This article introduces Harvey Brown and Oliver Pooley’s ‘dynamical approach’ to special relativity, and argues that it may be construed as a relationalist form of Einstein’s ‘practical geometry’. This construal of the dynamical approach is shown to be compatible with related chapters of Brown’s text and also with recent descriptions of the dynamical approach by Pooley and others.

The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in (...) relation to practical geometry, some of which are basically explicit in definitions, like that of segments in Euclid’s Elements. Then, we will address how in pure geometry we, so to speak, “refer back” to practical geometry. This occurs in two ways. One, in the propositions of pure geometry. The other, when applying pure geometry. In this case, geometrical objects can represent practical figures like, e.g., a practical segment. (shrink)

While Einstein considered that sub specie astern the correct philosophical position regarding geometry was that of the conventionality of geometry, he felt that provisionally it was necessary to adopt a non-conventional stance that he called practical geometry. here we will make the case that even when adopting Einstein’s views we must conclude that practical geometry is conventional after all. Einstein missed the fact that the conventionality of simultaneity leads to a conventional element in the (...) chrono-geometry, since it corresponds to the possibility of different space-time metrics. (shrink)

It is well known that Felix Klein took a decisive step in investigating the invariants of transformation groups. However, less attention has been given to Klein’s considerations on the epistemological implications of his work on geometry. This paper proposes an interpretation of Klein’s view as a form of mathematical structuralism, according to which the study of mathematical structures provides the basis for a better understanding of how mathematical research and practice develop.

The purpose of this work is to address what notion of geometrical object and geometrical figure we have in different kinds of geometry: practical, pure, and applied. Also, we address the relation between geometrical objects and figures when this is possible, which is the case of pure and applied geometry. In practical geometry it turns out that there is no conception of geometrical object.

This paper examines the discrepancy between the attitudes of many historians of mathematics to sixteenth-century geometry and those of museum curators and others interested in practical mathematics and in instruments. It argues for the need to treat past mathematical practice, not in relation to timeless criteria of mathematical worth, but according to the agenda of the period. Three examples of geometrical activity are used to illustrate this, and two particular contexts are presented in which mathematical practice localised in (...) the sixteenth century takes on a special historical significance. (shrink)

This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege's Grundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes of Grundlagen are developed: the relationship Frege envisions between arithmetic and geometry (...) and the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer. (shrink)

In this paper, we will make explicit the relationship that exists between geometric objects and geometric figures in planar Euclidean geometry. That will enable us to determine basic features regarding the role of geometric figures and diagrams when used in the context of pure and applied planar Euclidean geometry, arising due to this relationship. By taking into account pure geometry, as developed in Euclid’s Elements, and practical geometry, we will establish a relation between geometric objects (...) and figures. Geometric objects are defined in terms of idealizations of the corresponding figures of practical geometry. We name the relationship between them as a relation of idealization. This relation, existing between objects and figures, is what enables figures to have a role in pure and applied geometry. That is, we can use a figure or diagram as a representation of geometric objects or composite geometric objects because the relation of idealization corresponds to a resemblance-like relationship between objects and figures. Moving beyond pure geometry, we will defend that there are two other ‘layers’ of representation at play in applied geometry. To show that, we will consider Euclid’s Optics. (shrink)

Klein’s Erlangen program contains the postulate to study thegroup of automorphisms instead of a structure itself. This postulate, takenliterally, sometimes means a substantial loss of information. For example, thegroup of automorphisms of the field of rational numbers is trivial. Howeverin the case of Euclidean plane geometry the situation is different. We shallprove that the plane Euclidean geometry is mutually interpretable with theelementary theory of the group of authomorphisms of its standard model.Thus both theories differ practically in the language (...) only. (shrink)

In the late sixteenth century a number of mathematicians tried to introduce geometrical methods into surveying practice, to be based on simplified astronomical instruments, angle measurement, and triangulation. A measure of success is indicated by the acceptance of the simple theodolite, but the surveyors resisted such complex instruments as the altazimuth theodolite, recipiangle, and trigonometer. Counter-proposals, in particular the plane table, threatened to undermine the geometrical programme, but by the mid-seventeenth century a stable compromise had evolved. Among other things, the (...) demise of the shadow square indicates that angle measurement was then part of surveying practice. (shrink)

The political integration of the European Union is fragile for many reasons, not least the reassertion of nationalism. That said, if we examine specific practices and infrastructures, a more complicated story emerges. We juxtapose the political fragility of the EU in relation to the ongoing formation of data infrastructures in official statistics that take part in postnational enactments of Europe’s populations and territories. We develop this argument by analyzing transformations in how European populations are enacted through new technological infrastructures that (...) seek to integrate national census data in “cubes” of cross-tabulated social topics and spatial “grids” of maps. In doing so, these infrastructures give meaning to what “is” Europe in ways that are both old and new. Through standardization and harmonization of social and geographical spaces, “old” geometries of organizing and mapping populations are deployed along with “new” topological arrangements that mix and fold categories of population. Furthermore, we consider how grids and cubes are generative of methodological topologies by closing the distances or differences between methods and making their data equivalent. By paying attention to these practices and infrastructures, we examine how they enable reconfiguring what is known and imagined as Europe and how it is governed. (shrink)

In this thesis, I investigate the nature of geometric knowledge and its relationship to spatial intuition. My goal is to rehabilitate the Kantian view that Euclid's geometry is a mathematical practice, which is grounded in spatial intuition, yet, nevertheless, yields a type of a priori knowledge about the structure of visual space. I argue for this by showing that Euclid's geometry allows us to derive knowledge from idealized visual objects, i.e., idealized diagrams by means of non-formal logical inferences. (...) By developing such an account of Euclid's geometry, I complete the "standard view" that geometry is either a formal system or an empirical science, which was developed mainly by the logical positivists and which is currently accepted by many mathematicians and philosophers. My thesis is divided into three parts. I use Hans Reichenbach's arguments against Kant and Edmund Husserl's genetic approach to the concept of space as a means of arguing that the "standard view" has to be supplemented by a concept of a geometry whose propositions have genuine spatial content. I then develop a coherent interpretation of Euclid's method by investigating both the subject matter of Euclid's geometry and the nature of geometric inferences. In the final part of this thesis, I modify Husserl's phenomenological analysis of the constitution of visual space in order to define a concept of spatial intuition that allows me not only to explain how Euclid's practice is grounded in visual space, but also to account for the apriority of its results. (shrink)

This work aim to analyse relations between Arithmetic and Geometry indicated by Piero della Francesca in his treatise Libellus de quinque corporibus regularibus. Piero della Francesca was a painter and scholar of perspective, geometry and arithmetic, in his time. He carried out investigations on pictorial, geometric and architectural issues. Of the treatises he wrote, only three are preserved, on perspective, Geometry and Arithmetic. The central document selected for this research was the manuscript Libellus de quinque corporibus regularibus, (...) deposited in the Vatican Apostolic Library in digitized copy. Throughout this work we have tried to understand the Libellus de quinque corporibus regularibus as a result of the work of an artist and scholar who proposed ways of relating Arithmetic and Geometry using both his practical knowledge and the works of authors of antiquity. (shrink)

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the (...) aforementioned areas. (shrink)

The article reflects on the relationship between pedagogy and body moving from the sense of tact. Firstly, the question of the body-mind relationship in contemporary pedagogy is presented. Starting from the cartesian division of mind and body, we expose the main issues related to a possible overcoming of such dualism. In addition, we maintain that cartesian dualism significantly contributed to a dominance of mind over body in education. Then, we reconstruct the history of “pedagogical tact”: this concept changed from an (...) original somatic significance to a more interpersonal sense, putting the question of the body aside. In conclusion, we maintain that re-considering tact in its somatic significance can help build innovative pedagogical theories and practices aimed at a broader reconsideration of bodies in education. (shrink)

The gods that guard the poles have been assigned the function of assembling the separate and unifying the manifold members of the whole, while those appointed to the axes keep the circuits in everlasting revolution around and around. And if I may add my own conceit, the centers and poles of all the spheres symbolize the wry-necked gods by imitating the mysterious union and synthesis which they effect; the axes represent the connectors of all the cosmic orders … and the (...) very spheres are likenesses of the perfecting divinities … 1 Proclus’s Commentary on the First Book of Euclid’s Elements contains one of the most important discussions of the nature of mathematical objects and preserves a wealth of historical information about mathematics from antiquity. However, large sections of the text have been neglected by scholars since the text’s first publication in the sixteenth century. Proclus expounds at length on correspondences between mathematical objects and the gods. At times he uses the language of “theurgy,” the ritual practices that later Neoplatonists insisted were necessary for the human soul to ascend to the intelligibles. The contention of this article is that the Commentary, taken as a whole, concerns the practice of “inner theurgy,” mental rituals used by advanced students to raise the soul beyond its assigned bounds. This practice can be traced back to Plotinus and the very beginnings of Neoplatonism; however, the particular methods presented by Proclus are his own development. This interpretation of the Commentary is supported by Proclus’s works Commentary on the Parmenides and Platonic Theology. Parts of the Commentary that have been studied carefully by modern scholars, especially the theory of the mathematical phantasia or imagination, will be shown to be crucial elements in Proclus’s theory and practice of inner theurgy. (shrink)

Against Thomas Mormann's argument that differential topology does not support Carnap's conventionalism in geometry we show their compatibility. However, Mormann's emphasis on the entanglement that characterizes topology and its associated metrics is not misplaced. It poses questions about limits of empirical inquiry. For Carnap, to pose a question is to give a statement with the task of deciding its truth. Mormann's point forces us to introduce more clarity to what it means to specify the task that decides between competing (...) hypotheses and in what way such a task may be both in practice and/or in principle impossible to carry out. (shrink)

J. H. Lambert proved important results of what we now think of as non-Euclidean geometries, and gave examples of surfaces satisfying their theorems. I use his philosophical views to explain why he did not think the certainty of Euclidean geometry was threatened by the development of what we regard as alternatives to it. Lambert holds that theories other than Euclid's fall prey to skeptical doubt. So despite their satisfiability, for him these theories are not equal to Euclid's in justification. (...) Contrary to recent interpretations, then, Lambert does not conceive of mathematical justification as semantic. According to Lambert, Euclid overcomes doubt by means of postulates. Euclid's theory thus owes its justification not to the existence of the surfaces that satisfy it, but to the postulates according to which these "models" are constructed. To understand Lambert's view of postulates and the doubt they answer, I examine his criticism of Christian Wolff's views. I argue that Lambert's view reflects insight into traditional mathematical practice and has value as a foil for contemporary, model-theoretic, views of justification. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

The paper attempts to summarize the debate on Kant’s philosophy of geometry and to offer a restricted area of mathematical practice for which Kant’s philosophy would be a reasonable account. Geometrical theories can be characterized using Wittgenstein’s notion of pictorial form . Kant’s philosophy of geometry can be interpreted as a reconstruction of geometry based on one of these forms — the projective form . If this is correct, Kant’s philosophy is a reasonable reconstruction of such theories (...) as projective geometry; and not only as they were practiced in Kant’s time, but also as architects use them today. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

In their publications during the 1820s, Jakob Steiner and Julius Plücker frequently derived the same results while claiming different methods. This paper focuses on two such results in order to compare their approaches to constructing figures, calculating with symbols, and representing geometric magnitudes. Underlying the repetitive display of similar problems and theorems, Steiner and Plücker redefined synthetic and analytic methods in distinctly personal practices.

In this paper I offer a theory about the nature of diagrammatic representation in geometry. On my view, diagrammatic representaiton differs from pictorial representation in that neither the resemblance between the diagram and its object nor the experience of such a resemblance plays an essential role. Instead, the diagrammatic representation is arises from the role the components of the diagram play in a diagramatic practice that allows us to draws inferences based on them about the ojbects they represent.

In this paper I offer a theory about the nature of diagrammatic representation in geometry. On my view, diagrammatic representaiton differs from pictorial representation in that neither the resemblance between the diagram and its object nor the experience of such a resemblance plays an essential role. Instead, the diagrammatic representation is arises from the role the components of the diagram play in a diagramatic practice that allows us to draws inferences based on them about the ojbects they represent.

In my dissertation, I argue that contemporary interpretive work on Kant's philosophy of geometry has failed to understand properly the diagrammatic aspects of Euclidean reasoning. Attention to these aspects is amply repaid, not only because it provides substantial insight into the role of intuition in Kant's philosophy of mathematics, but also because it brings out both the force and the limitations of Kant's philosophical account of geometry. ;Kant characterizes the predecessors with which he was engaged as agreeing that (...) mathematical judgments are analytic a priori. This is an inadequate account of geometrical judgments for Kant, because it cannot be reconciled with the evident success of the Euclidean practice in producing a priori knowledge of the nature of space. The ubiquitous presence of diagrams in Euclidean proofs suggests that, at least in Kant's time, any plausible account of the establishment of geometrical judgments must provide for the role of quasi-perceptual representations of individuals. Kant's claim that mathematical judgments are synthetic a priori reflects, in its positive aspect, his recognition of the importance of these 'intuitive' representations to Euclidean reasoning. Furthermore, Kant's doctrine that geometrical knowledge is established by the construction of concepts in intuition can be usefully understood as an attempt to accommodate the role of diagrams in the establishment of synthetic a priori geometrical judgments. ;Both Kant's rejection of the analytic a priori account of mathematics and his positive account of geometry depend on the plausibility of his appeals to Euclidean practice. Attention to the role of diagrams in Euclidean proof supports Kant's contention that intuitive representations are an essential component of Euclidean reasoning; however, the role of diagrams in both reductio proofs and in proofs by cases brings out tensions in Kant's positive account. The use of diagrams to represent impossible geometrical situations in reductio proofs indicates that Kant's notion of 'construction in intuition' can have a broader application than the construction of instances of geometrical concepts . Similarly, proofs by cases undermine Kant's attempts to explain the generality of judgments established through the consideration of individuals by appeal to the rules for the construction of those individuals. ;Contemporary interpretive debates about Kant's philosophy of geometry have centered on the philosophical role played by his appeal to intuition. This appeal cannot be primarily understood as an attempt to establish that the fundamental assumptions of Euclidean geometry are necessary truths about space. Nor can the role for intuition be regarded as merely licensing moves in geometrical proofs like an inference rule in modern logic. Instead, by understanding Kant's appeal to intuition as, at least in part, an attempt to accommodate the diagrammatic features of Euclidean reasoning, a much richer role for intuition is revealed. Intuition is required to play a logical role by expressing spatial relationships and establishing the existence of geometrical entities. In addition, it plays a quasi-perceptual role by supporting the truth of certain claims appealed to during the course of Euclidean proof. (shrink)

How can we invent new certain knowledge in a methodical manner? This question stands at the heart of Salomon Maimon's theory of invention. Chikurel argues that Maimon's contribution to the ars inveniendi tradition lies in the methods of invention which he prescribes for mathematics. Influenced by Proclus' commentary on Elements, these methods are applied on examples taken from Euclid's Elements and Data. Centering around methodical invention and scientific genius, Maimon's philosophy is unique in an era glorifying the artistic genius, known (...) as Geniezeit. Invention, primarily defined as constructing syllogisms, has implications on the notion of being given in intuition as well as in symbolic cognition. Chikurel introduces Maimon's notion of analysis in the broader sense, grounded not only on the principle of contradiction but on intuition as well. In philosophy, ampliative analysis is based on Maimon's logical term of analysis of the object, a term that has yet to be discussed in Maimonian scholarship. Following its introduction, a new version of the question quid juris? arises. In mathematics, Chikurel demonstrates how this conception of analysis originates from practices of Greek geometrical analysis. (shrink)

There is little agreement about Aristotle’s philosophy of geometry, partly due to the textual evidence and partly part to disagreement over what constitutes a plausible view. I keep separate the questions ‘What is Aristotle’s philosophy of geometry?’ and ‘Is Aristotle right?’, and consider the textual evidence in the context of Greek geometrical practice, and show that, for Aristotle, plane geometry is about properties of certain sensible objects—specifically, dimensional continuity—and certain properties possessed by actual and potential compass-and-straightedge drawings (...) qua quantitative and continuous. For their part, the objects of stereometry are potential sensible three-dimensional figures qua quantitative and continuous. (shrink)

If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...) recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (shrink)

The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive (...) conception of rigor and intuition. Unlike what is often assumed today, from their perspective, rigor is neither opposed to intuition nor understood as a unitary phenomenon – Enriques distinguishes between small-scale rigor and large-scale rigor and Severi between formal rigor and substantial rigor. Finally, we turn to the notion of mathematical objectivity. We draw from our case study in order to advance a multi-dimensional analysis of objectivity. Specifically, we suggest that various types of rigor may be associated with different conceptions of objectivity: namely, objectivity as faithfulness to facts and objectivity as intersubjectivity. (shrink)

The paper investigates Ernst Cassirer’s structuralist account of geometrical knowledge developed in his Substanzbegriff und Funktionsbegriff. The aim here is twofold. First, to give a closer study of several developments in projective geometry that form the direct background for Cassirer’s philosophical remarks on geometrical concept formation. Specifically, the paper will survey different attempts to justify the principle of duality in projective geometry as well as Felix Klein’s generalization of the use of geometrical transformations in his Erlangen program. The (...) second aim is to analyze the specific character of Cassirer’s geometrical structuralism formulated in 1910 as well as in subsequent writings. As will be argued, his account of modern geometry is best described as a “methodological structuralism”, that is, as a view mainly concerned with the role of structural methods in modern mathematical practice. (shrink)

Not simply set out in accompaniment of the Greek geometrical text, the diagram also is coaxed into existence manually (using straightedge and compasses) by commands in the text. The marks that a diligent reader thus sequentially produces typically sum, however, to a figure more complex than the provided one and also not (as it is) artful for being synoptically instructive. To provide a figure artfully is to balance multiple desiderata, interlocking the timelessness of insight with the temporality of construction. Our (...) account of the diagram complements those of Manders and Macbeth by more strongly emphasizing practical synthesis. (shrink)

Gothic cathedrals like Chartres were built in a discontinuous process by groups of masons using their own local knowledge, measures, and techniques. They had neither plans nor knowledge of structural mechanics. The success of the masons in building such large complex innovative structures lies in the use of templates, string, constructive geometry, and social organization to assemble a coherent whole from the messy heterogeneous practices of diverse groups of workers. Chartres resulted from the ad hoc accumulation of the work (...) of many men. (shrink)

In the preface to the Principia (1687) Newton famously states that “geometry is founded on mechanical practice.” Several commentators have taken this and similar remarks as an indication that Newton was firmly situated in the constructivist tradition of geometry that was prevalent in the seventeenth century. By drawing on a selection of Newton's unpublished texts, I hope to show the faults of such an interpretation. In these texts, Newton not only rejects the constructivism that took its birth in (...) Descartes's Géométrie (1637); he also presents the science of geometry as being more powerful than his Principia remarks may lead us to believe. (shrink)

Since the application of Postulate I.2 in Euclid's Elements is not uniform, one could wonder in what way should it be applied in Euclid's plane geometry. Besides legitimizing questions like this from the perspective of a philosophy of mathematical practice, we sketch a general perspective of conceptual analysis of mathematical texts, which involves an extended notion of mathematical theory as system of authorizations, and an audience-dependent notion of proof.

A common view is that, whether taught in philosophy departments or elsewhere, practical ethics should include some introduction to philosophical ethics. But even an entire course cannot afford much time for this and expect to do justice to ethical concerns in the practical area . The concern is that ethical theories would need to be “watered down,” or over-simplified. So, we should not expect that this will be in good keeping with either the theories or the practical (...) concerns.In addressing this problem, we turn to philosopher Thomas Reid . He insisted that, because morality is for everyone, one needn’t be a philosopher to understand its requirements. Although it can be useful to organize our moral thinking around a few basic principles, a system of morality is more like a system of botany or mineralogy than geometry. Noting this can guide us in constructing effective courses in practical ethics. (shrink)

Drawing on work done by Helmholtz, I argue that Reid was in no position to infer that objects appear as if projected on the inner surface of a sphere, or that they have the geometric properties of such projections even though they do not look concave towards the eye. A careful consideration of the phenomena of visual experience, as further illuminated by the practice of visual artists, should have led him to conclude that the sides of visible appearances either look (...) straight, in which case their angles appear to approximate Euclidean measures, or their angles do not appear to approximate Euclidean measures for straight line figures, in which case their sides do not look straight. (shrink)

This paper examines the ethical and religious dimensions of mathematical practice in the early modern era by offering an interpretation of Antoine Arnauld and Pierre Nicole’s Nouveaux éléments de géométrie. According to these important figures of seventeenth-century French philosophy and theology, mathematics could achieve extra-mathematical or non-mathematical goals; that is, mathematics could foster practices of moral self-improvement, deepen the mathematician’s piety and cultivate epistemic virtues. The Nouveaux éléments de géométrie, which I contend offers the most robust account of the virtues (...) cultivated by mathematics in the period, was envisaged by its authors to cultivate moral, Christian and epistemic virtues that could serve in the fulfilment of moral and Christian obligations. In this paper, I set out the goals of mathematical inquiry for the Port-Royalists and describe the specific virtues they believed a revised edition of the Elements of Euclid could foster. I show that Arnauld and Nicole believed that an acquaintance with mathematics could render a student of Euclid more just, truth-loving, attentive and humble, and better able to discern truth from falsity. (shrink)

Part 1 of this article exposed a tension between Poincaré’s views of arithmetic and geometry and argued that it could not be resolved by taking geometry to depend on arithmetic. Part 2 aims to resolve the tension by supposing not merely that intuition’s role is to justify induction on the natural numbers but rather that it also functions to acquaint us with the unity of orders and structures and show practices to fit or harmonize with experience. I argue (...) that in this manner, intuition serves the epistemological function of warranting generalizations and justifying practices. In particular, it justifies the application of group-theoretic notions in geometry but not the use of set-theoretic notions in arithmetic. (shrink)

Graduate Papers from the 2022 Joint Session. It is often said that Aristotle takes geometrical objects to be absolutely unmovable and unchangeable. However, Greek geometrical practice does appeal to motion and change, and geometers seem to consider their objects apt to be manipulated. In this paper, I examine if and how Aristotle’s philosophy of geometry can account for the geometers’ practices and way of talking. First, I illustrate three different ways in which Greek geometry appeals to change. Second, (...) I examine Aristotle’s ontology of geometrical objects and argue that although the truth-makers of geometrical statements are in fact unmovable because they are properties of sensible objects, geometers ‘separate them in thought’ and treat them as substances apt to be modified. Finally, I examine whether allowing for the possibility of manipulating these semi-fictional geometrical individuals creates problems for the applicability of geometry. I find that it does not, insofar as one accepts that geometry is not meant to track physical change but merely to study the instantaneous geometrical configuration of sensible bodies, and is thus only applicable at the instant. (shrink)

When ancient mathematical treatises lack expositions of numerical techniques, what purposes could ancient mathematical theories be expected to serve? Ancient writers only rarely address questions of this sort directly. Possible answers are suggested by surveying geometry, mechanics, optics, and spherics to discover how the mathematical treatments imply positions on this issue. This survey shows the ways in which these ancient theoretical inquiries reflect practical activity in their fields. This account, in turn, suggests that the authors may have intended (...) their theorems not to predict, but to explain phenomena. We may then consider what kind of explanations they were seeking. (shrink)

This article is primarily concerned with Aristotle’s theory of the syllogistic, and the investigation of the hypothesis that logical symbolism and methodology were in these early stages of a geometrical nature; with the gradual algebraization that occurred historically being one of the main reasons that some of the earlier passages on logic may often appear enigmatic. The article begins with a brief introduction that underlines the importance of geometric thought in ancient Greek science, and continues with a short exposition of (...) Aristotle’s views and methods in regard to logic. We then offer an interpretation of syllogisms, as well as of the main proof methods that are utilized by Aristotle, in terms of diagrams that can be seen as analogous to those of Euclidean geometry; where the various proofs proceed by appropriately manipulating an initially drawn diagram, in a way parallel to constructions that occur in Euclid’s Elements. In this way, logic is presented as following, to a large degree, methodological tenets established by the practice of geometry. Finally, we present a diagrammatic decision procedure that allows one to directly read off the validity of syllogisms from a corresponding diagram, in a way such that no further manipulation of it is necessitated. (shrink)

Since the application of Postulate I.2 in Euclid’s Elements is not uniform, one could wonder in what way should it be applied in Euclid’s plane geometry. Besides legitimizing questions like this from the perspective of a philosophy of mathematical practice, we sketch a general perspective of conceptual analysis of mathematical texts, which involves an extended notion of mathematical theory as system of authorizations, and an audience-dependent notion of proof.

This essay explores the research practice of French geometer Michel Chasles, from his 1837 Aperçu historique up to the preparation of his courses on ‘higher geometry’ between 1846 and 1852. It argues that this scientific pursuit was jointly carried out on a historiographical and a mathematical terrain. Epistemic techniques such as the archival search for and comparison of manuscripts, the deconstructive historiography of past geometrical methods, and the epistemologically motivated periodization of the history of mathematics are shown to have (...) played a crucial role in the shaping of Chasles's own theories. In particular, we present Chasles's approach to the ‘material history’ of algebraic symbolism and argue that it motivated and informed his subsequent invention of a novel notational technology for the writing of geometrical proofs and propositions. In return, this technology allowed Chasles to carry out a programme for the modernization of geometry in keeping with epistemic requirements he had also delineated via a form of historical writing. (shrink)

Open Court began publishingThe Monistin 1890 as a journal“devotedto the philosophy of science”that regularly included mathematics. The audiencewas understood to be“cultured people who have not a technical mathematicaltraining”but nevertheless“have a mathematical penchant.”With these constraints,the mathematical content varied from recreations to logical foundations, but every-one had something to say about non-Euclidean geometry, in debates that rangedfrom psychology to semantics. The focus in this essay is on the contested value ofmathematical expertise in legitimating what should be considered as mathematics.While some mathematicians (...) urgedThe Monistto uphold disciplinary standards ofgeometrical reasoning, authors opposed tonon-Euclidean geometry aligned theirreasoning with practical applications, universal know-how, and nonhierarchicaldemocracy. As one contributorinquired,“How isthe professional expert betterfit-ted to see more lucidly in dealing with the elements of geometry than any otherperson of good geometric faculty?”. (shrink)

In the middle part of his Brouillon Project on conics, Girard Desargues develops the theory of the traversale, a notion that generalizes the Apollonian diameter and allows to give a unified treatment of the three kinds of conics. We showed elsewhere that it leads Desargues to a complete theory of projective polarity for conics. The present article, which shall close our study of the Brouillon Project, is devoted to the last part of the text, in which Desargues puts his theory (...) of the traversal into practice by giving a very elegant tratment of the classical theory of parameters and foci. This will lead us to show that Desargues’ proofs can only be understood if one accepts that he reasons in a resolutely projective framework, completely assimilating elements at infinity to those at finite distance in his proofs. (shrink)

The literature of the past century produced an historical reconstruction of statics theory applied to mechanical structures coinciding–starting with Le Mecaniche (1634) and Discorsi e dimostrazioni matematiche sopra a due nuove scienze (1638) by Galileo Galilei (1564–1642). Based on previous research (RP) and our historical and historiographical line of research [37], in this paper we briefly analyse Buonaiuto Lorini (1540–1611) Le Fortificationi ([1596] 1609) as a bridge between the science of weights and early mechanical science, including the graphical scale and (...) machines. This is a fundamental and advanced military fortification–engineering treatise with an evident important relationship between theory & practice, including designs–projects of both fortresses and machines, understood as tools, to manufacture them. Lorini also presents an excursus on how fortifications developed according to the advent of new artillery/weapons, comprising the crucial passage from the use of square to circular towers up to the description of modern fortresses in all their details, e.g., bulwarks and cavaliers, curtains, orillons, star–shaped plans. Basing on Archimedes’ (fl. 287–212 BCE) techniques (law of the lever) and Guidobaldo del Monte's (1545–1607; Le Mecaniche (1577–1581) he successfully engaged in advanced mechanical considerations. His position (at the end of the first chapter) on the distinction between the mathematician (scientist) and the mechanician (architect–engineer or even machine expert) is historiographically very interesting within the difference between abstract Euclidean geometry and the application of its contents to imperfect and heterogeneous reality. The translations are ours. (shrink)