Astrology: Early Applied Mathematics?

June 17, 2000

Book Review
Philip J. Davis

Cardano's Cosmos: The Worlds and Works of a Renaissance Astrologer. By Anthony Grafton, Harvard University Press, 1999, 352 pages, $35.00.

Seer to client: "Your astrological chart tells me you need to lose a few pounds. . . . Your moon has been in the House of Pancakes."
---Jeff MacNelly, "Shoe" (comic strip), February 28, 2000

Girolamo Cardano (1501-1576) is known to the community of mathematicians as the leading mathematician of his day. Though not the discoverer of the solution to the general cubic equation, he was the first to publish the formula and, as a consequence, one of the first to puzzle over the existence and nature of complex imaginary numbers. He is also known among mathematicians as one of the founding fathers of probability theory.

But Cardano was much more: He was a practicing physician, a professor, a natural philosopher, an experimental gambler, a prolific author and autobiographer, and, by no means least, a practicing astrologer and an interpreter of dreams. As a minor accomplishment, he was the inventor of the citation index, a contribution for which, I suppose, today's selection committees are most grateful.

He was volatile, outspoken in his criticisms of standard medical practices; he was a heretic, condemned to house arrest by the Inquisition. His complete works in ten volumes, first published in Lyons in 1663, were reprinted in 1966. His autobiography, which is remarkable for its candidness, is available in English and still bears reading.

What were some of Cardano's accomplishments as a practicing astrologer? He established genitures (times of birth or of other key personal events and the related stellar configurations) and interpreted them; he wrote, published, experimented (often using his own medical problems and moods); he read, criticized, and "improved" on the established procedures. He collected, collated, and cast horoscopes for the Great: Charles V, Sultan Süleyman, Francesco Il Sforza, Pope Paul III, Jesus, among many others. He wrote a code of ethics for astrologers. (Will we shortly need such a code for mathematicians?) He admitted that he had his own "batting averages" and advised caution for fellow and prospective astrologers, stating that at the bottom line, the practice of astrology was (in today's lingo) part algorithm and part a personal art; he spent a lifetime fine-tuning both parts. Thus, Cardano was a Renaissance man, not in the sense of today's puffed up assessments but in the true sense of the word.

"Cardano's palette glowed with many colors, not just the black and gold of astrology, but the red of medicine." So writes Anthony Grafton, Dodge Professor of History at Princeton, who has given us a most excellent book on Cardano in his persona as astrologer, a book that is at once scholarly and eminently readable. Grafton gives Cardano and his astrologic practice sympathetic placement and understanding within the mindset of the 16th century. This is astrology in its most advanced, most theoretical, most computational hour, set forth as a belief and a social practice. To write sympathetically, of course, is not the same as to agree.

Other biographies of Cardano exist; Yale mathematician Oystein Ore, for example, wrote of Cardano and probability theory (Cardano, the Gambling Scholar, 1953). As far as I am aware, Grafton's is the only English-language biography that deals with the man in the context of an age when astrology was a standard part of the furniture of the mind. It is something of a pity that Grafton does not expound in any depth on the nature of the computations that Cardano performed to arrive at a horoscope, or on the debt that mathematics and astronomy owe to astrology because of the insistence in astronomy on the production of accurate planetary tables. But then Grafton is fundamentally a classics scholar, although a "great-grandstudent" of the renowned historian of mathematics Otto Neugebauer.


I would like to think that my epigraph summarizes the average American opinion today about astrology, but I doubt that it does. The zillions of dollars that people all around the world spend annually on astrology make me conclude otherwise. A small fraction of this money is spent on sophisticated Web programs that, to the average person, contain deep and arcane mathematical computations.

The basic premise of astrology is that the world is decipherable and that the celestial bodies and our relation to them hold the key to our personal history and to world history. The language of astrology is still embedded in common expressions: We "thank our lucky stars"; we speak of jovial, mercurial, saturnine, martial personalities; Aquarians are thus and so; Sagittarians are quite different. Relating the planets to parts of the human body, to human thoughts and intentions, and to such material objects as precious stones and metals, or to the constellations and the "houses" in which they occur; considering carefully the conjunctions and oppositions, quadratures, trines, and sextiles of the planets; regarding the exact moments of a person's birth and the location of the planets at that moment---these are all constituent parts of astrologic recipes for interpretation.

Astrology has been suspect during all of its existence; it has been exploded as mere superstition, and a century ago any historian of science would have regarded it with nothing but distaste and would then have proceeded to ignore it. Its ideas have come to be regarded as an embarrassment, particularly when its practitioners---as in the case of Cardano---also had accomplishments that are now regarded as fundamental.

More than distaste and avoidance, there has been denial. Indeed, the authenticity of the Tetrabiblos (four books concerning the influence of the stars), highly regarded for fifteen hundred years and written by the great Claudius Ptolemy, the most famous astronomer of antiquity, was questioned in the early 19th century. How is it possible for such a great scientist to have been so self-deceived? Somewhat later, through textual comparison with Ptolemy's Almagest, Franz Boll re-established Ptolemy as the author.

"Trash is trash," a scholar once remarked to me, "but the study of trash can be scholarship." What we have in Grafton's book is a shift from astrology as a discredited superstition to consideration of astrology as a social and cultural arrangement. Why the turnaround?

Until the advent of philosophical relativism, astrology, like the doctrines of kabbala or the practice of talismans, was not regarded as a set of notions that was worth recreating or reanimating. These systems were just random, confused charlatanry in which no one ever really believed. The idea, now popular, that there are many possible world-views, or that each person may have an individual world-view, has opened up these and many other topics for scholarly study. It is part of today's multiculturalism. Depending on the notions discussed, this shift has occurred at different times in the 20th century. The introduction of ethnomathematics into discourses on mathematical education may be a late instance of such a shift. Grafton's splendid book raises another question in my mind: How are we now to regard astrology as practiced by someone like Cardano? Is it trash or is it science?

Well, what are the hallmarks of a science? A sound metaphysical basis? Observation? Experimentation? Reproducibility? Empirical verification? The back and forth of fine-tuning? A theory that is expressible in mathematical terms and within which manipulations lead to conclusions, verifiable or not? The potential falsifiability of its statements? Acceptance by a knowledgeable elite?

Although a final and definitive definition of science is impossible, all of the above, in various combinations, have been proposed. And all are manifested in Cardano (including the idea that we are affected by the stars; that's what Newton is all about!). If such is the case, one can argue with some cogency that astrology as practiced by Cardano was a science. A failed science, certainly, and a social practice with certain consequences, but not a pseudo-science.

If this judgment runs counter to the prejudice of some readers, let me propose a conclusion that is closer to our own interests by asking the following question: What is applied mathematics? An acceptable answer might be: mathematics in the service of matters that are exterior to mathematics itself. In that service, the question of logical proof or truth or finality is not really an issue; what matters is utility or significance, in some shape or form, to a sufficiently large group of people. With this criterion in mind, one can assert that Cardano's astrology constituted a non-negligible chapter in the history of applied mathematics.

It has been half a millennium since Cardano lived. The thought of looking ahead five hundred years and asking what of our present applied mathematics will survive boggles my imagination.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island.

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