How many Cardanos were there? As mathematicians,
we are mostly familiar with the Cardano who wrote the Ars Magna, the
first book on algebra to give a complete solution for algebraic equations
of degree three and four. (As far as I can tell, his book is the only one
of Cardano's books that is still in print in English.) Some of us may also
know about Cardano's book On Games Of Chance, one of the earliest
attempt to construct what was later called the theory of
probability. Others may know about his autobiography, The Book of my
Life, a book that was such a literary success that "late in the
nineteenth century, Jacob Burkhardt still took it as a given (...) that he
did not need to describe or analyze De Vita Propria, since all his
readers would be familiar with it." (Grafton, p. 198) We learn from this
book that Cardano thought of himself as primarily a physician, and that he
put great effort into his study of astrology. In fact, astrological ideas
provide one the organizing principles for Cardano's account of his own
life. Finally, on top of all this, in his time Cardano was also of some
note as a philosopher. Capturing such a multifaceted man in a book is no easy task. A half century ago, Oystein Ore wrote a biography of Cardano, The Gambling Scholar (Princeton, 1953, also out of print). Since Ore was a mathematician, this book presumably (I have never had a chance to look at it) treated Cardano's mathematical work carefully. More recently, a couple of books attest to a new interest in Cardano, but now for very different reasons. In 1997, Princeton University Press published The Clock and the Mirror, by Nancy Siraisi, which is about Cardano as a significant, and in many ways representative, figure in Renaissance medicine. And now we have Anthony Grafton's Cardano's Cosmos, which is about Cardano the astrologer.
Grafton discusses Cardano's astrological work partly in order to emphasize the significant role of astrology in sixteenth-century thought. There is a lot here about astrology and how it was done, and Grafton seems to delight in mentioning names we recognize but that we may not connect with astrological ideas. The main focus, however, is in the complex and sometimes confusing thought-world of Cardano. It was certainly a capacious world, where astrology and medicine sat side-by-side and where a mathematical and scientific attitude towards the world was accompanied by a strong belief in the significance, sometimes even the supernatural significance, of small details of everyday life. For Cardano, everything was an omen, everything carried meaning... even though, for the most part, this meaning could only be completely understood after the fact. The predictive value of dreams, for example, plays a large role in his autobiography. While Grafton is perhaps a bit too quick to accuse Cardano of self-contradiction, the book does give a fascinating picture of a very complicated man.
It is, unfortunately, a partial picture only. Grafton generally pays little attention to anything that is not directly relevant to astrology. He mentions Cardano's mathematical work only in passing, and shows no appreciation of its significance and no insight on how it fit into the overall scheme of Cardano's thought. So after books on Cardano the medical man and Cardano the astrologer, we're still awaiting a reappraisal of Cardano the mathematician. Perhaps, when we have it, someone will come along, put all the pieces together, and give us Cardano as a whole. For now, Grafton's book is interesting if you'd like to know about the other sides of the man and some aspects of his intellectual world. (Fernando Q. Gouvêa)
There's a kind of science fiction novel that is
sometimes described as "intensively recomplicated". One might also, in a
less generous mood, describe these as "everything but the kitchen sink"
books: idea is piled upon idea, new plot elements are introduced at every
moment, until a towering structure is built. Such structures often have
little internal coherence; done well, however, the result can be quite
effective.
I'm not sure if it's completely fair to list Stephen Baxter's Manifold: Time as an instance of this subgenre, but it certainly comes close. It starts off with a talented entrepreneur interested in space exploration, his ex-wife who still works for him, and a brilliant mathematician who, in a somewhat sinister way, is one of the movers of the story. Then we get a mathematical "proof" that our civilization is headed for disaster, alien artifacts, intelligent squids, mutant children who become brilliant mathematicians themselves, portals to other space-times (or sometimes to other points in our space-time), space battles, dazzle-your-eye high tech, and more. It ends with a dizzying romp through various alternative space-times, most of which fail to satisfy the stringent initial conditions which make our universe so well-suited for intelligent life, and finally leaves us hanging, because, yes, this is part of a trilogy, and there is More To Come.
Baxter does know at least to some extent what he's talking about. He has a mathematics degree from Cambridge and a doctorate in aerospace engineering, and his books have always displayed his deep knowledge of the possibilities and the tragedies of the American space program, coupled with an intense desire that space exploration continue. At least in part, this book grows out of discussions surrounding the "anthropic principle." In particular, it reflects the proposal that ours is one of many universes (which collectively make up what he annoyingly calls "the manifold"). But what is most interesting to me is the role of mathematics in this book.
First of all, there's that proof. Basically, it asks you to consider yourself as a random sample picked from among all possible human beings that ever existed and ever will exist, and argues that the fact that you live now argues strongly for the probability that most human beings that ever will exist are alive now, and therefore for the likelihood that some huge catastrophe is about to destroy the human race. (I'm pretty sure I've oversimplified, but that's the essence of the idea.) This seems pretty clearly fallacious to me, since any human being at any time could have made the same argument, but maybe I'm missing the point. In any case, many people in this book do take it seriously, most notably by Cornelius Taine, the strange mathematician that sets the story in motion. Taine (the name is clearly a reference to E. T. Bell) is initially presented as strange, perhaps a little bit unbalanced. He becomes more and more a central character, but he remains distant and sinister.
This is even more so about those super-intelligent mutant children. Baxter works very hard to convince us that these kids are really different, and one of the ways he does it is by making some of them be natural mathematical geniuses. In fact, early in the book a group of them produce a proof of the Riemann hypothesis. These super-intelligent children are the target of unrelenting hostility from the societies surrounding them, and finally end up using their abilities to escape from Earth entirely.
So what does it all mean? In this book, mathematics is a distancer of people. Those who are talented at mathematics are obsessive-compulsive, strange, dangerous, scary. Taine, at one point, describes himself as almost autistic, so intense is his concentration on mathematics, and the children take this one level further and become truly alien.
In the context of the book, it all makes sense, but it did leave a bad taste in my mouth. I hope many of us mathematicians, though perhaps strange and quirky, are not quite as anti-social as Baxter paints us in this novel. I hope mathematics itself is not as threatening. (Fernando Q. Gouvêa)
Cardano's Cosmos: The Worlds and Works of a Renaissance Astrologer, by Anthony Grafton. Harvard University Press, 1999. Hardcover, 284 pages, $36.00. ISBN 0-674-09555-3.
Manifold: Time, by Stephen Baxter. Del Rey Books, 2000. Hardcover, 442 pages, $24.00. ISBN 0-345-43075-1.
Fernando Gouvêa (fqgouvea@colby.edu) is Associate Professor of Mathematics at Colby College in Waterville, Maine. His special interests include number theory, history of mathematics, science fiction, and Christian theology.
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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouvêa, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.
MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu). Last modified: Sun Mar 19 22:12:15 -0500 2000