The Geometrical Method The Geometrical Method is the style of proof that was used in Euclid’s proofs in geometry, and that was used in philosophy in Spinoza’s proofs in his Ethics. The term appeared first in 16th century Europe when mathematics was on an upswing due to the new science of mechanics. … Continue reading Geometrical Method →.

The object of this paper is to show the predominance of the influence of geometrical ideas in Aristotle's account of first principles in the Posterior Analytics— to show that his analysis of first principles is in its essentials an analysis of the first principles of geometry as he conceived them. My proof of this falls into two parts. I. A consideration of the parallel between Aristotle's and Euclid's account of first principles. II. A comparison between the general movement of (...) thought in the Aristotelian παγωγ and πδειξις and in the δινοια and νησις of the Platonic dialectic. (shrink)

The object of this paper is to show the predominance of the influence of geometrical ideas in Aristotle's account of first principles in the Posterior Analytics— to show that his analysis of first principles is in its essentials an analysis of the first principles of geometry as he conceived them. My proof of this falls into two parts. I. A consideration of the parallel between Aristotle's and Euclid's account of first principles. II. A comparison between the general movement of (...) thought in the Aristotelian παγωγ and πδειξις and in the δινοια and νησις of the Platonic dialectic. (shrink)

This book is the fruit of twenty-five years of study of Spinoza by the editor and translator of a new and widely acclaimed edition of Spinoza's collected works. Based on three lectures delivered at the Hebrew University of Jerusalem in 1984, the work provides a useful focal point for continued discussion of the relationship between Descartes and Spinoza, while also serving as a readable and relatively brief but substantial introduction to the Ethics for students. Behind the Geometrical Method (...) is actually two books in one. The first is Edwin Curley's text, which explains Spinoza's masterwork to readers who have little background in philosophy. This text will prove a boon to those who have tried to read the Ethics, but have been baffled by the geometrical style in which it is written. Here Professor Curley undertakes to show how the central claims of the Ethics arose out of critical reflection on the philosophies of Spinoza's two great predecessors, Descartes and Hobbes. The second book, whose argument is conducted in the notes to the text, attempts to support further the often controversial interpretations offered in the text and to carry on a dialogue with recent commentators on Spinoza. The author aligns himself with those who interpret Spinoza naturalistically and materialistically. (shrink)

This book is the fruit of twenty-five years of study of Spinoza by the editor and translator of a new and widely acclaimed edition of Spinoza's collected works.

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Rhetoric 34.4 (2001) 292-307 [Access article in PDF] The Rhetoric of the Geometrical Method Spinoza's Double Strategy Christopher P. Long A double strategy may be apprehended in the first definitions, axioms and propositions of Spinoza's Ethics: the one is rhetorical, the other, systematic. Insofar as these opening passages constitute a geometrical argument that leads ultimately to the strict monism that lies at the heart (...) of Spinoza's philosophy, they function systematically as grounding principles. However, insofar as they are designed to uncover errors at the heart of the Cartesian system, they function rhetorically as a critique of Descartes's equivocal theory of substance. The geometrical mode of presentation lends an aura of necessity to the systematic strategy and tends to eclipse the rhetorical strategy that functions as a critique of Cartesian dualism. 1 To this extent, the systematic strategy itself functions rhetorically. On the other hand, to the extent that any seventeenth-century attempt to establish a monistic system would inevitably have had to involve a critique of Cartesian dualism, the rhetorical strategy itself functions systematically. Thus, although two distinct strategies are simultaneously discernible in the opening passages of the Ethics, the two work together, each arguing on a different front, to establish Spinoza's monism. By focusing on the rhetorical side of this double strategy, the underlying nature of Spinoza's critique of Descartes can be clarified and certain gaps in the systematic, geometrical argument for monism can be explained. Descartes's theory of substance The rhetorical strategy Spinoza employs at the outset of the Ethics involves a brilliant use of equivocation and ambiguity designed to draw a reader dedicated to Cartesian dualism into the heart of an argument designed to [End Page 292] undermine this dualism itself. The impetus behind this rhetorical critique is surely Spinoza's dissatisfaction with Descartes's theory of substance. 1. Descartes's equivocation Descartes's theory rests upon a fundamental equivocation: on the one hand, the term "substance" is meant strictly to designate that which is absolutely self-sufficient, namely, God; on the other hand, Descartes allows for a weaker understanding of substance, which pertains to those things (specifically res cogitans and res extensa) that require only the concurrence of God in order to exist (Principles I, 51-52; AT VIIIA24-25). 2 Thus, for Descartes, although God is, strictly speaking, the only substance, there remains a sense in which both res cogitans and res extensa may be understood as substances as well.Ultimately, this equivocal understanding of substance may be traced to a tension internal to Descartes between the Aristotelian notion of substance as underlying subject and the seventeenth-century mutation of that notion, which came to be defined in terms of causal self-sufficiency. 3 Jonathan Bennett traces the source of this mutation back to the view held by Aristotle that the existence of some property depends upon its being instantiated--that is, there can be no courage unless there exists someone courageous. He continues his interesting argument thus: This suggests that there is a relation of dependence running from properties to the things that have them and not conversely. The dependence would be logical: the thesis must be that "existent but uninstantiated property" is conceptually defective, not merely contrary to physics. Thus, one mark of a genuine thing, or substance, is its not being logically dependent for its existence on anything else. (Bennett 1984, 56) The development of this line of thinking leads to the conception of substance we find both in Descartes's Principles and in Spinoza's Ethics, namely, that a substance must not causally depend on anything outside of itself for its existence. 4 However, with Descartes, what emerges out of this tradition is a hybrid understanding of substance in which the term "substance" is not reserved simply for that which is utterly self-sufficient (God), but extends to those things that depend only upon the concurrence of God for their existence (rescogitans and resextensa). This bifurcated understanding of substance seems to stem from the distinction at work in [End Page 293] Aristotle... (shrink)

As a generalization of both single-valued neutrosophic element and hesitant fuzzy element, single-valued neutrosophic hesitant fuzzy element is an efficient tool for describing uncertain and imprecise information. Thus, it is of great significance to deal with single-valued neutrosophic hesitant fuzzy information for many practical problems. In this paper, we study the aggregation of SVNHFEs based on some normalized operations from geometric viewpoint. Firstly, two normalized operations are defined for processing SVNHFEs. Then, a series of normalized aggregation operators which fulfill some (...) basic conditions of a valid aggregation operator are proposed. Additionally, a decision-making method is developed for resolving multiattribute decision-making problems based on the proposed operators. Finally, a numerical example is provided to illustrate the feasibility and effectiveness of the method. (shrink)

We give a brief overview of the evolution of mathematics, starting from antiquity, through Renaissance, to the 19th century, and the culmination of the train of thought of history’s greatest thinkers that lead to the grand unification of geometry and algebra. The goal of this paper is not a complete formal description of any particular theoretical framework, but to show how extremisation of mathematical rigor in requiring everything be drivable directly from first principles without any arbitrary assumptions actually leads to (...) relaxing the computational difficulty along with maximal conceptual clarity. With this, we consider a revision of the foundations of elementary geometry and algebra based on the work of Grassmann and Clifford and apply it to conceptual and practical problems of past and present modern mathematics and mathematical physics. (shrink)

Packing optimization problems have a wide spectrum of real-word applications. One of the applications of the problems is problem of placement of containers with spent nuclear fuel on the storage platform. The solution of the problem can be reduced to the solution of the problem of finding the optimal placement of a given set of congruent circles into a multiconnected domain taking into account technological restrictions. A mathematical model of the prob-lem is constructed and its peculiarities are considered. Our approach (...) is based on the mathematical modelling of rela-tions between geometric objects by means of phi-function technique. That allowed us to reduce the problem solving to nonlinear programming. Today, an important scientific problem is the problem of creating conditions for safe storage of spent nuclear fuel. In the process of creating any dry spent nuclear fuel storage, the following main stages can be identified: site selection, storage design, construction, operation and decommissioning. A full check for compliance of the repository and its elements with these standards usually begins at the design stage. At the stage of site selection, the inspection for compliance with safety standards is carried out only in terms of the impact of the repository as a whole on the environment. This approach cannot be considered fully appropriate, because, taking into account, for example, all the climatic features of the future storage site, it is possible to adjust the thermal storage regimes of spent nuclear fuel. Similarly, it can be considered necessary to analyze and select the shape of the storage site in order to accommo-date the maximum possible number of spent fuel containers. Such a choice, obviously, should be made taking into ac-count the norms of nuclear, radiation and thermal safety, as well as in compliance with technological limitations. The problem of finding the optimal placement of containers taking into account the given technological limitations can be formulated in the form of the problem of optimization of geometric design. Therefore, the purpose of the study is to build a mathematical model of the problem and study its characteristics to develop effective methods of solution. The proposed approach is based on mathematical modeling of relations between geometric objects using the method of phi-functions. This allowed to reduce the solution of the problem to the problem of nonlinear programming. (shrink)

Charlotte Angas Scott (1858–1932) was an internationally renowned geometer, the first British woman to earn a doctorate in mathematics, and the chair of the Bryn Mawr mathematics department for forty years. There she helped shape the burgeoning mathematics community in the United States. Scott often motivated her research as providing a “geometric treatment” of results that had previously been derived algebraically. The adjective “geometric” likely entailed many things for Scott, from her careful illustration of diagrams to her choice of references (...) and citations. This article will focus on Scott’s striking and consistent use of geometric to describe a reality of dynamic points, lines, planes, and spaces that could be manipulated analogously to physical objects. By providing geometric interpretations of algebraic derivations, Scott committed to an early-nineteenth-century aesthetic vision of a “whole” analytical geometry that she adapted to modern research areas. (shrink)

Much of our information comes to us indirectly, in the form of conclusions others have drawn from evidence they gathered. When we hear these conclusions, how can we modify our own opinions so as to gain the benefit of their evidence? In this paper we study the method known as geometric pooling. We consider two arguments in its favour, raising several objections to one, and proposing an amendment to the other.

In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF₀ given by Pierce and Pillay. Moreover, it provides a geometric alternative to the axiomatizations obtained (...) by Tressl and Guzy/Point in separate papers where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that, in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 63.4 (2002) 599-617 [Access article in PDF] Thinking Geometrically in Pierre-Daniel Huet's Demonstratio evangelica (1679) April G. Shelford Sometime after 1679, Pierre-Daniel Huet (1630-1721) indulged an author's vanity by comparing his Demonstratio evangelica with works whose authors are far better known today. He recorded his judgments on a scrap of paper. 1First, he contrasted the Demonstratio to Antoine Arnauld's Les nouveaux élémens de (...) géometrie (1664). While Arnauld had assumed that all the principles of geometry were true and all geometrical demonstrations certain, and had considered religious questions not amenable to methodical demonstration, Huet noted, "I made clear the contrary by proving religion through a methodical sequence of propositions similar to those one finds in geometry." For Arnauld religious belief fell into the class of things better proven through a series of mutually reinforcing propositions than through proofs modeled on geometry; Huet had actually accomplished the latter. Arnauld did not require geometric proofs in religious works; Huet wrote the Demonstratio "to satisfy those who do." Indeed, Arnauld forbade such proofs, "but I permitted them... and I proposed something to satisfy those who do [demand them]."Huet then considered Blaise Pascal's Pensées (1670). Pascal had also rejected demonstration in proving religious truths, Huet noted, opting instead for "moral proofs, which are directed more to the heart than to the mind. In other words he desired to work harder at moving and disposing the heart than at convincing and persuading the intellect." Apparently, "M. Pascal did not believe that one could prove the truth of the Christian religion by geometric proofs; much less that he had any hope of doing so." [End Page 599]Huet's jottings precisely situate the Demonstratio evangelica (1679)between the claims of Cartesian rationalism and Pascal's fideism and religion of the heart, inviting us to take another look at an imperfectly understood work. Scholarly treatment of the Demonstratio has generally been cursory or partial because it has been considered less for its own merits than for how it fits into larger research agendas. 2 Some scholars have offered important insights but either do not develop them or do not consider the Demonstratio in a sufficiently broad context. 3 A consensus on the Demonstratio's general thrust thus masks puzzling questions. Everyone acknowledges, for example, that Huet's apologetic responded to Baruch Spinoza's Tractatus theologico-politicus (1670). Like other Christian apologists, he had to maintain "the Mosaic authorship" that was "the supposed guarantee of the truth of the text," 4 devoting "long chapters to the authenticity of the books of the Old Testament." 5 Yet Huet began the Demonstratio with an assault on geometrical knowledge and an epistemological discussion—an odd place to begin refuting the Tractatus. 6 Why? The question deserves a good answer, as Huet was a scholar of international reputation, and his contemporaries judged the Demonstratio an important work. 7More important for us, the evolution of the Demonstratio superbly reflects a vanished intellectual world and scholarly culture and in fact records change [End Page 600] in that world over four decades. Its genesis was the rash promise of a young man inspired by the possibilities of geometry and aspiring to recognition in the Republic of Letters, but before fulfilling that promise Huet developed a lively interest in such diverse fields as biblical criticism and natural philosophy, and he would apply the insights gained there in his apologetic. 8 A central question confronting Huet and his contemporaries was how to achieve certainty in matters religious, historical, and scientific. 9 The Demonstratio reveals Huet's conclusion that certainty in all three was profoundly interrelated. Finally, he composed the Demonstratio during the 1670s, having grasped with much alarm the consequences of the spread of Cartesianism. Thus, the Demonstratio is an intellectual biography not just of the author but of an era.Of course, Huet had to contend with Spinoza's challenge to Mosaic authorship, which fatally undermined the historical certainty of the Bible, but he also wrote... (shrink)

Most of what is told in this paper has been told before by the same author, in a number of publications of various kinds, but this is the first time that all this material has been brought together and treated in a uniform way. Smaller errors in the earlier publications are corrected here without comment. It has been known since the 1920s that quadratic equations played a prominent role in Babylonian mathematics. See, most recently, Høyrup (Hist Sci 34:1–32, 1996, and (...) Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. Springer, New York, 2002). What has not been known, however, is how quadratic equations came to play that role, since it is difficult to think of any practical use for quadratic equations in the life and work of a Babylonian scribe. One goal of the present paper is to show how the need to find solutions to quadratic equations actually arose in Mesopotamia not later than in the second half of the third millennium BC, and probably before that in connection with certain geometric division of property problems. This issue was brought up for the first time in Friberg (Cuneiform Digit Lib J 2009:3, 2009). In this connection, it is argued that the tool used for the first exact solution of a quadratic equation was either a clever use of the “conjugate rule” or a “completion of the square,” but that both methods ultimately depend on a certain division of a square, the same in both cases. Another, closely related goal of the paper is to discuss briefly certain of the most impressive achievements of anonymous Babylonian mathematicians in the first half of the second millennium BC, namely recursive geometric algorithms for the solution of various problems related to division of figures, more specifically trapezoidal fields. For an earlier, comprehensive (but less accessible) treatment of these issues, see Friberg (Amazing traces of a Babylonian origin in Greek mathematics. WorldScientific, Singapore 2007b, Ch. 11 and App. 1). (shrink)

Hobbes' philosophy of geometry was eccentric to contemporary movements and worsted in specific controversy. But he laid down stipulations defining geometry and its method which might provide a significant and workable alternative "meta-geometry". Some of these are isolated and reinterpreted here, especially those concerned with describing magnitudes, motions and quantities, and with his use of proportions. Rather than refutation of commentaries and historical rehash, the effort here is to isolate definitive texts and to offer a reinterpretation of their arguments (...) in succinct form showing their coherence and their interdependence with his other philosophic principles. (shrink)

The first article in this issue attempts to refute the concept of Altruism and calls it akin to Selfishness. The arguments are logically set in the way like that of Spinoza’s method of demonstration, with Axioms, Definitions, Propositions and Notes: so as to make them exact and precise. Interestingly, the writer introduces a new concept of Credit and through various other original propositions and examples rebuts the altruistic nature which is generally ascribed to humans.

As a continuation of ideas initiated in [19], we study bi-colored (generic) expansions of geometric theories in the style of the Fraïssé–Hrushovski construction method. Here we examine that the properties $NTP_{2}$, strongness, $NSOP_{1}$, and simplicity can be transferred to the expansions. As a consequence, while the corresponding bi-colored expansion of a red non-principal ultraproduct of p-adic fields is $NTP_{2}$, the expansion of algebraically closed fields with generic automorphism is a simple theory. Furthermore, these theories are strong with $\operatorname {\mathrm (...) {bdn}}(\text {"}x=x\text {"})=(\aleph _0)_{-}$. (shrink)