This book offers an historically-informed critical assessment of Dummett's account of abstract objects, examining in detail some of the Fregean presuppositions whilst also engaging with recent work on the problem of abstract entities.

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what (...) does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (shrink)

I argue that we find the articulation of a problem concerning bodily agency in the early works of the Merleau-Ponty which he explicates as analogous to what he explicitly calls the problem of perception. The problem of perception is the problem of seeing how we can have the object given in person through it perspectival appearances. The problem concerning bodily agency is the problem of seeing how our bodily movements can be the direct manifestation of a person’s intentions in the (...) world. In both cases what, according to Merleau-Ponty, obscures a recognition of the phenomenon in question is a conception of our bodily capacities, i.e. our sensibility and our motility, which reduces these to the workings of mechanisms that are blind to meaning. I argue that both the problem concerning perception and the problem concerning bodily agency can be properly called transcendental. The problem of perception is transcendental because it concerns the very intelligibility of appearances and judgements with empirical content. The problem of bodily agency is transcendental in the sense that it concerns the very intelligibility of our bodily capacity to carry out intentions and by implication the intelligibility of our intentions as such. (shrink)

The goal of this paper is to provide an account of the role played by logic in the context of what Husserl names the “crisis of European sciences.” Presupposing the analyses offered in the Krisis, I look at Formale und Transzendentale Logik to demonstrate that the crisis of logic stems from the deviation of its original meaning as a “theory of science” and from its restriction to a mere “theoretical technique.” Through a comparison between Aristotelian syllogistic and modern logic, I (...) show why the modern discovery of the purely formal dimension of knowledge which makes possible such a mathematical technization is a positive achievement that hinders at the same time the disclosure of the truly philosophical nature of logic. The correct appraisal of this ambiguous phenomenon will explain why the rise of modern logic represents a decisive challenge for the success of Husserl’s late phenomenological project. (shrink)

John Locke’s treatment of quantity in the Essay Concerning Human Understanding is not nearly as extensive or as well-known as his treatment of quality and his distinction between primary and secondary qualities. Yet I contend that a close examination of Locke’s comments on quantity in the Essay reveals that he endorses a general theory of quantity that not only distinguishes quantities from qualities, but also plays several other important roles in his overall philosophy—particularly in his treatments of infinity and demonstrative (...) knowledge. Moreover, I argue that this theory does not appear out of thin air, but is in certain respects one and the same as the approach to quantity found in Aristotle and developed by medieval Aristotelian thinkers. Identifying the core tenets of this approach in the texts of Aristotle and medieval Aristotelians, and formulating them clearly enough to shed light on Locke’s theory of quantity in the Essay, is the principal task of this paper. (shrink)

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...) today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

This paper explores the political import of Husserl’s critical discussion of the epistemic effects of the formalization of rational thinking. More specifically, it argues that this discussion is of direct relevance to make sense of the pervasive processes of ‘technization’, that is, of a mechanistic and superficial generation and use of knowledge, to be observed in current contexts of governance. Building upon Husserl’s understanding of formalization as a symbolic technique for abstraction in the thinking with and about numbers, I argue (...) that processes of technization, while being necessary and legitimate procedures for the reduction of complexities, also may give rise to politically unresponsive and ultimately dysfunctional ‘economies of thinking.’ This paper is structured in three parts. In the first part I outline Husserl’s account of the formalization and technization of thought and knowledge. In the second part I make my case for the political import of this account, departing in this context from positions that (a) regard Husserl’s discussions of formalization and its effects as merely epistemological, or that (b) try to mobilize Husserl for a one-sided critique of instrumental reason. In the final part I address a major shortcoming of Husserl’s account, namely its neglect of the concrete and historically evolving technological infrastructures of processes of formalization/technization. (shrink)

This paper reconstructs and critically analyzes Husserl’s philosophical engagement with symbolic technologies—those material artifacts and cultural devices that serve to aid, structure and guide processes of thinking. Identifying and exploring a range of tensions in Husserl’s conception of symbolic technologies, I argue that this conception is limited in several ways, and particularly with regard to the task of accounting for the more constructive role these technologies play in processes of meaning-constitution. At the same time, this paper shows that a critical (...) examination of Husserl’s account of symbolic technologies, particularly as developed in his mature, genetic phenomenology, can be enduringly fruitful—if some of the specific conceptual weakness of this account are identified and properly accounted for. My discussion will proceed as follows. In the first part I briefly analyze the early Husserl’s account of the role the ‘method of sensible signs’ plays in arithmetic cognition. In the second, main part I critically examine the bearing the genetic-phenomenological concepts of sedimentation and technization have on the conceptualization of symbolic technologies in Husserl’s work. In the final part I summarize the major strengths and weaknesses of Husserl’s account of symbolic technologies, and in the process make a case for the ongoing relevance of some of the crucial elements of this account. (shrink)

This paper explores some of the constructive dimensions and specifics of human theoretic cognition, combining perspectives from (Husserlian) genetic phenomenology and distributed cognition approaches. I further consult recent psychological research concerning spatial and numerical cognition. The focus is on the nexus between the theoretic development of abstract, idealized geometrical and mathematical notions of space and the development and effective use of environmental cognitive support systems. In my discussion, I show that the evolution of the theoretic cognition of space apparently follows (...) two opposing, but in truth, intrinsically aligned trajectories. On the epistemic plane, which is the main focus of Husserl’s genetic phenomenological investigations, theoretic conceptions of space are progressively constituted by way of an idealizing emancipation of spatial cognition from the concrete, embodied intentionality underlying the human organism’s perception of space. As a result of this emancipation, it ultimately becomes possible for the human mind to theoretically conceive of and posit space as an ideal entity that is universally geometrical and mathematical. At the same time, by synthesizing a range of literature on spatial and mathematical cognition, I illustrate that for the theoretic mind to undertake precisely this emancipating process successfully, and further, for an ideal and objective notion of geometrical and mathematical space to first of all become fully scientifically operative, the cognitive support provided by a range of specific symbolic technologies is central. These include lettered diagrams, notation systems, and more generally, the technique of formalization and require for their functioning various cognitively efficacious types of embodiment. Ultimately, this paper endeavors to understand the specific symbolic-technological dimensions that have been instrumental to major shifts in the development of idealized, scientific conceptions of space. The epistemic characteristics of these shifts have been previously discussed in genetic phenomenology, but without devoting sufficient attention to the constructive role of symbolic technologies. At the same time, this paper identifies some of the irreducible phenomenological and epistemic dimensions that characterize the functioning of the historically situated, embodied and distributed theoretic mind. (shrink)

This paper offers a conceptually novel contribution to the understanding of the distinctive governance challenges arising from the increasing reliance on formalized knowledge in the governance of research activities. It uses the current Australian research governance system as an example – a system which exhibits a comparatively strong degree of formalization as to its knowledge mechanisms. Combining theoretical reflections on the political-administrative and epistemic dimensions of processes of formalization with analyses of interview data gathered at Australian universities, it is suggested (...) that such a strong reliance on formalized knowledge has rather ambivalent governance ramifications. On the one hand, it allows for a seemingly rational and efficient form of the control and coordination of research activities. Yet on the other hand, it also increases the risk that knowledge is used in governance contexts in superficial, unconsidered and ultimately unreasonable ways. It is further suggested that there are a range of indications that precisely such use elicits and reinforces a range of dysfunctional behaviors on part of relevant individual and organizational actors in the public science system. (shrink)

In book 3 of his Republic, Plato has Socrates undertake an assessment of the educational curriculum that the city (which is being constructed by him in speech) will implement for its youth. Consequently we see that Socrates assigns to poetry a crucial importance; by their imitation of it, poetry shapes the citizens with an initial formation, casts them within a certain orientation, and places them on a path leading in an already conceived direction, toward some unarticulated good. Thus, in forming (...) this city and the souls of its citizens, Socrates first conducts a censorship of the content of the formative myths of the city in an attempt to orchestrate a certain fail-safe against ambiguity and against falling off .. (shrink)

Kant held that under the concept of √2 falls a geometrical magnitude, but not a number. In particular, he explicitly distinguished this root from potentially infinite converging sequences of rationals. Like Kant, Brouwer based his foundations of mathematics on the a priori intuition of time, but unlike Kant, Brouwer did identify this root with a potentially infinite sequence. In this paper I discuss the systematical reasons why in Kant's philosophy this identification is impossible.

When the Association of German Scientists and Physicians last met in Düsseldorf exactly twenty-eight years ago on September 24, a debate took place following lectures by Felix Klein and Alfred Pringsheim on roughly the same topic to which I would like to direct your attention today. The printed report of the Düsseldorf debate only remarked that, “It is not possible to go into details here,” so one can only guess how two of the most powerful teacher personalities among German mathematicians (...) of that time had confronted one another with their diametrically opposed views on this topic and how they did so with their characteristically lively spirit. (shrink)

Diophantos' solutions to the problems of Arithmetica have been the object of extensive reading and interpretation in modern times, especially from the point of view of identifying ``hidden steps'' or ``general methods''. In this paper, after examining the relevance of various interpretations given for the famous problem II 8 in the context of modern algebra or geometry, we focus on a close reading of the ancient text of some problems of Arithmetica in order to investigate Diophantos' solving practices. This inquiry (...) reveals certain pointers, which enable us to create a framework for defining the generality of Diophantos' methods. (shrink)

In recent work, Sören Stenlund (2015) contextualizes Wittgenstein’s philosophy of mathematics as aligned with the tradition of symbolic mathematics. In the early modern era, mathematicians began using purely formal methods disconnected from any obvious empirical applications, transforming their subject into a symbolic discipline. With this, Stenlund argues, they were freeing themselves of ancient ontological presuppositions and discovering the ultimately autonomous nature of mathematical symbolism, which eventually formed the basis for Wittgenstein’s thinking. A crucial premise of Wittgenstein’s philosophy of mathematics, on (...) this view, is that the development of mathematical concepts is independent of any ontological implications and occurs in principle without normative connections to empirical applicability. This paper critically examines this narrative and arrives at the conclusion that Stenlund’s view of mathematical progress is in stark contrast to the later Wittgenstein’s writing, which emphasizes links between symbolisms and their applications. (shrink)

The paper analyzes the final proof with Greek mathematics and the possibility of intermediates in the Phaedo. The final proof in Plato’s Phaedo depends on a claim at 105c6, that μονάς, ‘unit’, generates περιττός ‘odd’ in number. So, ψυχή ‘soul’ generates ζωή ‘life’ in a body, at 105c10-11. Yet commentators disagree how to understand these mathematical terms and their relation to the soul in Plato’s arguments. The Greek mathematicians understood odd numbers in one of two ways: either that which is (...) not divisible into two equal parts, or that which differs from an even number by a unit. Plato uses the second way in the final proof. This paper argues that a proper understanding of these mathematical terms within Greek mathematics shows that the argument for the final proof is better than previously thought. Such an interpretation of the final proof lends credence to Platonic intermediates. (shrink)

The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also (...) central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity. (shrink)

La filosofía de las matemáticas de Aristóteles es una investigación acerca de tres asuntos diferentes pero complementarios: el lugar epistemológico de las matemáticas en el organigrama de las ciencias teoréticas o especulativas; el estudio del método usado por el matemático para elaborar sus doctrinas, sobre todo la geometría y la aritmética; y la averiguación del estatuto ontológico de las entidades matemáticas. Para comprender lo peculiar de la doctrina aristotélica es necesario tener en cuenta que su principal interés está en poner (...) en relación la matemática con la filosofía primera u ontología y la física, y salvaguardar el campo de conocimiento propio de cada una de ellas. (shrink)

ArgumentThe aim of this paper is to employ the newly contextualized historiographical category of “premodern algebra” in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on “geometrical algebra.” Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related toElem.II.5 as (...) a case study, we discuss Euclid's geometrical proofs, the so-called “semi-algebraic” alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing “premodern algebra,” and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition. (shrink)

Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of (...) the other. I suggest that ‘i’ functions like a parameter in natural deduction systems. I gave an early version of this paper at a workshop on structuralism in mathematics and science, held in the Autumn of 2006, at Bristol University. Thanks to the organizers, particularly Hannes Leitgeb, James Ladyman, and Øystein Linnebo, to my commentator Richard Pettigrew, and to the audience there. The paper also benefited considerably from a preliminary session at the Arché Research Centre at the University of St Andrews. I am indebted to my colleagues Craige Roberts, for help with the linguistics literature, and Ben Caplan and Gabriel Uzquiano, for help with the metaphysics. Thanks also to Hannes Leitgeb and Jeffrey Ketland for reading an earlier version of the manuscript and making helpful suggestions. I also benefited from conversations with Richard Heck, John Mayberry, Kevin Scharp, and Jason Stanley. CiteULike Connotea Del.icio.us What's this? (shrink)

…all men begin… by wondering that things are as they are…as they do about…the incommensurability of the diagonal of the square with the side; for it seems wonderful to all who have not yet seen the reason, that there is a thing which cannot be measured even by the smallest unit. But we must end in the contrary and, according to the proverb, the better state, as is the case in these instances too when men learn the cause; for there (...) is nothing which would surprise a geometer so much as if the diagonal turned out to be commensurable. (shrink)

Ever since its foundation in 1540, the Society of Jesus had had one mission—to restore order where Luther, Calvin and the other instigators of the Reformation had brought chaos. To stop the hemorrhage of believers, the Jesuits needed to form a united front. No signs of internal disagreement could to be shown to the outside world, lest the congregation lose its credibility. But in 1570s two prominent Jesuits, Cristophorus Clavius and Benito Perera, had engaged in a bitter controversy. The issue (...) at stake had apparently nothing to do with the values on which Ignazio of Loyola had built the Society of Jesus. And yet the dispute between Clavius and Perera was matter of concern for the entire Jesuit community. They were... (shrink)

ArgumentFrom the time of al-Khwārizmī in the ninth century to the beginning of the sixteenth century algebraists did not allow irrational numbers to serve as coefficients. To multiply$\sqrt {18} $byx, for instance, the result was expressed as the rhetorical equivalent of$\sqrt {18{x^2}} $. The reason for this practice has to do with the premodern concept of a monomial. The coefficient, or “number,” of a term was thought of as how many of that term are present, and not as the scalar (...) multiple that we work with today. Then, in sixteenth-century Europe, a few algebraists began to allow for irrational coefficients in their notation. Christoff Rudolff was the first to admit them in special cases, and subsequently they appear more liberally in Cardano, Scheubel, Bombelli, and others, though most algebraists continued to ban them. We survey this development by examining the texts that show irrational coefficients and those that argue against them. We show that the debate took place entirely in the conceptual context of premodern, “cossic” algebra, and persisted in the sixteenth century independent of the development of the new algebra of Viète, Decartes, and Fermat. This was a formal innovation violating prevailing concepts that we propose could only be introduced because of the growing autonomy of notation from rhetorical text. (shrink)

Françios Viète was a geometer in search of better techniques for astronomical calculation. Through his theorem on angular sections he found a use for higher-dimensional geometric magnitudes which allowed him to create an algebra for geometry. We show that unlike traditional numerical algebra, the knowns and unknowns in Viète’s logistice speciosa are the relative sizes of non-arithmetized magnitudes in which the “calculations” must respect dimension. Along with this foundational shift Viète adopted a radically new notation based in Greek geometric equalities. (...) His letters stand for values rather than types, and his given values are undetermined. Where previously algebra was founded in polynomials as aggregations, Viète became the first modern algebraist in working with polynomials built from operations, and the notations reflect these conceptions. Viète’s innovations are situated in the context of sixteenth-century practice, and we examine the interpretation of Jacob Klein, the only historian to have conducted a serious inquiry into the ontology of Viète’s “species”. (shrink)

ABSTRACT In a recent essay, Sören Stenlund tries to align Wittgenstein’s approach to the foundations and nature of mathematics with the tradition of symbolic mathematics. The characterization of symbolic mathematics made by Stenlund, according to which mathematics is logically separated from its external applications, brings it closer to the formalist position. This raises naturally the question whether Wittgenstein holds a formalist position in philosophy of mathematics. The aim of this paper is to give a negative answer to this question, defending (...) the view that Wittgenstein always thought that there is no logical separation between mathematics and its applications. I will focus on Wittgenstein’s remarks about arithmetic during his middle period, because it is in this period that a formalist reading of his writings is most tempting. I will show how his idea of autonomy of arithmetic is not to be compared with the formalist idea of autonomy, according to which a calculus is “cut off” from its applications. The autonomy of arithmetic, according to Wittgenstein, guarantees its own applicability, thus providing its own raison d’être. RESUMO Em um recente artigo, Sören Stenlund procura alinhar a abordagem de Wittgenstein em relação aos fundamentos e à natureza da matemática com a tradição da matemática simbólica. A caracterização da matemática simbólica feita por Stenlund, de acordo com a qual a matemática é logicamente separada de suas aplicações externas, a aproxima da posição formalista. Isto naturalmente levanta a questão de se Wittgenstein defende uma posição formalista em filosofia da matemática. O objetivo deste artigo é dar uma resposta negativa a esta questão, ao defender que Wittgenstein sempre pensou não haver separação lógica entre a matemática e suas aplicações. Atenção especial será dada às observações de Wittgenstein sobre a aritmética pertencentes ao seu período intermediário, pois é neste período que uma leitura formalista de seus escritos é mais sedutora. Mostrarei como sua ideia de autonomia da aritmética não deve ser comparada à ideia formalista de autonomia, segundo a qual um cálculo é “cortado” de suas aplicações. A autonomia da aritmética, para Wittgenstein, garante ela própria sua aplicabilidade, provendo, assim, sua própria raison d’être. (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 his critical study of Speusippus Leonardo Tarán (T.) expounds an interpretation of a considerable part of the controversial books M and N of Aristotle's Metaphysics. In this essay I want to consider three aspects of the interpretation, the account of Plato's ‘ideal numbers’ (section I), the account of Speusippus’ mathematical ontology (section II), and the account of the principles of that ontology (section III). T. builds his interpretation squarely on the work of Harold Cherniss (C.), to whom I will (...) also refer. I concentrate on T. because he has brought the ideas in which I am interested together and given them a concise formulation; he is also meticulous in indicating the secondary sources with which he agrees or disagrees, so that anyone interested in pursuing particular points can do so easily by consulting his book. (shrink)

This paper analyzes and explains certain parts of Syrianus’s Commentary on book M of Aristotle’s Metaphysics, which details Syrianus’s response to Aristotle’s attack against the Platonic position of the separate existence of numbers. Syrianus defends the separate existence not only of eidetic but also of mathematical numbers, following a line of argumentation which involves a hylomorphic approach to the latter. He proceeds with an analysis of the mathematical number into matter and form, but his interpretation entails that form is the (...) constituent of number, which has the status and role of a Platonic Form. This solution allows him not only to explain and justify the unity of number, but also to apply the Platonic thesis of the separate existence of numbers, to the mathematical or monadic numbers themselves. It also betrays its tendency to combine theses of the Platonic Ontology with fundamental Aristotelian doctrines. (shrink)

The purpose of this paper is to reassess some mathematical examples in Plato’s dialogues which at a first glance may appear to be nothing more than trivial puzzles. In order to provide the necessary background for this analysis, I shall begin by sketching a brief overview of Plato’s mathematical passages and discuss the criteria for aptly selecting them. Second, I shall explain what I mean by ‘mathematical examples,’ and reflect on their function in light of the discussion on παραδείγματα outlined (...) in theSophistand theStatesman. Against this framework, I shall move to a close examination of specific examples drawn from theTheaetetus(154c–55d), theRepublic(523c–24d), and thePhaedo(96d–97b, 100e–101d). By placing these examples in the broader context of pre-Euclidean mathematics, I shall show that their mathematical content is often less elementary than might appear at first sight. Moreover, by placing emphasis on the specific philosophical concerns that motivated their introduction, I shall argue that the examples are not merely nonsensical jokes. Even if their illustrative purpose is not always immediately clear, and even if they can sound playful and bizarre, they in fact fulfil an important function: by virtue of their power to generate wonder or confusion, they serve as exercises and act as a trigger with respect to the difficult philosophical issues they are intended to clarify. (shrink)

This paper discerns two types of mathematization, a foundational and an explorative one. The foundational perspective is well-established, but we argue that the explorative type is essential when approaching the problem of applicability and how it influences our conception of mathematics. The first part of the paper argues that a philosophical transformation made explorative mathematization possible. This transformation took place in early modernity when sense acquired partial independence from reference. The second part of the paper discusses a series of examples (...) from the history of mathematics that highlight the complementary nature of the foundational and exploratory types of mathematization. (shrink)

This paper offers an epistemological reconstruction of the historical development of algebra from al-Khwrizm, Cardano, and Descartes to Euler, Lagrange, and Galois. In the reconstruction it interprets the algebraic formulas as a symbolic language and analyzes the changes of this language in the course of history. It turns out that the most fundamental epistemological changes in the development of algebra can be interpreted as changes of the pictorial form of the symbolic language of algebra. Thus the paper develops further the (...) method of reconstruction which the author introduced for the analysis of the development of geometry. (shrink)

In this paper I consider the relation between Descartes' psychology of vision and the cognitive science approach to psychology (henceforth CS). In particular, I examine Descartes' the Optics (1637) in the light of David Marr's (1982) position in CS. My general claim is that CS can be seen as a rediscovery of Descartes' psychology of vision. In the first section, I point to a parallel between Descartes' epistemological revolution, which created the modem version of the problem of perception, and the (...) cognitive revolution. These fundamental revolutions in theoretical psychology were both inspired and legitimated by a revolution in mathematics. They took place in accordance with one of Marr's maxims: “To the desirable via the possible”. In the second section, I demonstrate that in the Optics, Descartes explains perception of metrical properties in a way that — on a detailed level — is in accordance with how Man argues that complex information processing systems have to be explained: both Descartes and Man emphasize the co-ordination of logical and physical analysis. In the third section, I claim that Descartes' arguments for a sharp distinction between mechanical transmission of sense data (sensation) and non-mechanical inferences on those sense data (thinking) are sound arguments seen from Man's position in CS. Descartes' arguments are based on his logical and physical analysis. Malebranche's radicalized version of Cartesian dualism turns Descartes' empirically-based assumption that mechanisms cannot realize inferences into a metaphysical assumption. In the final section, I argue that this metaphysical assumption contributes to an understanding of perception as a non-symbolic, non-inferential bottom-up process in mainstream monistic and mechanistic scientific psychology until the cognitive revolution. (shrink)

Frege argues that number is so unlike the things we accept as properties of external objects that it cannot be such a property. In particular, (1) number is arbitrary in a way that qualities are not, and (2) number is not predicated of its subjects in the way that qualities are. Most Aristotle scholars suppose either that Frege has refuted Aristotle's number theory or that Aristotle avoids Frege's objections by not making numbers properties of external objects. This has led some (...) to conclude that Aristotle's accounts of arithmetical and geometrical objects differ substantially. I close this supposed gap by showing that Aristotle's arithmetical objects, like geometrical objects, are just certain sensible things qua certain properties they in fact possess. Specifically, numbers are pluralities qua quantitative or relational properties like ten units or ten. I show that this view is resistant to the Fregean concerns about arbitrariness and numerical predication. (shrink)

This article examines Hobbes’s conception of mathematical method, situating his methodological writings in the context of disputed mathematical issues of the seventeenth century. After a brief exposition of the Hobbesian philosophy of mathematics, it investigates Hobbes’s attempts to resolve three important mathematical controversies of the seventeenth century: the debates over the status of analytic geometry, disputes over the nature of ratios, and the problem of the “angle of contact” between a curve and tangent. In the course of these investigations, Hobbes’s (...) account of mathematics and its method is contrasted with the those of Descartes, Isaac Barrow, John Wallis, and Christopher Clavius. (shrink)

Dallas Willard’s contribution to phenomenology is presented in terms of his articles on, and translations into English of, Edmund Husserl’s early philosophical writings, which single-handedly prevented them from falling into oblivion, both literally and philosophically. Willard’s account of Husserl’s “negative critique” of formalized logic in those writings, and argument for its contemporary relevance, is presented and largely endorsed.

Dallas Willard’s contribution to phenomenology is presented in terms of his articles on, and translations into English of, Edmund Husserl’s early philosophical writings, which single-handedly prevented them from falling into oblivion, both literally and philosophically. Willard’s account of Husserl’s “negative critique” of formalized logic in those writings, and argument for its contemporary relevance, is presented and largely endorsed.