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Orthogonality versus Opposite Directionby Donald Mills
The following article could be subtitled "A Simple Mathematical Explanation for Why Our Side Is Different From Theirs." Here, "our side" is the side of anti-militarism (not pacifism), of non-interventionism (not isolationism), the advocates of a way of thinking that espouses the notions of individual and proprietary primacy, and the right of others to live as they will so long as they do not infringe upon others’ rights. On the other side are the militarists, the interventionists, the proponents of a philosophy that advocates the fulfillment of America’s "destiny" via the most aggressive of means. What follows is a simple exposition that highlights quite clearly the vast differences that exist between the two groups, and moreover illustrates why former leftists such as David Horowitz, Ronald Radosh, William Kristol, and others cannot be considered to have shifted paradigmatically, but only directionally, in their views and objectives. In mathematics, a vector is a quantity having both magnitude and direction. A good example that illuminates the difference between vectors and scalars, quantities possessing only magnitude, is the example given by an object that moves on a two-dimensional surface with a given velocity v. For example, say an object (suppose it’s a ball – this is both the mathematician’s, and the physicist’s, favorite object in motion problems, as a ball is just a "really big" point) has velocity vector v = [2, 1], that is, at the moment in time you observe the ball, the ball is heading in the direction specified by v, namely the direction that requires you to take one step in the upward direction for every two steps you take to the right. The speed of the ball, on the other hand, is just the magnitude of the velocity vector, namely the square root of the sum of the squares of the components of v, or the square root of 4 + 1 = 5 (here we use Pythagoras’ Rule). Now suppose I told you that the ball’s velocity vector was given by w = [-2, -1]. In other words, the ball is headed in the opposite direction that it was headed before, but has the same speed, namely the square root of five. So, even though the velocity vector w is the opposite of the velocity vector v, the speed remains the same and, more importantly for our discussion, the line of motion specified by w is the same as that for v. In that sense, then, the directions specified by v and w are two sides of the same coin. With respect to the line of motion, no fundamental change has occurred; you are still moving along the same path that you were before, but in the opposite direction. Contrast this with the notion of orthogonality, or perpendicularity, if you will. Let’s keep our discussion in the realm of two-dimensional vectors. Two vectors v = [a, b] and w = [c, d] are said to be orthogonal to each other if what is known as their dot product equals zero. In other words, ac + bd = 0. Geometrically speaking, the lines along which v and w lie are perpendicular to each other, or at right angles to each other. If we think of these as lines of motion for two different balls, then we can say that the balls’ paths have nothing in common, with the possible exception of the unique point lying on both lines. Of course, if you wanted to extend this analogy further by making the altogether legitimate point that the velocity vectors of the balls need not be constant, you could do so, by saying that the curves (paths) along which the two balls travel are orthogonal to each other, where orthogonality is defined in much the same way we have defined it above. The examples given thus far, however, will suffice for the main point. What is the thrust of this exercise? Namely this: The words and actions of a person over the course of a lifetime can be thought of in reference to a path that one treads, which we might think of as a two-dimensional graph. At any point in time, a rational being can, if he is willing, gauge the direction in which he is heading, and decide to (1) continue on his current path in the same direction, with his worldviews, objectives, and methods of either enforcing his worldview within his sphere of influence, or expanding said sphere by various avenues of proselytization, all intact; (2) remain on the current path, but proceed now in the opposite direction, with a worldview that is the polar opposite (but likely little, if any, more nuanced) of that which he formerly espoused, redefining his objectives to fit his altered worldview but, as befits someone who is still on the same path, embracing and employing the same methods to achieve his newfound ends; or (3) strike out on a new path altogether, one that diverges to a significant extent from the path that he formerly walked, so that his worldview, goals, and methods all change to some extent. Under this logic, the most radical change that one strikes out upon is not one that requires a "180-degree turnaround," but a "90-degree turnaround," an orthogonal shift that puts the person on a path that is, forever after, inimical to the path formerly trod. Nowhere is such a characterization more clearly seen than in the personage of David Horowitz, editor-in-chief of the online magazine FrontPage, and co-founder, with Peter Collier, of the Center for the Study of Popular Culture. An avowed Marxist who helped edit Ramparts magazine and was intimately acquainted with several in the Black Panthers movement, Horowitz experienced a change of heart during the 1970’s due to, among other things, the murder of a friend. Observe that I said a change of heart, as opposed to a fundamental change in thinking – Horowitz’s actions since then, particularly as they relate to American foreign policy, betray his unswervingly expansionistic mindset, befitting his spiritual antecedents Leon Trotsky and Michael Bakunin, the former in regards to the violent conversion of others to his way of thinking, and the latter with respect to the agitation of pliable people against those who would disagree with him. Horowitz paints his opponents, not with a highliner or a pointed round brush so as to carefully highlight areas of disagreement while leaving the image intact, but with the one-stroke brush that knows only the colors of black and white, divine good versus depraved evil, ministers of grace standing athwart the very demons of hell. His favorite tool is that of "guilt by association," and he never misses an opportunity to paint even the most thoughtful rock-ribbed Republican with the charge of "sleeping with the enemy," should said person be caught at an antiwar rally sponsored by A.N.S.W.E.R., for example. Never mind that Horowitz, given his many statements on the Israeli-Palestinian conflict, would likely have little, if any problem in being lumped together with the most extremist, and most radically Zionistic, factions of the struggle. It’s one thing to support the existence of the state of Israel, and another thing altogether to declare a "holy war" against millions of Arabs in pursuit of a "Greater Israel." But in this, as with the question of how America should conduct itself in the post-9/11 era, one is left with the distinct impression that, in Horowitz’s world, the ends justify the means, that American hegemony is a laudable objective that should be pursued with all due vigor around the globe, and that American and Israeli interests should coincide, with the former giving way to the latter, so far as the Middle East is concerned. Anybody who doubts Horowitz’s slavish obeisance to Mars should scroll halfway down the main page of FrontPage, where for only $18, one can purchase a tee shirt with a bomber on the front and the words "Peace Through Superior Firepower" underneath. His approach to issues, like his apparel, is "one size fits all." Or rather, one size fits all, whether you like it or not! At the close of Chapter 8 of Trotsky’s Revolution Betrayed, the former head of the Red Army had this to say:
Substitute "American" for "European" and "Earth" for "Europe," and you get a good idea of what Horowitz’s, Kristol’s, and Norman Podhoretz’s true ambitions are. For them, the Revolution never ended; it just took a detour through the halls of conservatism, and involved the marriage of socialism to the language of conservatism, espousing equality at the expense of opportunity, fraternity at the expense of free association, and a definition of liberty that subjugates the individual to the State, which is no true liberty at all, but slavery under a different name. In the process, those such as Podhoretz, William Buckley and Irving Kristol staged their own French Revolution, and in so doing overthrew the remnants of the American Revolution. So – where do you stand? Do you travel along the curve of true liberty, which is at all times orthogonal to the curve of statism, welfare, and warfare? Do you choose to be "situationally orthogonal," as do certain mainstream libertarians such as members of the Cato Institute, traveling along the path of anti-statism only when it suits your purposes? Or, are you borne on the wings of "good intentions," traveling along the same path carved by countless men over the years, the path of interventionism and do-goodism that inevitably causes more problems than it solves? Which path will you tread?
February 14, 2003 Donald Mills [send him mail] is an assistant professor of mathematics at Southern Illinois University – Carbondale. He has a new web site, http://www.donaldmills.com. Copyright © 2003 LewRockwell.com
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