The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also (...) central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity. (shrink)

ArgumentThis article offers a social history of the “Galois Affair,” which arose in 1831 when the French Academy of Sciences decided to reject a paper presented by an aspiring mathematician, Évariste Galois. In order to historicize the meaning of Galois's work at the time he tried to earn recognition for his research on the algebraic solution of equations, this paper explores two interrelated questions. First, it analyzes scholarly algebraic practices and the way mathematicians were trained in the nineteenth century to (...) reconstruct the scientific meaning of Galois's research. By emphasizing the historically contingent nature of scientific debates, the paper argues that to historicize the work of a mathematician, we must historicize the scientific world as well. The second part of this article goes further to seek a fresh perspective on Galois's short career by analyzing the cultural, social, and institutional dynamics that governed how the mathematical community came to recognize an aspiring mathematician as one of its own. (shrink)

Much writing on scientific biography focuses on the legitimacy and utility of this genre. In contrast, this essay discusses a variety of genre conventions and imperatives which continue to exert a powerful influence on the selection of biographical subjects, and to control the plot and structure of the ensuing biographies. These imperatives include the following: the plot templates of the Bildungsroman (the realistic novel of individual self-development), the life trajectories of Weberian ideal types, and the functional elements and personae of (...) the folkloric tale of the "hero's quest." The essay discusses the nature and application of these genre conventions in some detail, with the conclusion that biography, however useful, exerts a powerfully distorting influence on the image of how most science gets done. (shrink)

Definitions of nature and transcendence are given, and the framework of Hindu thought is presented. The levels of reality as discovered by physics are then discussed, which leads us to revise our notions of reality and objectivity. Transcendence is defined as something beyond matter‐energy in space‐time and is explored in several contexts of modern science, as in pre‐Big‐Bang state, negative entropy, information, complexity, and others. Finally, a philosophical reflection on consciousness is presented.

200 years ago, on August 5, 1802, Niels Henrik Abel was born on Finnøy near Stavanger on the Norwegian west coast. During a short life span, Abel contributed to a deep transition in mathematics in which concepts replaced formulae as the basic objects of mathematics. The transformation of mathematics in the 1820s and its manifestation in Abel’s works are the themes of the author’s PhD thesis. After sketching the formative instances in Abel’s well-known biography, this article illustrates two aspects of (...) the transformation which concern the introduction of concept based mathematics and the related shift in standards of mathematical rigor. Furthermore, the article outlines some of the many bicentennial celebrations in Norway and gives a short, thematic introduction to the literature on Abel and his work. (shrink)

This essay argues that through a new and radical relationship with digital technologies that are oriented towards networking and automaticity, humans have become estranged from what philosopher Arnold Gehlen termed the ‘circle of action’ that expressed our ancient adaptation to tool use and constituted the basis for our capacity for reflective consciousness. The objectification of the material and analogue relationship that enabled humans to ‘act’ upon the world and to construct the basis for our collective endeavours, this paper shows, is (...) beginning to redirect the social orientation that tool use enabled, back to a form of individualism I term a ‘post-modern vanity’. Computer-driven automation on a networked and global scale is engendering a new illusion regarding our relationship with technology. Greek philosophy, later religion, and then the Enlightenment all saw our relation to technology as evidence of either our ‘specialness’ or our inherent ‘mastery over nature’. These perspectives resulted in our acquiring a ‘reasoned self-delusion’ wherein for the individual life had meaning and was ontologically tolerable—we thought ourselves to be of a realm beyond raw nature. Automation, however, to an unprecedented degree, objectifies and exteriorizes the processes of technology use, alienating the individual from the ‘circle of action’. This process is throwing us back to another form of what this essay terms a ‘reasoned self-delusion’ where we increasingly believe that we no longer need each other so much. The cult of computer efficiency and the virtual sphere of the network that it has spawned, I argue, is a largely barren sphere of radical individualism, a post-modern vanity, where we exist at a borderland closer to nature and to the animals than we have been for thousands of years. (shrink)

Two recently published scholarly biographies of leading nineteenth-century British mathematicians, pioneers in abstract algebra and personal friends, provide an informed comparison of their lives and work.

Since the discovery of incommensurability in ancient Greece, arithmeticism and geometricism constantly switched roles. After ninetieth century arithmeticism Frege eventually returned to the view that mathematics is really entirely geometry. Yet Poincaré, Brouwer, Weyl and Bernays are mathematicians opposed to the explication of the continuum purely in terms of the discrete. At the beginning of the twenty-first century ‘continuum theorists’ in France (Longo, Thom and others) believe that the continuum precedes the discrete. In addition the last 50 years witnessed the (...) revival of infinitesimals (Laugwitz and Robinson—non-standard analysis) and—based upon category theory—the rise of smooth infinitesimal analysis and differential geometry. The spatial whole-parts relation is irreducible (Russell) and correlated with the spatial order of simultaneity. The human imaginative capacities are connected to the characterization of points and lines (Euclid) and to the views of Aristotle (the irreducibility of the continuity of a line to its points), which remained in force until the ninetieth century. Although Bolzano once more launched an attempt to arithmetize continuity, it appears as if Weierstrass, Cantor and Dedekind finally succeeded in bringing this ideal to its completion. Their views are assessed by analyzing the contradiction present in Grünbaum’s attempt to explain the continuum as an aggregate of unextended elements (degenerate intervals). Alternatively a line-stretch is characterized as a one-dimensional spatial subject, given at once in its totality (as a whole) and delimited by two points—but it is neither a breadthless length nor the (shortest) distance between two points. The overall aim of this analysis is to account for the uniqueness of discreteness and continuity by highlighting their mutual interconnections exemplified in the nature of a line as a one-dimensional spatial subject, while acknowledging that points are merely spatial objects which are always dependent upon an extended spatial subject. Instead of attempting to reduce continuity to discreteness or discreteness to continuity, a third alternative is explored: accept the irreducibility of number and space and then proceed by analyzing their unbreakable coherence. The argument may be seen as exploring some implications of the view of John Bell, namely that the “continuous is an autonomous notion, not explicable in terms of the discrete.” Bell points out that initially Brouwer, in his dissertation of 1907, “regards continuity and discreteness as complementary notions, neither of which is reducible to each other.”. (shrink)

Lichtenberg, the German philosopher and physicist, once remarked: “It is almost impossible to carry the torch of truth through a crowd without singeing someone's beard.” Goethe, coeval with Lichtenberg, possessed by the conviction that he too was the bearer of truth, recked not whose beard was singed. Between his scientific attitude and his philosophic insight, however, a contradiction was patent, as revealed in his maxim that truth is a torch and that it is only with blinking eyes that we try (...) to get past it, whilst in actual terror of being burnt. (shrink)