Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that (...) this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge. (shrink)

Descartes held that practicing mathematics was important for developing the mental faculties necessary for science and a virtuous life. Otherwise, he maintained that the proper uses of mathematics were extremely limited. This article discusses his reasons which include a theory of education, the metaphysics of matter, and a psychologistic theory of deductive reasoning. It is argued that these reasons cohere with his system of philosophy.

No processo do conhecimento a imaginação desempenha para Hegel o estágio no qual a mente humana dissocia o objeto em dois diferentes conteúdos, o conteúdo-coisa do mundo externo e o conteúdo interno da própria mente, de tal modo que ambas as versões do mesmo devem corroborar-se mutuamemente ao modo de uma síntese simples de elementos heterogéneos que apenas em seu cotejamento reconhecem sua identidade. Na atividade de compreensão, ao contrário, este dualismo é suprassumido e, com ele, o empirismo e a (...) teoria correspondista da verdade que, aos olhos de Hegel, está em sua base. (shrink)

This paper asks how we find our way in the hitherto unknown. In search of an answer, the author returns to the three Critiques of Immanuel Kant, noting especially their grounding in the geometric mode of (re)presentation and the thingification processes connected therewith. It is argued that Kant's choice of metaphors in effect makes him more of a geographer than of a philosopher. To understand the taken-for-granted of thought-and-action, the time has therefore come for the writing of a fourth volume (...) entitled A Critique of Cartographical Reason. The focus must there be on the relations between personal pronouns and prepositions, on the one hand, and the powers of tautology, on the other. (shrink)

It is often maintained that one insight of Kant's Critical philosophy is its recognition of the need to distinguish accounts of knowledge acquisition from knowledge justification. In particular, it is claimed that Kant held that the detailing of a concept's acquisition conditions is insufficient to determine its legitimacy. I argue that this is not the case at least with regard to geometrical concepts. Considered in the light of his pre-Critical writings on the mathematical method, construction in the Critique can be (...) seen to be a form of concept acquisition, one that is related to the modal phenomenology of geometrical judgement. (shrink)

At 435c-d and 504b ff., Socrates indicates that there is a "longer and fuller way" that one must take in order to get "the best possible view" of the soul and its virtues. But Plato does not have him take this "longer way." Instead Socrates restricts himself to an indirect indication of its goals by his images of sun, line, and cave and to a programmatic outline of its first phase, the five mathematical studies. Doesn't this pointed restraint function as (...) a provocation, moving us to want to begin the "longer way" and to make use of its conceptual resources to rethink Socrates' images? I begin by finding a double movement in the complex trajectory of the five studies: they both guide the soul in the "turn away from what is coming to be ... [to] what is" (518c) and, at the same time, lead the soul back, albeit in the medium of pure intelligibility, to the sensible world; for the pure figures and ratios that they disclose constitute the core structures of sensible things. I then draw on what Socrates says about geometry and harmonics to address three fundamental questions that he leaves open: the nature of the Good in its responsibility for truth and for the being of the forms; the relations of forms, mathematicals, and sensibles as these are disclosed by dialectic; and the bearing of the philosopher's discovery of the Good on his disposition towards his community and the task of ruling. I close by marking six sets of further questions that these reflections bequeath for dialogues to come. (shrink)

Tarski’s conceptual analysis of the notion of logical consequence is one of the pinnacles of the process of defining the metamathematical foundations of mathematics in the tradition of his predecessors Euclid, Frege, Russell and Hilbert, and his contemporaries Carnap, Gödel, Gentzen and Turing. However, he also notes that in defining the concept of consequence “efforts were made to adhere to the common usage of the language of every day life.” This paper addresses the issue of what relationship Tarski’s analysis, and (...) Béziau’s further generalization of it in universal logic, have to reasoning in the everyday lives of ordinary people from the cognitive processes of children through to those of specialists in the empirical and deductive sciences. It surveys a selection of relevant research in a range of disciplines providing theoretical and empirical studies of human reasoning, discusses the value of adopting a universal logic perspective, answers the questions posed in the call for this special issue, and suggests some specific research challenges. (shrink)

This paper explores Simon Stevin’s l’Arithmétique of 1585, where we find a novel understanding of the concept of number. I will discuss the dynamics between his practice and philosophy of mathematics, and put it in the context of his general epistemological attitude. Subsequently, I will take a close look at his justificational concerns, and at how these are reflected in his inductive, a postiori and structuralist approach to investigating the numerical field. I will argue that Stevin’s renewed conceptualisation of the (...) notion of number is a sort of “existential closure” of the numerical domain, founded upon the practice of his predecessors and contemporaries. Accordingly, I want to make clear that l’Aritmetique have to be read not as an ontological analysis or exploration of the numerical field, but as an explication of a mathematical ethos. In this sense, this article also intends to make a specific contribution to the broader issue of the “ethics of geometry.”. (shrink)

In this essay I try to give some overall statements in order to show that new realism is to be understood as a kind of positive philosophy. Against constructivism, I argue that there is a prevalence of the objects themselves on our understanding of them because reality offers a resistance to our attempt to grasp it depending on its level of dependence from our own understanding, which is different in the case of natural objects, ideal object and social object. This (...) negativity leads to an ontological necessity of facticity. In the second part of this paper I will describe – through the exposition of three key concepts of new realism, i.e. “unamendability”, “interaction” and “affordance” – an ascending path for which the negative and the resistance turn into positivity. (shrink)

The problem of ‘collective unity’ in the transcendental philosophies of Kant and Husserl is investigated on the basis of number’s exemplary ‘collective unity’. To this end, the investigation reconstructs the historical context of the conceptuality of the mathematics that informs Kant’s and Husserl’s accounts of manifold, intuition, and synthesis. On the basis of this reconstruction, the argument is advanced that the unity of number – not the unity of the ‘concept’ of number – is presupposed by each transcendental philosopher in (...) their accounts of the transcendental foundation of manifold, intuition, and synthesis. This presupposition is ultimately traced to Kant’s and Husserl’s responses to Hume’s philosophy of human understanding and the critical limits of what Kant calls the ‘qualitative’ unity of transcendental consciousness. These critical limits are exposed in both philosophers’ attempts to account for that ‘qualitative’ unity on the basis of the ‘quantitative’ unity of number. (shrink)

We like Euclidean geometry because we are men [sic], and have eyes and hands, and need to operate a concept of space that will be independent of orientation, distance and size. Lucas, A Treatise on Time and Space.

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such (...) as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

The current global economic crisis concerns the way in which contemporary capitalism has turned to financialisation as a double cure for both a falling rate of profit and a deficiency of demand. Although this turning is by no means unprecedented, policies of financialisation have depressed demand (in part as a result of the long-term stagnation of average wages) while at the same time not proving adequate to restore profits and growth. This paper argues that the current crisis is less the (...) ‘normal’ one that has to do with a constitutive need to balance growth of abstract wealth with demand for concrete commodities. Rather, it marks a meta-crisis of capitalism that is to do with the difficulties of sustaining abstract growth as such. This meta-crisis is the tendency at once to abstract from the real economy of productive activities and to reduce everything to its bare materiality. By contrast with a market economy that binds material value to symbolic meaning, a capitalist economy tends to separate matter from symbol and reduce materiality to calculable numbers representing ‘wealth’. Such a conception of wealth rests on the aggregation of abstract numbers that cuts out all the relational goods and the ‘commons’ on which shared prosperity depends. (shrink)

This paper asks how we find our way in the hitherto unknown. In search of an answer, the author returns to the three Critiques of Immanuel Kant, noting especially their grounding in the geometric mode of (re)presentation and the thingification processes connected therewith. It is argued that Kant's choice of metaphors in effect makes him more of a geographer than of a philosopher. To understand the taken-for-granted of thought-and-action, the time has therefore come for the writing of a fourth volume (...) entitled A Critique of Cartographical Reason. The focus must there be on the relations between personal pronouns and prepositions, on the one hand, and the powers of tautology, on the other. (shrink)

Discusses the importance of the early history of Greek mathematics to education and civic life through a study of the Parthenon and dialogues of Plato.

What is the relationship between intellectual and cultural history? An answer to this question may be found in the area between the two poles of inquiry commonly known as internalist and externalist methods. The first of these deals with old-fashioned `ideas' (in Lovejoy's sense) and the second with social and political context and the sociology and anthropology of knowledge. This article reviews this question in the light of the earlier historiography of philosophy, literature and science, and debates over the role (...) of context in determining historical meaning. Within the horizon-structure of experience and interpretation the short answer is that cultural history is the outside of intellectual history and intellectual history the inside of cultural history. Ideally, historians ought to work both sides of the historical street. (shrink)

There has been no shortage of readers who take Nietzsche as educator (cf., for a by no means exhaustive list: Allen, 2017; Aviram, 1991; Bell, 2007; Cooper 1983; Fairfield, 2017; Fitzsimons, 2007;...

The question of nationalism as spoken about in contem porary circles is structurally the same as Marx's 'Jewish Question'. Through a reading of Marx's early writings, particularly the 'Jewish Question' essay, guided by Derrida's Specters of Marx and Benedict Anderson's Imagined Communities, it is possible to begin to rethink the nationalist question. In this light, nationalism emerges as the byproduct of the reduction of heterogeneous 'people' into a homo geneous 'state'; such 'excessive' voices occupy an ontological space outside of the (...) categories within which the state operates, and thus return, in Derrida's terms, to 'haunt' it. When the nationalist ques tion is viewed in this way, contemporary solutions that depend on reproducing this reduction are called into question. Key Words: B. Anderson • Derrida • G. Gottlieb • R. Kaplan • Marx • nationalism. (shrink)

The ArgumentIn writing the history of science, the fluctuations between two meanings of the concept of style are of special interest: a simple or direct meaning of this concept referring to a means of expression and of presentation, and a philosophical interpretation of this term referring to “a world of objective spiritual order.” The last two chapters of this paper consider the perspective of the simple meaning of the concept, the first two chapters take the philosophical meaning as their starting (...) point.The concept of style in its general epistemological meaning emerges within a conceptual space that becomes effective as a totality at the end of the eighteenth century and which is built up of further notions such as: individual, genius, expression, symbol, education, creativity, and others.The individual and, as believed, the nevertheless infinitely creative subject has taken the place that the concept of god had occupied within rationalism. But it is not only the subject as construction and will, but also the subject who reflected in a new way about the objective foundations of his conscience and tried to bring the object and the means of knowledge into a new relation. (shrink)