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)

Explications of the reconstruction of Leibniz’s metaphysics that Deleuze undertakes in 'The Fold: Leibniz and the Baroque' focus predominantly on the role of the infinitesimal calculus developed by Leibniz.1 While not underestimat- ing the importance of the infinitesimal calculus and the law of continuity as reflected in the calculus of infinite series to any understanding of Leibniz’s metaphysics and to Deleuze’s reconstruction of it in The Fold, what I propose to examine in this paper is the role played by other (...) developments in mathematics that Deleuze draws upon, including those made by a number of Leibniz’s near contemporaries – the projective geometry that has its roots in the work of Desargues (1591–1661) and the ‘proto-topology’ that appears in the work of Du ̈rer (1471–1528) – and a number of the subsequent developments in these fields of mathematics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. What is provided in this paper is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

The role of mathematics in the development of Gilles Deleuze's (1925-95) philosophy of difference as an alternative to the dialectical philosophy determined by the Hegelian dialectic logic is demonstrated in this paper by differentiating Deleuze's interpretation of the problem of the infinitesimal in Difference and Repetition from that which G. W. F Hegel (1770-1831) presents in the Science of Logic . Each deploys the operation of integration as conceived at different stages in the development of the infinitesimal calculus in his (...) treatment of the problem of the infinitesimal. Against the role that Hegel assigns to integration as the inverse transformation of differentiation in the development of his dialectical logic, Deleuze strategically redeploys Leibniz's account of integration as a method of summation in the form of a series in the development of his philosophy of difference. By demonstrating the relation between the differential point of view of the Leibnizian infinitesimal calculus and the differential calculus of contemporary mathematics, I argue that Deleuze effectively bypasses the methods of the differential calculus which Hegel uses to support the development of the dialectical logic, and by doing so, sets up the critical perspective from which to construct an alternative logic of relations characteristic of a philosophy of difference. The mode of operation of this logic is then demonstrated by drawing upon the mathematical philosophy of Albert Lautman (1908-44), which plays a significant role in Deleuze's project of constructing a philosophy of difference. Indeed, the logic of relations that Deleuze constructs is dialectical in the Lautmanian sense. (shrink)

The aim of this paper is to explore the uses made of the calculus by Gilles Deleuze and G. W. F. Hegel. I show how both Deleuze and Hegel see the calculus as providing a way of thinking outside of finite representation. For Hegel, this involves attempting to show that the foundations of the calculus cannot be thought by the finite understanding, and necessitate a move to the standpoint of infinite reason. I analyse Hegel’s justification for this introduction of dialectical (...) reason by looking at his responses to Berkeley’s criticisms of the calculus. For Deleuze, instead, I show that the differential must be understood as escaping from both finite and infinite representation. By highlighting the sub-representational character of the differential in his system, I show how the differential is a key moment in Deleuze’s formulation of a transcendental empiricism. I conclude by dealing with some of the common misunderstandings that occur when Deleuze is read as endorsing a modern mathematical interpretation of the calculus. (shrink)

The main thesis of this paper is that, Contrary to general belief, George berkeley did in fact express a coherent philosophy of mathematics in his major published works. He treated arithmetic and geometry separately and differently, And this paper focuses on his philosophy of arithmetic, Which is shown to be strikingly similar to the 19th and 20th century philosophies of mathematics known as 'formalism' and 'instrumentalism'. A major portion of the paper is devoted to showing how this philosophy of mathematics (...) follows directly from berkeley 's theory of signs and his philosophy of language, And from his discovery of an alternative to the correspondence theory of truth used by most of his predecessors. (shrink)

I extract some philosophical morals from some aspects of Lagrangian mechanics. One main moral concerns methodology: Lagrangian mechanics provides a level of description of phenomena which has been largely ignored by philosophers, since it falls between their accustomed levels---``laws of nature'' and ``models''. Another main moral concerns ontology: the ontology of Lagrangian mechanics is both more subtle and more problematic than philosophers often realize. The treatment of Lagrangian mechanics provides an introduction to the subject for philosophers, and is technically elementary. (...) In particular, it is confined to systems with a finite number of degrees of freedom, and for the most part eschews modern geometry. (shrink)

In this paper we introduce a paraconsistent reasoning strategy, Chunk and Permeate. In this, information is broken up into chunks, and a limited amount of information is allowed to flow between chunks. We start by giving an abstract characterisation of the strategy. It is then applied to model the reasoning employed in the original infinitesimal calculus. The paper next establishes some results concerning the legitimacy of reasoning of this kind - specifically concerning the preservation of the consistency of each chunk (...) and concludes with some other possible applications and technical questions. (shrink)

What is the relationship between the world and logic, between intuition and language, between objects and their quantitative determinations? Rationalists, on the one hand, hold that the world is structured in a rational way. Representationalists, on the other hand, assume that language, logic, and mathematics are only the means to order and describe the intuitively given world. In World and Logic, Jens Lemanski takes up three surprising arguments from Arthur Schopenhauer’s hitherto undiscovered Berlin Lectures, which concern the philosophy of language, (...) logic, and mathematics. Based on these arguments, Lemanski develops a new position entitled ‘rational representationalism’: the world is always structured by human beings according to linguistic, logical, and mathematical principles, but the basic vocabulary of these structural descriptions already contains metaphors taken from the world around us. (shrink)

In this paper I present an interpretation of Deleuze's concept of the virtual. I argue that this concept is best understood in relation to the problematic of individuation or differentiation, which Deleuze inherits from Duns Scotus. After analysing Scotus' critique of Aristotelian or hylomorphic approaches to the problem of individuation, I turn to Deleuze's account of differentiation and his interpretation of the calculus in chapter 4 of Difference and Repetition. The paper seeks thereby to explicate Deleuze's dialectics or theory of (...) Ideas, his approach to the principle of sufficient reason, and the concept of the virtual itself. (shrink)

This work examines Gilles Deleuze’s concept of intensity and the role of this concept in his philosophy of technology. The work has two main objectives. First, it analyses the role of Deleuze’s theory of intensity in his metaphysical system and in his philosophy of technology. Second, on the basis of this theory, it presents an original analysis of the creative potential of technology. The importance of the concept of intensity in Deleuze’s philosophy has been acknowledged, but so far, his views (...) on intensity have not been analysed in terms of a consistent theory. Furthermore, the implications of Deleuze’s theory of intensity for his and Félix Guattari’s analysis of technology have been entirely overlooked. This thesis shows for the first time the pivotal role of intensity in Deleuze’s philosophy and demonstrates the importance of the concept of intensity for the analysis of technicity. Part I explicates Deleuze’s theory of intensity. For Deleuze, intensity is the nature of the gradual variation of material, heterogeneous wholes. Within such wholes, intensity characterises variation in processes of individuation and systemic transformation. Moreover, these intensive processes and transformations take place within a sphere of material immediacy. Part II fleshes out the implications of this theory for a philosophy of technics. It proposes a definition of technics as immediate interaction with material variation. Technics always has an established role as a component of a particular social system, but it also bears a constant undercurrent of creative and transformative potential. Finally, the current dominance of information technology and the potential for technical creativity in contemporary society are examined on the basis of the analysis of intensive technics. The work offers an original and comprehensive analysis of Deleuze’s theory of intensity and demonstrates his contribution to the philosophy of technology. It also clarifies the processual nature of materiality and the nature of systemic transformation. Finally, it offers conceptual tools for fathoming the constraints as well as the creative potential of technics in contemporary culture. (shrink)

The paper contends that neoclassical ideology stems, to a great extent, from mathematical analysis. It is suggested that mainstream economic thought can be comprehensively revisited if both histories of mathematical and economic thought are to be taken collaboratively into account. Ideology is understood as a 'social construction of reality' that prevents us from evaluating our own standpoint, and impedes us from realising our value judgments as well as our theories of society and nature. However, the mid-19th century's intellectual controversies about (...) the validity of mathematical thought, truth and knowledge can procure new interesting insights concerning the ideological stance of the first marginalists. In this respect, the methodological categories of analysis and synthesis serve as the basis for the crucial distinction between old geometry and mathematical analysis, indicating that the discipline of mathematics has its own history of fundamentally unresolved disputes. Lastly, this may also shed some light on Alfred Marshall's peculiarly reluctant attitude towards the use of mathematical analysis in his work. (shrink)

As a philosophical paradigm, differential heterogenesis offers us a novel descriptive vantage with which to inscribe Deleuze’s virtuality within the terrain of “differential becoming,” conjugating “pure saliences” so as to parse economies, microhistories, insurgencies, and epistemological evolutionary processes that can be conceived of independently from their representational form. Unlike Gestalt theory’s oppositional constructions, the advantage of this aperture is that it posits a dynamic context to both media and its analysis, rendering them functionally tractable and set in relation to other (...) objects, rather than as sedentary identities. Surveying the genealogy of differential heterogenesis with particular interest in the legacy of Lautman’s dialectic, I make the case for a reading of the Deleuzean virtual that departs from an event-oriented approach, galvanizing Sarti and Citti’s dynamic a priori vis-à-vis Deleuze’s philosophy of difference. Specifically, I posit differential heterogenesis as frame with which to examine our contemporaneous epistemic shift as it relates to multi-scalar computational modeling while paying particular attention to neuro-inferential modes of inductive learning and homologous cognitive architecture. Carving a bricolage between Mark Wilson’s work on the “greediness of scales” and Deleuze’s “scales of reality”, this project threads between static ecologies and active externalism vis-à-vis endocentric frames of reference and syntactical scaffolding. (shrink)

RésuméCet article est consacré à l’histoire de la théorie locale des courbes “à double courbure”. Initiée par Clairaut en 1731, cette théorie se développe en parallèle à la théorie des surfaces et trouve son achèvement avec les formules de Serret et Frenet et leur interprétation par Darboux, en 1887. Au delà de l’analyse des contributions de nombreux mathématiciens, parmi lesquels Monge bien sûr mais aussi Fourier, Lagrange et Cauchy, notre étude donne un regard particulier sur l’évolution conjointe de l’Analyse et (...) de la Géométrie, dans une longue période riche de nombreuses remises en cause théoriques. (shrink)

The aim of this article is to shed light on an understudied aspect of Giordano Bruno's intellectual biography, namely, his career as a mathematical practitioner. Early interpreters, especially, have criticized Bruno's mathematics for being “outdated” or too “concrete”. However, thanks to developments in the study of early modern mathematics and the rediscovery of Bruno's first mathematical writings (four dialogues on Fabrizio's Mordente proportional compass), we are in a position to better understand Bruno's mathematics. In particular, this article aims to reopen (...) the question of whether Bruno anticipated the concept of infinitesimal quantity. It does so by providing an analysis of the dialogues on Mordente's compass and of the historical circumstances under which those dialogues were written. (shrink)

A theorem from Archimedes on the area of a circle is proved in a setting where some inconsistency is permissible, by using paraconsistent reasoning. The new proof emphasizes that the famous method of exhaustion gives approximations of areas closer than any consistent quantity. This is equivalent to the classical theorem in a classical context, but not in a context where it is possible that there are inconsistent innitesimals. The area of the circle is taken 'up to inconsistency'. The fact that (...) the core of Archimedes's proof still works in a weaker logic is evidence that the integral calculus and analysis more generally are still practicable even in the event of inconsistency. (shrink)

In Book II of The Geometry, Descartes distinguishes some special lines, which he calls geometrical curves. From the mathematical perspective, these curves are identified with polynomials of two variables. In this way, curves, which were understood as continuous quantities in Greek mathematics, turned into objects composed of points in The Geome- try. In this article we present assumptions which led Descartes to this radical change of the concept of curve.

One influential interpretation of Newton's formulation of his calculus has regarded his work as an organized, cohesive presentation, shaped primarily by technical issues and implicitly motivated by a knowledge of the form which a "finished" calculus should take. Offered as an alternative to this view is a less systematic and more realistic picture, in which both philosophical and technical considerations played a part in influencing the structure and interpretation of the calculus throughout Newton's mathematical career. This analysis sees the development (...) of Newton's calculus not principally as a calculated movement toward "rigorous" justification (in the sense of Cauchy, Weierstrass, and their 19th century contemporaries) and refinement of techniques, but as an evolution in the light of his involvement in debates over philosophical and scientific issues central to the rise of modern philosophy and science in the 17th and early 18th centuries. (shrink)

In this paper, I advance an original view of the structure of space called Infinitesimal Gunk. This view says that every region of space can be further divided and some regions have infinitesimal size, where infinitesimals are understood in the framework of Robinson’s nonstandard analysis. This view, I argue, provides a novel reply to the inconsistency arguments proposed by Arntzenius and Russell, which have troubled a more familiar gunky approach. Moreover, it has important advantages over the alternative views these authors (...) suggested. Unlike Arntzenius’s proposal, it does not introduce regions with no interior. It also has a much richer measure theory than Russell’s proposal and does not retreat to mere finite additivity. (shrink)

Alexander’s "Infinitesimal. How a dangerous mathematical theory shaped the modern world"(London: Oneworld Publications, 2015) is right to argue that the Jesuits had a chilling effect on Italian mathematics, but I question his account of the Jesuit motivations for suppressing indivisibles. Alexander alleges that the Jesuits’ intransigent commitment to Aristotle and Euclid explains their opposition to the method of indivisibles. A different hypothesis, which Alexander doesn’t pursue, is a conflict between the method of indivisibles and the Catholic doctrine of the Eucharist. (...) This is a pity, for the conflict with the Eucharist has advantages over the Jesuit commitment to Aristotle and Euclid. The method of indivisibles was a method that developed in the course of the seventeenth century, and those who developed ‘beyond the Alps’ relied upon Aristotelian and Euclidean ideals. Alexander’s failure to recognize the importance of Aristotle and Euclid for the development of the method of indivisibles arises from an unwarranted conflation of indivisibles and infinitesimals. Once indivisibles and infinitesimals are distinguished, we observe that the development of the method of indivisibles exhibits an unmistakable sympathy for Aristotle and Euclid. Thus, it makes sense to consider an alternative explanation for the Jesuit abhorrence of indivisibles. And indeed, indivisibles but not infinitesimals conflict with the doctrine of the Eucharist, the central dogma of the Church. (shrink)

One of the most famous critiques of the Leibnitian calculus is contained in the essay “The Analyst” written by George Berkeley in 1734. His key argument is those on compensating errors. In this article, we reconstruct Berkeley's argument from a systematical point of view showing that the argument is neither circular nor trivial, as some modern historians think. In spite of this well-founded argument, the critique of Berkeley is with respect to the calculus not a fundamental one. Nevertheless, it highlights (...) central aspects of the calculus that are characteristic of modern scientific theories. (shrink)

On this occasion of honouring the achievement of Philip McShane, I would like to recall his earliest and, in my judgment, most important work, Randomness, Statistics and Emergence. In particular, I will recall how that work situated Lonergan’s important breakthrough on statistical method in relation to the major currents of thought on the subject, many of which remain influential still today.

The ubiquitous assertion that the early calculus of Newton and Leibniz was an inconsistent theory is examined. Two different objects of a possible inconsistency claim are distinguished: (i) the calculus as an algorithm; (ii) proposed explanations of the moves made within the algorithm. In the first case the calculus can be interpreted as a theory in something like the logician’s sense, whereas in the second case it acts more like a scientific theory. I find no inconsistency in the first case, (...) and an inconsistency in the second case which can only be imputed to a small minority of the relevant community. (shrink)

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)

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)

I propose a theory of space with infinitesimal regions called smooth infinitesimal geometry based on certain algebraic objects, which regiments a mode of reasoning heuristically used by geometricists and physicists. I argue that SIG has the following utilities. It provides a simple metaphysics of vector fields and tangent space that are otherwise perplexing. A tangent space can be considered an infinitesimal region of space. It generalizes a standard implementation of spacetime algebraicism called Einstein algebras. It solves the long-standing problem of (...) interpreting smooth infinitesimal analysis realistically, an alternative foundation of spacetime theories to real analysis, 277–392, 1980). SIA is formulated in intuitionistic logic and is thought to have no classical reformulations. Against this, I argue that SIG is such a reformulation. But SIG has an unorthodox mereology, in which the principle of supplementation fails. (shrink)

Este trabajo estudia la teoría de los diferenciales de Salomon Maimon como respuesta a la cuestión quid juris y a la vez como clave para la fundamentación de la posibilidad de la metafísica como ciencia. Se reconstruye primero la crítica de Maimon al tratamiento kantiano de la pregunta quid juris. Luego, se analiza la respuesta del propio Maimon a esa pregunta, para establecer finalmente cómo tal respuesta abre el camino para la explicación de la posibilidad de la metafísica como ciencia.

El objetivo de este artículo es señalar si el cálculo infinitesimal creado por Leibniz a finales del siglo XVII supone un abandono de la metafísica: mientras que la metafísica se encontraba en la base del fundamento conceptual del cálculo leibniziano, es posible que el éxito de la aplicación del cálculo incitase a los científicos de la época a desgajar la metafísica del corpus científico.

The paper presents a comparison of the theories of truth, and the solutions of the liar paradox, proposed by Thomas Bradwardine , Jean Buridan , and Alfred Tarski.

The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts without caution, as the use (...) of anachronistic notation tends to impede, rather than enhance, our ability to recognize the emergent nature of mathematical objects. (shrink)

Bošković’s distinction between two kinds of velocities – velocity in the first act, or potential velocity, and velocity in the second act, or actual velocity – is considered in respect to the concept of instantaneous velocity as defined by calculus differentialis. Contrary to the seeming inconsistency of Bošković’s duality of velocities and the concept of instantaneous velocity, due to a critical examination of logical and methodological foundations of the calculus, the article shows that the duality of velocities is consistent with (...) the interpretations of instantaneous velocity given by Oresme, Euler and Maclaurin, as with the definition of instantaneous velocity according to the rigorous Cauchyan founding of the calculus. Bošković’s duality of velocities is also shown to be consistent with Aristotelian-scholastic doctrine of potentiality and actuality, especially in its domain related to the nature of continuous motion. (shrink)

The infinitesimal methods commonly used in the 17th and 18th centuries to solve analytical problems had a great deal of elegance and intuitive appeal. But the notion of infinitesimal itself was flawed by contradictions. These arose as a result of attempting to representchange in terms ofstatic conceptions. Now, one may regard infinitesimals as the residual traces of change after the process of change has been terminated. The difficulty was that these residual traces could not logically coexist with the static quantities (...) traditionally employed by mathematics. The solution to this difficulty, as it turns out, is to regard these quantities asalso being subject to (a form of) change, for then they will have the same nature as the infinitesimals representing the residual traces of change, and will become,ipso facto, compatible with these latter.In fact, the category-theoretic models which realize the Principle of Infinitesimal Linearity may themselves be regarded as representations of a general concept of variation (cf. Bell (1986)). While the static set-theoretical models represent change or motion by making a detour through the actual (but static) infinite, the varying category-theoretic models enable such change to be representeddirectly, thus permitting the introduction of geometric infinitesimals and, as we have attempted to demonstrate in this paper, the virtually complete incorporation of the methods of the early calculus.It is surely a remarkable — even an ironic — fact that the contradiction between the flux of the objective world and the stasis of mathematical entities has found its resolution in category theory, a branch of mathematics commonly, and, as one now sees, mistakenly, regarded as the summit of gratuitous abstraction. (shrink)

The aim of this paper is to analyze Dimitry Gawronsky’s doctrine of actual infinitesimals. I examine the peculiar connection that his critical idealism establishes between transcendental philosophy and mathematics. In particular, I reconstruct the relationship between Gawronsky’s differentials, Cantor’s transfinite numbers, Veronese’s trans-Archimedean numbers and Robinson’s hyperreal numbers. I argue that by means of his doctrine of actual infinitesimals, Gawronsky aims to provide an interpretation of calculus that eliminates any alleged given element in knowledge.

The art of memory started with Aristotle’s questions on memory. During its long evolution, it had important contributions from alchemists, was transformed by Ramon Llull and apparently ended with Giordano Bruno, who was considered the best known representative of this art. This tradition did not disappear, but lives in the formulations of our modern scientific theories. From its initial form as a method of keeping information via associations, it became a principle of classification and structuring of knowledge. This principle, which (...) we here name differentiation with stratification, is a structural design behind classical mechanics. Integrating two different traditions of science in one structure, this physical theory became the modern paradigm of science. In this paper, we show that this principle can also be formulated as a set of questions. This is done via an analysis of theories, based on the epistemology of observational realism. A combination of Rudolph Carnap’s concept of theory as a system of observational and theoretical languages, with a criterion for separating observational languages, based on analytical psychology, shapes this epistemology. The ‘nuclear’ role of the observational laws and the differentiations from these nucleus, reproducing the general cases of phenomena, reveals the memory art’s heritage in the theories. Here in this paper we argue that this design is also present in special relativity and in quantum mechanics. (shrink)

The notion of indivisibles and atoms arose in ancient Greece. The continuum—that is, the collection of points in a straight line segment, appeared to have paradoxical properties, arising from the ‘indivisibles’ that remain after a process of division has been carried out throughout the continuum. In the seventeenth century, Italian mathematicians were using new methods involving the notion of indivisibles, and the paradoxes of the continuum appeared in a new context. This cast doubt on the validity of the methods and (...) the reliability of mathematical knowledge which had been regarded as established by the axiomatic method in geometry expounded by Aristotle’s younger contemporary Euclid. The teaching of indivisibles was banned within the Society of Jesus, the Jesuits. In England, indivisibles were used by the mathematician John Wallis, and there was an acrimonious and extended feud between Wallis and the philosopher Thomas Hobbes over legitimate methods of argument in mathematics. Notions of the infinitesimal were used by Isaac Newton and Gottfried Leibniz, and were attacked by Bishop Berkeley for the vagueness of the concept and the illegitimate reasoning applied to it. This article discusses aspects of these events with reference to the book Infinitesimal by Amir Alexander and to other sources. Also discussed are wider issues arising from Alexander’s book including: the changes in cultural sensibility associated with the growth of new mathematical and scientific knowledge in the seventeenth century, the changes in language concomitant with these changes, what constitutes valid methods of enquiry in various contexts, and the question of authoritarianism in knowledge. More general aims of this article are to widen the immediate mathematical and historical contexts in Alexander’s book, to bridge a gap in conversations between mathematics and the humanities, and to relate mathematical ideas to wider human and contemporary issues. (shrink)