Ever since its foundation in 1540, the Society of Jesus had had one mission—to restore order where Luther, Calvin and the other instigators of the Reformation had brought chaos. To stop the hemorrhage of believers, the Jesuits needed to form a united front. No signs of internal disagreement could to be shown to the outside world, lest the congregation lose its credibility. But in 1570s two prominent Jesuits, Cristophorus Clavius and Benito Perera, had engaged in a bitter controversy. The issue (...) at stake had apparently nothing to do with the values on which Ignazio of Loyola had built the Society of Jesus. And yet the dispute between Clavius and Perera was matter of concern for the entire Jesuit community. They were... (shrink)

Françios Viète was a geometer in search of better techniques for astronomical calculation. Through his theorem on angular sections he found a use for higher-dimensional geometric magnitudes which allowed him to create an algebra for geometry. We show that unlike traditional numerical algebra, the knowns and unknowns in Viète’s logistice speciosa are the relative sizes of non-arithmetized magnitudes in which the “calculations” must respect dimension. Along with this foundational shift Viète adopted a radically new notation based in Greek geometric equalities. (...) His letters stand for values rather than types, and his given values are undetermined. Where previously algebra was founded in polynomials as aggregations, Viète became the first modern algebraist in working with polynomials built from operations, and the notations reflect these conceptions. Viète’s innovations are situated in the context of sixteenth-century practice, and we examine the interpretation of Jacob Klein, the only historian to have conducted a serious inquiry into the ontology of Viète’s “species”. (shrink)

This article will rework the classical question ‘Can a machine think?’ into a more specific problem: ‘Can a machine think anything new?’ It will consider traditional computational tasks such as prediction and decision-making, so as to investigate whether the instrumentality of these operations can be understood in terms of the creation of novel thought. By addressing philosophical and technoscientific attempts to mechanise thought on the one hand, and the philosophical and cultural critique of these attempts on the other, I will (...) argue that computation’s epistemic productions should be assessed vis-à-vis the logico-mathematical specificity of formal axiomatic systems. Such an assessment requires us to conceive automated modes of thought in such a way as to supersede the hope that machines might replicate human cognitive faculties, and to thereby acknowledge a form of onto-epistemological autonomy in automated ‘thinking’ processes. This involves moving beyond the view that machines might merely simulate humans. Machine thought should be seen as dramatically alien to human thought, and to the dimension of lived experience upon which the latter is predicated. Having stepped outside the simulative paradigm, the question ‘Can a machine think anything new?’ can then be reformulated. One should ask whether novel behaviour in computing might come not from the breaking of mechanical rules, but from following them: from doing what computers do already, and not what we might think they should be doing if we wanted them to imitate us. (shrink)

This article will rework the classical question ‘Can a machine think?’ into a more specific problem: ‘Can a machine think anything new?’ It will consider traditional computational tasks such as prediction and decision-making, so as to investigate whether the instrumentality of these operations can be understood in terms of the creation of novel thought. By addressing philosophical and technoscientific attempts to mechanise thought on the one hand, and the philosophical and cultural critique of these attempts on the other, I will (...) argue that computation’s epistemic productions should be assessed vis-à-vis the logico-mathematical specificity of formal axiomatic systems. Such an assessment requires us to conceive automated modes of thought in such a way as to supersede the hope that machines might replicate human cognitive faculties, and to thereby acknowledge a form of onto-epistemological autonomy in automated ‘thinking’ processes. This involves moving beyond the view that machines might merely simulate humans. Machine thought should be seen as dramatically alien to human thought, and to the dimension of lived experience upon which the latter is predicated. Having stepped outside the simulative paradigm, the question ‘Can a machine think anything new?’ can then be reformulated. One should ask whether novel behaviour in computing might come not from the breaking of mechanical rules, but from following them: from doing what computers do already, and not what we might think they should be doing if we wanted them to imitate us. (shrink)

Los famosos sueños de Descartes han constituido siempre para la historia de la filosofía un incómodo problema, pues dejaban la puerta abierta a la posibilidad de que el origen de la moderna racionalidad hubiese nacido de lo irracional. Superar este problema no fue fácil para el propio Descartes pasados los años ni lo ha sido para quienes desde entonces han mantenido que sus ideas marcaron el origen de una etapa de la filosofía y la ciencia caracterizada por el rechazo de (...) cualquier tipo de saber basado en la iluminación, la revelación o la inspiración mística. La mayoría de los autores han intentado marginar tal carácter irracional de los orígenes de la racionalidad realizando una lectura de los sueños cartesianos a la luz del posterior desarrollo del pensamiento cartesiano. Aquí se propone, en cambio, hacer una valoración de aquellos sueños en el contexto de las inquietudes intelectuales que el francés tenía en 1619. Hay elementos en la descripción de los sueños cartesianos, como la relevancia otorgada al «entusiasmo», o la revelación de una nueva ciencia «desde lo alto», que la vinculan estrechamente con una concepción de la verdad y la razón muy cercana a la de platónicos, filósofos y poetas que posteriormente la historia de la filosofía ha considerado en las antípodas de la moderna racionalidad. (shrink)

The gods that guard the poles have been assigned the function of assembling the separate and unifying the manifold members of the whole, while those appointed to the axes keep the circuits in everlasting revolution around and around. And if I may add my own conceit, the centers and poles of all the spheres symbolize the wry-necked gods by imitating the mysterious union and synthesis which they effect; the axes represent the connectors of all the cosmic orders … and the (...) very spheres are likenesses of the perfecting divinities … 1 Proclus’s Commentary on the First Book of Euclid’s Elements contains one of the most important discussions of the nature of mathematical objects and preserves a wealth of historical information about mathematics from antiquity. However, large sections of the text have been neglected by scholars since the text’s first publication in the sixteenth century. Proclus expounds at length on correspondences between mathematical objects and the gods. At times he uses the language of “theurgy,” the ritual practices that later Neoplatonists insisted were necessary for the human soul to ascend to the intelligibles. The contention of this article is that the Commentary, taken as a whole, concerns the practice of “inner theurgy,” mental rituals used by advanced students to raise the soul beyond its assigned bounds. This practice can be traced back to Plotinus and the very beginnings of Neoplatonism; however, the particular methods presented by Proclus are his own development. This interpretation of the Commentary is supported by Proclus’s works Commentary on the Parmenides and Platonic Theology. Parts of the Commentary that have been studied carefully by modern scholars, especially the theory of the mathematical phantasia or imagination, will be shown to be crucial elements in Proclus’s theory and practice of inner theurgy. (shrink)

The ArgumentIn this paper, I argue that the most significant contribution of the Jesuits to early modern science consists in the introduction of a new “image of knowledge.”In contradistinction to traditional Scholasticism, this image of knowledge allows for the possibility of a science of hypothetical entities.This problem became crucial in two specific areas. In astronomy, knowledge of mathematical entities of unclear ontological status was nevertheless proclaimed certain. In theology, God's knowledge of the future acts of man, logically considered as future (...) contingents, was also proclaimed certain. In both cases the concept of certain knowledge of hypothetical entities was problematic and challenged a central premise of the accepted canons of logic, i.e., that the objects of true knowledge must be real objects.The main argument of this paper is that the practical orientation of the Jesuit cultural milieu enabled Jesuit scientists and theologians to ignore accepted logical considerations and to modify traditional Thomist images of knowledge. Nevertheless, this modification was not so radical as to change the contemporary organization of knowledge. This was due to the peculiar status of the Jesuits within the church establishment, which exposed them to harsh criticism and created a deep need for legitimation. Thus, the limitations of Jesuit scientific culture are accounted for in institutional, rather than in logical terms. (shrink)

Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that (...) this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge. (shrink)

Adriaan van Roomen published an outline of what he called a Mathesis Universalis in 1597. This earned him a well-deserved place in the history of early modern ideas about a universal mathematics which was intended to encompass both geometry and arithmetic and to provide general rules valid for operations involving numbers, geometrical magnitudes, and all other quantities amenable to measurement and calculation. ‘Mathesis Universalis’ (MU) became the most common (though not the only) term for mathematical theories developed with that aim. (...) At some time around 1600 van Roomen composed a new version of his MU, considerably different from the earlier one. This second version was never effectively published and it has not been discussed in detail in the secondary literature before. The text has, however, survived and the two versions are presented and compared in the present article. Sections 1–6 are about the first version of van Roomen’s MU the occasion of its publication (a controversy about Archimedes’ treatise on the circle, Sect. 2), its conceptual context (Sect. 3), its structure (with an overview of its definitions, axioms, and theorems) and its dependence on Clavius’ use of numbers in dealing with both rational and irrational ratios (Sect. 4), the geometrical interpretation of arithmetical operations multiplication and division (Sect. 5), and an analysis of its content in modern terms. In his second version of a MU van Roomen took algebra into account, inspired by Viète’s early treatises; he planned to publish it as part of a new edition of Al-Khwarizmi’s treatise on algebra (Sect. 7). Section 8 describes the conceptual background and the difficulties involved in the merging of algebra and geometry; Sect. 9 summarizes and analyzes the definitions, axioms and theorems of the second version, noting the differences with the first version and tracing the influence of Viète. Section 10 deals with the influence of van Roomen on later discussions of MU, and briefly sketches Descartes’ ideas about MU as expressed in the latter’s Regulae. (shrink)