This paper studies internal (or intra-)mathematical explanations, namely those proofs of mathematical theorems that seem to explain the theorem they prove. The goal of the paper is a rigorous analysis of these explanations. This will be done in two steps. First, we will show how to move from informal proofs of mathematical theorems to a formal presentation that involves proof trees, together with a decomposition of their elements; secondly we will show that those mathematical proofs that are regarded as having (...) explanatory power all display an increase of conceptual complexity from the assumptions to the conclusion. (shrink)

In “Reconstructing Bergson’s Critique of Intensive Magnitude” John Bagby defends Bergson against the criticism that I develop in “Bergson and Intensive Magnitude: Dismantling...

This paper examines the Leibnizian influence in Deleuze's theory of the spatium. Leibniz's critique of Cartesian extension and Newtonian space leads him to a conception of space in terms of internal determination and internal difference. Space is thus understood as a structure of individual relations internal to substances. Making some Nietzschean corrections to Leibniz, Deleuze understands the spatium in terms of individuating differences instead of individual relations. Leibnizian space is thus transformed into a genetic space producing both extension (quantity) and (...) quality. (shrink)

Why does Deleuze rely so heavily on Spinoza and Leibniz? At first glance, his critique of representation and the traditional ‘image of thought’ seems to oppose him to rationalism. However, Deleuze says that when the ‘cry of rationalism’ is pursued until the end, rationalism becomes ‘delirious’. In such a state, it undermines the model of representation. This delirium is found in Spinoza and Leibniz's critique of generality and their conflation of essence and existence, through which they ruin the traditional mediation (...) of difference by identity and generality. However, for this rationalism to completely break with representationalism it must undergo some Deleuzian corrections. (shrink)

In this paper I argue that inconsistencies in scientific theories may arise from the type of causality relation they—tacitly or explicitly—embody. All these seemingly different causality relations can be subsumed under a general strategy developed to defeat the paradoxes which inevitably occur in our experience of the real. With respect to this, scientific theories are just a subclass of the larger class of metaphysical theories, construed as theories that attempt to explain a (part of) the world consistently. All metaphysical theories (...) share a common structural backbone specificially designed to defeat paradoxes, their often wildly diverging ontological claims notwithstanding. This common structure shapes the procedures which govern the invention of ideas in the context of such theories, by codifying some onto-logical a priori assumptions regarding the consistency of reality into its bare conceptual framework. Causality plays a key rôle here, because it implies conservation of identity, itself a far from simple notion. It imposes strong demands on the universalising power of the theories concerned. These demands are often met by the introduction of a metalevel which encompasses the notions of ‘system’ and ‘lawful behaviour’. In classical mechanics, the division between universal and particular leaves its traces in the separate treatment of cinematics and dynamics. The fundamental backbone’s specific gestalt thus functions as a theory’s individual signature and paves the way to a comparative historical approach towards their study. An important part of my paper therefore explores the strong connections between paradoxes as they appear and are dealt with in ancient philosophy and their re-appearance in early modern natural philosophy and science. This analysis is applied to the mechanical theories of Newton and Leibniz, with some surprising results. (shrink)

The question “Is Hegel a pragmatist?” has been the subject of much debate among specialists in Hegelian philosophy and pragmatism. In this article, I intend to clarify the terms of the problem, and to assess the strengths and weaknesses of two ways of answering this question: the first is to make explicit pragmatic themes in Hegelian thought, and the second is to identify a minimal and negative criterion of rapprochement between Hegelianism and pragmatism. In light of this critical examination, it (...) appears that any general approach to this question is doomed to failure. (shrink)

Leibniz propôs mais de um conceito para descrever filosoficamente a substância. Ironia instrutiva, esta pluralidade que tem por fim uma explicação unificada da realidade culminou em uma definição dos componentes fundamentais do mundo – as mônadas – como unidades que abrigam a multiplicidade. Estas substâncias, bem como a sua função essencial de se exprimirem mutuamente, apenas se tornam plenamente inteligíveis quando observamos os conceitos anteriores, de que a filosofia madura de Leibniz herdou algumas intuições. Guiando-nos pelas obras-chave do filósofo e (...) físico alemão, procuraremos retraçar esse percurso intelectual, a fim não de esboçar um hipotético sistema seu, mas de _unificar_ o tratamento que ele dispensou à _unidade_. Tal hipótese de leitura concorda fundamentalmente com aquela sustentada por Michel Fichant, mas traz pontuais acréscimos ou desacordos, que serão indicados neste artigo. (shrink)

I argue that Descartes intended the so-called ontological "argument" as a self-validating intuition, rather than as a formal proof. The textual evidence for this view is highly compelling, but the strongest support comes from understanding Descartes's diagnosis for why God's existence is not 'immediately' self-evident to everyone and the method of analysis that he develops for making it self-evident. The larger aim of the paper is to use the ontological argument as a case study of Descartes's nonformalist theory of deduction (...) and his method of analysis, showing how he conceives the latter as a form of philosophical therapy. (shrink)

This article offers a reconstruction of Derrida’s critique of Leibniz. It suggests that in attempting to fit Leibniz into his conception of the history of metaphysics and the all-embracing notion of logocentrism that underwrites it, Derrida presupposes two regimes of logocentrism: one subjective, the other theological. Subsumed into this second mode, Derrida casts Leibniz as a progenitor of structuralism and the new sciences and technologies of information in order to expose their logocentric foundations. However, in doing so, he ends up (...) sidelining the aspect of Leibniz’s ideas that anticipates the specificity of the new epoch of writing he claims to historicise. The article goes on to argue that to grasp this specificity one must look beyond the “expressivist” schema of exterior representation, which Derrida draws from Husserl’s theory of meaning, and instead look to Leibniz’s own formal-combinatorial concept of expression, which exceeds Derrida’s critical framework. (shrink)

In this paper I show that proofs by contradiction were a serious problem in seventeenth century mathematics and philosophy. Their status was put into question and positive mathematical developments emerged from such reflections. I analyse how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programmes of providing a development of parts of geometry free of proofs by contradiction. The logical part shows how the traditional Aristotelean doctrine that perfect demonstrations (...) are causal demonstrations influenced the reflection on proofs by contradiction. The main protagonist of this part is Wallis. Finally, I analyse some epistemological developments arising from the Cartesian tradition. In particular, I look at Arnauld's programme of providing an epistemologically motivated reformulation of Geometry free of proofs by contradiction. The conclusion explains in which sense these epistemological reflections can be compared with those informing contemporary intuitionism. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

Leibniz’s universal characteristic is a fundamental aspect of his theory of cognition. Without symbols or characters it would be difficult for the human mind to define several concepts and to achieve many demonstrations. In most disciplines, and particularly in mathematics, the mind must then focus on symbols and their combinatorial rules rather than on mental contents. For Leibniz, mental perception is most of the time too confused for attaining distinct notions and valid deductions. In this paper, I argue that the (...) functions of symbolization differ depending upon the kind of concepts that are replaced with characters. In my view, most commentators did not sufficiently underline the distinction between two main functions of formal substitution in Leibniz’s characteristic: either increasing our knowledge or simply structuring it. In the first case, we complete our knowledge because formal substitution makes sensible and imaginary concepts more distinct. In the second case, symbolization helps to organize contents that are already conceived of by reason. Thus the process of substitution is not always identically applicable, for symbols replace different types of concepts. (shrink)

O verdadeiro infinito, afirma Leibniz em seus Novos ensaios, não é um modo da quantidade, é anterior a qualquer composição e não é formado pela adição de partes. O infinito, para Leibniz, é atual e é propriedade de todas as coisas. Como criaturas finitas conhecem o infinito? Neste artigo, investigamos que tipo de relação pode ter o infinito matemático, quantitativo, para o conhecimento da infinitude divida e do infinito atual que existe no mundo. A ordem ideal da matemática instrui sobre (...) a ordem real do mundo? E quais os limites da ordem ideal na explicação do infinito atual que caracteriza Deus e o mundo? (shrink)

O caráter protocolar desempenhado pelas matemáticas na formulação do conceito cartesiano de ciência é amplamente difundido e frequentemente reinvocado na literatura especializada quando se trata de abordar a exigência apodítica inerente a este conceito. No entanto, pouco se explora o que a diversidade das disciplinas matemáticas bem como a relação entretida por elas permite trazer de elucidação à noção cartesiana de ciência. Nosso propósito consiste, aqui, em tomar posição quanto a um debate acerca do estatuto da álgebra e da geometria (...) como disciplinas que fixam os lineamentos da teoria cartesiana do conhecimento ao exibirem o que Descartes reputaria como a aplicação ideal do intelecto. Trata-se de insistir, por vias que assinalam uma decisão teórica em favor de uma concepção mais intelectualista e menos imaginativa da ciência, que não há hesitação quanto ao papel primitivo e fundacional da álgebra se comparada à geometria. (shrink)

This work is devoted to clear how the foundation of modern thought takes place as a reinforced reiteration of metaphysics which is provoked by a defensive fold of reason against the assault of transrational. We begin showing the way this asault is experimented by Descartes in a decisive moment of his life and how reagently replies to him. The article continues showing how physics and metaphysics are being moved in Descartes on the wake of a try to build seawalls against (...) the incessant return of what has been repressed. (shrink)

L’articolo si propone di analizzare la presenza di Descartes nella Critica della ragion pura, con particolare attenzione ai paralogismi della psicologia razionale e alla confutazione dell’idealismo. L’aspetto più rilevante dell’analisi kantiana non concerne l’interpretazione del pensiero cartesiano fornita dal filosofo tedesco, data la scarsa conoscenza che Kant possedeva dei testi di Descartes. Ad essere interessante, al contrario, soprattutto se considerata alla luce della sua successiva ricezione, è l’immagine di Descartes che emerge dalle pagine della Critica. A questo riguardo, è possibile (...) affermare che Kant abbia contribuito al ritorno sulla scena del pensiero tedesco del Descartes metafisico, una maschera filosofica che svolgerà un ruolo centrale nella ricostruzione della storia della filosofia proposta dall’idealismo post-kantiano. (shrink)

Leibniz frequently argued that reasons are to be weighed against each other as in a pair of scales, as Professor Marcelo Dascal has shown in his article "The Balance of Reason." In this kind of weighing it is not necessary to reach demonstrative certainty – one need only judge whether the reasons weigh more on behalf of one or the other option However, a different kind of account about rational decision-making can be found in some of Leibniz's writings. In his (...) article "Was Leibniz's Deity an Akrates?" Professor Jaakko Hintikka has argued that Leibniz developed a new vectorial model for rational decisions which is better suited to complicated decisions, where values are complementary to each other. This model, related closely to his work in metaphysics and the philosophy of mind, is a heuristic device which helps in finding rational combinations - and in an ideal case an optimum - between plural inclinations to the good. I shall argue that Leibniz applies more or less implicitly both of these models in his practical rationality. In simple situations he applied the pair of scales model and in more complicated situations he applied the vectorial model. (shrink)

If we had a balance of reasons, where the arguments presented in favor and against the case were weighed precisely and the verdict could be pronounced in favor of the most inclined scale ... [we would have] a more valuable art than that miraculous science of producing gold.