Mathesis Universalis Research Articles

Mathesis Universalis and the Cartesian Unification of Philosophy, Science, and Religion

In this paper we present the manner in which René Descartes discovered the principle of the autonomy of the spirit as a Mathesis Universalis, as a universal science, his perception being much different from the medieval scholastic one, where the intellect corresponded with the sensible reality. Descartes reversed this suitability, considering that the intellect should not be guided by things and build judgments according to them, but, on the contrary, things are analyzed according to the intellect's abilities to give them meanings and sense, to make them intelligible. Firstly, we will demonstrate that, for Descartes, the very existence, the reality of a thing depends on this light of the intellect that unifies all knowledge through Mathesis Universalis. For the French philosopher, order and measure, captured by Mathesis Universalis, become the qualities by which God, the only perfect Being, created the Universe that obeys a coherent mathematical model. Secondly, we will highlight the fact that the starting point of this new metaphysics could be found in Descartes’s view that God cannot be considered deceptive, that the world is not the creation of an evil and cunning Genius. In conclusion, knowledge through Mathesis Universalis leads the spirit to that place where the reasons of universal peace can be founded, whose purpose is to overcome those structures of the imaginary that wing the irrational drives dressed in the clothes of war and death.

Prince(ss) Charming of the Japanese Popular Theatre

Prince(ss) Charming of the Japanese Popular Theatre

Entre Husserl y Tomás de Aquino. Aspectos metafísicos y epistemológicos del diálogo entre fenomenología y escolástica según Edith Stein

The fundamental program of Stein’s philosophy after his conversion was the relationship between modern philosophy and scholasticism. This article deals with new metaphysical and epistemological aspects of Stein’s work that reflect the encounter between both philosophical currents, starting from her reading of Thomas Aquinas’ De veritate, which have not yet been treated in the Edith Stein Forschung. Thus, from the metaphysical point of view, the question of the relationship between unity and multiplicity and the problem of the conditions of possibility of organic development are identified. And from the epistemological point of view, the following relevant aspects are raised: the starting point of philosophizing; the principles of knowledge; the extent to which habit influences knowledge; the question of whether angelic knowledge can serve to clarify human knowledge; the relationship between mathesis universalis and divine knowledge; the relationship between abstraction and intuition; and the critique of the Heideggerian interpretation of the concept of truth in the philosophical tradition.

Introduction: The Idiosyncratic Nature of Renaissance Mathematics

Introduction: The Idiosyncratic Nature of Renaissance Mathematics

Quantity as Limit

This paper deals with the metaphysics of the notion of quantity in the philosophy of Leibniz, and its aim is to defend the following bi-conditional: for any object x, x has a certain quantity if and only if x has a (metaphysical) limit or a bound. The direction from left to right is justified in §3, while in §4 I develop an argument to justify the direction from right to left. Since the bi-conditional links the metaphysical notion of limit to the mathematical notion of quantity (and in this way it links Leibniz’s metaphysics with his conception of Mathesis Universalis), it allows the use of metaphysics to clarify the features of the mathematical notion of quantity. This task is accomplished in §5 and §6. Finally, §7 discusses a possible objection.

Mathesis Universalis and Husserl’s Phenomenology

The paper’s central theme is the link between phenomenology and the notion of the mathesis universalis, a link articulated by Husserl in the third volume of the Ideas: “My way to phenomenology was essentially determined by the mathesis universalis (Bolzano did not see anything of this).” The paper suggests three interpretations of the phenomenology—mathesis universalis nexus: the first is related to the development of Husserl’s conception of the foundations of arithmetic; the second is based on the role of the theory of manifolds in Husserl’s Logical Investigations; and the third reflects the importance of the distinction between “generalization” and “formalization” for phenomenology. After examining these interpretations, the paper explores which is most helpful for understanding why Husserl distanced himself from Bolzano, arguing that the third interpretation provides the most edifying answer.

El pensamiento cartesiano y la fundación de la sociología

espanolResulta de interes para el presente trabajo analizar los fundamentos de la denominada mathesis universalis y como esta nocion, propuesta por Descartes en pleno siglo XVII, ha tenido implicancias significativas en el origen y constitucion (muy ulterior) de dos de las versiones mas importantes de la sociologia: el positivismo sociologico y la sociologia comprensiva. Sosteniendo que la disciplina que estudia los fenomenos sociales ha estado influenciada constantemente por el pensamiento cartesiano, se analizan algunos postulados de ambos enfoques con base en el principio indicado. Se concluye que, pese a ser diametralmente opuestas desde el punto de vista de los principios ontologicos sobre los que observan la realidad social, en ambas lineas de analisis sociologico el principio de la mathesis universalis sigue vigente. Esto da cuenta de una evidente logica de continuidad critica del cartesianismo en los enfoques fundacionales de la sociologia, todo lo cual ha resultado ser epistemologicamente significativo para la disciplina que se encamina a estudiar cientificamente los hechos sociales. EnglishThis paper analyses the foundations of the so-called mathesis universalis, and how this notion, proposed by Descartes in the 17th century, has had significant implications in the origin and constitution (much later) of two of the most important versions of sociology: sociological positivism and comprehensive sociology. Maintaining that the discipline that studies social phenomena has been constantly influenced by Cartesian thought, some postulates of both approaches are analysed based on the indicated principle. It is concluded that, despite being diametrically opposed from the point of view of the ontological principles on which they observe social reality, in both lines of sociological analysis the principle of mathesis universalis remains in force. This shows an evident logic of critical continuity of Cartesianism in the foundational approaches of sociology, all of which has turned out to be epistemologically significant for the discipline that aims to scientifically study social facts.

Fenomenologia polska przełomu XX i XXI wieku na obczyźnie. Kluczowe założenia fenomenologii życia Anny Teresy Tymienieckiej

Artykuł jest krótką prezentacją głównych założeń koncepcji fenomenologii życia współczesnej polskiej myślicielki Anny Teresy Tymienieckiej (1923-2014). Są to te założenia, które Tymieniecka wypunktowała w jednej ze swoich wypowiedzi zaprezentowanej na Kongresie Kultury Polskiej w Londynie w 1985 roku. Celem obrad sekcji filozoficznej było nakreślenie sytuacji filozofii polskiej oraz polskiej kultury na emigracji w kontekście kryzysu kultury europejskiej. Tymieniecka, działająca naukowo w Stanach Zjednoczonych, przedstawiła własny punkt widzenia tego kryzysu, jako mającego źródło w braku komunikacji między różnymi dziedzinami nauki, ale także między nauką w ogóle a innymi formami ludzkiej działalności kulturotwórczej. Przedstawiła też własną wizję przezwyciężenia tego kryzysu w postaci fenomenologii twórczej kondycji ludzkiej jako nowej mathesis universalis.

МАТЕМАТИЧЕСКИЙ ХАРАКТЕР НЬЮТОНИАНСКОЙ ФИЗИКИ В КОНТЕНТЕ НОВОЕВРОПЕЙСКОЙ НАУКИ

The article is devoted to interpreting the modern age science as mathesis universalis. To clarify the importance of the intellectual revolution provoked by Newton’s First Law, the author analyses Aristotle’s, Plato’s, Galilei’s and Descartes’s fundamental provisions and tackles the problem of applying Newton’s First Law, considering the fact that it introduces a new conceptual approach, a mathematical one. The paper reveals the mathematical nature of the new European science. The findings indicate that Newton’s revolution in natural science promoted the formation of a new fundamental metaphysical approach, which determined a new attitude towards the matter and the truth.

"Mathesis universalis" en Proclo

The author shows how Proclo is a precursor of 'Mathesis universalis' concept, without admiting the aporetic method of mathematics which is in Plato, Aristotle and Euclides thought. Today, his paradigm is rejected but it is a decisive factor to understand the sources of western thought. This study deals with the works of Brisson, Cleary, Trudeau, Beierwaltes and Schmitz.

El método husserliano en las "Logische Untersuchungen"

Our proposal is that there exists already a method in Husserl’s Logische Untersuchungen. Nevertheless, this method is operative, but not thematized. We characterize it as follows: analytic, descriptive, pure, reductive, intuitive, reflexive, variational and functional. Concerning Logische Untersuchungen, analysis of significations fulfills a strategic role of mediation between mathesis universalis’ analysis and phenomenology’s analysis. Finally, one of the leading features of Husserlian method is reflexion considered as anábasis of pneuma.

데카르트 『정신지도규칙』에서 ‘보편수리학’(mathesis universalis)과 순수 지성의 자율성

데카르트는 학문들의 통일된 체계를 보편적 방법을 통해 확립하고자 하였다. 이러한 그의 기획은 그의 초기 미완의 저술 『정신지도규칙』에 이미 주제로 나타난다. 본 논문은 「규칙4-B」의 보편수리학과 「규칙4-A」의 보편 방법사이의 관계를 살펴봄으로써, 보편 방법의 일차적인 관심이 존재론적인 것이라기보다 논리학적인 것임을 보일 것이다. 보편 방법은 학문적 증명들을 순수 지성의 근원적 자율성 위에 정초함으로써 학문들의 통일된 체계를 확립하고자 하는 기획이다. 데카르트는 이러한 기획을 실현하기 위해 두 가지 중요한 방법을 사용한다. 첫째, 「규칙6」에서 그는 관계들을 정신활동들로 이해한다는 전제 하에서, 학문적 물음들을 관계들을 통해 정식화할 것을 권한다. 둘째, 「규칙7」에서 그는 연속적이고 부단한 사유의 반복을 통해서, 학문적 증명이 정신의 자율성 위에 정초되는 방식을 동시적으로 직관하라고 권한다. 그렇지만 『규칙』에서의 자율성은 데카르트가 『성찰』에서 주장하게 될 “자유의지”와 구별되어야 한다. 왜냐하면 『규칙』의 자율성은 규칙을 생산하는 근원적 능력을 의미하지 아직 절대적 선택의 능력을 의미하지는 않기 때문이다.

Can a machine think (anything new)? Automation beyond simulation

This article will rework the classical question ‘Can a machine think?’ into a more specific problem: ‘Can a machine think anything new?’ It will consider traditional computational tasks such as prediction and decision-making, so as to investigate whether the instrumentality of these operations can be understood in terms of the creation of novel thought. By addressing philosophical and technoscientific attempts to mechanise thought on the one hand (e.g. Leibniz’s mathesis universalis and Turing’s algorithmic method of computation), and the philosophical and cultural critique of these attempts on the other, I will argue that computation’s epistemic productions should be assessed vis-à-vis the logico-mathematical specificity of formal axiomatic systems. Such an assessment requires us to conceive automated modes of thought in such a way as to supersede the hope that machines might replicate human cognitive faculties, and to thereby acknowledge a form of onto-epistemological autonomy in automated ‘thinking’ processes. This involves moving beyond the view that machines might merely simulate humans. Machine thought should be seen as dramatically alien to human thought, and to the dimension of lived experience upon which the latter is predicated. Having stepped outside the simulative paradigm, the question ‘Can a machine think anything new?’ can then be reformulated. One should ask whether novel behaviour in computing might come not from the breaking of mechanical rules, but from following them: from doing what computers do already, and not what we might think they should be doing if we wanted them to imitate us.

LAJBNICOV POJAM NAUKE – POČETAK I KRAJ OPTIMIZMA

<p>Vilhelm Gotfrid Lajbnic bio je jedan od najvažnijih utemeljitelja modernog pojma nauke. Svojim određenjem moderne nauke, njegovo ime i njegova fi lozofi ja su višestruko povezani sa najpoznatijom katastrofom na početku moderne epohe, Lisabonskim zemljotresom iz 1755. godine. Lajbnicov ideal nauke iskazan je kroz ideju <em>mathesis universalis </em>(univerzalna matematika). Prema njegovoj interpretaciji, sve stvari u prirodi trebalo bi da se razumeju i objasne preko veličina (<em>magnitudo</em>), oblika (fi gure) i pokreta (<em>motus</em>). Mogli bismo reći da prema ovakvoj ideji moderne nauke svi posebni prirodni fenomeni treba da budu objašnjeni matematički. Lajbnicova ideja nauke je potpuni izraz moderne epohe. Matematika ne bi trebalo da samo nauci donese izvesnost (<em>certitude</em>), već i našim životima. Lajbnicova matematička interpretacija prirodnih fenomena vodi nas do „optimizma” i ideje o <em>najboljem od svih mogućih svetova</em>. Nažalost, Lajbnicov optimistički pogled na svet, tipičan za kraj 17. i početak 18. veka, bio je dramatično prekinut Lisabonskim zemljotresom. Sposobnost merenja i računanja snage zemljotresa i visine cunamija nisu u duše Portugalaca i drugih Evropljana uneli mir i sigurnost. Upravo suprotno, računajući odnos prema Lisabonskom zemljotresu uneo je u njihove živote ogroman strah od ponavljanja katastrofe. Prethodna katastrofa je shvatana kao potencijalna mogućnost novih, koje tek treba da se dogode. Stoga autor ovog teksta zastupa stav da sâm Lisabonski zemljotres iz 1755. godine nije ono što je označilo kraj optimizma, već moderni pojam nauke koji je, preko računanja i predikcije verovatnoće novih katastrofa u budućnosti, u duše ljudi uneo strah.</p>

LAJBNIC I DEKART: OD UNIVERZALNOG JEZIKA DO UNIVERZALNOG ZNANJA

Tekst se bavi problemom odnosa Dekartovog projekta <em>Mathesis universalis </em>i Lajbnicovog univerzalnog jezika<em>.</em><em> </em>Iako je u Lajbnicovo vreme postojalo već mnoštvo pokušaja da se razvije jedan takav jezik, ono što Lajbnica izdvaja u odnosu na njih jeste pokušaj da se taj jezik utemelji na filozofski način – koreni Lajbnicovog filozofskog utemeljenja <em>lingue universalis </em>imaju kartezijanske osnove. Premda je Dekartovu potragu za univerzalnim znanjem iz <em>Regulae </em>Lajbnic kritikovao, on je preuzeo osnovne motive takve potrage kada 1666. razvija svoju univerzalnu karakteristiku u <em>Ars combinatoria. </em>On daje nov odgovor na potrebu utemeljivanja nauka – umesto metodologije i percepcije, Lajbnic kriterijume naučnosti vidi u mogućnosti jezičke formalizacije. Ovde se javlja ono što je strano Dekartovom, a blisko Lajbnicovom sistemu i što će ostaviti traga i na dvadesetovekovne mislioce poput Borhesa, Huserla, Fukoa i Deleza – naime, pojam znaka kao osnovnog izraza misli. U daljem tekstu autor razvija implikacije ove promene obzirom na osnovne motive Lajbnicove filozofije – pojmove kombinatorike, monade te odnosa univerzalnog jezika i naučnosti.

La ciencia nueva de vico: entre la mathesis universalis y la lengua heroica

Against the traditional interpretation of Vico as an anticartesian philosopher of civil society’s history, the first part of this essay depicts the New Science as an extension to the historical world of the mathesis universalis. This is proved by the axiomatic method present in the first Book(the so called Dignities) and by the division in three ages —of gods, of heroes and of men— of the necessary historical order: Vico’s «eternal ideal history», above which the nations’ histories «pass by within time». The second part of the article deals with the «heroic language» and shows the limitations of this mathesis, unable to express —as Vico itself admits— through an abstract and reflexive language the first age of the world: the arousal from nature’s ingens sylva of the historical human kind. In this way, Vico restores history and historiography with all their problematic complexity.

The relativity and universality of logic

After recalling the distinction between logic as reasoning and logic as theory of reasoning, we first examine the question of relativity of logic arguing that the theory of reasoning as any other science is relative. In a second part we discuss the emergence of universal logic as a general theory of logical systems, making comparison with universal algebra and the project of mathesis universalis. In a third part we critically present three lines of research connected to universal logic: logical pluralism, non-classical logics and cognitive science.

Horkheimer's Criticism of Husserl

This article focuses on Max Horkheimer’s criticism of Husserl’s phenomenology in basic philosophical matters such as method, theory, logic, truth, metaphysics, etc. Horkheimer objects to Husserl’s conception of philosophy as a mathesis universalis and of science as relativistic research. However, he finds Husserl’s criticism of scientific rationalism the most important step for the legitimacy of philosophy. According to him, Husserl’s method is intended to be a science of apriority. But his understanding of apriority is static, is radically abstract, and overlooks the dialectical relation. Therefore, his method is ahistorical and undialectical. Horkheimer does not interpret Husserl’s idealism in the sense of classical idealism. However, he believes that the positivistic and Cartesian implications in Husserl’s philosophy made his method less fruitful in concrete situations. Consequently, he calls Husserl’s phenomenology abstract positivism, traditional theory and a bourgeois ideology. Horkheimer’s critique focuses on Husserl’s early period of phenomenology.

Le bon usage de l'imagination: Malebranche lecteur des Regulae ad directionem ingenii

Malebranche's conception of imagination is often thought to be entirely contained in Book II of La Recherche de la vérité, which aims at destroying the beliefs and false theories resulting from the untempered use of that faculty. And yet Malebranche shows, in Book VI of the same work, that imagination has an essential function in the construction of science, particularly when it comes to producing geometrical models for phenomena; and it is, from the point of view of cognition, a singular cunning of reason which associates the presence of corporeal affects in the mind, and ideas, which enlighten it without touching it. This conception rests upon unacknowledged quotations from the Regulae ad directionem ingenii, which Malebranche must have had at hand while working on the Recherche. At stake is, for us, the precise understanding of the relation between mathematics and imagination, first in the Regulae and then in the Recherche, in the general context of the project of a universal science or mathesis universalis.

Lines and Crimes of Demarcation: Mathematizing Nature in Heidegger, Pynchon, and Kehlmann

In his bestselling novel from 2005, Die Vermessung der Welt (Measuring the World), Daniel Kehlmann brings together the explorer Alexander von Humboldt and the mathematician Carl Friedrich Gauss in an effort to show that the scientific progress preached by the Enlightenment is a double-edged sword. Hence the double entendre of the title: Humboldt's and Gauss's “surveying” (Vermessung) enterprises typify human “arrogance” (Vermessenheit) toward nature. The ideas presented in Kehlmann's book were no doubt influenced by Thomas Pynchon's Mason & Dixon (1997), a metafictional history in which the Mason and Dixon survey is represented as a rationalist endeavor that transforms the primitive American landscape into an organized grid of human control. Both authors are critiquing, from a postmodern vantage, eighteenth-century instrumental reason, which manifests itself in the urge to measure, chart, and demarcate the world. At various points in his narrative, Pynchon sums up this basic anthropocentric view as mathesis, a term that recalls the early modern scientific project of mathesis universalis or the attempt to “mathematize” nature. According to Heidegger, mathesis, which sums up our modern technological attitude toward the world, developed as a result of Galilean geometry, Newtonian science, and Cartesian philosophy, all of which conceptualize reality according to purely speculative mathematical laws. This article brings Heidegger, Pynchon, and Kehlmann into dialogue with one another in an effort to show how they each critique, in remarkably similar fashion, the conceptual as well as physical lines of demarcation that mathesis inflicts upon the world.