Joseph Kevin Cosgrove

The assimilation of Cartesian algebra into mathematical physics met with significant opposition, engendered by basic issues of physical intelligibility. In this chapter Cosgrove considers more closely those historical developments by which Cartesian algebra came to be the principal, and by now the exclusive, mathematical and conceptual language of theoretical physics. Utilizing the pre-algebraic example of Galileo and a short case study of Newton’s treatment of quantity of motion in the Principia, in addition to more general observations on physics and algebra in the eighteenth century, Cosgrove concludes that algebraic equations in physics are physically intelligible solely as symbolic abbreviations for Euclidean proportions.

Cosgrove here applies the historical findings of the previous two chapters to the concept of Minkowski spacetime. While the Minkowski spacetime interval is often called a “generalization” of the Pythagorean Theorem, Einstein himself always more correctly referred to a formal analogy between the four-dimensional spacetime continuum and the three-dimensional continuum of Euclidean space. Thus the physical reality of Minkowski spacetime depends on whether the squared terms in the expression c2dt2 − dx2 designate actual geometrical quantities. Cosgrove concludes that the aforementioned algebraic terms do not designate geometrical quantities but rather represent symbolically abbreviated compound ratios. The Minkowski spacetime interval and all four-vectors constructed upon it are thereby revealed as symbolic artifacts.

This chapter is a critical reading of the published version of Minkowski’s 1908 Cologne address, in which the concept of four-dimensional spacetime was publically unveiled. Cosgrove places Minkowski’s endeavors in relativity theory in the context of formalism-driven “Göttingen science” of the early twentieth century. Turning to Minkowski’s four-dimensional kinematics, Cosgrove sees the latter as simply a formal-mathematical representation of Einstein’s 1905 special relativity with no physical coherence or additional physical insight beyond Einstein’s theory. Cosgrove finds Minkowski’s four-dimensional vector formulation of relativistic mechanics to be once again merely of formal-mathematical significance with no physical intelligibility.

Cosgrove demonstrates both that Minkowski spacetime taken as a physical concept is unnecessary to general relativity and that the mathematical apparatus of four-vectors is superfluous as well. He addresses the two principal respects in which Minkowski’s theory enters standard formulations of general theory of relativity: the field equation itself and the law of geodesic motion. Cosgrove concludes with the suggestion that the stress-energy tensor in the general field equation lacks physical intelligibility and might best be discarded in favor of the pure (“vacuum”) gravitational field equation.

Cosgrove shows that the conceptual obstacles to the assimilation of algebra into mathematical physics were successfully overcome from the late seventeenth through the eighteenth century, yielding a new mathematical language for the science of physics and a new conception of nature inseparable from symbolic mathematics. He traces some salient features of this development, first in the received Greek mathematical tradition of Euclid and Diophantus and then in the writings of Vieta and Descartes, the principal architects of modern symbolic algebra in the seventeenth-century.

In this chapter Cosgrove subjects the concept of Minkowski spacetime to critique in three principal respects: (1) the theory of spacetime articulates no clear meaning to the concept of a single continuum of space and time; (2) the theory of spacetime conflates two quite different types of geometrical representation, graphs and images, and so takes visual features of graphs as if they were direct images of the physical world; and third, the theory of spacetime misconstrues the significance of invariance in special relativity, by way a false analogy with geometry.

Cosgrove lays out the overall task of the book, which is to subject the concept of Minkowski spacetime in relativity theory to a comprehensive critique from a conceptual and historical perspective. Cosgrove first distinguishes the concept of Minkowski spacetime from other senses of the term spacetime and then highlights the essential role of algebraic representation in the constitution of Minkowski spacetime. Finally, Cosgrove delineates the historical task of “desedimenting” the concept of spacetime by recovering its historically constituted sense-structure.

Life increasingly is understood in terms of information. I consider two attempts to formulate life in terms of mathematical information theory. G. J. Chaitin proposes to define life in terms of the relation between order and algorithmic compressibility in biological information. More recently, William Dembski, Winston Ewart, and Robert J. Mark’s suggest that Dembski’s notion of specified complexity can be mathematically expressed in information-theoretic terms through the concept of algorithmic specified complexity. The mathematical approaches are similar and in both cases essentially dependent on the concept of randomness deficiency in algorithmic information theory: to subtract out the degree of randomness from an informational measure, yielding a remainder of organization (Chatin) or specified complexity (Dembski, Ewart, and Marks). I argue that these attempts must fail due the information-theoretic indistinguishability of organization and randomness. The failure is instructive, however, because it illustrates the inability of mathematical information theory to register biological organization.

The philosophy of physics literature contains conflicting claims on the heuristic significance of general covariance. Some authors maintain that Einstein's general relativity distinguishes itself from other theories in that it must be generally covariant, for example, while others argue that general covariance is a physically vacuous and trivial requirement applicable to virtually any theory. Moreover, when general covariance is invested with heuristic significance, that significance as a rule is assigned to so-called “active” general covariance, underwritten by the principle of background independence, rather than to general covriance as Einstein understood it. While agreeing with the latter group of commentators that general covariance indeed carries heuristic significance, I argue that a background independent theory need not be generally covariant and that instead the Principle of Equivalence as Einstein understood it provides the key to understanding the heuristic power of general covariance as a “mathematical sieve” for determining the gravitational field law. However, heuristic significance accrues to general covariance only indirectly, by its encompassing the relativity of frames underwritten by Einstein's principle of equivalence.

In 1908, three years after Einstein first published his special theory of relativity, the mathematician Hermann Minkowski introduced his four-dimensional “spacetime” interpretation of the theory. Einstein initially dismissed Minkowski’s theory, remarking that “since the mathematicians have invaded the theory of relativity I do not understand it myself anymore.” Yet Minkowski’s theory soon found wide acceptance among physicists, including eventually Einstein himself, whose conversion to Minkowski’s way of thinking was engendered by the realization that he could profitably employ it for the formulation of his new theory of gravity. The validity of Minkowski’s mathematical “merging” of space and time has rarely been questioned by either physicists or philosophers since Einstein incorporated it into his theory of gravity. Physicists often employ Minkowski spacetime with little regard to the whether it provides a true account of the physical world as opposed to a useful mathematical tool in the theory of relativity. Philosophers sometimes treat the philosophy of space and time as if it were a mere appendix to Minkowski’s theory. In this critical study, Joseph Cosgrove subjects the concept of spacetime to a comprehensive examination and concludes that Einstein’s initial assessment of Minkowksi was essentially correct.

The tension between beauty and technology is evinced in the modern distinction within technē itself between technology and “fine art.” Yet while beauty,as Kant observes, is never a means to an end, neither is it an “end in itself.” Beauty points beyond itself while refusing subordination to human interests. Both its noninstrumentality and its self-transcending character I trace to the intrinsic necessity of the beautiful, which is essentially impersonal while paradoxically being an object of love. I suggest that we conceive of beauty as an “anonymous voice,” and I relate the latter to Heidegger’s critique of modern technology as a projection upon nature of “resource being.” I conclude that technology can be creative rather than destructive of beauty when it lets natural ends, which are inescapably in conflict with one another, transcend themselves through self-sacrifice.

Simone Weil is widely recognized today as one of the profound religious thinkers of the twentieth century. Yet while her interpretation of natural science is critical to Weil's overall understanding of religious faith, her writings on science have received little attention compared with her more overtly theological writings. The present essay, which builds on Vance Morgan's Weaving the World: Simone Weil on Science, Necessity, and Love (2005), critically examines Weil's interpretation of the history of science. Weil believed that mathematical science, for the ancient Pythagoreans a mystical expression of the love of God, had in the modern period degenerated into a kind of reification of method that confuses the means of representing nature with nature itself. Beginning with classical (Newtonian) science's representation of nature as a machine, and even more so with the subsequent assimilation of symbolic algebra as the principal language of mathematical physics, modern science according to Weil trades genuine insight into the order of the world for symbolic manipulation yielding mere predictive success and technological domination of nature. I show that Weil's expressed desire to revive a Pythagorean scientific approach, inspired by the "mysterious complicity" in nature between brute necessity and love, must be recast in view of the intrinsically symbolic character of modern mathematical science. I argue further that a genuinely mystical attitude toward nature is nascent within symbolic mathematical science itself.

The goal of the modern scientific project, as defined by such thinkers as Descartes and Bacon, is "mastery of nature." Martin Heidegger, in an interpretation of mastery of nature that has left its imprint on post-modern critique of science, maintains that the essence of modern science lies in a projection of "technological being" upon nature. This projective "assault" has its origin in the "self-grounding" project of modern metaphysics, in which the human subject attempts to secure a self-sufficient position over against all possible objects. ;The dissertation examines Heidegger's "technological thesis" from the vantage point of the history and philosophy of science. Adopting the vis viva debate between Leibniz and the Cartesians as a point of departure, it identifies and develops, historically and systematically, two themes upon which hinges the question of the technological essence of modern science and its relationship to the "self-grounding" project: the "rationalism" of modern science, and modern science's view of natural teleology. The main argument of the dissertation is that examination of the relationship between these two themes reveals the "teleology of reason" in modern science. ;Chapter's One and Two focus on Leibniz' disagreement with Descartes over the ontology of corporeal substance. From that discussion emerge our two metaphysical themes, which are addressed in Chapter's Three and Four respectively. On the basis of the results of those chapters it is argued that the problem of "mastery of nature" resides in a clash, indigenous to reason, between the intrinsic teleology of nature, through which nature is rendered intelligible to reason, and the teleology of reason itself, which is freedom. ;The conclusion shows that Heidegger ignores tendencies within science, and within the very self-grounding project itself, which run counter to his pessimistic portrayal of science as an "assault" upon nature. Finally, the prospect for a non-coercive resolution to the clash of teleologies in modern science through the "ravishing of nature" is considered.

This essay is a contribution to the historical phenomenology of science, taking as its point of departure Husserl’s later philosophy of science and Jacob Klein’s seminal work on the emergence of the symbolic conception of number in European mathematics during the late sixteenth and seventeenth centuries. Sinceneither Husserl nor Klein applied their ideas to actual theories of modern mathematical physics, this essay attempts to do so through a case study of the conceptof “spacetime.” In §1, I sketch Klein’s account of the emergence of the symbolic conception of number, beginning with Vieta in the late sixteenth century. In §2,through a series of historical illustrations, I show how the principal impediment to assimilating the new symbolic algebra to mathematical physics, namely, thedimensionless character of symbolic number, is overcome via the translation of the traditional language of ratio and proportion into the symbolic language of equations. In §§3–4, I critically examine the concept of “Minkowski spacetime,” specifically, the purported analogy between the Pythagorean distance formula and the Minkowski “spacetime interval.” Finally, in §5, I address the question of whether the concept of Minkowski spacetime is, as generally assumed, indispensable to Einstein’s general theory of relativity.