This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view (...) that geometry is the study of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them. (shrink)

I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at (...) Göttingen. Concurring with Tappenden, I hold that Frege’s logicism should not be understood as a continuing a project of reductionist arithmetization. However, Frege does not quite take up the Riemannian banner either. His logicism supports a hierarchical understanding of the structure of mathematical knowledge, according to which arithmetic is applicable to geometry but not vice versa because the former is more general, as revealed by the strictly logical nature of its objects in comparison to the intuitional nature of geometric objects. I suggest, in particular, that Frege intended that foundational work would show the use of geometric intuition in complex analysis, a source of error for Riemann that Weierstrass was proud to have uncovered, to be inessential. (shrink)

We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which (...) the brain can transform and organize its perceptual intake. It is not necessary for a geometrical form to be perfectly instantiated in order for perception of such a form to be the basis of a geometrical concept. (shrink)

In the philosophy of perception, direct realism has come into vogue. Philosophical authors assert and assume that what their readers want, and what anyone should want, is some form of direct realism. There are disagreements over precisely what form this direct realism should take. The majority of positions in favor now offer a direct realism in which objects and their material or physical properties constitute the contents of perception, either because we have an immediate or intuitive acquaintance with those (...) objects and properties, or because our perceptual states have informational content that represents the properties of those objects (and which is not itself an object of perception and has no specifically subjective aspect). This paper considers various forms of perceptual realism, including, for purposes of comparison, the largely abandoned indirect or representative realism. After surveying the variety of perceptual realisms and considering their various commitments, I introduce some considerations concerning the phenomenology of visual space that cause trouble for most forms of direct realism. These considerations pertain to the perception of objects in the distance and, secondarily, to the perception of shapes at a slant. I argue that one of the lesser known varieties of perceptual realism, critical direct realism, can meet the challenges offered by the facts of spatial perception. (shrink)

This paper investigates the nature of reality by looking at the philosophical debate between realism and idealism and at scientific investigations in quantum physics and at recent studies of animal senses, neurology and cognitive psychology. The concept of perceptual relativity is examined and this involves looking at sense perception in other animals and various examples of perceptual relativity in science. It will be concluded that the universe is observer dependent and that there is no reality independent of the observer, which (...) is knowable to the observer. The paper concludes by an investigation of what an observer dependent universe would be like and that recognition of an observer dependent world would lead to a more open minded and tolerant world. (shrink)

Descartes holds that the tell-tale sign of a solid proof is that its entailments appear clearly and distinctly. Yet, since there is a limit to what a subject can consciously fathom at any given moment, a mnemonic shortcoming threatens to render complex geometrical reasoning impossible. Thus, what enables us to recall earlier proofs, according to Descartes, is God’s benevolence: He is too good to pull a deceptive switch on us. Accordingly, Descartes concludes that geometry and belief in God must (...) go hand in hand. However, I argue that, while theism adds a layer of psychological reassurance, the mind-independent reality of God would ensure the preservation of past demonstrations for atheists as well. (shrink)

David Hyder.The Determinate World: Kant and Helmholtz on the Physical Meaning of Geometry. viii + 229 pp., bibl., index. Berlin/New York: Walter de Gruyter, 2009.

This paper examines Hobbes’s criticisms of Robert Boyle’s air-pump experiments in light of Hobbes’s account in _De Corpore_ and _De Homine_ of the relationship of natural philosophy to geometry. I argue that Hobbes’s criticisms rely upon his understanding of what counts as “true physics.” Instead of seeing Hobbes as defending natural philosophy as “a causal enterprise … [that] as such, secured total and irrevocable assent,” 1 I argue that, in his disagreement with Boyle, Hobbes relied upon his (...) understanding of natural philosophy as a mixed mathematical science. In a mixed mathematical science one can mix facts from experience with causal principles borrowed from geometry. Hobbes’s harsh criticisms of Boyle’s philosophy, especially in the _Dialogus Physicus, sive De natura aeris_, should thus be understood as Hobbes advancing his view of the proper relationship of natural philosophy to geometry in terms of mixing principles from geometry with facts from experience. Understood in this light, Hobbes need not be taken to reject or diminish the importance of experiment/experience; nor should Hobbes’s criticisms in _Dialogus Physicus_ be understood as rejecting experimenting as ignoble and not befitting a philosopher. Instead, Hobbes’s viewpoint is that experiment/experience must be understood within its proper place – it establishes the ‘that’ for a mixed mathematical science explanation. (shrink)

my goal here is to provide a detailed analysis of the methods of inference that are employed in De prospectiva pingendi. For this purpose, a method of natural deduction is proposed. the treatise by Piero della Francesca is a manifestation of a union between the ne arts and the mathematical sciences of arithmetic and geometry. He de nes painting as a part of perspective and, speaking precisely, as a branch of geometry, which is why we nd advanced geometrical (...) exercises here. (shrink)

Since antiquity well into the beginnings of the 20th century geometry was a central topic for philosophy. Since then, however, most philosophers of science, if they took notice of topology at all, considered it as an abstruse subdiscipline of mathematics lacking philosophical interest. Here it is argued that this neglect of topology by philosophy may be conceived of as the sign of a conceptual sea-change in philosophy of science that expelled geometry, and, more generally, mathematics, (...) from the central position it used to have in philosophy of science and placed logic at center stage in the 20th century philosophy of science. Only in recent decades logic has begun to loose its monopoly and geometry and topology received a new chance to find a place in philosophy of science. (shrink)

Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation (...) or theorem and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers. (shrink)

In the article, an argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modelling, not the world, but our activities in the world.

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us, all directions are the same to us and all units of length we use to create geometric figures are the same to us. On the other hand, through the process of algebraic simplification, this system of axioms directly provides the (...) Weyl’s system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: it supports the thesis that Euclidean geometry is a priori, it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics. (shrink)

Linear Algebra is an extremely important field that extends everyday concepts about geometry and algebra into higher spaces. This text serves as a gentle motivating introduction to the principles (and philosophy) behind linear algebra. This is aimed at undergraduate students taking a linear algebra class - in particular engineering students who are expected to understand and use linear algebra to build and design things, however it may also prove helpful for philosophy majors and anyone else interested in (...) the ideas behind linear algebra. Linear algebra is used extensively in statistics, mechanical engineering, circuit theory, signal processing, economics and data science/machine learning. I have found in my teaching career that students tend to struggle the most with motivating the "why" behind the abstract concepts discussed in linear algebra. This can be difficult for professors to understand because we are so used to using the language of linear algebra we take for granted why these questions are important in the first place. In other words, we may understand the “why” (after years of using these tools), but our students may not. This text seeks to explain this “why” in an accessible, engaging way. (shrink)

This work examines the unique way in which Benedict de Spinoza combines two significant philosophical principles: that real existence requires causal power and that geometrical objects display exceptionally clearly how things have properties in virtue of their essences. Valtteri Viljanen argues that underlying Spinoza's psychology and ethics is a compelling metaphysical theory according to which each and every genuine thing is an entity of power endowed with an internal structure akin to that of geometrical objects. This allows Spinoza to offer (...) a theory of existence and of action - human and non-human alike - as dynamic striving that takes place with the same kind of necessity and intelligibility that pertain to geometry. This fresh and original study will interest a wide range of readers in Spinoza studies and early modern philosophy more generally. (shrink)

Two seemingly contradictory tendencies have accompanied the development of the natural sciences in the past 150 years. On the one hand, the natural sciences have been instrumental in effecting a thoroughgoing transformation of social structures and have made a permanent impact on the conceptual world of human beings. This historical period has, on the other hand, also brought to light the merely hypothetical validity of scientific knowledge. As late as the middle of the 19th century the truth-pathos in the natural (...) sciences was still unbroken. Yet in the succeeding years these claims to certain knowledge underwent a fundamental crisis. For scientists today, of course, the fact that their knowledge can possess only relative validity is a matter of self-evidence. The present analysis investigates the early phase of this fundamental change in the concept of science through an examination of Hermann von Helmholtz's conception of science and his mechanistic interpretation of nature. Helmholtz (1821-1894) was one of the most important natural scientists in Germany. The development of this thoughts offers an impressive but, until now, relatively little considered report from the field of the experimental sciences chronicling the erosion of certainty. (shrink)

Hermann Weyl was one of the most important figures involved in the early elaboration of the general theory of relativity and its fundamentally geometrical spacetime picture of the world. Weyl’s development of “pure infinitesimal geometry” out of relativity theory was the basis of his remarkable attempt at unifying gravitation and electromagnetism. Many interpreters have focused primarily on Weyl’s philosophical influences, especially the influence of Husserl’s transcendental phenomenology, as the motivation for these efforts. In this article, I argue both that (...) these efforts are most naturally understood as an outgrowth of the distinctive mathematical-physical tradition in Göttingen and also that phenomenology has little to no constructive role to play in them. (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)

This paper examines explanations that turn on non-local geometrical facts about the space of possible configurations a system can occupy. I argue that it makes sense to contrast such explanations from "geometry of motion" with causal explanations. I also explore how my analysis of these explanations cuts across the distinction between kinematics and dynamics.

The role of conventions in the formulation of Thomas Reid’s theory of the geometry of vision, which he calls the “geometry of visibles”, is the subject of this investigation. In particular, we will examine the work of N. Daniels and R. Angell who have alleged that, respectively, Reid’s “geometry of visibles” and the geometry of the visual field are non-Euclidean. As will be demonstrated, however, the construction of any geometry of vision is subject to a (...) choice of conventions regarding the construction and assignment of its various properties, especially metric properties, and this fact undermines the claim for a unique non-Euclidean status for the geometry of vision. Finally, a suggestion is offered for trying to reconcile Reid’s direct realist theory of perception with his geometry of visibles. (shrink)

Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, she writes that things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of material things. After situating Du Châtelet in (...) this debate, this chapter argues, first, that her account of the origins of mathematical objects is less subjectivist than it might seem. Mathematical objects are non-arbitrary, public entities. While mathematical objects are partly mind-dependent, so are material things. Mathematical objects can approximate the material. Second, it is argued that this moderate metaphysical position underlies Du Châtelet’s persistent claims that mathematics is required for certain kinds of general knowledge, including in natural science. The chapter concludes with an illustrative example: an analysis of Du Châtelet’s argument that matter is continuous. A key premise in the argument is that mathematical representations and material nature correspond. (shrink)

The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical (...) responses to these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics. (shrink)

Despite the efforts undertaken to separate scientific reasoning and metaphysical considerations, despite the rigor of construction of mathematics, these are not, in their very foundations, independent of the modalities, of the laws of representation of the world. The OdC shows that the logical Facts Exist neither more nor less than the Facts of the world which are Facts of Knowledge. Mathematical facts are representation facts. The primary objective of this article is to integrate the subject into mathematics as a mode (...) of emergence of meaning and then to evaluate its consequences. (shrink)

When national borders in the modern sense first began to be established in early modern Europe, non-contiguous and perforated nations were a commonplace. According to the conception of the shapes of nations that is currently preferred, however, nations must conform to the topological model of circularity; their borders must guarantee contiguity and simple connectedness, and such borders must as far as possible conform to existing topographical features on the ground. The striving to conform to this model can be seen at (...) work today in Quebec and in Ireland, it underpins much of the rhetoric of the P.L.O., and was certainly to some degree involved as a motivating factor in much of the ethnic cleansing which took place in Bosnia in recent times. The question to be addressed in what follows is: to what extent could inter-group disputes be more peacefully resolved, and ethnic cleansing avoided, if political leaders, diplomats and others involved in the resolution of such disputes could be brought to accept weaker geometrical constraints on the shapes of nations? A number of associated questions then present themselves: What sorts of administrative and logistical problems have been encountered by existing non contiguous nations and by perforated nations, and by other nations deviating in different ways from the received geometrical ideal? To what degree is the desire for continuity and simple connectedness a rational desire, and to what degree does it rest on species of political rhetoric which might be countered by, for example, philosophical argument? These and a series of related questions will form the subject- matter of the present essay. (shrink)

This paper aims to clarify Merleau-Ponty’s contribution to an embodied-enactive account of mathematical cognition. I first identify the main points of interest in the current discussions of embodied higher cognition and explain how they relate to Merleau-Ponty and his sources, in particular Husserl’s late works. Subsequently, I explain these convergences in greater detail by more specifically discussing the domains of geometry and algebra and by clarifying the role of gestalt psychology in Merleau-Ponty’s account. Beyond that, I explain how, for (...) Merleau-Ponty, mathematical cognition requires not only the presence and actual manipulation of some concrete perceptible symbols but, more strongly, how it is fundamentally linked to the structural transformation of the concrete configurations of symbolic systems to which these symbols appertain. Furthemore, I fill a gap in the literature by explaining Merleau-Ponty’s claim that these structural transformations are operated through motor intentionality. This makes it possible, in turn, to contrast Merleau-Ponty’s approach to ontologically idealistic and realistic views on mathematical objects. On Merleau-Ponty’s account, mathematical objects are relational entities, that is, gestalts that necessarily imply situated cognizers to whom they afford a specific type of engagement in the world and on whom they depend in their eventual structural transformations. I argue that, by attributing a strongly constitutive role to phenomenal configurations and their motor transformation in mathematical thinking, Merleau-Ponty contributes to clarifying the worldly, historical, and socio-cultural aspects of mathematical truths without compromising what we perceive as their universality, certainty, and necessity. (shrink)

In searching for the origin of Europe and the cultural region/continent that we call “Europe”, at first glance we have to consider at least a double view: on the one hand the geographical understanding which indicates a region or a continent. On the other a certain form of identity and culture described and defined as European. Rodolphe Gasché taking hint from Husserl’s passage ‘Europe is not to be construed simply as a geographical and political entity’ states that a rigorous engagement (...) with what we understand by “Europe” requires that we acknowledge it as involving ‘something else as well’. With regard to the many bequests of Europe, founded in ancient Greece, in this essay I will attempt to elucidate some essential features of its cultural identity such as science and philosophy, and reflect upon several specific aspects: on the origin of Europe, on its roots and heritage, on the concept of culture, and especially on the foundation of sciences (Geometry), which contains a large part of European spirit and civilization. In particular I will address some European historical moments with reference to Husserl, Heidegger, the concept of Thaumazein… In the second part of the essay, I will shift my interest to Ancient Greece in order to access to the value of the Esprit de géométrie as defined in Proclus on the Commentary on the First Book of Euclid’s. (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)

John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion follows from the (...) premises: there must be something deductions can show. Corcoran calls a proposition that follows from given premises a hidden consequence of those premises if it is not obvious that the proposition follows from those premises. By a Euclidean geometry we mean an extended discourse beginning with basic premises—axioms, postulates, definitions—1) treating a universe of geometrical figures and 2) resembling Euclid’s Elements. There were Euclidean geometries before Euclid (fl. 300 BCE), even before Aristotle (384–322 BCE). Bochenski, Lukasiewicz, Patzig and others never new this or if they did they found it inconvenient to mention. Euclid shows no awareness of Aristotle. It is obvious today—as it should have been obvious in Euclid’s time, if anyone knew both—that Aristotle’s logic was insufficient for Euclid’s geometry: few if any geometrical theorems can be deduced from Euclid’s premises by means of Aristotle’s deductions. Aristotle’s writings don’t say whether his logic is sufficient for Euclidean geometry. But, there is not even one fully-presented example. However, Aristotle’s writings do make clear that he endorsed the goal of a sufficient system. Nevertheless, incredible as this is today, many logicians after Aristotle claimed that Aristotelian logics are sufficient for Euclidean geometries. This paper reviews and analyses such claims by Mill, Boole, De Morgan, Russell, Poincaré, and others. It also examines early contrary statements by Hintikka, Mueller, Smith, and others. Special attention is given to the argumentations pro or con and especially to their logical, epistemic, and ontological presuppositions. What methodology is necessary or sufficient to show that a given logic is adequate or inadequate to serve as the underlying logi of a given science. (shrink)

This paper argues that Frege's notoriously long commitment to Kant's thesis that Euclidean geometry is synthetic _a priori_ is best explained by realizing that Frege uses ‘intuition’ in two senses. Frege sometimes adopts the usage presented in Hermann Helmholtz's sign theory of perception. However, when using ‘intuition’ to denote the source of geometric knowledge, he is appealing to Hermann Cohen's use of Kantian terminology. We will see that Cohen reinterpreted Kantian notions, stripping them of any psychological connotation. Cohen's defense (...) of his modified Kantian thesis on the unique status of the Euclidean axioms presents Frege's own views in a much more favorable light. (shrink)

A book review of Valtteri Viljanen's "Spinoza’s Geometry of Power".

Using as a starting point recent and apparently incompatible conclusions by Saunders and Knox, I revisit the question of the correct spacetime setting for Newtonian physics. I argue that understood correctly, these two versions of Newtonian physics make the same claims both about the background geometry required to define the theory, and about the inertial structure of the theory. In doing so I illustrate and explore in detail the view—espoused by Knox, and also by Brown —that inertial structure is (...) defined by the dynamics governing subsystems of a larger system. This clarifies some interesting features of Newtonian physics, notably the distinction between using the theory to model subsystems of a larger whole and using it to model complete universes, and the scale-relativity of spacetime structure. _1_ Introduction _2_ Newtonian Mechanics and Galilean Spacetime _3_ Vector Relationism and Maxwellian Spacetime _4_ Recovering the Galilei Group: Dynamics of Subsystems _5_ Knox on Inertial Structure _6_ Connections on Maxwellian Spacetime _7_ Knox on Newtonian Gravity _8_ Vector Relationism and Newton–Cartan Theory _9_ Inertial Structure in Newton–Cartan Gravity _10_ Reconciling Knox and Saunders _11_ Conclusions. (shrink)

This paper aims to deal with the disputes on transferring demonstration between the various sciences in the context of the medicine-geometry relationship. According to Aristotle’s metabasis-prohibition, these two sciences should be located in separate compartments due to the characteristics of their subject-matter. However, a thorough analysis of the critical passage in Aristotle’s Posterior Analytics on circular wounds forces a revision of the boundaries of the interactions between sciences, since subsequently Avicenna, on the grounds of this passage, would widen the (...) area of the transference of demonstration. Furthermore, the fact that Avicenna and Ibn al-Nafīs continued to use geometrical demonstrations in their anatomical investigations shows the need to understand kind-crossing prohibition as a reminder to take into account the present scientific infrastructure and logical rules before proceeding onto a scientific investigation instead of accepting it as a mere nominal doctrine. Therefore, whether kind-crossing was possible or not depended on the extent to which the conclusion derived at the end of the scientific investigation, using a different method after taking into account all these reminders, had contributed to the solution of a particular proposition or the achievement of an approximate truth. (shrink)

“Nature” is one of the most challenging concepts in philosophy, and notoriously difficult to define. In ancient Greece, two strategies for coming to terms with nature were developed. On the one hand, nature was seen as a perfect geometrical order, analysable with the help of geometry and deductive reasoning. On the other hand, a more Dionysian view emerged, stressing nature’s unpredictability, capriciousness and fluidity. This view was exemplified by De Rerum Natura, a philosophical masterpiece in verse. In a (...) philosophy course for science students, participants use both approaches. They are asked to give a definition of nature, and subsequently to capture nature in a poem. Quite consistently, their poetry proves more convincing than their definitions. In this paper, an anthology of student poetry is presented and analysed. To what extent may verse-writing as a philosophical assignment enable science students to come to terms with their understanding of nature? (shrink)

The aim of this article is to shed light on an understudied aspect of Giordano Bruno's intellectual biography, namely, his career as a mathematical practitioner. Early interpreters, especially, have criticized Bruno's mathematics for being “outdated” or too “concrete”. However, thanks to developments in the study of early modern mathematics and the rediscovery of Bruno's first mathematical writings (four dialogues on Fabrizio's Mordente proportional compass), we are in a position to better understand Bruno's mathematics. In particular, this article aims to reopen (...) the question of whether Bruno anticipated the concept of infinitesimal quantity. It does so by providing an analysis of the dialogues on Mordente's compass and of the historical circumstances under which those dialogues were written. (shrink)

How is knowledge of geometry developed and acquired? This central question in the philosophy of mathematics has received very different answers. Spelke and colleagues argue for a “core cognitivist”, nativist, view according to which geometric cognition is in an important way shaped by genetically determined abilities for shape recognition and orientation. Against the nativist position, Ferreirós and García-Pérez have argued for a “culturalist” account that takes geometric cognition to be fundamentally a culturally developed phenomenon. In this paper, I (...) argue that when understood as moderate versions supported by the state-of-the-art research, the nativist and culturalist views are in fact possible to reconcile. While Ferreirós and García-Pérez present the work of Spelke and colleagues as implying that geometric cognition is genetically determined, I argue that they fail to appreciate the role that Spelke and colleagues see for cultural factors. On this basis, I provide theoretical and terminological clarifications and show that moderate versions of the nativist and culturalist view are in fact consistent with each other. I then propose a unifying theoretical framework for future study that can integrate the two accounts in ontogeny by moving beyond the crude nature (nativism) vs. nurture (culturalism) dichotomy. (shrink)

This paper examines Helmholtz's attempt to use empirical psychology to refute certain of Kant's epistemological positions. Particularly, Helmholtz believed that his work in the psychology of visual perception showed Kant's doctrine of the a priori character of spatial intuition to be in error. Some of Helmholtz's arguments are effective, but this effectiveness derives from his arguments to show the possibility of obtaining evidence that the structure of physical space is non-Euclidean, and these arguments do not depend on his theory of (...) vision. Helmholtz's general attempt to provide an empirical account of the "inferences" of perception is regarded as a failure. (shrink)

After a brief review of the golden ratio in history and our previous exposition of the fine-structure constant and equations with the exponential function, the fine-structure constant is studied in the context of other research calculating the fine-structure constant from the golden ratio geometry of the hydrogen atom. This research is extended and the fine-structure constant is then calculated in powers of the golden ratio to an accuracy consistent with the most recent publications. The mathematical constants associated with the (...) golden ratio are also involved in both the calculation of the fine-structure constant and the proton-electron mass ratio. These constants are included in symbolic geometry of historical relevance in the science of the ancients. (shrink)

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least (...) according to some speculative research programs. (shrink)

Kant's arguments for the synthetic a priori status of geometry are generally taken to have been refuted by the development of non-Euclidean geometries. Recently, however, some philosophers have argued that, on the contrary, the development of non-Euclidean geometry has confirmed Kant's views, for since a demonstration of the consistency of non-Euclidean geometry depends on a demonstration of its equi-consistency with Euclidean geometry, one need only show that the axioms of Euclidean geometry have 'intuitive content' in (...) order to show that both Euclidean and non-Euclidean geometry are bodies of synthetic a priori truths. Michael Friedman has argued that this defence presumes a polyadic conception of logic that was foreign to Kant. According to Friedman, Kant held that geometrical reasoning itself relies essentially on intuition, and that this precludes the very possibility of non-Euclidean geometry. While Friedman's characterization of Kant's views on geometrical reasoning is correct, I argue that Friedman's conclusion that non-Euclidean geometries are logically impossible for Kant is not. I argue that Kant is best understood as a proto-constructivist and that modern constructive axiomatizations (unlike Hilbert-style axiomatizations) of both Euclidean and non-Euclidean geometry capture Kant's views on the essentially constructive nature of geometrical reasoning well. (shrink)

The following text is divided in four parts. The first presents the inner relation between the phenomenological concept of intentionality and space in a general mathematical sense. Following this train of though the second part brie_ly characterizes the use of the geometrical concept of manifold (Mannigfaltigkeit) in Husserl’s work. In the third part we present some examples of the use of the concept in Husserl’s analyses of space, time and intersubjectivity, pointing out some dif_iculties in his endeavor. In the fourth (...) and _inal part we offer some points of coincidence between phenomenology and category theory suggesting that the latter can work as a formal frame for ontology in the former. Our thesis is that intentionality operates in different levels as a morphism, functor and natural transformation. (shrink)

The following text is divided in four parts. The first presents the inner relation between the phenomenological concept of intentionality and space in a general mathematical sense. Following this train of though the second part brie_ly characterizes the use of the geometrical concept of manifold (Mannigfaltigkeit) in Husserl’s work. In the third part we present some examples of the use of the concept in Husserl’s analyses of space, time and intersubjectivity, pointing out some dif_iculties in his endeavor. In the fourth (...) and _inal part we offer some points of coincidence between phenomenology and category theory suggesting that the latter can work as a formal frame for ontology in the former. Our thesis is that intentionality operates in different levels as a morphism, functor and natural transformation. (shrink)

Art plays a significant role in Iris Murdoch’s moral philosophy, a major part of which may be interpreted as a proposal for the revision of religious belief. In this paper, I identify within Murdoch’s philosophical writings five distinct but related ways in which great art can assist moral/religious belief and practice: art can reveal to us “the world as we were never able so clearly to see it before”; this revelatory capacity provides us with evidence for the existence of (...) the Good, a metaphor for a transcendent reality of which God was also a symbol; art is a “hall of reflection” in which “everything under the sun can be examined and considered”; art provides us with an analogue for the way in which we should try to perceive our world; and art enables us to transcend our selfish concerns. I consider three possible objections: that Murdoch’s theory is not applicable to all forms of art; that the meaning of works of art is often ambiguous; and that there is disagreement about what constitutes a great work of art. I argue that none of these objections are decisive, and that all forms of art have at least the potential to furnish us with important tools for developing the insight required to live a moral/religious life. (shrink)

When beliefs are quantified as credences, they are related to each other in terms of closeness and accuracy. The “accuracy first” approach in formal epistemology wants to establish a normative account for credences based entirely on the alethic properties of the credence: how close it is to the truth. To pull off this project, there is a need for a scoring rule. There is widespread agreement about some constraints on this scoring rule, but not whether a unique scoring rule stands (...) above the rest. The Brier score equips credences with a structure similar to metric space and induces a “geometry of reason.” It enjoys great popularity in the current debate. I point out many of its weaknesses in this article. An alternative approach is to reject the geometry of reason and accept information theory in its stead. Information theory comes fully equipped with an axiomatic approach which covers probabilism, standard conditioning, and Jeffrey conditioning. It is not based on an underlying topology of a metric space, but uses a non-commutative divergence instead of a symmetric distance measure. I show that information theory, despite initial promise, also fails to accommodate basic epistemic intuitions; and speculate on its remediation. (shrink)

Accounts of Hobbes’s ‘system’ of sciences oscillate between two extremes. On one extreme, the system is portrayed as wholly axiomtic-deductive, with statecraft being deduced in an unbroken chain from the principles of logic and first philosophy. On the other, it is portrayed as rife with conceptual cracks and fissures, with Hobbes’s statements about its deductive structure amounting to mere window-dressing. This paper argues that a middle way is found by conceiving of Hobbes’s _Elements of Philosophy_ on the model of (...) a mixed-mathematical science, not the model provided by Euclid’s _Elements of Geometry_. I suggest that Hobbes is a test case for understanding early-modern system-construction more generally, as inspired by the structure of the applied mathematical sciences. This approach has the additional virtue of bolstering, in a novel way, the thesis that the transformation of philosophy in the long seventeenth century was heavily indebted to mathematics, a thesis that has increasingly come under attack in recent years. (shrink)

The article deals with the problem of the divine light in the mystical works of St Symeon the New Theologian in the context of the Eastern Christian ascetical tradition. The author focuses on the passages referring to the divine light in the works of Evagrios Pontikos, St Isaac the Syrian, St Maximus the Confessor, and in the Makarian corpus. As is shown in the present contribution, none of these authors created a fully-developed theory of the vision of the divine light. (...) Being close to these writers in many ideas, St Symeon was generally independent of any of them in his treatment of the theme of vision of light, always basing himself primarily upon his own experience. (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)

This article accomplishes two goals. First, the paper clarifies Edmund Husserl’s investigation of the historical inception of the number system from his early works, Philosophy of Arithmetic and, “On the Logic of Signs (Semiotic)”. The article explores Husserl’s analysis of five historical developmental stages, which culminated in our ancestor’s ability to employ and enumerate with number signs. Second, the article reveals how Husserl’s conclusions about the history of the number system from his early works opens up a fusion point (...) with his investigations from his mature texts, The Crisis of the European Sciences and “The Origin of Geometry”. On the one hand, the essay shows that Husserl’s methodology was similar, as he sought in both his early and late writings to uncover the essence of the history of the formal sciences and was not executing mere intellectual history. On the other hand, the article discloses that Husserl’s insights from both time periods are strikingly analogous. Already in his early texts, Husserl saw that the sciences emerged from pre-theoretical experiences of the world and that the sciences are the result of a historical process, which involves the psychic activities of past individuals and the maintaining of discoveries over time by intersubjective communities. I conclude by showing how, in light of the analysis of this paper, we can rethink the evolution of Husserl’s philosophy. (shrink)