Online research in philosophy

In the New Essays on Human Understanding, Leibniz argues chapter by chapter with John Locke's Essay Concerning Human Understanding, challenging his views about knowledge, personal identity, God, morality, mind and matter, nature versus nurture, logic and language, and a host of other topics. The work is a series of sharp, deep discussions by one great philosopher of the work of another. Leibniz's references to his contemporaries and his discussions of the ideas and institutions of the age make this a fascinating and valuable document in the history of ideas. The work was originally written in French, and the version by Peter Remnant and Jonathan Bennett, based on the only reliable French edition (published in 1962), first appeared in 1981 and has become the standard English translation. It has been thoroughly revised for this series and provided with a new and longer introduction, a chronology on Leibniz's life and career and a guide to further reading.

In the paper “Math Anxiety,” Aden Evens explores the manner by means of which con- cepts are implicated in the problematic Idea according to the philosophy of Gilles Deleuze. The example that Evens draws from Difference and Repetition in order to demonstrate this relation is a mathematics problem, the elements of which are the differentials of the differential calculus. What I would like to offer in the present paper is an historical account of the mathematical problematic that Deleuze deploys in his philosophy, and an introduction to the role that this problematic plays in the develop- ment of his philosophy of difference. One of the points of departure that I will take from the Evens paper is the theme of “power series.”2 This will involve a detailed elaboration of the mechanism by means of which power series operate in the differential calculus deployed by Deleuze in Difference and Repetition. Deleuze actually constructs an alternative history of mathematics that establishes an historical conti- nuity between the differential point of view of the infinitesimal calculus and modern theories of the differential calculus. It is in relation to the differential point of view of the infinitesimal calculus that Deleuze determines a differential logic which he deploys, in the form of a logic of different/ciation, in the development of his proj- ect of constructing a philosophy of difference.

In The Logic of Sense (section 33) Gilles Deleuze defines novelists/artists as "clinicians of civilisation". Great authors are more like doctors than their patients - in that, like great clinicians, they create a set of disorders out of disorder, a table or grouping of symptoms out of disparate symptoms, so that "it is not the [Freudian Oedipus] complex which provides us with information about Oedipus and Hamlet, but rather Oedipus and Hamlet who provide us with information about the complex". For Deleuze, this creation of disorders takes on a particular character. In truth, for him, these structurings are not created from "disorder", since that would be to define the "choasmos" of differences - "disorder" in common parlance - by means of the notion of order; that is, it would be to define differences in terms of sameness - something he had spent the whole of his preceding book, Difference and Repetition, battling against. Thus the creation of disorders can only occur within a field where originary difference has been proclaimed and acknowledged, and where every notion of the same or the one is derived from, or "said of" (as he puts it) that which always and from the start differs. The exemplary novelist - Deleuze cites Joyce's Ulysses, Proust and Robbe-Grillet - disposes within this original difference two heterogeneous series of signifier and signified (section 6). These two series resonate through a single homogenous series of names where each term can be seen to relate to the preceding one and the next one, thus: n1 - n2 - n3 - n4... The first name, or signifier, relates to the second name/signifier, relates to the third etc in the familiar continuous chain of signifiers. But it is the novelist's task to consider this homogenous chain instead from the point of view of "that which alternates in this succession" - ie the alternation of signified and signifier through the terms - and to allow these to resonate. In the case of Joyce, for instance, there is a series surrounding "Bloom" which is given as the signifying set; and a corresponding signified series "Ulysses"; between which the author establishes a resonance and relation by various means. In the case of Robbe-Grillet, the two series operate on the smaller scale of descriptions of tiny "states of affairs" against "rigorous designations". In all cases, it is for Deleuze the differences between the series and their terms which "become [through the auspices of the author] primary", not the resemblances. This paper will attempt a preliminary transposition of these Deleuzian strategies onto a creative spatial field. Can a place be disposed according to this strategy of primary difference, homogenous chain of signifiers and the creative diagnosis of two resonating, heterogeneous series? In this case, what would constitute the two aspects of sense? How does this relate to the stoic disjunctive logic of the state of affairs of bodies and the entirely other order of "events", which hover over stat

Wes Morriston argues that even if we take an endless series of events to be merely potentially, rather than actually, infinite, still no distinction between a beginningless and an endless series of events has been established which is relevant to arguments against the metaphysical possibility of an actually infinite number of things: if a beginningless series is impossible, so is an endless series. The success of Morriston’s argument, however, comes to depend on rejecting the characterization of an endless series of events as a potential infinite. It turns out that according to his own analysis it is vitally relevant whether the series of events is potentially, as opposed to actually, infinite. If it is reasonable to maintain that an endless series of events is potentially infinite while a beginningless series is actually infinite, then a relevant distinction has been established for any person who thinks that an actual infinite cannot exist.

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.

In 1675, Leibniz elaborated his longest mathematical treatise he everwrote, the treatise ``On the arithmetical quadrature of the circle, theellipse, and the hyperbola. A corollary is a trigonometry withouttables''. It was unpublished until 1993, and represents a comprehensive discussion of infinitesimalgeometry. In this treatise, Leibniz laid the rigorous foundation of thetheory of infinitely small and infinite quantities or, in other words,of the theory of quantified indivisibles. In modern terms Leibnizintroduced `Riemannian sums' in order to demonstrate the integrabilityof continuous functions. The article deals with this demonstration,with Leibniz's handling of infinitely small and infinite quantities,and with a general theorem regarding hyperboloids.

For Deleuze, the exemplary novelist (Joyce, Proust, Robbe-Grillet…) disposes within original difference two heterogeneous series of signifier and signified (Sixth Series on Serialization ). These two series resonate through a single homogenous series of names where each term can be seen to relate to the preceding one and the next one, thus: n1→ n2→ n3→ n4→…. The first name, or signifier, relates to the second name/signifier, relates to the third etc in the familiar continuous chain of signifiers. It is the novelist’s task to consider this homogenous chain from the point of view of “that which alternates in this succession” – ie the alternation of signified and signifier through the terms - and to allow these to resonate. In what way is the architect a novelist, and in what way is exemplary architecture disposed according to this strategy of primary difference, homogenous chain of signifiers and the creative diagnosis of two resonating, heterogeneous series?

Abstract : In this paper, it is argued that Leibniz’s view that necessity is grounded in the availability of a demonstration is incorrect and furthermore, can be shown to be so by using Leibniz’s own examples of infinite analyses. First, I show that modern mathematical logic makes clear that Leibniz’s "infinite analysis" view of contingency is incorrect. It is then argued that Leibniz's own examples of incommensurable lines and convergent series undermine, rather than bolster his view by providing examples of necessary mathematical truths that are not demonstrable. Finally, it is argued that a more modern view on convergent series would, in certain respects, help support some claims he makes about the necessity of mathematical truths, but would still not yield a viable theory of necessity due to remaining problems with other logical, mathematical, and modal claims.

Throughout all of Deleuze’s work one finds an extended encounter with the Event of Difference. Deleuze’s extraordinary work on Leibniz is no exception. In the ‘later’ work, and regarding Leibniz, Deleuze remarks, “no philosophy has ever pushed to such an extreme the affirmations of one and the same world, and of an infinite difference and variety in this world”. This positive identification with Leibniz is not found in the ‘earlier’ wave of Deleuzian texts from the sixties where Leibniz is captured hesitating over the possible and the virtual. Any such hesitation over the possible and the virtual is “disastrous” for a philosophy of the event and difference since it abolishes the reality of the virtual and subordinates it to the identical, replacing pure immanence with a ‘theological model’ of creation. Is the Leibniz of Deleuze’s early texts compossible with the later? What is the significance of the event of difference or fold that joins and separates Deleuze’s continuing encounter with Leibniz? We will examine what is at stake in these differing understandings of Leibniz to Deleuze’s philosophy of events of difference.

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.