The natures of numbers in and around Bombelli’s L’algebra

The purpose of this article is to analyse the mathematical practices leading to Rafael Bombelli’s L’algebra (1572). The context for the analysis is the Italian algebra practiced by abbacus masters and Renaissance mathematicians of the fourteenth to sixteenth centuries. We will focus here on the semiotic aspects of algebraic practices and on the organisation of knowledge. Our purpose is to show how symbols that stand for underdetermined meanings combine with shifting principles of organisation to change the character of algebra.     

The coming-to-be of Hansen’s method

This article by Curtis Wilson is an account of the origin of Hansen’s powerful systematic method for finding contributions of higher order perturbations in celestial mechanics. Hansen’s method was developed in the course of improving on Laplace’s treatment of the mutual perturbations of Jupiter and Saturn. This method, an entirely new way of doing celestial mechanics when it first appeared, later made possible the successful treatment of the complicated motions of our moon (see Wilson 2010). In this paper Wilson gives a brief historical introduction followed by an account of relevant technical details of the Laplacian background, an account illustrating technical details in Hansen’s initial development in his Disquisitions of 1829, and a treatment illustrating details contributing to the achievement of Hansen’s more refined development in his Untersuchung of 1831. These details include conditional equations Hansen provides for checking the accuracy of calculations. Wilson also includes a detailed assessment showing the extraordinary improvement in empirical accuracy of Hansen’s treatment over the best earlier treatment of the Jupiter-Saturn  interactions.     

The role of inflammation in sporadic and familial Parkinson’s disease

The etiology of Parkinson’s disease (PD) is complex and most likely involves numerous environmental and heritable risk factors. Interestingly, many genetic variants, which have been linked to familial forms of PD or identified as strong risk factors, also play a critical role in modulating inflammatory responses. There has been considerable debate in the field as to whether inflammation is a driving force in neurodegeneration or simply represents a response to neuronal death. One emerging hypothesis is that inflammation plays a critical role in the early phases of neurodegeneration. In this review, we will discuss emerging aspects of both innate and adaptive immunity in the context of the pathogenesis of PD. We will highlight recent data from genetic and functional studies that strongly support the theory that genetic susceptibility plays an important role in modulating immune pathways and inflammatory reactions, which may precede and initiate neuronal dysfunction and subsequent neurodegeneration. A detailed understanding of such cellular and molecular inflammatory pathways is crucial to uncover pathogenic mechanisms linking sporadic and hereditary PD and devise tailored neuroprotective interventions.     

On the equivalence of Goodman’s and Hempel’s paradoxes

Historically, Nelson Goodman’s paradox involving the predicates ‘grue’ and ‘bleen’ has been taken to furnish a serious blow to Carl Hempel’s theory of confirmation in particular and to purely formal theories of confirmation in general. In this paper, I argue that Goodman’s paradox is no more serious of a threat to Hempel’s theory of confirmation than is Hempel’s own paradox of the ravens. I proceed by developing a suggestion from R. D. Rosenkrantz into an argument for the conclusion that these paradoxes are, in fact, equivalent. My argument, if successful, is of both historical and philosophical interest. Goodman himself maintained that Hempel’s theory of confirmation was capable of handling the paradox of the ravens. And Hempel eventually conceded that Goodman’s paradox showed that there could be no adequate, purely syntactical theory of confirmation. The conclusion of my argument entails, by contrast, that Hempel’s theory of confirmation is incapable of handling Goodman’s paradox if and only if it is incapable of handling the paradox of the ravens. It also entails that for any adequate solution to one of these paradoxes, there is a corresponding and equally adequate solution to the other.     

The semantic conception and the structuralist view of theories: A critique of Suppe’s criticisms

Different conceptions of scientific theories, such as the state spaces approach of Bas van Fraassen, the phase spaces approach of Frederick Suppe, the set-theoretical approach of Patrick Suppes, and the structuralist view of Joseph Sneed et al. are usually put together into one big family. In addition, the definite article is normally used, and thus we speak of the semantic conception (view or approach) of theories and of its different approaches (variants or versions). However, in The Semantic Conception of Theories and Scientific Realism (Urban and Chicago: University of Illinois Press, 1989), starting from certain remarks already made in “Theory Structure” (in P. Asquith and H. Kyburg (Eds.), Current Research in Philosophy of Science, East Lansing: Philosophy of Science Association, 1979, pp. 317–338), Frederick Suppe excludes the structuralist view as well as other “European” versions from the semantic conception of theories. In this paper I will critically examine the reasons put forward by Suppe for this decision and, later, I will provide a general characterization of the semantic family and of the structuralist view of theories in such a way as to justify the inclusion of the structuralist view (as well as other “European” versions) as a member of this family.     

The Principia’s second law (as Newton understood it) from Galileo to Laplace

Archive for History of Exact Sciences - Newton certainly regarded his second law of motion in the Principia as a fundamental axiom of mechanics. Yet the works that came after the Principia, the...     

The professor and the pea: Lives and afterlives of William Bateson’s campaign for the utility of Mendelism

As a defender of the fundamental importance of Mendel’s experiments for understanding heredity, the English biologist William Bateson (1861–1926) did much to publicize the usefulness of Mendelian science for practical breeders. In the course of his campaigning, he not only secured a reputation among breeders as a scientific expert worth listening to but articulated a vision of the ideal relations between pure and applied science in the modern state. Yet historical writing about Bateson has tended to underplay these utilitarian elements of his program, to the extent of portraying him, notably in still-influential work from the 1960s and 1970s, as a type specimen of the scientist who could not care less about application. This paper offers a corrective view of Bateson himself—including the first detailed account of his role as an expert witness in a courtroom dispute over the identity of a commercial pea variety—and an inquiry into the historiographic fate of his efforts in support of Mendelism’s productivity. For all that a Marxian perspective classically brings applied science to the fore, in Bateson’s case, and for a range of reasons, it did the opposite during the Cold War.     

Borelli’s edition of books V–VII of Apollonius’s Conics,and Lemma 12 in Newton’s Principia

To solve the direct problem of central forces when the trajectory is an ellipse and the force is directed to its centre, Newton made use of the famous Lemma 12 (Principia, I, sect. II) that was later recognized equivalent to proposition 31 of book VII of Apollonius’s Conics. In this paper, in which we look for Newton’s possible sources for Lemma 12, we compare Apollonius’s original proof, as edited by Borelli, with those of other authors, including that given by Newton himself. Moreover, after having retraced its editorial history, we evaluate the dissemination of Borelli's edition of books V-VII of Apollonius’s Conics before the printing of the Principia.

     

The foundational aspects of Gauss’s work on the hypergeometric,factorial and digamma functions

In his writings about hypergeometric functions Gauss succeeded in moving beyond the restricted domain of eighteenth-century functions by changing several basic notions of analysis. He rejected formal methodology and the traditional notions of functions, complex numbers, infinite numbers, integration, and the sum of a series. Indeed, he thought that analysis derived from a few, intuitively given notions by means of other well-defined concepts which were reducible to intuitive ones. Gauss considered functions to be relations between continuous variable quantities while he regarded integration and summation as appropriate operations with limits. He also regarded infinite and infinitesimal numbers as a façon de parler and used inequalities in order to prove the existence of certain limits. He took complex numbers to have the same legitimacy as real quantities. However, Gauss’s continuum was linked to a revised form of the eighteenth-century notion of continuous quantity: it was not reducible to a set of numbers but was immediately given.     

The collaboration of Emil Artin and George Whaples: Artin’s mathematical circle extends to America

In his biography of Emil Artin, Richard Brauer describes the years from 1931–1941 as a time when “Artin spoke through his students and through the members of his mathematical circle” rather than through written publications. This paper explores these seemingly quiet years when Artin immigrated to America and disseminated ideas about algebraic number theory during this time in his collaboration with George Whaples, a young American mathematician who had just completed his Ph.D. at the University of Wisconsin. The main result of their work is the use of the product formula for valuations to come up with an axiomatic characterization of both algebraic number fields and algebraic function fields with a finite field of constants. These two families of fields are exactly the fields for which class field theory is known to hold. We situate their mathematical work in the broader context of algebraic number theory and their lives within the broader historical context.     

Lagrange’s theory of analytical functions and his ideal of purity of method

We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and we concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem—especially on the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of the discipline.     

The making of an intrinsic property: “Symmetry heuristics” in early particle physics

Mathematical invariances, usually referred to as “symmetries”, are today often regarded as providing a privileged heuristic guideline for understanding natural phenomena, especially those of micro-physics. The rise of symmetries in particle physics has often been portrayed by physicists and philosophers as the “application” of mathematical invariances to the ordering of particle phenomena, but no historical studies exist on whether and how mathematical invariances actually played a heuristic role in shaping microphysics. Moreover, speaking of an “application” of invariances conflates the formation of concepts of new intrinsic degrees of freedom of elementary particles with the formulation of models containing invariances with respect to those degrees of freedom. I shall present here a case study from early particle physics (ca. 1930–1954) focussed on the formation of one of the earliest concepts of a new degree of freedom, baryon number, and on the emergence of the invariance today associated to it. The results of the analysis show how concept formation and “application” of mathematical invariances were distinct components of a complex historical constellation in which, beside symmetries, two further elements were essential: the idea of physically conserved quantities and that of selection rules. I shall refer to the collection of different heuristic strategies involving selection rules, invariances and conserved quantities as the “SIC-triangle” and show how different authors made use of them to interpret the wealth of new experimental data. It was only a posteriori that the successes of this hybrid “symmetry heuristics” came to be attributed exclusively to mathematical invariances and group theory, forgetting the role of selection rules and of the notion of physically conserved quantity in the emergence of new degrees of freedom and new invariances. The results of the present investigation clearly indicate that opinions on the role of symmetries in fundamental physics need to be critically reviewed in the spirit of integrated history and philosophy of science.     

Today’s and tomorrow’s imaging and circulating biomarkers for pulmonary arterial hypertension

The pathobiology of pulmonary arterial hypertension (PAH) involves a remodeling process in distal pulmonary arteries, as well as vasoconstriction and in situ thrombosis, leading to an increase in pulmonary vascular resistance, right heart failure and death. Its etiology may be idiopathic, but PAH is also frequently associated with underlying conditions such as connective tissue diseases. During the past decade, more than welcome novel therapies have been developed and are in development, including those increasingly targeting the remodeling process. These therapeutic options modestly increase the patients' long-term survival, now approaching 60% at 5 years. However, non-invasive tools for confirming PAH diagnosis, and assessing disease severity and response to therapy, are tragically lacking and would help to select the best treatment. After exclusion of other causes of pulmonary hypertension, a final diagnosis still relies on right heart catheterization, an invasive technique which cannot be repeated as often as an optimal follow-up might require. Similarly, other techniques and biomarkers used for assessing disease severity and response to treatment generally lack specificity and have significant limitations. In this review, imaging as well as current and future circulating biomarkers for diagnosis, prognosis, and follow-up are discussed.