This paper concerns late 1920 s attempts to construct unitary theories of gravity and electromagnetism. A first attempt using a non-standard connection—with torsion and zero-curvature—was carried out by Albert Einstein in a number of publications that appeared between 1928 and 1931. In 1929, Tullio Levi-Civita discussed Einstein’s geometric structure and deduced a new system of differential equations in a Riemannian manifold endowed with what is nowadays known as Levi-Civita connection. He attained an important result: Maxwell’s (...) electromagnetic equations and the gravitational equations were obtained exactly, while Einstein had deduced them only as a first order approximation. A main feature of Levi-Civita’s theory is the essential use of the Ricci’s rotation coefficients, introduced by Gregorio Ricci Curbastro many years before. We trace the history of Ricci’s coefficients that are still used today, and highlight their geometric and mechanical meaning. (shrink)

The current literature on history of science reports that Levi-Civita’s parallel transport was motivated by his attempt to provide the covariant derivative of the absolute differential calculus with a geometrical interpretation (For instance, see Scholz in ''The intersection of history and mathematics'', Birkhäuser, Basel, pp 203-230, 1994, Sect. 4). Levi-Civita’s memoir on the subject was explicitly aimed at simplifying the geometrical computation of the curvature of a Riemannian manifold. In the present paper, we wish to point (...) out the possible role implicitly played by the principle of virtual work in Levi-Civita’s conceptual reasoning to formulate parallel transport. (shrink)

79. This study of the interaction between mechanics and differential geometry does not pretend to be exhaustive. In particular, there is probably more to be said about the mathematical side of the history from Darboux to Ricci and Levi Civita and beyond. Statistical mechanics may also be of interest and there is definitely more to be said about Hertz (I plan to continue in this direction) and about Poincaré's geometric and topological reasonings for example about the three body (...) problem [Poincaré 1890] (cf. also [Poincaré 1993], [Andersson 1994] and [Barrow-Green 1994]). Moreover, it would be interesting to find out how the 19th century ideas discussed here influenced the developments in the 20th century. Einstein himself is a hotly debated case. Yet, despite these shortcommings, I hope that this paper has shown that the interactions between mechanics and differential geometry is not a 20th century invention. Klein's view (see my Introduction) that Riemannian geometry grew out of mechanics, more specifically the principle of least action, cannot be maintained. On the other hand, when Riemannian geometry became known around 1870 it was immediately used in mechanics by Lipschitz. He began a continued tradition in this field, which had several elements in common with the new view of mechanics conceived by the physicists and explicitly carried out by Hertz. Before 1870 we found only scattered interactions between differential geometry and mechanics and only direct ones for systems of two or three degrees of freedom. For more degrees of freedom the geometrical ideas were in some interesting cases taken over by analogy, but these analogies did not lead to formal introduction of geometries of more than three dimensions. (shrink)

The article investigates the different conceptions of stability found in qualitative studies on the solutions of differential equations. We start from the definitions proposed by Poincaré and criticized by Birkhoff for not being fully qualitative, and show that the clarification of the criterion for stating that a property is qualitative comes precisely with Birkhoff. In addition, we note that the stability conceptions of Lyapunov and Levi-Civita are also important in this transition from the appearance of the first qualitative (...) tools in the study of differential equations to the definition of stability in use in dynamical systems theory. The history of stability can help to explain the meaning of the word “qualitative” in this context. (shrink)

The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev’s claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi-Civita fields and with the hyperreal framework of A. Robinson, and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev’s grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within a (...) stronger and clearer system. Lou Kauffman, who published an article on a grossone, places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals. (shrink)

The present paper attempts to show that a 1915 article by Erich Kretschmann must be credited not only for being the source of Einstein’s point-coincidence remark, but also for having anticipated the main lines of the logical-empiricist interpretation of general relativity. Whereas Kretschmann was inspired by the work of Mach and Poincaré, Einstein inserted Kretschmann’s point-coincidence parlance into the context of Ricci and Levi-Civita’s absolute differential calculus. Kretschmann himself realized this and turned the point-coincidence argument against Einstein in (...) his second and more famous 1918 paper. While Einstein had taken nothing from Kretschmann but the expression “point-coincidences”, the logical empiricists, however, instinctively dragged along with it the entire apparatus of Kretschmann’s conventionalism. Disappointingly, in their interpretation of general relativity, the logical empiricists unwittingly replicated some epistemological remarks Kretschmann had written before General Relativity even existed. (shrink)

This paper attempts to explain the emergence of the logical empiricist philosophy of space and time as a collision of mathematical traditions. The historical development of the ``Riemannian'' and ``Helmholtzian'' traditions in 19th century mathematics is investigated. Whereas Helmholtz's insistence on rigid bodies in geometry was developed group theoretically by Lie and philosophically by Poincaré, Riemann's Habilitationsvotrag triggered Christoffel's and Lipschitz's work on quadratic differential forms, paving the way to Ricci's absolute differential calculus. The transition from special to general relativity (...) is briefly sketched as a process of escaping from the Helmholtzian tradition and entering the Riemannian one. Early logical empiricist conventionalism, it is argued, emerges as the failed attempt to interpret Einstein's reflections on rods and clocks in general relativity through the conceptual resources of the Helmholtzian tradition. Einstein's epistemology of geometry should, in spite of his rhetorical appeal to Helmholtz and Poincaré, be understood in the wake the Riemannian tradition and of its aftermath in the work of Levi-Civita, Weyl, Eddington, and others. (shrink)

In many systems, the tensors used to describe physical properties must acquire their structure from one vector. Knowledge of that fact alone leads to an interesting line of analysis for such systems. The analysis begins with a discussion of the types of dyadics that can be constructed from one vector. Attention is focused on certain exemplary dyadic operators, which, because of their geometrical properties, would appear particularly basic; the algebra of these dyadics is developed in detail. The algebra is then (...) used in a derivation of the expansion of general functions of these dyadics. Several applications of the general expansion are presented: the inverse of a general dyadic operator (when it exists), a general rotation operator that includes strains and reflections, the determinant of a general dyadic operator, and operators, including some new ones, used in constructing Lorentz boosts. Finally, a coordinate-free representation of the Levi-Civita tensor is provided. (shrink)

The main features of how to build a Born’s Reciprocal Gravitational theory in curved phase-spaces are developed. By recurring to the nonlinear connection formalism of Finsler geometry a generalized gravitational action in the 8D cotangent space (curved phase space) can be constructed involving sums of 5 distinct types of torsion squared terms and 2 distinct curvature scalars ${\mathcal{R}}, {\mathcal{S}}$ which are associated with the curvature in the horizontal and vertical spaces, respectively. A Kaluza-Klein-like approach to the construction of the curvature (...) of the 8D cotangent space and based on the (torsionless) Levi-Civita connection is provided that yields the observed value of the cosmological constant and the Brans-Dicke-Jordan Gravity action in 4D as two special cases. It is found that the geometry of the momentum space can be linked to the observed value of the cosmological constant when the curvature in $\mathit{momentum}$ space is very large, namely the small size of P is of the order of $( 1/R_{\mathit{Hubble}})$ . Finally we develop a Born’s reciprocal complex gravitational theory as a local gauge theory in 8D of the $\mathit{deformed}$ Quaplectic group that is given by the semi-direct product of U(1,3) with the $\mathit{deformed}$ (noncommutative) Weyl-Heisenberg group involving four $\mathit{noncommutative}$ coordinates and momenta. The metric is complex with symmetric real components and antisymmetric imaginary ones. An action in 8D involving 2 curvature scalars and torsion squared terms is presented. (shrink)

A connection viewed from the perspective of integration has the Bianchi identities as constraints. It is shown that the removal of these constraints admits a natural solution on manifolds endowed with a metric and teleparallelism. In the process, the equations of structure and the Bianchi identities take standard forms of field equations and conservation laws.The Levi-Civita (part of the) connection ends up as the potential for the gravity sector, where the source is geometric and tensorial and contains an (...) explicit gravitational contribution.Nonlinear field equations for the torsion result. In a “low-energy” approximation (linearity andlow energy-momentumtransfer), the postulate that only charge and velocities contribute to the source transforms these equations into the Maxwell system. Moreover, the affine geodesics become the equations of motion of special relativity with Lorentz force in the same approximation [J. G. Vargas,Found. Phys. 21, 379 (1991)]. The field equations for the torsion must then be viewed as applying to an electromagnetic/strong interaction.A classical unified theory thus arises where the underlying geometry confers their contrasting characters to Maxwell-Lorentz electrodynamics and to an Einstein's-like theory of gravity. The highly compact field equations must, however, be developed in phase-spacetime, since the connection is velocity-dependent, i.e., Finsler-like.Further opportunities for similarities with present-day physics are discussed: (a) teleparallelism allows for the formulation of the torsion sector of the theory as a flat space theory with concomitant point-dependent transformations; (b) spinors should replace Lorentz frames in their role as the subjects to which the connection refers; (c) the Dirac equation consistent with the frame bundle for a velocity-dependent metric with Lorentz signature generates a weak-like interaction in the torsion sector. (shrink)

An analysis of the classical-quantum correspondence shows that it needs to identify a preferred class of coordinate systems, which defines a torsionless connection. One such class is that of the locally-geodesic systems, corresponding to the Levi-Civita connection. Another class, thus another connection, emerges if a preferred reference frame is available. From the classical Hamiltonian that rules geodesic motion, the correspondence yields two distinct Klein-Gordon equations and two distinct Dirac-type equations in a general metric, depending on the connection used. (...) Each of these two equations is generally-covariant, transforms the wave function as a four-vector, and differs from the Fock-Weyl gravitational Dirac equation (DFW equation). One obeys the equivalence principle in an often-accepted sense, whereas the DFW equation obeys that principle only in an extended sense. (shrink)

Is Einstein's metric theory of gravitation to be quantized to yield a complete and logically consistent picture of the geometry of the real world in the presence of quantized material sources? To answer this question, we give arguments that there is a consistent way to extend general relativity to small distances by incorporating further geometric quantities at the level of the connection into the theory and introducing corresponding field equations for their determination, allowing thereby the metric and the Levi- (...) class='Hi'>Civita connection to remain classical quantities. The dualism between matter and geometry is extended to quantized fields with the help of a Hibert bundle ℋ raised over a Riemann-Cartan spacetime. Quantized subnuclear matter fields (generalized quantum mechanical wave functions) are sections on ℋ which determine generalized bilinear currents acting as sourc currents for the bundle geometry at small distances. The established dualism between matter and the underlying bundle geometry contains general relativity as a classical part. (shrink)

The boundary of a boundary principle in field theories is described. The difference in treatment of the principle in electrodynamics and general relativity is pointed out and reformulated in terms of underlying mathematical structure of the theories. The problem of unifying the treatment is formulated and solved. The role of E. Cartan's concept of the moment of rotation associated with the curvature of a Levi-Civita connection on a frame bundle is shown to be crucial for the unification. The (...) analysis of the boundary of a boundary principle in Kaluza-Klein theory is performed and the recipe for a unified treatment of the principle in electrodynamics and general relativity is shown to follow from the analysis. It is pointed out that the unification cand be extended to Yang-Mills fields easily. (shrink)

Some concepts that are now part and parcel of mathematics used to be, at least until the beginning of the twentieth century, a central preoccupation of mathematicians and philosophers. The concept of continuity, or the continuous, is one of them. Nowadays, many philosophers of mathematics take it for granted that mathematicians of the last quarter of the nineteenth century found an adequate conceptual analysis of the continuous in terms of limits and that serious philosophical thinking is no longer required, except (...) perhaps when the question of the continuum is transferred to the arena of set theory where it takes the form of the infamous continuum hypothesis. As Philip Ehrlich has recently shown, this conviction goes back to the early writings of Russell who, in 1903 and then again in later writings, forcefully and eloquently pushed the view that mathematicians had given the final answer to immemorial conundrums arising from the continuous and infinitesimals . This proclamation of victory came with what was announced as the necessary defeat of the notion of the infinitesimal, despite the fact that mathematicians like Thomae, Du Bois-Reymond, Stolz, Bettazi, Veronese, Levi-Civita, and Hahn were investigating mathematical structures containing infinitesimals in a mathematically rigorous and logically consistent manner. In this respect Russell was merely walking in the footsteps of Cantor, and many of Russell's contemporaries were only too keen to keep infinitesimals out of Cantor's paradise. However, although Cantor certainly wanted to send the notion of infinitesimal to Hell, the notion kept a low profile in the mathematical purgatory, making its way in the study of non-Archimedean ordered algebraic systems. Of course, nowadays everyone has heard of Robinson's attempt at resurrecting infinitesimals in analysis in the form of non-standard analysis, but Robinson's work, despite the fact that it had, in …. (shrink)

While the agricultural knowledges and practices of Black, Indigenous, and People of Color and women have shaped agriculture in the US, these knowledges have been colonized, exploited, and appropriated, cleaving space for the presently dominant white male agricultural narrative. Simultaneously, these knowledges and practices have been transformed to fit within a society that values individualism, production, efficiency, and profit. The authors use a decolonial Feminist Political Ecology framework to highlight the ways in which the knowledges of Indigenous, Black, and women (...) farmers have been and are being colonized; a tradition that makes alternative agriculture a predominantly white space. The authors interviewed 10 BIPOC and women farmers in Colorado to understand what values and knowledges were shaping their often-appropriated agricultural practices. Three themes emerged: people, place, and patterns. By centering these values, farmers create relational agricultural practices that support the well-being of human and more-than-human beings. To support the widespread implementation of these practices, food systems practitioners must elevate the voices and knowledges of historically excluded farmers. Only then can truly just and equitable alternative agricultural practices be realized in the US. (shrink)

'At last one of the most famous generalizing works in anthropology by the field's most stimulating and controversial contemporary figure has been translated, beautifully, and with the enlightening preface of the second French edition.

Continental philosophy has entered a new period of ferment. The long deconstructionist era was followed with a period dominated by Deleuze, which has in turn evolved into a new situation still difficult to define. However, one common thread running through the new brand of continental positions is a renewed attention to materialist and realist options in philosophy. Among the leaders of the established generation, this new focus takes numerous forms. It might be hard to find many shared positions in the (...) writings of Badiou, DeLanda, Laruelle, Latour, Stengers, and Žižek, but what is missing from their positions is an obsession with the critique of written texts. All of them elaborate a positive ontology, despite the incompatibility of their results. Meanwhile, the new generation of continental thinkers is pushing these trends still further, as seen in currents ranging from transcendental materialism to the London-based speculative realism movement to new revivals of Derrida. As indicated by the title The Speculative Turn, the new currents of continental philosophy depart from the text-centered hermeneutic models of the past and engage in daring speculations about the nature of reality itself. This anthology assembles authors, of several generations and numerous nationalities, who will be at the centre of debate in continental philosophy for decades to come. (shrink)

"Every word, like a sacred object, has its place. No _précis_ is possible. This extraordinary book must be read."—Edmund Carpenter, _New York Times Book Review _ "No outline is possible; I can only say that reading this book is a most exciting intellectual exercise in which dialectic, wit, and imagination combine to stimulate and provoke at every page."—Edmund Leach, _Man _ "Lévi-Strauss's books are tough: very scholarly, very dense, very rapid in argument. But once you have mastered him, human history (...) can never be the same, nor indeed can one's view of contemporary society. And his latest book, _The Savage Mind_, is his most comprehensive and certainly his most profound. Everyone interested in the history of ideas _must_ read it; everyone interested in human institutions _should _read it."—J. H. Plumb, _Saturday Review_ "A constantly stimulating, informative and suggestive intellectual challenge."—Geoffrey Gorer, _The Observer_, London. (shrink)

For three centuries, philosophers have mounted defenses against the melan genie with an obsessive intensity comparable to the Reaganite determination to squander American wealth on defenses against a Communist threat. And for three centuries, skeptics have argued for the futility of the expenditure of conceptual effort with no more success than critics of the Pentagon have had in stemming the flow of funds to the military and its industrial minions. My own sympathies are with the skeptics. However, their own intense (...) involvement with the obsessions of the epistemological fear mongers and their failure to address the analogue of the problem of conversion from military to civilian industry in an acceptable manner often deprive the skeptical position of the subversive force it ought to have. (shrink)

How does valuing something for its own sake differ from valuing an entity for the sake of other things? Although numerous answers come to mind, many of them rule out substantive views about what is valuable for its own sake. I therefore seek to provide a more neutral way to distinguish the two valuing attitudes. Drawing from existing accounts of valuing, I argue that the two can be distinguished in terms of a conative-volitional feature. Focusing first on “non-final valuing”—i.e. valuing_ (...) x_ for the sake of something else—I argue that it involves adopting certain reasons on account of a desire for _x_ to contribute to other things. I then show how this contrasts final valuing. The result, I argue, is a plausible account of how the two modes of valuing differ that leaves open substantive views about what all can be valued for its own sake. This is helpful because it develops a popular methodology used to explore the value of a wide range of things, including natural entities, family heirlooms, and artworks, as well as, more broadly, entities that might have “extrinsic final value.”. (shrink)

Trust and Governance asks several important questions: Is trust really essential to good governance, or are strong laws more important? What leads people either to trust or to distrust government, and what makes officials decide to be trustworthy? Can too much trust render the public vulnerable to government corruption, and if so what safeguards are necessary? In approaching these questions, the contributors draw upon an abundance of resources to offer different perspectives on the role of trust in government. Enriched by (...) perspectives from political science, sociology, psychology, economics, history, and philosophy, Trust and Governance opens a new dialogue on the role of trust in the vital relationship between citizenry and government. (shrink)

There are two uses of randomization in efforts to control systematic bias in experimental design: (a) Alchemical uses seek to convert unavoidable systematic errors into random errors. (b) Hygienic uses seek to reduce the prospect of the experimenter's involvement with the implementation of the experiment contributing to bias. A few remarks are made at the end of the paper about the hygienic use of randomization as a preventative against sticky fingers. The bulk of the discussion addresses the alchemical applications. The (...) thesis is that attitudes towards the cogency of Fisher's alchemical use of randomization ought to depend on views concerning statistical deduction or direct inference. (shrink)

A sketch of a way of characterizing multidimensional value commitments and the way they can come into conflict derived from my book Hard Choices is presented and applied to the question of how to characterize the relevance of identity to value commitments and conflict. The views of A.K. Sen and A. Bilgrami are examined in the light of these ideas.

This major two volume reference work focuses on the works of contemporary Marxism which take as their inspiration the classical Marxian political economy, especially that of Marx, Engels, Lenin, Luxemburg and Gramsci. The authors reprinted here are engaged in the common enterprise of attempting to understand the world in a manner that might facilitate its transformation for the better, or at least help prevent the worst outcomes from predictable and inevitable changes. Committed to the critical, scientific and explanatory project of (...) Marxism, the authors represented in these volumes tend to be structuralist and determinist but they also acknowledge the role of voluntarism, chance or untheorized circumstance. All have borrowed from the philosophical, political, and most importantly, political economic strands of the classical Marxist legacy to create a new and contemporary Marxian political economy. These authoritative volumes will be an essential reference point for an analysis of one of the most influential political ideologies of the 20th century. (shrink)

Anthropology cannot be distinguished from other social and human sciences by its own particular object of study. Apparently concerned with the so-called “primitive” peoples, or peoples “without writing,” it developed into a science at the same time that these peoples were declining, or at least losing their distinctive characteristics. For the last ten years or so, some anthropologists have turned to studying the so-called civilized societies. Clearly, then, anthropology issues less from the existence of a specific object of study than (...) from an original way of formulating problems which are shared by all the sciences of man. Anthropology acquired its importance by studying social phenomena which, because of their strangeness and difference from those of the observer's own society—and not because they were any simpler—afforded an insight into certain properties which are at once general and basic to all social life. We could compare the anthropologist's position in the social sciences to that of the astronomer in the natural sciences: man is apprehended through his remotest manifestations, over a distance which acquires temporal, spatial, and moral value. The distance which separates the anthropologist from his object of study reduces the complexity of what he can see, but, making a virtue out of this constraint, it forces him to perceive only those phenomena which may be considered essential. (shrink)