It is a common belief that science ought to be an open, public activity. Creative scientists should publish their theories and discoveries so that all researchers have easy access to new ideas and developments. Popular belief might begrudgingly admit a few exceptions. We may wish, for instance, to keep methods for making bombs secret for the sake of public safety. Some may argue further, though this is more controversial, that national security sometimes justifies secret research. Even more controversially, some argue that business practice justifies trade secrecy. The tendency, however, is to explain away these exceptions as belonging to applied technology. Thus the demand for openness in the theoretic sciences is saved.

I suggest that the high value placed on openness in science ought to be qualified, even beyond those situations mentioned above. I do this by looking at four famous anecdotes concerning secrecy in mathematics. These are chosen in part because they involve important theoretic results in perhaps the "most abstract of the sciences" and cannot be dismissed as simply applied technology. The four anecdotes concern Hippasus, Tartaglia, Newton, and Gauss.

(1) Some version of the Pythagorean theorem, as we learned it in school, was common knowledge in ancient Greece. The Pythagoreans, however, tried to keep certain consequences secret, including the fact that the length of the diagonal of a square cannot be measured in units used to measure the sides of the square. Legend has it that Hippasus, who divulged the secret, either was executed by drowning or willfully perished at sea rather than let a ship on which the secret was known make a safe port.

(2) The solution by radicals to second degree equations of the form ax2+bx+c=0 is now derived as a lower school exercise. A solution to a third degree equation ax3+bx2+cx+d=0 is, however, much more difficult to derive. In the 15311's Tartaglia found a general solution to a significant class (perhaps all) of these equations. He then showed his solution to Cardano, swearing him to secrecy. Cardano's publication of a solution in 1545 created an acrimonious dispute that remains notorious to this day.

(3) An equally nasty debate broke out between the supporters of Newton and the supporters of Leibniz as to who discovered the infinitesimal calculus in the late 17th century. Regardless of the merits of this silly debate, it is clear that Newton had certain techniques in the 1660's that he did not make public for 40 years. It is often claimed that he withheld the techniques in order to maintain a scientific advantage over (to be able to find new results faster than) other researchers of his time.

(4) In the 1830's John Holyai worked out the principles of hyperbolic geometry. This is the first truly non-Euclidian geometry in the sense that it falsifies Euclid's parallel postulate (that there is only one line parallel to a given line through a given point). This, then, resolved a theoretic problem that had been famous since the time of Euclid. When informed of this result, Gauss applauded Holyai's ingenuity, but admitted that he had known the result since 1792. Gauss had kept the secret, he said, to avoid "the howling of the Boetians." That is (since Hoetian jokes were the ethnic jokes of the time), he wished to avoid the controversy that the new crazy geometries would create among his narrow-minded colleagues.

Although the details of these anecdotes are not historically clear, the stories do tell us something about the tendency of leading mathematicians to withhold important theoretic results. I will now briefly develop these stories in such a way as to justify the secrecy. Although somewhat controversial, these sketches are all historically plausible. I argue that although the apparent reasons for secrecy differ from case to case,yet in each case the need for secrecy may be viewed as following from the nature of mathematical research. These are not cases of aberrant mathematicians holding back results for immoral reasons, but of leading mathematicians working on famous results.

I. Hippasus
The story of Hippasus' death is more legend than history. It is, however, entirely plausible given the nature of Pythagorean mathematics. It is fairly clear that the Pythagoreans withheld their theoretic results. (A small number of commentators doubt this. See Thomas Heath, History of Greek Mathematics, ch. III.) It seems reasonable that the incommensurability of the diagonal would have been kept secret because of its significance to the Pythagorean blend of moral reform, mathematics, and philosophical cosmology. The Pythagoreans had worked out a complex theory of numerical relations and ratios, and developed a theory on which those ratios underlay the structure of the universe. Moreover, the cosmos was seen as built on a geometric basis, with an emphasis on certain elementary shapes such as cubes and pyramids. That the diagonal of a square is incommensurable shows that a basic aspect of an elementary shape cannot be expressed by a recognized numeric relation. This rift between geometry and number theory either destroys the cosmology or becomes a religious mystery. In any event, we can easily see why the result may have became esoteric doctrine studied only by the inner circle of Pythagorean mathematicians.

To modern commentators it may look as if Pythagorean secrecy follows from a confusion of pure mathematics with religious mysticism. Hut the view fails to appreciate the motivation for research in pure mathematics. The Pythagoreans were among the first to do pure, as opposed to applied, mathematics. (There were some applied mathematicians: Archytus, a friend of Plato, is said to have applied Pythagorean methods to pragmatic mechanics. This was viewed by both the Pythagoreans and Plato as an improper turn away from abstract study.) The Pythagoreans arguedthattheoretic proof should be preferred to observational verification. They placed a new emphasis on pure science, of the sort that popular belief now says should be kept open. Since their study is not simply pragmatic, it became a puzzle why one would be interested in those proofs.

It remains, as always, difficult to explain to a non-mathematician why certain abstract results are significant while others, equally difficult to establish, are of no interest. In antiquity this is explained by the fact that mathematics is part of the philosophical cosmology. The philosophical interest in cosmology justifies both the interest in pure mathematics and the choice of which parts of mathematics are worth pursuing. It turns out that the justification presented here for keeping the incommensurability of the diagonal secret is closely tied to the justification of the whole Pythagorean study into pure mathematics. The secrecy is not abberant, but follows from the nature of the study.

II. Tartaglia
The general outline of Tartaglia's dispute with Cardano is clear. There is no doubt that Tartaglia attempted to keep his solutions secret. (We may question the extent of Cardano's culpability. He may have believed either that he had an independent source for the results, or that he had significantly improved and changed Tartaglia's methods.) To understand the point of Tartaglia's secrecy, we must recognize that in 16th century Italy, even theoretic knowledge without technical application had special professional significance for those privy to it. In 1535, Tartaglia had enhanced his reputation in a mathematical duel in which the contestants, without divulging method, solved cubic equations presented by their opponents. This sort of challenge only makes sense in an intellectual tradition that emphasizes secret knowledge passed on to select students. Although the custom was changing, Tartaglia was responding to a tradition in which a master's reputation depended on the amount of secret information he could pass on.

Later commentators have ridiculed the "ill-bred" Tartaglia (the destitute son of a mailman) for lacking the true spirit of mathematics and selfishly refusing to promote open research. That criticism shows an historical parochialism. Both abstract science and applied technology are now viewed as creative endeavors in which researchers are applauded more for innovation than for mastery of established principles. But in the earlier "guild" tradition, both scholars and technicians were viewed as the preservers of knowledge. Knowledge accumulated, not discovery, was the basis of reputation. In that tradition, Tartaglia's secrecy is reasonable.Still it may be argued that we ought not now be impressed by a misguided tradition of secrecy in 16th century Italy. But from one point of view, Tartaglia's attitude is still relevant. He viewed his results much as we may now consider trade secrets. His attitude follows from a refusal to draw a distinction between pure mathematics which is only of value as knowledge for its own sake and applied mathematics which may be withheld as a trade secret. In this Tartaglia may be right. It is notoriously difficult to maintain a clear distinction between applied and pure research. Tartaglia's attitude demonstrates the fact that we cannot distinguish between pure and applied research by looking at the topic of research but only by looking at how industry views it: it is applied if it is called applied. Although the practice of trade secrecy is itself controversial, Tartaglia's secrecy is no more to be faulted than that practice in general.

III. Newton
Newton's reason for withholding his theory of fluxions is far from clear. He was, in fact, just generally slow to publish. He may simply have been shy and have wished to avoid the controversies that his work would inevitably create. But he did show his work to selected colleagues. (And according to Newton's followers, Leibniz too saw this work.) Any discussion of Newton's motives for withholding the theory of fluxions is highly controversial. I consider, however, two possible motives. (1) As suggested above, he may have wished to maintain an unfair advantage over his uninformed competitors. Or (2) he may have hesitated to publish his work in anything less than final form. We tend to see the first as unscientific selfishness and the second as, at worst, over-timidity. But I suggest that the difference is not all that great. They may both follow from fear that others may pre-empt your work by carrying your preliminary results to completion before you.

Mathematics is highly competitive. And those mathematicians who, as Newton did, keenly feel the competitive pressure are often placed in a quandary: publish partial results and risk letting others carry them to completion before you, or hold back partial results while seeking elegant or important geDeralizations and risk letting others step in to take the credit for that work you have completed. Not long ago, it was common practice to publish intermediate results in code while doing further work. Then the decoded paper could be used as evidence to establish priority. Since competition in scientific research is so central to present scientific practice, we should recognize this need for secrecy as a part of scientific practice and not a perversion of it.

Newton is said to have been secretive, jealous of his reputation and scientific authority, possessive about his results, incapable of accepting criticism, and unwilling personally to enter into a public debate over intellectual issues. On this view, his failure to publicize the theory of fluxions is symptomatic of a refusal to enter into the spirit of the scientific enterprise. But there is a much more favorable view of the situation. Newton's early work on fluxions is crude, speculative, and certainly begs several scientific questions that were seen as important in the 17th century. We may view Newton as holding back his speculations while seeking an acceptable justification for them. That Newton felt the need for this exercise is clear from his later attempts to provide a justification for the cosmology expressed in his Principia. Even in published form, Newton's theory of fluxions now seems sloppy (and certainly differed from Leibniz's calculus). We should then praise Newton's attempts to clear up the foundations before publication. My point here is that these two views of Newton's reticence merge into each other. It is then misleading to deplore his reticence without also seeing it as part of good scientific practice.

IV. Gauss
The story of Gauss's secret discovery of hyperbolic geometry is the best documented of the four anecdotes. His early work is recorded in his diaries. And his justification for the secrecy appears in his correspondence with Bolyai's father, a respected mathematician in his own right. On Gauss's assessment, early publication of work on the crazy new geometry would either have been rejected as "non-geometric" or have created a mathematical controversy that would only have detracted from his other research. He may well have been right. As early as the 1730's, Saccheri had described such geometries and then rejected them as absurd. There is some reason to believe that Saccheri formulated his results this way in a sly attempt to present non-Euclidian geometry without arousing the wrath of those "Boetians" that scared off Gauss. But whether Saccheri accepted the non-Euclidian geometry or presented it as part of a reductio ad absurdum, his attitude adds credence to Gauss's fears.

There are many examples of theories rejected as absurd at one time and later accepted as dogma, even in the areas of supposedly undoubtable, provable mathematics. Before the 19th century, for instance, the notion of an infinite number was rejected as absurd, since it entails that a set may have the same number of elements as some of its proper subsets. In the 19th century, however, this property of infinite cardinals became not only the starting point for modern set theory, but definitional of infinite cardinality. Such reversals and the occasional ostracism of those who press radical reversals before the scientific community is ready to accept them, have become a cliche in recent philosophy and history of science. Since Kuhn's studies, reversals have been seen as unavoidable events in the history of science. Although in Kuhn's terminology they are not part of "normal" science, we certainly cannot dismiss them as mistakes or avoidable aberations in the advancement of science. Therefore, although we may be disappointed by Gauss's cowardice, we can understand and perhaps even applaud his foresight to hold back the radical results while he paved the way to their acceptance with his other brilliant work.

I have suggested various reasons for withholding innovative results in mathematics ranging from recognition of their cosmological significance to a political sense of how to present radical results to a conservative community of scholars. These are good reasons, well founded in scientific practice.

Still I do not think we should be overly impressed by these examples. They are more popular anecdote than rigorous argument. And there are excellent arguments that can be presented against them in favor of openness in science. Even Tartaglia must have felt the appeal of openness when he did publish his results not long after Cardano. All the same, the values of openness are not obvious and are too often affirmed complacently and unreflectively. Although openness may be a general value in science, there are also conflicting values that lead to secrecy.