Has the emergence of post-positivism in philosophy of science changed the terms of the “is/ought” dichotomy? If it has demonstrated convincingly that there are no “facts” apart from the theoretical frames and evaluative standards constructing them, can such a cordon sanitaire really be upheld between “facts” and values? The point I wish to stress is that philosophy of science has had a central role in constituting and imposing the fact/value dichotomy and a revolution in the philosophy of science should not (...) leave the dichotomy unaffected. The connection between post-positivism and naturalism will be my guiding thread in considering this “last dogma of positivism.” First this essay will specify the sense of naturalism that it will take to be essential to the post-positivist philosophy of science: the deflation of the notion of the “purity” of scientific knowledge. Then it will turn to the question of the implications that follow for the “autonomy” of ethics, including the danger posed by a new form of scientistic reductionism. (shrink)

We provide an overview of three ways in which the expression “Historical epistemology” (HE) is often understood: (1) HE as a study of the history of higher-order epistemic concepts such as objectivity, observation, experimentation, or probability; (2) HE as a study of the historical trajectories of the objects of research, such as the electron, DNA, or phlogiston; (3) HE as the long-term study of scientific developments. After laying out various ways in which these agendas touch on current debates within both (...) epistemology and philosophy of science (e.g., skepticism, realism, rationality of scientific change), we conclude by highlighting three topics as especially worthy of further philosophical investigation. The first concerns the methods, aims and systematic ambitions of the history of epistemology. The second concerns the ways in versions of HE can be connected to versions of naturalized and social epistemologies. The third concerns the philosophy of history, and in particular the level of analysis at which a historical analysis should aim. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of \. Here (...) Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? As well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an “unrigorous” practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory’s teacher degli Angeli, and Gregory’s books at Venice, in the late 1660s. (shrink)

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

Scientific concepts and conceptual systems (theories) are particular forms of higher mental activity. They are cognitive organs that provide the ability of systematic cognition of phenomena, which are not available to the grasp of ordinary sense organs. They are tools of scientific “groping” of phenomena. Scientific concepts free perceptual and cognitive activity from determination of ordinary sense organs by providing a high degree of cognitive abstraction and generalization. Scientific cognition, like perceptual activity, is actualized by consciousness but outside the consciousness.