Our brains make up a series of signs and are engaged in making or manifesting or reacting to a series of signs: through this semiotic activity they are at the same time engaged in “being minds” and so in thinking intelligently. An important effect of this semiotic activity of brains is a continuous process of “externalization of the mind” that exhibits a new cognitive perspective on the mechanisms underling the semiotic emergence of abductive processes of meaning formation. To illustrate this (...) process I will take advantage of the analysis of some aspects of the cognitive interplay between internal and external representations. I consider this interplay critical in analyzing the relation between meaningful semiotic internal resources and devices and their dynamical interactions with the externalized semiotic materiality suitably stocked in the environment. Hence, minds are material, “extended” and artificial in themselves. A considerable part of human abductive thinking is occurring through an activity consisting in a kind of reification in the external environment and a subsequent re-projection and reinterpretation through new configurations of neural networks and chemical processes. I also illustrate how this activity takes advantage of hybrid representations and how it can nicely account for various processes of creative and selective abduction, bringing up the question of how multimodal aspects involving a full range of sensory modalities are important in hypothetical reasoning. (shrink)

Technology moves us to a better world. We contend that through technology people can simplify and solve moral tasks when they are in presence of incomplete information and possess a diminished capacity to act morally. Many external things, usually inert from the moral point of view, can be transformed into the so-called moral mediators. Hence, not all of the moral tools are inside the head, many of them are shared and distributed in “external” objects and structures which function as ethical (...) devices. (shrink)

What I call semiotic brains are brains that make up a series of signs and that are engaged in making or manifesting or reacting to a series of signs: through this semiotic activity they are at the same time engaged in “being minds” and so in thinking intelligently. An important effect of this semiotic activity of brains is a continuous process of disembodiment of mind that exhibits a new cognitive perspective on the mechanisms underling the semiotic emergence of meaning processes. (...) Indeed at the roots of sophisticated thinking abilities there is a process of disembodiment of mind that presents a new cognitive perspective on the role of external models, representations, and various semiotic materials. Taking advantage of Turing’s comparison between “unorganized” brains and “logical” and “practical” machines” this paper illustrates the centrality to cognition of the disembodiment of mind from the point of view of the interplay between internal and external representations, both mimetic and creative. The last part of the paper describes the concept of mimetic mind I have introduced to shed new cognitive and philosophical light on the role of computational modeling and on the decline of the so-called Cartesian computationalism. (shrink)

Philosophers have usually offered a number of ways of describing hypotheses generation, but all aim at demonstrating that the activity of generating hypotheses is paradoxical, illusory or obscure, and then not analysable. Those descriptions are often so far from Peircian pragmatic prescription and so abstract to result completely unknowable and obscure. The “computational turn” gives us a new way to understand creative processes in a strictly pragmatic sense. In fact, by exploiting artificial intelligence and cognitive science tools, computational philosophy allows (...) us to test concepts and ideas previously conceived only in abstract terms. It is in the perspective of these actual computational models that I find the central role of abduction in the explanation of creative reasoning in science. Creativity and discovery are no more seen as a mysterious irrational process, but, thanks to constructive accounts, as a complex relationship among different inferential steps that can be clearly analysed and identified. I maintain that the computational philosophy analysis of model-based and manipulative abduction and of external and epistemic mediators is important not only to delineate the actual practice of abduction, but also to further enhance the development of programs computationally adequate in rediscovering, or discovering for the first time, for example, scientific hypotheses or mathematical theorems. (shrink)

We explore the potential of Simon Stevin’s numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, (...) Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity. (shrink)

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, (...) the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered. (shrink)

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum (...) with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others. (shrink)

Classically, an osculating circle at a point of a planar curve is introduced technically, often with formula giving its radius and the coordinates of its center. In this note, we propose a new and intuitive definition of this concept: among all the circles which have, on the considered point, the same tangent as the studied curve and thus seem equal to the curve through a microscope, the osculating circle is this that seems equal to the curve through a microscope within (...) microscope. (shrink)