Let no one ignorant of geometry pass through these gates.
--Inscription at the entrance of the Athenian Academy

As was demonstrated in "Hus, Luther, and Gutenberg," technological advances in the area of communications, ranging from papyrus scrolls to electronic mail, have also contributed greatly to intellectual life. But science and technology do not relate to intellectual advances in other fields only in this indirect manner. The approaches of scientists and engineers have always reflected and influenced those of their societies. In fact, many societal revolutions have been spurred on by intellectual ferment in the scientific sphere; Darwin's views alone have spawned several upheavals in the last century and a half.

But science can only affect the way people think if they know science. A public whose scientific literacy is limited is not only impaired in making wise decisions regarding science-related issues, such as the safety of a new power plant. Equally importantly, such a public is incapable of using scientific ideas and approaches in order to understand the non-scientific questions which confront them.

In part, widespread scientific illiteracy, even among otherwise educated individuals, is due to the disturbing extent of academic overspecialization. To be sure, specialization is itself inevitable, and as our pool of information grows, so too do subspecialties appear inexorably. But these trends in no way justify the pervasive intellectual narrowness to be found in the academy today. Even people within the same department frequently find themselves unable to communicate due to their lack of understanding of one another's areas of research. This calls into question their ability to contextualize their work, to amalgamate it with that of their colleagues, or to explain its significance when teaching it. The British academic C.P. Snow wrote earlier in this century that scientists and professors of humanities were increasingly coming to represent two different cultures, speaking altogether different languages; in our own day, this situation has degenerated into a virtual Babel.

Such overspecialization also disadvantages scholars by limiting their exposure to useful analytical approaches outside their own subspecialties. There are considerable intellectual synergisms within and between disciplines, and often the most powerful ones are between seemingly disparate fields. Not only does this challenge those who emphasize specialization at the expense of intellectual breadth; it also pierces the arguments of those who take a perverse pride in their scientific and mathematical illiteracy, proclaiming the purity of their chosen fields from concepts raised in scientific thought.

For example, historians perennially debate the significance of individuals and small forces in the shaping of history. At one extreme are the "Cleopatra's nose" theorists, so called because they hold that if Cleopatra had had a less attractive nose, then Mark Antony would not have pursued her, and the subsequent history of the Roman Empire and ultimately, the Mediterranean basin would have been considerably different. That is, history is transformed by the smallest of human circumstances, actions, and idiosyncrasies. At the opposite extreme, including but not limited to the self- proclaimed "scientific socialists," are believers in history as an inevitable sequence of events. Adherents of this school see human actions in the aggregate, if not on the individual level, as highly predictable. Members of this camp are frequently utopianists who see the world as gradually progressing towards the world of their imaginations. Even an obvious setback (say, the collapse of a regime which claims to share this utopian vision, such as the Soviet Union) is perceived as movement along the road of historical evolution.

As is usually the case in human affairs, the truth lies somewhere in between. Seemingly small decisions by people of no apparent consequence do sometimes affect the course of history; one might think of the driver who hit and nearly killed Winston Churchill long before the Second World War. On the other hand, social stresses and trends can often be perceived and short-to-medium term predictions made. For example, some form of upheaval was bound to convulse both Russia and China in the first quarter of the twentieth century. Accepting this dichotomy between influences great and small is anathema to most historians, who live in a world so disorderly that they make dangerous attempts to oversimplify it in their quest to tidy it up. It is a lesser challenge for students of fluid mechanics and chaos theory, who are made comfortable with the idea that causes induce effects which, while generally on the order of the causes themselves, are sometimes disproportionately large.

A comparable synergy can be seen in combining our understanding of LeChatelier's Principle and Hegel's interpretation of historical trends. (The reader will forgive me if many of the examples in this essay and throughout the book are related to chemical engineering. This is not because it is markedly different from other scientific fields, but simply because it is the area which I know best.) LeChat�lier perceived that chemical systems are frequently in a state of stable equilibrium, which they seek to retain; when we displace them from that equilibrium, they respond by moving back to a state intermediate between the original one and the point of displacement. Similarly, Hegel perceived that in the face of a displacement from historical equilibrium, which he called a thesis (e.g., a popular revolution) the very opposite tendency would emerge, called the antithesis (e.g., monarchism) and that a subsequent system would arise which incorporated elements of both, which he termed the synthesis. On some level, both of these discoveries are recognitions of the central fact that we live in a world of relative stability. Change occurs, but the fact that both our chemical structures and our societies exist for a long enough period of time for meaningful observation to take place means that our universe and our social structures both tend towards some sort of steady state.

Another example of conceptual links between scientific and non-scientific fields lies in connecting organic chemistry and international relations. Molecular behavior in reactions is often influenced most strongly by the immediate atomic neighborhood of the reaction site. That is, the ability of an atom to influence reactions falls off with distance, just as that of nation-states does. A strongly electronegative atom, such as fluorine, will have a strong impact in its immediate area, but its influence will diminish farther afield. Likewise, India is a powerful figure in South Asia, but its influence in most other areas of the world is much smaller. It is easy to carry such analogies too far, but I think that the point is made. Nor am I the first to note such isomorphisms. Primo Levi's insightful book, The Periodic Table, explores analogies between human behavior and that of chemical elements.

One of the central principles of physical chemistry, which turns out to be surprisingly relevant to political and economic theorists, is: "Thermodynamics is not kinetics." Thermodynamics, which analyzes the characteristics of chemical equilibrium, says nothing whatsoever about the rapidity with which levels of greater stability are attained. Kinetics, on the other hand, looks at the mechanisms whereby systems move from one state to another, and the speed with which such processes take place. The fact that one state may be more stable than another (that is, thermodynamically favored) does not imply that the transition to the more stable state will take place swiftly, if at all. Those who envision much-improved, self-stabilizing political and societal systems should be conscious of how this fact relates to their work. Reformers in former Communist states have already learned that the transition to a more prosperous future entails a mechanism so complex that it would make an organic chemist blush.

Even that most abstract of scientific disciplines, mathematics, can make dramatic differences in the way in which its students deal with non-scientific problems. Though students with little interest in the sciences frequently complain of the alleged "uselessness" of required classes in calculus, the two fundamental concepts of this discipline are of critical importance in many areas of human endeavor. Differentiation is a powerful mechanism for scrutinizing the rate at which functions change, a crucial concept for anyone who deals with business, the weather, production processes, or indeed, with any area of life that involves changing quantities. Integration, conversely, is the study of how to sum changing quantities (whose instantaneous values are known) into a meaningful total which measures their overall impact.

Likewise, the areas of geometry and topology can make meaningful contributions to the field of international relations. As the Palestinians gain increasing autonomy over the West Bank and Gaza, a troubling issue looms ahead: these two regions have no common border. In order to move people or goods between the two enclaves, one must pass through Israeli territory. Border incidents--in which Palestinians and Israelis clash as the former attempt to travel from one part of their country to another, while the latter try to control movement of foreigners within their territory--are a virtual certainty.

Nor can this problem be resolved by some simple exchange of territories. Intuitively, with a look at the map, it is easy to see that there is no way in which to connect Gaza to the West Bank without dividing Israel itself into two. Mathematicians have spent thousands of hours confirming this fact in the abstract, calling it the "four-color theorem," and the naked geometry of the situation triumphs over the best of intentions.

The original partition plan of Palestine in 1947 tried to resolve this problem by having Jewish and Arab territories intersect at mathematical points, although the practical mechanism for moving people and goods between zones was never developed. Even if it had been, conflicts at these crossover points would have inevitably exacerbated tensions between Jews and Arabs. Another solution might be to create a neutral zone under the control of the United Nations, which would allow for traffic from both countries to pass through. Such a plan is a recipe for future conflict; one envisions headlines such as "Israelis, Palestinians, Clash in Demilitarized Zone." (Aside: Why are "demilitarized zones" always a focal point of conflict? I hold that it is not merely because they are created in order to separate combatants, but also because they represent a tempting target for aggressors and infiltrators. Armies abhor a military vacuum.) Moreover, Israel would, sensibly, never allow itself to be split in two, not even by a netural zone under UN supervision.

Cartographers and diplomats could play with constructs for the next fifty years, and yet they would never resolve the crux of the issue: one cannot connect Gaza to Hebron on the map without severing the link between the Negev and the rest of Israel. The problem is made immensely easier, however, when we recall that the two-dimensional map is merely a representation of our three- dimensional world. Given this geometric and cartographic caveat, the problem nearly solves itself: an elevated rail line should be built from the West Bank to Gaza, and completely enclosed so as to form a twenty-mile Palestinian tunnel passing through Israeli territory.

In the interest of both communities, the rail link would need to be completely enclosed by thick, impenetrable walls. Each side would then be able to maintain its privacy, with Palestinians passing between their two enclaves without being stopped at Israeli checkpoints, and Israelis unconcerned that Palestinians would be using the railway as a base for espionage or terrorism. Israel's security concerns would be allayed by simply setting up an array of cameras and motion detectors along the tunel route, which would instantly detect any attempt by a terrorist to break out of the rail tunnel into Israeli territory.

A durable Middle Eastern peace requires a great number of contributions, from all of the parties involved and from interested outsiders. At least one key problem, though, would be resolved if political leaders are willing to incorporate an understanding of both geometry and cartographic representation into their thought processes.

To cite yet another example of intellectual synergy between seemingly disparate fields, the issue of mass transfer in chemical systems appears remote from macroscopic concerns, but it explains much about the nature of warfare. A cell or catalytic particle immersed in a fluid is "leaky," meaning that only a minuscule portion of its surface need be permeable to a given substance for that substance to enter or exit the cell quite rapidly. In fact, this is the basis for antibiotics, which simply make bacteria more porous, causing their contents to flow out so that they die.

But the astounding penetration of external materials into even small orifices is likewise a great lesson for military strategists. Edward Gibbon refers to it in his Decline and Fall of the Roman Empire:

The experience of the world, from China to Britain, has exposed the vain attempt of fortifying any extensive tract of country. An active enemy, who can select and vary his points of attack, must, in the end, discover some feeble spot or unguarded moment. (Volume 1, chapter 12)

When Gibbon mentions China, he is of course referring to the Great Wall, a vast barrier built along China's northern frontier to keep out the pastoralists who raided and sometimes conquered that country. But the builders' well-laid plans were thwarted by invaders who slipped through decaying parts of the wall, or found weak points around its edges. So well-penetrated was the Great Wall that two of the last three Chinese dynasties consisted of northern peoples who conquered China, despite this physical obstacle.

Similar observations about the penetration of barriers were made early in the twentieth century, when the strategist B.H. Liddell Hart noted that despite all efforts to forestall attack from the air, "the bomber always gets through." The French did not recognize the ease of penetration of any barrier when they built the Maginot Line to shield themselves from Germany; in 1940, the Germans maneuvered around the line and took Paris in weeks. Nor have American strategists understood this concept well. The U.S. tried to stop supplies and soldiers from entering South Vietnam via the Ho Chi Minh Trail. Despite massive bombing, deforestation, and endless ground patrols, it was ultimately determined that American forces were incapable of preventing Viet Cong guerrillas from resupplying themselves. As we have seen, this was not a reflection on American courage or intelligence, but rather on the nature of porosity. Similar mistakes were evident in that 1980s concept, the Strategic Defense Initiative (Star Wars), whereby the Reagan administration proposed to shield America from nuclear attack.

The tragedy of scientific illiteracy of otherwise knowledgeable individuals, together with academic overspecialization, is that so few people (including professional scholars) are able to develop the types of connections suggested above. These heuristic links are both practically useful and intellectually fascinating; but even more powerfully, they serve to reaffirm, in this highly specialized age, the essentially unitary nature of human thought.