Spinoza: Mathematics and Logic

This chapter argues that the standard conception of Spinoza as a fellow-travelling mechanical philosopher and proto-scientific naturalist is misleading. It argues, first, that Spinoza’s account of the proper method for the study of nature presented in the Theological-Political Treatise (TTP) points away from the one commonly associated with the mechanical philosophy. Moreover, throughout his works Spinoza’s views on the very possibility of knowledge of nature are decidedly sceptical (as specified below). Third, in the seventeenth-century debates over proper methods in the (...) sciences, Spinoza sided with those that criticized the aspirations of those (the physico-mathematicians, Galileo, Huygens, Wallis, Wren, etc) who thought the application of mathematics to nature was the way to make progress. In particular, he offers grounds for doubting their confidence in the significance of measurement as well as their piece-meal methodology (see section 2). Along the way, this chapter offers a new interpretation of common notions in the context of treating Spinoza’s account of motion (see section 3). (shrink)

Like most, if not all, of his contemporaries, Spinoza never developed a full-fledged philosophy of mathematics. Still, his numerous remarks about mathematics attest not only to his deep interest in the subject (a point which is also confirmed by the significant presence of mathematical books in his library), but also to his quite elaborate and perhaps unique understanding of the nature of mathematics. At the very center of his thought about mathematics stands a paradox (or, at least, an apparent paradox): (...) mathematics provides Spinoza with an epistemic model. Mathematical knowledge is certain (II/138/9 and II/138/9), clear (IV/261/8), and free from teleological thinking (II/79/33), but the objects of mathematical knowledge – i.e., mathematical entities – are nothing but “auxila imaginationis [aids of the imagination],” (IV/57/16 and II/83/15), entities that are not real and merely assist the imagination in carving the world in manner that is suitable to our limited and distortive cognitive capacities. (shrink)

In the present paper we propose to approach the Spinoza`s methodological project from a philosophical and historical perspective broad enough to adequately understand the reasons that led him to adopt geometric method to expose his philosophy. Even if the topic has been widely discussed by Spinoza´s commentators in the four centuries since the Ethics was published, we believe that the approaches are either inadequate or suffer from some fragmentation, in the sense that they address this or that aspect, but don`t (...) put together all pieces that make up the puzzle. Likewise, we consider that a full understanding of such application of geometric method transcends the mere question of whether it was successful or not in terms of logical rigor. So that, the fundamental task in the present work is unite various aspects that help us understand Spinoza`s methodological choice in a synoptic vision, in which both historical and systematic aspects can be included. To this end, we start from a hypothesis which allow us glimpse a certain unity behind the reasons that lead Spinoza to choose the geometric method, namely: our philosopher tries to reach an ideal of science, which combines some conceptual elements of Aristotelian tradition with others from Cartesian philosophy. (shrink)

This book interrogates the ontology of mathematical entities in Spinoza as a basis for addressing a wide range of interpretive issues in Spinoza’s epistemology—from his antiskepticism and philosophy of science to the nature and scope of reason and intuitive knowledge and the intellectual love of God. Going against recent trends in Spinoza scholarship, and drawing on various sources, including Spinoza’s engagements with optical theory and physics, Matthew Homan argues for a realist interpretation of geometrical figures in Spinoza; illustrates their role (...) in a Spinozan hypothetico-deductive scientific method; and develops Spinoza’s mathematical examples to better illuminate the three kinds of knowledge. The result is a portrait of Spinoza’s epistemology as sanguine and distinctive yet at home in the new Cartesian and Galilean scientific-philosophical paradigm. (shrink)

This paper examines Spinoza’s view on the consistency of mental representation. First, I argue that he departs from Scholastic tradition by arguing that all mental states—whether desires, intentions, beliefs, perceptions, entertainings, etc.—must be logically consistent. Second, I argue that his endorsement of this view is motivated by key Spinozistic doctrines, most importantly the doctrine that all acts of thought represent what could follow from God’s nature. Finally, I argue that Spinoza’s view that all mental representation is consistent pushes him to (...) a linguistic account of contradiction. (shrink)

This essay offers an in-depth reading of the geometrical illustration of Ethics IIP8S and shows how it can be used to explicate the whole architecture of Spinoza’s system by specifying the way in which all the key structural features of his basic ontology find their analogies in the example. The illustration can also throw light on Spinoza’s ontology of finite things and inform us about what is at stake when we form universal ideas. In general, my reading of IIP8S thus (...) elucidates what it means, for Spinoza, to think geometrically or to consider geometry as a model: fundamentally, geometricity is not a form of exposition but the way in which reality itself is structured. (shrink)

Spinoza is commonly perceived as the great metaphysician of coherence. The Euclidean manner in which he presented his philosophy in the _Ethics _has led readers to assume they are facing a strict and consistent philosophical system that necessarily follows from itself. As opposed to the prevailing understanding of Spinoza and his work, _The Role of Contradictions in Spinoza's Philosophy_ explores an array of profound and pervasive contradictions in Spinoza’s system and argues they are deliberate and constitutive of his philosophical thinking (...) and the notion of God at its heart. Relying on a meticulous and careful reading of the _Theological-Political Treatise_ and the _Ethics_, this book reconstructs Spinoza's philosophy of contradictions as a key to the ascending three degrees of knowledge leading to the _Amor intellectualis Dei_. Offering an exciting and clearly-argued interpretation of Spinoza’s philosophy, this book will interest students and scholars of modern philosophy and philosophy of religion, as well as Jewish studies. Yuval Jobani is Assistant Professor at the Department of Hebrew Culture Studies and the School of Education at Tel-Aviv University. (shrink)

The seventeenth century was an important period in the conceptual development of the notion of the infinite. In 1643, Evangelista Torricelli (1608-1647)—Galileo’s successor in the chair of mathematics in Florence—communicated his proof of a solid of infinite length but finite volume. Many of the leading metaphysicians of the time, notably Spinoza and Leibniz, came out in defense of actual infinity, rejecting the Aristotelian ban on it, which had been almost universally accepted for two millennia. Though it would be another two (...) centuries before the notion of the actually infinite was rehabilitated in mathematics by Dedekind and Cantor (Cauchy and Weierstrass still considered it mere paradox), their impenitent advocacy of the concept had significant reverberations in both philosophy and mathematics. In this essay, I will attempt to clarify one thread in the development of the notion of the infinite. In the first part, I study Spinoza’s discussion and endorsement, in the Letter on the Infinite (Ep. 12), of Hasdai Crescas’ (c. 1340-1410/11) crucial amendment to a traditional proof of the existence of God (“the cosmological proof” ), in which he insightfully points out that the proof does not require the Aristotelian ban on actual infinity. In the second and last part, I examine the claim, advanced by Crescas and Spinoza, that God has infinitely many attributes, and explore the reasoning that motivated both philosophers to make such a claim. Similarities between Spinoza and Crescas, which suggest the latter’s influence on the former, can be discerned in several other important issues, such as necessitarianism, the view that we are compelled to assert or reject a belief by its representational content, the enigmatic notion of amor Dei intellectualis, and the view of punishment as a natural consequent of sin. Here, I will restrict myself to the issue of the infinite, clearly a substantial topic in itself. (shrink)

The article presents Leibniz's preoccupation (in 1675?6) with the difference between the notion of infinite number, which he regards as impossible, and that of the infinite being, which he regards as possible. I call this issue ?Leibniz's Problem? and examine Spinoza's solution to a similar problem that arises in the context of his philosophy. ?Spinoza's solution? is expounded in his letter on the infinite (Ep.12), which Leibniz read and annotated in April 1676. The gist of Spinoza's solution is to distinguish (...) between three kinds of infinity and, in particular, between one that applies to substance, and one that applies to numbers, seen as auxiliaries of the imagination. The rest of the paper examines the extent to which Spinoza's solution solves Leibniz's problem. The main thesis I advance is that, when Spinoza and Leibniz say that the divine substance is infinite, in most contexts it is to be understood in non-numerical and non-quantitative terms. Instead, for Spinoza and Leibniz, a substance is said to be infinite in a qualitative sense stressing that it is complete, perfect and indivisible. I argue that this approach solves one strand of Leibniz's problem and leaves another unsolved. (shrink)

Georg Cantor, the founder of set theory, cared much about a philosophical foundation for his theory of infinite numbers. To that end, he studied intensively the works of Baruch de Spinoza. In the paper, we survey the influence of Spinozean thoughts onto Cantor’s; we discuss Spinoza’s philosophy of infinity, as it is contained in his Ethics; and we attempt to draw a parallel between Spinoza’s and Cantor’s ontologies. Our conclusion is that the study of Spinoza provides deepening insights into Cantor’s (...) philosophical theory, whilst Cantor can not be called a ‘Spinozist’ in any stricter sense of that word.Keywords: Georg Cantor; Baruch de Spinoza; Infinity in philosophy and mathematics; Transfinite numbers; Set theory; Infinity and pantheism. (shrink)

Approaching the central themes of Spinoza's thought from both a historical and analytical perspective, this book examines the logical-metaphysical core of Spinoza's philosophy, its epistemology and its ramifications for his much disputed attitude towards religion. Opening with a discussion of Spinoza's historical and philosophical location as the appropriate context for the interpretation of his work, the book goes on to present a non-'logical' reading of Spinoza's metaphysics, a consideration of Spinoza's radical repudiation of Cartesian subjectivism and an examination of how (...) Spinoza wanted religion to be understood in the context of his wider thinking and the influence of his non-Christian background. Mason also assesses Spinoza's significance and importance for philosophy now. (shrink)

In Hegel ou Spinoza,1 Pierre Macherey challenges the influence of Hegel’s reading of Spinoza by stressing the degree to which Spinoza eludes the grasp of the Hegelian dialectical progression of the history of philosophy. He argues that Hegel provides a defensive misreading of Spinoza, and that he had to “misread him” in order to maintain his subjective idealism. The suggestion being that Spinoza’s philosophy represents, not a moment that can simply be sublated and subsumed within the dialectical progression of the (...) history of philosophy, but rather an alternative point of view for the development of a philosophy that overcomes Hegelian idealism. Gilles Deleuze also considers Spinoza’s philosophy to resist the totalising effects of the dialectic. Indeed, Deleuze demonstrates, by means of Spinoza, that a more complex philosophy antedates Hegel’s, which cannot be supplanted by it. Spinoza therefore becomes a significant figure in Deleuze’s project of tracing an alternative lineage in the history of philosophy, which, by distancing itself from Hegelian idealism, culminates in the construction of a philosophy of difference. It is Spinoza’s role in this project that will be demonstrated in this paper by differentiating Deleuze’s interpretation of the geometrical example of Spinoza’s Letter XII (on the problem of the infinite) in Expressionism in Philosophy, Spinoza,2 from that which Hegel presents in the Science of Logic.3. (shrink)

Sooner or later there comes a time in the history of a science when it pauses from dealing with its innumerable special problems and returns to the study of first principles and the foundations that delimit its particular field of inquiry. The result is usually a radical revision of a number of established basic ideas, the discovery of some hitherto unknown dimension in the field, and the emergence of an appropriate paradigm for investigating the field. It was the fate of (...) logic to be one of the first fields to undergo such a reformation, which lasted more than two centuries. Its instigators were Rene Descartes and Benedict Spinoza. (shrink)

An accurate analysis of the text shows that the small treatises have a logical structure and a style which is in all aspects unspinozistic. The main points of difference are : a formalistic interpretation of mathematics‚ the opposition between mathematics and physics‚ slavish cartesianism‚ the presence of numerous pleonasms‚ carelessness of expression‚ parade of learning‚ prolixity‚ attention for irrelevant qualities of authors quoted‚ educational purpose. Together with De Vet’s demonstration that the author of SRR and RK is still alive in (...) 1693 (see Tijdschr. v. Filosofie 45‚ 1983) these arguments force us‚ to consign Spinoza’s authorship‚ recently still defended by Petry a.o.‚ to the realm of fancy. (shrink)