This article explores three distinct approaches to politics identified by Deleuze and Guattari. I argue that they consistently privilege a 'problematic' approach entailing individuals and associations establishing norms on the basis of the potential they possess within a concrete situation, and that this implies resistance to both the 'theorematic' politics they associate with statist philosophy and struggles aiming to alter the 'axiomatic' determination induced by global forces of capital. While this resistance necessarily proceeds in relation to established notions of identity (...) and value, it must also do more than aim for the establishment of particular institutional 'constants,' including particular democratic procedures and human rights. (shrink)

The paper aims to show how by elaborating the Peircean terms used in the title creativity in learning processes and in scientific discoveries can be explained within a semiotic framework. The essential idea is to emphasize both the role of external representations and of experimenting with those representations , and to describe a process consisting of three steps: First, looking at diagrams "from a novel point of view" offers opportunities to synthesize elements of these diagrams which have never been perceived (...) as connected before. Second, by forming those observed syntheses to "new objects" of thinking, and by signifying these objects through new signs , new means of thinking and acting are created . And finally, by applying these new means – in proofs, for instance – the "intelligibility" of new discoveries and their power to explain problematic facts must be tested. (shrink)

This article aims to examine the relationship between image and narrative by means of Peirce’s first trichotomy of qualisign-sinsign-legisign or, for the purposes of the current argument, image-diagram-metaphor. It is argued that narrative, as an extended metaphor, can be examined in three modes: in the image; schematically, in the imagination; and allegorically or in a thought experiment, through hypothetic interpretation. The article outlines two kinds of diagrammatic reasoning emphasized by Peirce: corollarial deduction in which the image is ‘literally seen’ and (...) the reasoning steps are manifest in its conclusion; and theorematic deduction where the conclusion in a diagram is subject to a hypothesis which transforms the image into something new. Demonstrating the breadth of diagrammatic reasoning with reference to the 2018 film, The Shape of Water, the article seeks to explore how allegory and diagram are mutually cooperative, based on three ontological modes: the expressive, the cognitive, and the symbolic. Its primary focus, then, is not so much on the story events of the narrative, as the way that they are visualized and characterized as the fairy story unfolds. It is suggested that the interpreting activity involved in allegory and diagram ties interpretation to metacognition, ultimately (re)recognizing the image in The Shape of Water in an attempt to ascertain the meaning of love. (shrink)

Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on rule-following skepticism. We argue, with the help of C. S. Peirce's distinction between corollarial and theorematic proofs, that his intuitions are better explained by resistance to what we call conceptual omniscience, treating meaning as fixed content specified in advance. We interpret the distinction in the context of modern epistemic logic (...) and semantic information theory, and show how removing conceptual omniscience helps resolve Wittgenstein's paradoxes and explain the puzzle of deduction, its ability to generate new knowledge and meaning. (shrink)

In this paper we will show Peirce’s distinction between deduction, induction and abduction. The aim of the paper is to show how Peirce changed his views on the subject, from an understanding of deduction, induction and hypotheses as types of reasoning to understanding them as stages of inquiry very tightly connected. In order to get a better understanding of Peirce’s originality on this, we show Peirce’s distinctions between qualitative and quantitative induction and between theorematical and corollarial deduction, passing then to (...) the distinction between mathematics and logic. In the end, we propose a sketch of a comparison between Peirce and Whitehead concerning the two thinkers’ view of mathematics, hoping that this could point to further inquiries. (shrink)

We examine Charles S. Peirce's mature views on the logic of science, especially as contained in his later and still mostly unpublished writings. We focus on two main issues. The first concerns Peirce's late conception of retroduction. Peirce conceived inquiry as performed in three stages, which correspond to three classes of inferences: abduction or retroduction, deduction, and induction. The question of the logical form of retroduction, of its logical justification, and of its methodology stands out as the three major threads (...) in his later writings. The other issue concerns the second stage of scientific inquiry, deduction. According to Peirce's later formulation, deduction is divided not only into two kinds but also into two sub-stages: logical analysis and mathematical reasoning, where the latter is either corollarial or theorematic. Save for the inductive stage, which we do not address here, these points cover the essentials of Peirce's latest thinking on the l.. (shrink)

This paper reconstructs the Peircean interpretation of Kant's doctrine on the syntheticity of mathematics. Peirce correctly locates Kant's distinction in two different sources: Kant's lack of access to polyadic logic and, more interestingly, Kant's insight into the role of ingenious experiments required in theorem-proving. In this second respect, Kant's analytic/synthetic distinction is identical with the distinction Peirce discovered among types of mathematical reasoning. I contrast this Peircean theory with two other prominent views on Kant's syntheticity, i.e. the Russellian and the (...) Beckian views, and show how Peirce's interpretation of Kant solves the dilemma that each of these two views faces. I also show that Hintikka's criterion for Kant's synthetic judgments, i.e. a new individual introduced by the -instantiation rule, does not capture the most important characteristic of Peirce's theorematic reasoning, i.e. the process of choosing a correct individual. (shrink)

Abductive reasoning takes place in forming``hypotheses'''' in order to explain ``facts.'''' Thus, theconcept of abduction promises an understanding ofcreativity in science and learning. It raises,however, also a lot of problems. Some of them will bediscussed in this paper. After analyzing thedifference between induction and abduction (1), Ishall discuss Peirce''s claim that there is a ``logic''''of abduction (2). The thesis is that this claim can beunderstood, if we make a clear distinction between inferential elements and perceptive elements of abductive reasoning. For (...) Peirce, the creative act offorming explanatory hypotheses and the emergence of``new ideas'''' belongs exclusively to the perceptive side of abduction. Thus, it is necessary to study the roleof perception in abductive reasoning (3). A furtherproblem is the question whether there is arelationship between abduction and Peirce''s concept of``theorematic reasoning'''' in mathematics (4). Both forms of reasoning could be connected, because both arebased on perception. The last problem concerns therole of instincts in explaining the success ofabductive reasoning in science, and the question whether the concept of instinct might be replaced bymethods of inquiry (5). (shrink)

According to Kant, pure intuition is an indispensable ingredient of mathematical proofs. Kant‘s thesis has been considered as obsolete since the advent of modern relational logic at the end of 19th century. Against this logicist orthodoxy Cassirer’s “critical idealism” insisted that formal logic alone could not make sense of the conceptual co-evolution of mathematical and scientific concepts. For Cassirer, idealizations, or, more precisely, idealizing completions, played a fundamental role in the formation of the mathematical and empirical concepts. The aim of (...) this paper is to outline the basics of Cassirer’s idealizational account, and to point at some interesting similarities it has with Kant’s and Peirce’s philosophies of mathematics based on the key notions of pure intuition and theorematic reasoning, respectively. (shrink)

: In his paper "Explanation of Curiosity the First" Charles Peirce describes Euclid's procedure in proving theorems. Euclid first presents his theorem in general terms and then translates it into singular terms. Peirce pays attention to the fact that the generality of the statement is not lost by that move. The next step is construction, which is followed by demonstration. Finally the ergo-sentence repeats the original general proposition. Peirce lays much emphasis on the distinction between corollarial and theorematic reasoning in (...) geometry. He takes an argument to be corollarial if no auxiliary construction is needed. For Peirce, construction is "the principal theoric step" of the demonstration. Peirce also stresses that it is the observation of diagrams that is essential to all reasoning and that even if no auxiliary constructions are made, there is always the step from a general to a singular statement in deductive reasoning; that means introducing a kind of diagram to reasoning.This paper seeks to argue for two theses. One is that the way of thinking of reasoning which stresses the role of observation is essential to Peirce's discovery of the new logic. The other is that Peirce wants to consider both logical reasoning and philosophical thinking in general via the model of geometry. However, it is argued in this paper that it is the model of problematic analysis in geometry that is the most important for Peirce when he discovers the new structure of propositions. Moreover, this paper seeks to show that this very feature in Peirce's thought connects him with philosophers and logicians who emphasise the role of praxis instead of theoretic contemplation and that one who wants to reconstruct Peirce's metaphilosophical views must pay special attention to the model of problematic analysis.Key-words: construction; demonstration; corollarial and theorematic reasoning; diagrams.Resumo: Em seu artigo "Explanation of Curiosity the First", Charles Peirce descreve o procedimento de Euclides para demonstrar teoremas. Euclides primeiramente apresenta seu teorema em termos gerais, e então o traduz para termos singulares. Peirce presta atenção ao fato de que a generalidade do enunciado não é perdida por esse movimento. O próximo passo é a construção, que é seguida pela demonstração. Finalmente, a sentença-ergo repete a proposição geral original. Peirce dá bastante ênfase à distinção entre o raciocínio corolarial e o teoremático em geometria. Ele considera um argumento como corolarial se nenhuma construção auxiliar se fizer necessária. Para Peirce, a construção é "o principal passo teórico" da demonstração. Peirce também enfatiza que a observação de diagramas é essencial para todo raciocínio e que, mesmo se não são feitas construções auxiliares, há sempre o passo de um enunciado geral para o particular no raciocínio dedutivo; o que significa introduzir um tipo de diagrama no raciocínio.Este artigo procura defender duas teses. A primeira, que a maneira de pensar o raciocínio que enfatiza o papel da observação é essencial para a descoberta da nova lógica por Peirce. A outra, que Peirce deseja considerar tanto o raciocínio lógico quanto o pensamento filosófico em geral pelo modelo da geometria. Entretanto, defende-se neste artigo que o modelo de análise problemática em geometria é o mais importante para Peirce, quando ele descobre a nova estrutura das proposições. Este artigo procura mostrar, ainda, que é exatamente essa característica do pensamento de Peirce que o liga aos filósofos e lógicos que enfatizam o papel da práxis em vez da contemplação teórica, e que quem deseje reconstruir as visões metafilosóficas de Peirce deve prestar atenção especial ao modelo da análise problemática.Palavras-chave: construção, demonstração, raciocínios corolarial e teoremático, diagrama. (shrink)

Previous chapter The Transformations of Capitalism and the Nation-State in the 20th Century Politics, Deleuze and Guattari emphasized, is based on “experimentation, groping in the dark, injection, withdrawal, advances, retreats.” It basically deals with a form of struggle similar to that existing in science between “intuitionism,” “problematic conception of science,” “working in the undecidable and the fugitive,” on the one hand, and “axiomatics,” “theorematic conception of geometry,” “reordering - Économie classique et marxiste – Nouvel article.

Da Costa's paraconsistent systems of the series Cm (for finite m) (see [C1], [C2], and esp. [C3], pp. 237ff.) share important features with transitive logic, TL (which has been gone into in [P1] and [P2]), namely, they all coincide in that: (c1) they possess a strong negation, `¬', a conditional, `⊃', a conjunction, `∧', and a disjunction, `∨', with respect to which they are conservative extensions of CL or Classical Logic; (c2) they possess a non strong negation, `N' (notations are (...) different for systems C) which does not possess all properties of classical negation, but for which the following schemata are theorematic (I use the letters `p', `q', etc as schematic letters; my notational conventions are basically Church's: associativity leftwards; a dot stands for.. (shrink)

In this article, I examine the reconstruction that Peirce does on analytic/synthetic Kantian division, supported by his phenomenology, semiotic and pragmatism. The analysis of Peirce’s writings on mathematic suggests a notion of a posteriori and necessary analytical truths, that is, propositions that express one belief justified in experience, but whose generalization is valid for all the possible worlds. This was a new idea the time that Peirce formulated it, in 19th Century, and it contrasts with semantic-analytical tradition from Frege and (...) was contested by Quine in 1950’s. Peirce’s notion of analyticity is a result of his discovery of the logic of relatives, which led the philosopher to revise the deductive reasoning, dividing it into corollarial and theorematic, which in turn allow us to understand why mathematics is, simultaneously, deductive and non-trivial. (shrink)

Like Leibniz, C. S. Peirce drew much of the inspiration for his philosophical work from a close study of logical and mathematical reasoning. Now what insights did this study reveal to Peirce? His own answer is formulated as follows: “My first real discovery about mathematical procedure was that there are two kinds of necessary reasoning, which I call the Corollarial and the Theorematic.…” The import of this discovery was lost on philosophers for a long time. The purpose of the present (...) paper is to show what Peirce's insight amounts to, to point out some of its applications, relate it to certain recent developments in the philosophy of logic, and thereby vindicate Peirce's distinction. This undoubtedly will also help to show its present day relevance. (shrink)

Like Leibniz, C. S. Peirce drew much of the inspiration for his philosophical work from a close study of logical and mathematical reasoning. Now what insights did this study reveal to Peirce? His own answer is formulated as follows: “My first real discovery about mathematical procedure was that there are two kinds of necessary reasoning, which I call the Corollarial and the Theorematic.…” The import of this discovery was lost on philosophers for a long time. The purpose of the present (...) paper is to show what Peirce's insight amounts to, to point out some of its applications, relate it to certain recent developments in the philosophy of logic, and thereby vindicate Peirce's distinction. This undoubtedly will also help to show its present day relevance. (shrink)

RésuméSelon un pragmatiste, la réflexion du philosophe porte sur le lien entre la construction et la description des objets mathématiques. Grâce à sa conception?un raisonnement dit ≫theorematique≪, Peirce a réussi à etablir ľesquisse?une interprétation pragmatique?une structure mathématique. Cette dernière reste néanmoins vague quant à la catégoricité de la structure. – lci, il est utile de recourir à Gonseth et & dcar;analyser sa reconstruction?un système axiomatique: la genèse logique?une structure selon les principes de ľidonéisme correspond en effet au programme de Peirce, (...) à savoir àľexplication de la signification?un concept àľaide?une séquence cumulative? interprétants.SummaryAccording to a pragmatist, philosophical thought bears on the link between the construction and the description of mathematical objects. Thanks to his development of ≫theorematic reasoning)), Peirce succeeded in bringing forward a pragmatic interpretation of a mathematical structure, exhibiting however an inherent vagueness with regard to the categoricalness of the structure. ‐ In this matter, Gonseth's model of reconstruction of an axiomatic system can be resorted to: the logical genesis of a structure considered in his ≫ldoneism≪ corresponds to Peirce's program of explaining the meaning of a concept by means of a cumulative sequence of interpretants. (shrink)