopher who asks you on what basis you, personally, believe these facts that you have stored away and wave around so blithely. After a moment of silence, you may come up with a variety of answers.

The book said so. My professor said so. Mathematica said so. I checked it out for the cases n = 2, 3, and 4 and it's really so. It fits in nicely with lots that I know about the world. It works out very well numerically. The statement has been proved. (Notice the passive voice here.) Apparently, you believe what you believe for many different reasons, which, taken together, add up to your firm belief.

But how often can you respond honestly that you have gone over a published proof carefully and believe that the proof is correct in all respects? Not very often, I suspect, because the number of facts you know is very large indeed. I used to teach a course in advanced matrix theory and used the Jordanization of a square matrix regularly both in the course and in some of my personal research. This central theorem is often the goal of a first course in matrix theory (a.k.a. linear algebra). I don't believe I ever took the time to follow a proof, and I never insisted that my students see a proof, even though I believe that we would all be better mathematicians if we did. There's just so much in the world that you care to stuff into your brains.

I suppose that a fairly wide group of mathematicians and others know that the roots of the general polynomial of degree five or higher cannot be expressed by a finite number of additions, multiplications, and root extractions operating on its coefficients. This impossibility, often called the Ruffini-Abel theorem, is one of the jewels of mathematics and is often the goal of a first course in abstract algebra. But how many have been exposed to a proof?

Oh, yes, I know that the proof of this impossibility was the work of the brilliant Abel. I know that the genius of Galois turned the proof of impossibility into an early chapter of group theory; that is why there are now courses in Galois theory. I know that there are different proofs, and I know where I could find one if I ever needed it. I know that many, many famous mathematicians had a hand in all this. And I know that I have never gone through a proof in all its fine detail.

In Abel's Proof, Peter Pesic, who is a tutor and musician-in-residence at St. John's College, Santa Fe, has given us more than the romance of the impossibility theorem. Yes, he tells us the history of the problem, starting from Pythagoras. He gives us the lives of the principal actors in the story. He gives us lessons in complex numbers and in the algebra of permutations and symmetries. He gives us---and here is the kernel of this slim volume---the key ideas and the main steps of Abel's proof. In fact, he presents his own translation of Abel's paper of 1824, augmented with copious notes to help us get into Abel's mental world. This is no easy task. Having myself dealt on occasion with original 200-year-old material, I doff my hat to him.

Yes, you can read a good fraction of the book in bed as a history of mathematical ideas; this part comes fairly easily and makes good reading. It is a tribute to the humane aspect of mathematics. But to follow the arguments at the level of detail and with the intensity required for a solid understanding is probably too much to expect of the average reader.

As for myself? As I've said, I know about the theorem. I once heard Saunders MacLane present a proof via Galois theory, and as he wrote symbols on the blackboard, they washed over me. But I have never pursued the deeper understanding of the theorem, an understanding in my own reformulation, issuing from my own experience.

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