This paper seeks to indicate some connections between a major philosophi- cal project of the seventeenth century, the conception of a mathesis universalis, and the practice of baroque poetry. I shall argue that these connections consist in a peculiar view of language and systems of notation which was particularly common in European baroque culture and which provided the necessary conceptual background for both poetry and the mathesis universalis.

The paper discusses some changes in Bolzano's definition of mathematics attested in several quotations from the Beyträge, Wissenschaftslehre and Grössenlehre: is mathematics a theory of forms or a theory of quantities? Several issues that are maintained throughout Bolzano's works are distinguished from others that were accepted in the Beyträge and abandoned in the Grössenlehre. Changes are interpreted as a consequence of the new logical theory of truth introduced in the Wissenschaftslehre, but also as a consequence of the overcome of Kant's (...) terminology, and of the radicalization of Bolzano's anti‐Kantianism. Bolzano's evolution is understood as a coherent move, once the criticism expressed in the Beyträge on the notion of quantity is compared with a different and larger notion of quantity that Bolzano developed already in 1816. This discussion is enriched by the discovery that two unknown texts mentioned by Bolzano in the Beyträge can be identified with works by von Spaun and Vieth respectively. Bolzano's evolution is interpreted as a radicalization of the criticism of the Kantian definition of mathematics and as an effect of Bolzano's unaltered interest in the Leibnizian notion of mathesis universalis. As a conclusion, the author claims that Bolzano never abandoned his original idea of considering mathematics as a scientia universalis, i.e. as the science of quantities in general, and suggests that the question of ideal elements in mathematics, apart from being a main reason for the development of a new logical theory, can also be considered as a main reason for developing a different definition of quantity. (shrink)

In this article, I deal with the role of hypothesis in the scientific methodology of Isaac Newton, Francis Bacon and René Descartes. The first paragraph is about hypothesis in Newton's lexicon, especially trying to understand the meaning of his famous hypotheses non fingo. The second paragraph deals with Bacon's methodology, arguing especially that his epis-temology was the first to propose an artificial way for inductive inferences, also giving up all hypothesis in science. The third paragraph shows how Descartes, following Bacon's (...) traces and reading his methodology in the light of the idea of a Mathesis Universalis based on mental evidence, structures a full hypothetical-deductive methodology, charging metaphysics with the analytical phase and depriving experiments of any role in it. The fourth paragraph, on Newton again, tries to understand Newton's specific account of hypothesis as an auxiliary phase in the scientific discovery, and what really differentiates Newton from his contemporaries, Boyle and Hooke. (shrink)

Husserl’s theory of manifolds was developed for the first time in a very short form in the Prolegomena to his Logical Investigations, §§ 69–70 (pp. 248–53), then repeatedly discussed in Ideas I, §§ 71–2 (pp. 148–53), in Formal and Transcendental Logic, §§ 51–4 (pp. 142–54), and finally in the Crisis, § 9 (pp. 20–60). Husserl never lost sight of it: it was his idée fixe. He discussed this theme over forty years, expressing the same, in principle, ideas on it in (...) different terms and versions. His discussions of it, however, were always cut short and inconclusive, so that he never developed his theory of manifolds in detail. Main claim in this essay is that Husserl’s theory of manifolds is twofold. It is (1) a theory of theories, advanced in the good German tradition of Wissenschaftslehre, launched first by Bernard Bolzano. The latter, in turn, is an offspring of the ancient tradition of mathesis universalis, explored in the modernity by Descartes and Leibniz. Further, (2) the theory of manifolds is also a formal theory of everything. Its objective is to provide a systematic account of how the things in the world (its parts) hang together in wholes of quite different kind. It is to be noted that Husserl explicitly speaks about (1) and only implicitly about (2). In this paper we shall also show that the theory of everything is intrinsically connected with the theory of theories. (shrink)

Descartes' methodical doubt is being criticized by naïve realists and others who don't find doubt as a good starting point for metaphysical thought, however, the philosophical achievements of his method have been absorbed in all later philosophies. The objective of this paper is to demonstrate how an inevitable question concerning the foundation of Descartes' mathesis universalis, which led him to investigate this foundation by applying this very method in Metaphysics, has finally enabled him to discover his most important (...) philosophical principle, i.e., the necessary relation between being and reason in the identity of the "I". To demonstrate this, after summarizing the fundamental ideas of mathesis uneversalis, and showing the inner necessity of introducing the methodical doubt to fortify these ideas, the decisive role of metaphysical doubt as the third step of the methodical doubt, and malicious demon as its second phase would be highighted. It would also become clear that the metaphysical doubt has a great importance as a turning point from different aspects: as a knot that ties mathesis universalis with its underlying Metaphysics; as a bridge from within the psychological "I" to the metaphysical "I". (shrink)

By extending Husserl’s own historico-critical study to include the conceptual mathematics of more contemporary times – specifically category theory and its emphatic development since the second half of the 20th century – this paper claims that the delineation between mathematics and philosophy must be completely revisited. It will be contended that Husserl’s phenomenological work was very much influenced by the discoveries and limitations of the formal mathematics being developed at Göttingen during his tenure there and that, subsequently, the rôle he (...) envisaged for his material a priori science is heavily dependent upon his conception of the definite manifold. Motivating these contentions is the idea of a mathematics which would go beyond the constraints of formal ontology and subsequently achieve coherence with the full sense of transcendental phenomenology. While this final point will be by no means proven within the confines of this paper it is hoped that the very fact of opening up for the possibility of such an idea will act as a supporting argument to the overriding thesis that the relationship between mathematics and phenomenology must be problematised. (shrink)

Recent work in formal philosophy has concentrated over-whelmingly on the logical problems pertaining to epistemic shortfall - which is to say on the various ways in which partial and sometimes incorrect information may be stored and processed. A directly depicting language, in contrast, would reflect a condition of epistemic perfection. It would enable us to construct representations not of our knowledge but of the structures of reality itself, in much the way that chemical diagrams allow the representation (at a certain (...) level of abstractness) of the structures of molecules of different sorts. A diagram of such a language would be true if that which it sets out to depict exists in reality, i.e. if the structural relations between the names (and other bits and pieces in the diagram) map structural relations among the corresponding objects in the world. Otherwise it would be false. All of this should, of course, be perfectly familiar. (See, for example, Aristotle, Metaphysics, 1027 b 22, 1051 b 32ff.) The present paper seeks to go further than its predecessors, however, in offering a detailed account of the syntax of a working universal characteristic and of the ways in which it might be used. (shrink)

Hermeneutic studies of science locate a circle at the heart of scientific practice: scientists only gain knowledge of what they, in some sense, already know. This may seem to threaten the rational validity of science, but one can argue that this circle is a virtuous rather than a vicious one. A virtuous circle is one in which research conclusions are already present in the premises, but only in an indeterminate and underdeveloped way. In order to defend the validity of science, (...) the hermeneuticist must describe a method by which a vague and confused initial knowledge of nature gets transformed into a clear and determinate knowledge of nature. I consider three such methods. The first is regressus demonstrativa, favoured by the physicians of Padua during the fifteenth and sixteenth centuries. The second is mathēsis, introduced by Martin Heidegger in his discussion of seventeenth-century science. The third is Denkstil, a key concept in Ludwik Fleck’s history of syphilology. I conclude by listing three desiderata for a hermeneutic science studies: that it be anti-metaphysical, historical, and sociological. --- Reprinted in: Erich Otto Graf, Martin Schmid & Johannes Fehr (eds.), Fleck and the Hermeneutics of Science (Collegium Helveticum Heft 14) (Zürich, 2016), pp. 85-93. (shrink)

The task of this paper is to give a new, catholic interpretation of Leibniz’s concept of characteristica universalis. In § 2 we shall see that in different periods of his development, Leibniz defined this concept differently. He introduced it as “philosophical characteristic” in 1675, elaborated it further as characteristica universalis in 1679, and worked on it at least until 1690. Secondly, we shall see (in § 3) that in the last 130 years or so, different philosophers have advanced (...) projects similar to that of Leibniz, not always referring to him, at that. These very projects, we claim, threw some light on what Leibniz idea of characteristica universalis could be; or, in more positive sense, on how could it be reconstructed in a more workable way. Unfortunately, they failed to answer the question what exactly Leibniz’s philosophy was. Finally, despite the fact that Leibniz’s concept of characteristica universalis impressed generations of philosophers who tried to make sense of it, the result is that in the more than 300 years after it was introduced, it was never used in the scale its author dreamed of. This fact sets out the next objective which we are going to pursue in this paper: we shall try to find out (in §§ 4 and 5) how this concept can be interpreted in more practical form. (shrink)

El punto de partida del calculus universalis leibniziano es la teoría aristotélica del silogismo, pero Leibniz se independiza de las ideas de Aristóteles para desarrollar su propio sistema lógico, mucho más general, aplicando el instrumento combinatorio a la silogística. Lo que propone es una importante modificación del modelo demostrativo axiomático, mediante la creación de cálculos lógico-simbólicos que no se limitan a los ámbitos tradicionales de la deducción, sino que admiten procedimientos discursivos más complejos que los de la lógica clásica, (...) ampliando con ello la noción de racionalidad. (shrink)

The text discusses Linnaeus’ binominal classification and the idea of mathesis unversalis of Leibniz in the ground of Michel Foucault’s conception of ‘epistema’ as well the connexion of XVIII century’s representation of universal order with viewpoints of Descartes (Rules for the Direction of the Mind) and Kant (The Critique of Pure Reason, The Critique of (the Power of) Judgment).

Existe una larga trayectoria en la relación entre matemáticas y narrativa; en este trabajo se muestran los modos específi cos en que se vinculan pese a lo disímiles que, en apariencia, pueden llegar a ser. A partir de un modelo semántico se construye un esquema de análisis narrativo de carácter diagramático y articulación relacional, que considera y se apoya en la noción de ‘Visibilidad’ como dimensión semiótica.

Nell'alto medioevo vi erano tre possibilità per il filosofo che volesse difendere l'esistenza degli universali: in primo luogo ante rem, seguendo la linea platonico-agostiniana degli universali trascendenti e questa è la via teologica; in secondo luogo la via post rem, cioè la via del concettualismo, conosciuta oggi come astrattismo, per cui gli universali sono il prodotto dell'astrazione della mente e questa è una via logica; la terza via è quella in re e difende gli universali immanenti che esistono negli universali (...) ed è una risposta ontologica. La seconda via fu difesa da Boezio nel secondo commento a Porfirio. La prima teoria è stata generalmente accettata da tutti i filosofi dell'alto medioevo eccetto Abelardo, mentre la terza via, quella degli universali immanenti, che ha come fonti antiche l'Isagoge di Porfirio e le Categoriae di Aristotele, fu difesa principalmente da quattro autori: Giovanni Scoto Eriugena, Anselmo d'Aosta, Odone di Cambrai e Guglielmo di Champeaux. L'A. prende in esame e confronta le soluzioni di Anselmo di Canterbury e Odone di Cambrai. (shrink)

The discussions which follow rest on a distinction, first expounded by Husserl, between formal logic and formal ontology. The former concerns itself with (formal) meaning-structures; the latter with formal structures amongst objects and their parts. The paper attempts to show how, when formal ontological considerations are brought into play, contemporary extensionalist theories of part and whole, and above all the mereology of Leniewski, can be generalised to embrace not only relations between concrete objects and object-pieces, but also relations between what (...) we shall call dependent parts or moments. A two-dimensional formal language is canvassed for the resultant ontological theory, a language which owes more to the tradition of Euler, Boole and Venn than to the quantifier-centred languages which have predominated amongst analytic philosophers since the time of Frege and Russell. Analytic philosophical arguments against moments, and against the entire project of a formal ontology, are considered and rejected. The paper concludes with a brief account of some applications of the theory presented. (shrink)

In this paper I argue that differences between the ‘new moral science’ of the seventeenth century and scholastic natural law theory originated primarily from the skeptical challenge the former had to face. Pufendorf’s project of a scientia practica universalis is the paramount expression of an anti-skeptical moral science, a ‘science’ that is both explanatory and normative, but also anti-dogmatic insofar as it tries to base its laws on those basic phenomena of human life which, supposedly, are immune to skeptical (...) doubt. The main scholastic legacy to the new moral science is the dichotomy between an ‘intellectualist’ and a ‘voluntarist’ view of natural law (or between lex immanens and lex imposita). Voluntarism lies at the basis of both theological views, such as Calvinism, and political views, such as those of Hobbes and Locke. The need to counterbalance the undesirable implications of extreme voluntarism may account for much of the developments in ethics and politics during the seventeenth and eighteenth centuries.Scottish natural jurisprudence, which tried to find a middle way between skepticism and extreme voluntarism, is less secular and more empirical than received wisdom admits. There emerged, as one of its ‘accidental’ outcomes, a systematic, self-contained and empirical economic theory from the search for an empirically based normative theory of social life. The basic assumption of such a theory, namely, the notion of societal laws as embedded in trans-individual mechanisms, derives from the voluntarist view of natural law as ‘imposed’ law.Later discussions of social issues in terms of ‘economic’ and ‘ethical’ reasons originated partly from a misreading ofthe Scottish natural jurisprudential framework of economic theory. Starting with this reconstruction, I try to shed some light on recent discussions about the role of ethics in economics. (shrink)

The disciplinary differentiation of sciences attracted Leibniz’s attention for a long period of time. From nowadays prospects it looks very well grounded as soon as in Leibniz’s manuscripts a modern scholar finds clue ideas of any research field which would tempt him to consider Leibniz as one of the founders of this particular discipline. We argue that this is possible only in retrospection and would significantly distort the essence of Leibniz’s epistemology. Our approach implies, in contrary, the investigation of the (...) Leibniz doctrine of signs on the background of the related philosophical problem, that of expression. The choice of semiotics is justified by the fact that it took a central place in his theoretical constructions, both those of natural sciences and of philosophy. In Leibniz system of knowledge the concept of notes (notae) and sings (signa) served a theoretical foundation of his most important and long-life aspiration to build up practical science of universal characteristics (characteristica universalis). In his eyes this practical science was the science of sciences (Scientia scientiarum), and we can consider it as the matrix for all possible scientific knowledge. (shrink)

I argue that the difference between the 17th century new moral science and Scholastic Natural Law Theory derived primarily from the skeptical challenge the former had to face. Pufendorf's project of a 'scientia practica universalis' was the paramount expression of an anti-skeptical moral science, a «science» both explanatory and normative, but also anti-dogmatic in so far as it tried to base its laws on those basic phenomena of human life that supposedly were outside the scope of skeptical doubt. Of (...) the Scholastic legacy to the new moral science, a dichotomy between an «intellettualistic» and a «voluntaristic» view of natural law (or between lex immanens and lex imposita). Voluntarism lays at the root both of theological views such as those of Calvinism and of political views such as those of Hobbes and Locke. A need to counterbalance undesirable implications of estreme voluntarism may account for much of 17th and 18th centuries developments in ethics and politics. Scottish natural jurisprudence, an expression of such quest for a third way between scepticism and extreme voluntarism, is less secular and more empirical than received wisdom admits of. One of its side-effects, namely a systematic, self-contained, and empirical economic theory, results from the search for a normative theory of social life on an empirical basis. The main tool for building such a theory, namely a view of societal laws as embedded in trans-individual mechanisms, derives from the voluntarist view of natural law as «imposed» law. Subsequent discussions of social issues based on the opposition of economic and ethical reasons originated partly from gross misreading of the Scottish natural jurisprudential framework for economic theory. (shrink)

Accused to be leading a plot against the Spanish government in Calabria in 1599, supposedly supported by a Turkish fleet, Campanella was almost labeled as a renegade. On the contrary, while in jail, he deepened his prophetic interpretation of the history, and of the future of the world, offered theological and political confutation of Islam, and began shaping a wider idea of the role of this religion in the ‘apocalyptic’ times. Not focusing on the Turkish menace only, he tries to (...) integrate some of the Muslim kingdoms in his appeal to the universal recognition of the Christian revelation as ‘natural reason’. In his main work about this topic, Quod reminiscentur, completed in 1618, but published only in the 20th Century, Campanella presents different strategies of conversion of Islamic peoples, depending on the various historical features of Islam, and adapted to the diverse societies involved. Giovanni Botero’s Relationi universali was a very important source of Campanella’s historical and geographic information about Islam, but the philosopher developed an original prophetic and political vision of Church, conversion and apocalypse, alternative to that dominating in the Counter Reformation. (shrink)

Questa è un'eccellente revisione delle interazioni gene/ambiente sul comportamento e, nonostante sia un po' datata, è una lettura facile e utile. Iniziano con studi gemelli che mostrano l'impatto travolgente della genetica sul comportamento. Notano gli studi sempre più noti di Judith Harris che estendono e riassumono i fatti che l'ambiente domestico condiviso non ha quasi alcun effetto sul comportamento e che i bambini adottati crescono fino ad essere diversi dai loro fratellastri e sorelle come le persone scelte a caso. Un (...) punto fondamentale che loro (e quasi tutti coloro che discutono di genetica comportamentale) non riescono a notare è che le centinaia (migliaia a seconda del tuo punto di vista) degli universali comportamentali umani, comprese tutte le basi delle nostre personalità, sono determinate al 100% dai nostri geni, senza variazioni nella normalità. Tutti vedono un albero come un albero e non come una pietra, cercano e mangiano cibo, si arrabbiano e geloso ecc. Quindi, quello di cui stanno parlando per lo più qui è quanto ambiente (cultura) può influenzare il grado in cui vengono mostrati i vari tratti, piuttosto che il loro aspetto. Infine, discutono di eugenetica nel solito modo politicamente corretto, non riuscendo a notare che noi e tutti gli organismi siamo i prodotti dell'eugenetica della natura e che i tentativi di sconfiggere la selezione naturale con la medicina, l'agricoltura e la civiltà nel suo complesso sono disastrosi per qualsiasi società che persiste nel farlo. Fino al 50% di tutte le concezioni, o circa 100 milioni/anno, finiscono in un aborto spontaneo precoce, quasi tutti senza che la madre ne sia a conoscenza. Questo abbattimento naturale dei geni difettosi guida l'evoluzione, ci mantiene relativamente geneticamente sani e rende possibile la società. La disgenica è sufficiente a distruggere la civiltà, ma la sovrappopolazione sarà doprima. Coloro che desiderano un quadro aggiornato completo per il comportamento umano dalla moderna vista a due systems possono consultare il mio libro 'La struttura logica dellafilosofia, psicologia, Mind e il linguaggio in Ludwig Wittgenstein e John Searle' 2nd ed (2019). Coloro che sono interessati a più dei miei scritti possono vedere 'TalkingMonkeys--Filosofia, Psicologia, Scienza, Religione e Politica su un Pianeta Condannato--Articoli e Recensioni 2006-2017' 3rd ed (2019). (shrink)

Esta coletânea reúne, organizados na forma de capítulos, sete artigos que exploraram uma ideia central: a de que, subjacente à Lógica de Hegel, há uma mathesis, isto é, uma estruturação matemática que organiza as ideias dessa Lógica, mathesis de que Hegel faz uso, de modo implícito, ao longo do texto da "Ciência da Lógica".

This book historically analyzes the art of memory, surveying multiple cultures, examining Chinese, Indian and Greek meditation, I then construct a new universal language using Aristotelian, and Daoist philosophy. The language utilizes Newtonian physics and a symbolic form of binary. -/- The inspiration for the language was Leibniz's charastica universalis, along with Bruno and Ramon lull wheels. The language attempts to mathematically define archetypes across the stated cultures. If the language hasn't made any worthwhile discoveries within mathematics and physics, (...) I hope it signifies my attempt to understand something just beyond my grasp. Bruno spoke extensively of signs. (shrink)

Dans un texte désormais célèbre, Ferdinand de Saussure insiste sur l’arbitraire du signe dont il vante les qualités. Toutefois il s’avère que le symbole, signe non arbitraire, dans la mesure où il existe un rapport entre ce qui représente et ce qui est représenté, joue un rôle fondamental dans la plupart des activités humaines, qu’elles soient scientifiques, artistiques ou religieuses. C’est cette dimension symbolique, sa portée, son fonctionnement et sa signification dans des domaines aussi variés que la chimie, la théologie, (...) les mathématiques, le code de la route et bien d’autres qui est l’objet du livre La Pointure du symbole. -/- Jean-Yves Béziau, franco-suisse, est docteur en logique mathématique et docteur en philosophie. Il a poursuivi des recherches en France, au Brésil, en Suisse, aux États-Unis (UCLA et Stanford), en Pologne et développé la logique universelle. Éditeur-en-chef de la revue Logica Universalis et de la collection Studies in Universal Logic (Springer), il est actuellement professeur à l’Université Fédérale de Rio de Janeiro et membre de l’Académie brésilienne de Philosophie. SOMMAIRE -/- PRÉFACE L’arbitraire du signe face à la puissance du symbole Jean-Yves BÉZIAU La logique et la théorie de la notation (sémiotique) de Peirce (Traduit de l’anglais par Jean-Marie Chevalier) Irving H. ANELLIS Langage symbolique de Genèse 2-3 Lytta BASSET -/- Mécanique quantique : quelle réalité derrière les symboles ? Hans BECK -/- Quels langages et images pour représenter le corps humain ? Sarah CARVALLO Des jeux symboliques aux rituels collectifs. Quelques apports de la psychologie du développement à l’étude du symbolisme Fabrice CLÉMENT Les panneaux de signalisation (Traduit de l’anglais par Fabien Shang) Robert DEWAR Remarques sur l’émergence des activités symboliques Jean LASSÈGUE Les illustrations du "Songe de Poliphile" (1499). Notule sur les hiéroglyphes de Francesca Colonna Pierre-Alain MARIAUX Signes de vie Jeremy NARBY Visualising relations in society and economics. Otto Neuraths Isotype-method against the background of his economic thought Elisabeth NEMETH Algèbre et logique symboliques : arbitraire du signe et langage formel Marie-José DURAND – Amirouche MOKTEFI Les symboles mathématiques, signes du Ciel Jean-Claude PONT La mathématique : un langage mathématique ? Alain M. ROBERT. (shrink)

Between the end of the nineteenth century and the first twenty years of the twentieth century, stimulated by the impetuous development of logical studies and taking inspiration from Leibniz's idea of a characteristica universalis, the three founding fathers of the analytic tradition in philosophy, i.e., Frege, Russell, and Wittgenstein, started to talk of a logically perfect language, as opposed to natural languages, all feeling that the latter were inadequate to their (different) philosophical purposes. In the second half of the (...) twentieth century, however, the very idea of a logically perfect language ceased for various reasons to seem attractive to analytic philosophers. Thus, it might appear that this idea could be classified together with the many other bizarre ideas that from time to time surface in the history of philosophy-an idea that perhaps had a beneficial impact on the development of twentieth century logic, but which can now be put to rest. In this brief note, I contend that this conclusion may be too hasty. Indeed, if a well-known empirical hypothesis advanced in 1975 by Jerry Fodor turns out to be true, then there is a logically perfect language, after all. More precisely, I argue that, if it exists, Fodor's language of thought possesses the main characteristics a logically perfect language is required to have. (shrink)

We show that we in ways related to the classical Square of Opposition may define a Cube of Opposition for some useful statements, and we as a by-product isolate a distinct directive of being inviolable which deserves attention; a second central purpose is to show that we may extend our construction to isolate hypercubes of opposition of any finite cardinality when given enough independent modalities. The cube of opposition for obligations was first introduced publically in a lecture for the Square (...) of Opposition Conference in the Vatican in May 2014. -/- A COLORFUL ERRATUM: The article Cubes and Hypercubes of Opposition, with Ethical Ruminations on Inviolability, suffered a couple of mishaps when it was published in Logica Universalis, ISSN 1661-8297, 10(2-3), p. 373–376. The worst of them was that the colorful cube was printed in shades of grey, instead of decorating the sides with two transparent sides, and then a red, a yellow, a blue and a green side. The text of the article is not quite understandable because of this mishap, so I avail a colorfully updated version of the original article here. -/- . (shrink)

Ce texte est la préface à la réédition d'un certain nombre de textes du philosophe français Jean-Toussaint Desanti, philosophe des mathématiques qui a été marqué par la phénoménologie husserlienne et la pensée de Jean Cavaillès. Nous nous efforçons d'y rassembler les thèmes et thèses centraux de la pensée de Desanti, telle qu’elle s’élabore dans ces textes publiés dans différentes revues : en premier lieu, le thème de l’historicité des idéalités mathématiques et la récusation du sujet comme instance productrice de ces (...) objets. (shrink)

L’osservazione della natura con l’intento di capire l’origine della varietà di forme e fenomeni in cui si manifesta ha origini remote. All’inizio il rapporto con i fenomeni naturali era dominato da sentimenti quali paura e stupore che conducevano a supporre l’esistenza di entità sfuggenti alla percezione diretta che permeavano gli elementi animandoli. Ecco dunque che la magia rappresenta l’elemento dominante della filosofia naturale primitiva caratterizzata da una unicità degli eventi e dalla impossibilità di capirli e dominarli in quanto frutto della (...) volontà di essenze a noi estranee e non governabili. Con il nascere della civiltà ed il suo progredire il tempo dedicato ai lavori necessari per il sostentamento e la sopravvivenza diminuì e nella ripartizione dei compiti alcuni individui potevano dedicare parte del loro tempo alla osservazione della natura ed alla sua interpretazione in termini non trascendenti. Nella natura, intesa come tutto ciò che ci circonda composto da esseri viventi e da materia inorganica nelle sua varie aggregazioni sulla terra e nel cosmo, ciò che attirò l’attenzione fin dall’inizio furono furono i fenomeni regolari e periodici come i moti della luna, dei pianeti e delle stelle. Nel contempo dopo una spinta iniziale dettata da esigenze pratiche come contare gli oggetti o misurare i campi, la matematica si era sviluppata autonomamente e si rivelò idonea a descrivere in termini quantitativi i moti dei corpi celesti. La terra era al centro dell’universo mentre il moto degli altri corpi celesti risultava da una composizione di moti circolari uniformi. Questa visione geocentrica e pitagorica (armonia delle sfere) dell’universo ha prevalso fino agli albori della scienza moderna, anche se una descrizione eliocentrica, basata su validi argomenti, era stata proposta. Per quanto riguarda la struttura della materia i presocratici avevano già proposto i quattro elementi mentre gli atomisti avevano ricondotto tutto ad entità elementari primigenie, il cui aggregarsi e disgregarsi da luogo a tutti gli stati e le molteplici forme della materia. Queste intuizioni si ritrovano nella fisica moderna che contempla quattro stati di aggregazione, che hanno come unico sostrato comune gli atomi. La fisica moderna nasce con Galileo e Newton, la cui dinamica si sviluppa a partire dalle leggi di Keplero che descrivono il moto dei pianeti nel sistema eliocentrico, per potersi poi applicare ad un qualunque sistema materiale. Pertanto nei due secoli successivi si ritenne che un modello meccanico potesse essere sviluppato per un qualunque sistema fisico e quindi per l’universo intero la cui evoluzione doveva essere matematicamente prevedibile. Per i fenomeni termici tuttavia vennero formulate leggi ad hoc come quelle della termodinamica che mostrano come i processi macroscopici siano irreversibili in contrasto con le leggi della meccanica. Si deve a Boltzmann1 il tentativo di ricondurre la termodinamica alla meccanica per un gran numero di particelle dei cui moti disordinati viene data una lettura di carattere statistico. L’aumento della entropia e la irreversibilità seguono dalla ipotesi di caos molecolare ossia che i moti siano così disordinati che si perde rapidamente memoria dello stato iniziale. L’idea di introdurre una misura di probabilità nel contesto della meccanica sembra antitetica con la natura stessa della teoria rivolta fino ad allora allo studio di sistemi con moti regolari, reversibili e prevedibili individualmente su tempi lunghi. Tuttavia l’analisi probabilistica diventa essenziale per lo studio di sistemi caratterizzati da forti instabilità, e da orbite irregolari per i quali la previsione richiede una conoscenza della condizioni iniziali con precisioni fisicamente non raggiungibili. Combinando la evoluzione deterministica della meccanica di Newton o di Hamilton con la descrizione statistica attraverso una opportuna misura invariante di probabilità nello spazio delle fasi, nasce la teoria dei sistemi dinamici2 che consente di descrivere non solo i sistemi ordinati o i sistemi caotici ma anche tutti quelli che vedono coesistere in diverse proporzioni ordine e caos e che presentano una straordinaria varietà di strutture geometriche e proprietà statistiche, tanto da fornire almeno se non proprio un quadro teorico per lo meno metafore utili per la descrizione dei sistemi complessi. Anche se non c’è unanime consenso ci sembra appropriato definire complessi non tanto sistemi caratterizzati da interazioni non lineari tra i suoi componenti e da proprietà emergenti, che rientrano a pieno titolo nel quadro dei sistemi dinamici, ma piuttosto i sistemi viventi o quelli di vita artificiale che ne condividono le proprietà essenziali3. Tra queste possiamo certamente annoverare la capacità di gestire la informazione e di replicarsi, consentendo tramite un meccanismo di mutazione e selezione di dare origine a strutture di crescente ricchezza strutturale e dotate di capacità cognitive sempre più elaborate. Una teoria dei sistemi complessi non esiste ancora, anche se la teoria degli automi sviluppata da Von Neumann4 e la teoria della evoluzione di Darwin5 ne possono fornire alcuni pilastri importanti. Recentemente la teoria delle reti è stata utilizzata con successo per descrivere le proprietà statistiche delle connessioni tra gli elementi costitutivi (nodi) di un sistema complesso6. Le connessioni che non sono né completamente casuali né completamente gerarchiche, consentono una sufficiente robustezza rispetto a malfunzionamenti o danneggiamenti dei nodi unita a un adeguato livello di organizzazione per consentirne un funzionamento efficiente. Nei sistemi fisici il modello base è un insieme di atomi o molecole interagenti, che danno luogo a strutture diverse quali un gas, un liquido o un cristallo, come risultato di proprietà emergenti. Nello stesso modo per i sistemi complessi possiamo proporre un sistema automi interagenti come modello base. Le molteplici forme che il sistema assume anche in questo caso vanno considerate come proprietà emergenti del medesimo sostrato al mutare delle condizioni esterne e frutto delle i replicazioni, ciascuna delle quali introduce piccole ma significative varianti. Questa è la grande differenza tra un sistema fisico ed un sistema complesso: il primo fissate le condizioni esterne ha sempre le medesime proprietà, il secondo invece cambia con il fluire del tempo perché la sua organizzazione interna muta non solo al cambiare di fattori ambientali ma anche con il succedersi delle generazioni. C’è dunque un flusso di informazione che cresce con il tempo e che consente agli automi costituenti ed alla intera struttura di acquisire nuove capacità. Questo aumento di ordine e ricchezza strutturale avviene naturalmente a spese dell’ambiente circostante, in modo che globalmente i la sua entropia cresce in accordo con la seconda legge della termodinamica. In assenza di una teoria formalizzata paragonabile a quella dei sistemi dinamici, per i sistemi complessi si possono fare osservazioni e misure, sia puntuali sui costituenti elementari e sulle loro connessioni, sia globali sull’intero sistema, oppure costruire modelli suscettibili di essere validati attraverso la simulazione. Se di un sistema si riesce infatti a fornire una descrizione sufficientemente dettagliata, è poi possibile osservare come questo si comporti traducendo le regole in algoritmi e costruendo quindi una versione virtuale, anche se semplificata del sistema stesso. Il passaggio più difficile è il confronto tra il sistema simulato ed il sistema vero, che passa necessariamente attraverso la valutazione di una numero limitato di parametri che ne caratterizzino e proprietà. La codifica del progetto è una proprietà cruciale dei sistemi complessi perché questa si realizza con un dispendio di massa ed energia incomparabilmente più piccolo rispetto a quello necessario per realizzare l’intera struttura; nello stesso tempo apportare piccole modifiche ad un progetto è rapido ed economico. In questo processo che comporta la continua introduzione di varianti si aprono molteplici strade e con lo scorrere del tempo si realizza una storia in modo unico e irripetibile. Anche il susseguirsi di eventi fisici caratterizzati da processi irreversibili e dalla presenza di molteplici biforcazioni da origine ad una storia che non si può percorrere a ritroso, né riprodurre qualora fossimo in grado ripartire dalle stesse condizioni iniziali. Tuttavia esiste una differenza profonda tra la storia di un sistema fisico come il globo terrestre e la storia della vita. La prima registra i molteplici cambiamenti che ha subito la superficie del nostro pianeta ove montagne e mari nascono e scompaiono senza un chiaro disegno soggiacente. La storia della vita è caratterizzata da una progressiva crescita della ricchezza strutturale e funzionale e accompagnata da una crescita della complessità progettuale. La rappresentazione di questa storia prende la forma di un albero con le sue ramificazioni che mostra la continua diversificazione delle strutture e la loro evoluzione verso forme sempre più avanzate. La direzione in cui scorre il tempo è ben definita: le strutture affinano le capacità sensoriali mentre cresce la potenza degli organi che elaborano la informazione. Un sistema complesso è anche caratterizzato da un molteplicità di scale, tanto più alta quanto più si sale sulla scala evolutiva. La ragione è che il procedere verso strutture sempre più elaborate avviene utilizzando altre strutture come mattoni per cui l’immagine che si può fornire è quella di una rete di automi a più strati: partendo dal basso una rete con le sue proprietà emergenti diventa il nodo di una rete di secondo livello, cioè un automa di secondo livello che interagisce con altri automi dello stesso tipo e così via. Nei sistemi inorganici, dove non esiste un progetto, si distinguono di norma solo due livelli, quello dei costituenti elementari e quello su scala macroscopica. I sistemi fisici sono riconducibili a poche leggi universali, che governano i costituenti elementari della materia, ma il passaggio dalla descrizione dalla piccola alla grande scala è impervio e consentito soltanto dalla simulazione numerica, quando ci allontaniamo dalle situazioni più semplici caratterizzate da un equilibrio statistico. I limiti che il disegno riduzionista incontra già per i sistemi fisici diventano decisamente più forti nel caso dei sistemi complessi. (shrink)

O presente artigo busca debater a posição de Charles Sanders Peirce e dos primeiros estudantes peirceanos de Lógica (Christine Ladd e O. H. Mitchell nos Studies in Logic, 1883) dentro do debate inspirador da visão da linguagem dentro da Filosofia Analítica, conhecido como “Lingua Universalis contra Calculus Ratiocinator”, cujos primórdios podem ser traçados desde a filosofia de Gottfried Leibniz. Para isso, comparamos esse campo do pensamento peirceano com o debate crucial entre a conceitografia de Gottlob Frege (Begriffsschrift, 1879) e (...) a lógica algébrica de George Boole (An investigation of the Laws of Thought on which are founded the mathematical theories of Logic and Probabilities, 1854). O nosso objetivo principal é observar que esse momento da filosofia peirceana pode ser comparado com o pensamento wittgensteiniano, especialmente em sua nova vertente, chamada New Wittgenstein, que tenta superar a tradicional divisão entre Tractatus Logico-Philosophicus e Investigações Filosóficas. Pela comparação entre pensadores influenciados por C. S. Peirce e a filosofia de Ludwig Wittgenstein, pretendemos abrir caminhos na compreensão do conceito de jogo de linguagem, especialmente em sua gramática e em seus operadores simbólicos, através da observação das funções de verdade [truth functions] e as tabelas de verdade [truth tables] enquanto lógica algébrica. (shrink)

El análisis de las propiedades geométricas de las configuraciones finitas ha sido uno de los objetivos fundamentales del estudio de las diversas geometrías discretas y de la geometría combinatoria. Este artículo propone plantear la posibilidad de una elucidación de lo matemático desde una perspectiva derivada de las ‘matemáticas en acción’ y no desde una concepción ‘analítico gramatical’ de sus fundamentos, y establecer, al menos, un mínimo umbral de validez, que articule una interpretación semiótica de los signos matemáticos, a través del (...) análisis de un par de ejemplos elementales de geometría configuracional, cuyas construcciones son propuestas por el autor. (shrink)

El presupuesto según el cual el contenido de una manifestación compleja está en función de los contenidos de sus partes componentes, expresa claramente una intuición que solemos tener sobre lo múltiple; implica una reflexión sobre la relación entre el todo y las partes que lo componen; involucra una teoría de las multiplicidades que entraña atributos de naturaleza matemática; presenta el problema de cómo los seres humanos nos relacionamos con los entornos del mundo para generar unidad de sentido. La significación es (...) un proceso de síntesis. Y desentrañar los mecanismos de funcionamiento de dicho proceso es un enigma de carácter eminentemente fenomenológico. La matemática misma, como acto de significación, como conjunto de actos de contenidos intencionales, comparte este carácter y es susceptible de tratamiento fenomenológico. Noesis y semiosis se conjugan para dotar al saber matemático de un contenido, de articulación compleja, que trasciende las perspectivas meramente logicistas, formalistas o intuicionistas. Una primera aproximación a esta problemática es el propósito del presente trabajo. (shrink)

A partir de la conformación de un modelo semántico fundado por la relación de presuposición se construye un entramado algebraico de carácter diagramático y articulación reticular, que considera y se apoya en la noción de orden como dimensión semiótica. Esta articulación relacional se realiza al abarcar diversos niveles de análisis, que van desde un nivel grafémico a un nivel discursivo. Se destacan de modo significativo los procesos complejos de composicionalidad semántica (asociados a modalidades de síntesis) en contraposición con procesos elementales (...) de aditividad (asociados a sumas analíticas). Asimismo, a lo largo del análisis, quedarán establecidos los elementos configuracionales para la articulación reticular de los sucesos pertenecientes a un relato, formando unidades discursivas complejas a través de esquemas narrativos. (shrink)

Building on the notational principles of C. S. Peirce’s graphical logic, Pietarinen has tried to develop a propositional logic unfolding in the medium of sound. Apart from its intrinsic interest, this project serves as a concrete test of logic’s range. However, I argue that Pietarinen’s inaugural proposal, while promising, has an important shortcoming, since it cannot portray double-negation without thereby portraying a contradiction.

. Each predicate of the Aristotelian square of opposition includes the word “is”. Through a twofold interpretation of this word the square includes both classical logic and non-classical logic. All theses embodied by the square of opposition are preserved by the new interpretation, except for contradictories, which are substituted by incommensurabilities. Indeed, the new interpretation of the square of opposition concerns the relationships among entire theories, each represented by means of a characteristic predicate. A generalization of the square of opposition (...) is achieved by not adjoining, according to two Leibniz’ suggestions about human mind, one more choice about the kind of infinity; i.e., a choice which was unknown by Greek’s culture, but which played a decisive role for the birth and then the development of modern science. This essential innovation of modern scientific culture explains why in modern times the Aristotelian square of opposition was disregarded. (shrink)

Some philosophers advance the claim that the phenomena of logical omniscience and of the indiscernibility of metaphysical statements, which arise in (certain) interpretations of normal modal logic, provide strong reasons in favour of impossible world approaches. These two specific lines of argument will be presented and discussed in this paper. Contrary to the recent much-held view that the characteristics of these two phenomena provide us with strong reasons to adopt impossible world approaches, the view defended here is that no such (...) ‘knock-down arguments’ do emanate on those grounds. This is not to rule out that there cannot be any other good reasons for assuming impossible world semantics. However, the discussion of a further argument for impossible worlds will suggest that different attempts to argue for them likely present intertwined problems. (shrink)

The paper contains some algebraic results on several varieties of algebras having an (interlaced) bilattice reduct. Some of these algebras have already been studied in the literature (for instance bilattices with conflation, introduced by M. Fit- ting), while others arose from the algebraic study of O. Arieli and A. Avron’s bilattice logics developed in the third author’s PhD dissertation. We extend the representation theorem for bounded interlaced bilattices (proved, among others, by A. Avron) to un- bounded bilattices and prove analogous (...) representation theorems for the other classes of bilattices considered. We use these results to establish categorical equivalences between these structures and well-known varieties of lattices. (shrink)

The “somewhat vague, intuitive” notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined even if it is not sufficiently formal and precise to belong to mathematics proper (in a narrow sense)—and even if (as many have asserted) for that reason the Church–Turing thesis is unprovable. It is proved logically that the class of effective procedures is not decidable, i.e., that there is no effective procedure for ascertaining whether a given procedure is effective. This (...) result is proved directly from the notion itself of an effective procedure, without reliance on any (partly) mathematical lemma, conjecture, or thesis invoking recursiveness or Turing-computability. In fact, there is no reliance on anything very mathematical. The proof does not even appeal to a precise definition of ‘effective procedure’. Instead, it relies solely and entirely on a basic grasp of the intuitive notion of an effective procedure. Though the result that effectiveness is undecidable is not surprising, it is also not without significance. It has the consequence, for example, that the solution to a decision problem, if it is to be complete, must be accompanied by a separate argument that the proposed ascertainment procedure invariably terminates with the correct verdict. (shrink)

This paper discusses the philosophical and logical motivations for rejectivism, primarily by considering a dialogical approach to logic, which is formalized in a Question–Answer Semantics. We develop a generalized account of rejectivism through close consideration of Mark Textor's arguments against rejectivism that the negative expression ‘No’ is never used as an act of rejection and is equivalent with a negative sentence. In doing so, we also shed light upon well-known issues regarding the supposed non-embeddability and non-iterability of force indicators.

The so-called theory of karma is one of the distinguishing aspects of Hinduism and other non-Hindu south-Asian traditions. At the same time that the theory can be seen as closely connected with the freedom of will and action that we humans supposedly have, it has many times been said to be determinist and fatalist. The purpose of this paper is to analyze in some deepness the relations that are between the theory of karma on one side and determinism, fatalism and (...) free-will on the other side. In order to do that, I shall use what has been described as the best formal approach we have to indeterminism: branching time theory. More specifically, I shall introduce a branching time semantic framework in which, among other things, statements such as “state of affairs e is a karmic effect of agent a”, “a wills it to be the case that e” and “e is inevitable” could be properly represented. (shrink)

Logical and philosophical literature provides different classifications of reasoning. In the Polish literature on the subject, for instance, there are three popular ones accepted by representatives of the Lvov-Warsaw School: Jan Łukasiewicz, Tadeusz Czeżowski and Kazimierz Ajdukiewicz (Ajdukiewicz in Logika pragmatyczna [Pragmatic Logic]. PWN, Warsaw (1965, 2nd ed. 1974). Translated as: Pragmatic Logic. Reidel & PWN, Dordrecht, 1975). The author of this paper, having modified those classifications, distinguished the following types of reasoning: (1) deductive and (2) non-deductive, and additionally two (...) types of them in each of the two, depending on the manner of combining their premises with the conclusion through the relation of classical logical entailment. Consequently, the four types of reasoning: unilateral deductive (incl. its sub-types: deductive inference and proof), bilateral deductive (incl. complete induction), and reductive (incl. the sub-types: explanation and verification), logically nonvaluable (incl. inference by analogy, statistic inference), correspond to four operators of derivability. They are defined formally on the ground of Tarski’s axiomatic theory of deductive systems, by means of the consequence operation _Cn_ (Tarski in Monatshefte Math Phys 37:361–404, 1930a, C R Soc Sci Lett Vars 23:22–29, 1930b). Also, certain metalogical properties of these operators are given, as well as their relations with Tarski’s consequence operations \(Cn^+\) ( \(Cn^+ = Cn\) ) and dual consequences \(Cn^{-1}\) (Słupecki in Zeszyty Naukowe Uniwersytetu Wrocławskiego Seria B Nr 3:33–40, 1959, Słupecki et al. in Stud Log 29:76–123, 1971, Wybraniec-Skardowska, in: Wybraniec-Skardowska, Bryll (eds) Z badań nad teorią zdań odrzuconych [Studies in the Theory of Rejected Propositions], Series B, Studia i Monografie, Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Opole, 1969), and \(Cn^-\) (Wójcicki in Bull Sect Log 2(2):54–57, 1973)). (shrink)