page 361 is where the most math stuff is that I don’t get. It is also the key criticism of the paper, so I want to understand it.
The use of symbolic algebra in physics would therefore appear problem-atic prima facie, because the science of physics claims to be “about” thephysical world itself rather than about a symbolic entity. Indeed, for acentury or so after Newton, mathematical physicists resisted the use ofalgebra, none more so perhaps than Newton himself, and for the very rea-son of the felt need to keep symbolic and physical quantities conceptuallydistinct. Mathematical physics instead employed the traditional languageof ratio and proportion. At some point in the nineteenth century, the equa-tions of physics generally ceased to be understood as abbreviated propor-tions and began to be taken instead as direct assertions about the physicalworld. Thus we today think of a body’s energy as itself “being” mc2, eventhough we have no intuitive conception of the product of mass and veloc-ity. The training of scientists undoubtedly encourages this kind of reifica-tion of symbolic mathematical entities, and Weil is therefore right to see init the danger of a science that does not actually think. Beyond that, how-ever, Weil seems to assume that any valid formula of physics should inprinciple be redeemable as a proportion involving intuitable physical quantities. “Science,” she maintains, “has as its object the study and the theoretical reconstruction of the order of the world—the order of the world in relation to the mental, psychic, and bodily structure of man. Contrary to the naïve illusions of certain scholars, neither the use of telescopes and microscopes, nor the employment of the most unusual algebraic formulae . . . will allow it to get beyond the limits of this structure” (1951, 169).But if this assumption was true for mathematical physics at least well into the nineteenth century, it is no longer true.
this is where it starts to lose me
In Hermann Minkowski’s “spacetime” formulation of Einstein’s specialtheory of relativity, first set forth in 1908, an invariant quantity designated the “spacetime interval” is defined for any two events: s2 = (ct)2 – x2 (wheres is the spacetime interval, t is the time interval between the events, x is the distance between the events, and c is a constant, the velocity of light in empty space).13 Of course, since ct has units of distance, the literal mean-ing of c2t2 – x2 is simply the difference between the distance separating two events and the distance light would travel in the time interval between the two events.14 But if we represent time in terms of ct, the expression can be understood as a “spacetime interval” via an analogy with a rotation of co-ordinate axes in Euclidean space, where the distance between two pointson a Cartesian plane is given by the Pythagorean theorem: s = x2+ y2.Representing ct and x on respective axes of a Cartesian plane, we can per-form a “hyperbolic rotation,” in which angles are measured on arcs of ahyperbola rather than arcs of a circle as in regular trigonometry (Fig. 1).The hyperbolic rotation does in fact yield the relativistic expressions2 = (ct)2 – x2.15 If we then set c equal to unity and drop its units (distanceper unit time), such that ct = 1t or simply t, substitution into the original expression yields t – x2 as the “spacetime interval.”
Clearly, because t now designates time, the subtraction is intuitively meaningless. The operation can still be carried out, however, if we perform it on symbolic, dimensionless numbers and then “plug” the result back into units of “spacetime.” Observe that to carry out the “spacetime” sub-traction, we first had to convert c, the velocity of light, into the symbolicand dimensionless number “1,” and then drop the dimensions of t and xrespectively.16 Consequently, save for the symbolic conception of number uncovered by Klein, the very mathematical operations by which the spacetime interval is defined would be impossible. Here, that is to say, the symbolic form of representation cannot be intuitively redeemed even indi-rectly. Rather, the “spacetime interval” is an irreducibly symbolic entity.
If four-dimensional “spacetime” is devoid of intuitive sense, by contrast with the three-dimensional space and one-dimensional time of experience,why should it be accepted as real? The answer, in short, is that the concept of space time is an inevitable result of recognition in mathematical physics of the relativity of particular reference frames for measuring space and time.Thus Minkowski characterizes the theory of space time as the “postulate of the absolute world” (1952, 83). Such reasoning, we shall see, cannot simply be dismissed as thoughtless manipulation of algebraic symbols. At the same time, there is no denying that the Minkowski approach leaves us with the sense that “things” have somehow been replaced by relations among symbols.
Weil evinces a clear sense for this aspect of modern science when shecomments, for instance, that in modern science order as expressed in alge-braic symbols has become “a thing instead of an idea” (Weil 1965, letter toAlain of 1933), and that algebra “puts everything on the same level,” since“things, once translated into letters, play an equal role in equations” (Weil1968, 54). Indeed, for Weil, the deleterious effects of “algebraic conscious-ness” extend beyond mathematical physics per se. For in the contemporary world such consciousness has become socially reified, as it were, such that thought itself is now essentially “without a thinker”:
“In all spheres, thought, the prerogative of the individual, is subordinated to the vast mechanisms which crystallize collective life, and that is so to such an extent that we have almost lost the notion of what real thought is . . . signs, words, and algebraic formulae in the field of knowledge, money and credit symbols in eco-nomic life, play the part of realities of which the actual things themselves consti-tute only the shadows, exactly as in Hans Anderson’s tale in which the scientistand his shadow exchange roles. . . . (Weil 1958, 93)”
The reification of method, so compelling in the lone Cartesian thinker or ego cogito, and so productive in mathematical physics, evidently has destructive consequences when embodied in social structures. The architects of seventeenth-century mechanics knew that algebraic formula could not be simply read into nature. However, to the degree that a symbolic system of thought becomes socially reified, such that the thoughts of individual thinkers themselves are essentially constituted by the system itself,there is no truly individual thought remaining by which such a reification of technique could be even recognized.
For Weil, the reification of technique represents an abdication of indi-vidual thought and responsibility. But she further suggests that algebra could be relegated in science itself to the status of a “mere instrument”(Weil 1965, 3), an aid to the imagination in conceiving analogies (propor-tions). In other words, if algebra is reified technique, we should dereify it.Here, however, Weil underestimates the hurdles for a reorientation of science along the lines of Pythagorean number mysticism. As we have seen,in the context of modern mathematical science, there is no way of dereifying algebraic technique, at least not in the way Weil suggests. That could bedone only by abandoning modern science per se, since in modern science algebraic entities are indeed “the thing.” To be sure, in some sense Weil did wish for contemporary science to be, if not abandoned, at least conceptually reformulated at its very root. Whether that desire is justifiable is a question to which we presently turn.
There are some images included on the actual document.