We argue against the claim that the employment of diagrams in Euclidean geometry gives rise to gaps in the proofs. First, we argue that it is a mistake to evaluate its merits through the lenses of Hilbert’s formal reconstruction. Second, we elucidate the abilities employed in diagram-based inferences in the Elements and show that diagrams are mathematically reputable tools. Finally, we complement our analysis with a review of recent experimental results purporting to show that, not only is the Euclidean diagram-based (...) practice strictly regimented, it is rooted in cognitive abilities that are universally shared. (shrink)

Why is a red face not really red? How do we decide that this book is a textbook or not? Conceptual spaces provide the medium on which these computations are performed, but an additional operation is needed: Contrast. By contrasting a reddish face with a prototypical face, one gets a prototypical ‘red’. By contrasting this book with a prototypical textbook, the lack of exercises may pop out. Dynamic contrasting is an essential operation for converting perceptions into predicates. The existence of (...) dynamic contrasting may contribute to explaining why lexical meanings correspond to convex regions of conceptual spaces. But it also explains why predication is most of the time opportunistic, depending on context. While off-line conceptual similarity is a holistic operation, the contrast operation provides a context-dependent distance that creates ephemeral predicative judgments that are essential for interfacing conceptual spaces with natural language and with reasoning. (shrink)

Frege writes in Numbers and Arithmetic about kindergarten-numbers and “an a priori mode of cognition” that they may have “a geometrical source.” This resembles recent findings on arithmetical cognition. In my paper, I explore this resemblance between Gottlob Frege’s later position concerning the geometrical source of arithmetical knowledge, and some current positions in the literature dedicated to arithmetical cognition, especially that of Stanislas Dehaene. In my analysis, I shall try to mainly see to what extent logicism is (...) compatible with intuitionism. (shrink)

Frege writes in Numbers and Arithmetic about kindergarten-numbers and “an a priori mode of cognition” that they may have “a geometrical source.” This resembles recent findings on arithmetical cognition. In my paper, I explore this resemblance between Gottlob Frege’s later position concerning the geometrical source of arithmetical knowledge, and some current positions in the literature dedicated to arithmetical cognition, especially that of Stanislas Dehaene. In my analysis, I shall try to mainly see to what extent (Frege’s) logicism (...) is compatible with (Dehaene’s) intuitionism. (shrink)

© 2017 by the authors. Aristotelian diagrams visualize the logical relations among a finite set of objects. These diagrams originated in philosophy, but recently, they have also been used extensively in artificial intelligence, in order to study various knowledge representation formalisms. In this paper, we develop the idea that Aristotelian diagrams can be fruitfully studied as geometrical entities. In particular, we focus on four polyhedral Aristotelian diagrams for the Boolean algebra B4, viz. the rhombic dodecahedron, the tetrakis hexahedron, (...) the tetraicosahedron and the nested tetrahedron. After an in-depth investigation of the geometrical properties and interrelationships of these polyhedral diagrams, we analyze the correlation between logical and geometrical distance in each of these diagrams. The outcome of this analysis is that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit the strongest degree of correlation between logical and geometrical distance; the tetraicosahedron performs worse; and the nested tetrahedron has the lowest degree of correlation. Finally, these results are used to shed new light on the relative strengths and weaknesses of these polyhedral Aristotelian diagrams, by appealing to the congruence principle from cognitive research on diagram design. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 63.4 (2002) 599-617 [Access article in PDF] Thinking Geometrically in Pierre-Daniel Huet's Demonstratio evangelica (1679) April G. Shelford Sometime after 1679, Pierre-Daniel Huet (1630-1721) indulged an author's vanity by comparing his Demonstratio evangelica with works whose authors are far better known today. He recorded his judgments on a scrap of paper. 1First, he contrasted the Demonstratio to Antoine Arnauld's Les nouveaux élémens de (...) géometrie (1664). While Arnauld had assumed that all the principles of geometry were true and all geometrical demonstrations certain, and had considered religious questions not amenable to methodical demonstration, Huet noted, "I made clear the contrary by proving religion through a methodical sequence of propositions similar to those one finds in geometry." For Arnauld religious belief fell into the class of things better proven through a series of mutually reinforcing propositions than through proofs modeled on geometry; Huet had actually accomplished the latter. Arnauld did not require geometric proofs in religious works; Huet wrote the Demonstratio "to satisfy those who do." Indeed, Arnauld forbade such proofs, "but I permitted them... and I proposed something to satisfy those who do [demand them]."Huet then considered Blaise Pascal's Pensées (1670). Pascal had also rejected demonstration in proving religious truths, Huet noted, opting instead for "moral proofs, which are directed more to the heart than to the mind. In other words he desired to work harder at moving and disposing the heart than at convincing and persuading the intellect." Apparently, "M. Pascal did not believe that one could prove the truth of the Christian religion by geometric proofs; much less that he had any hope of doing so." [End Page 599]Huet's jottings precisely situate the Demonstratio evangelica (1679)between the claims of Cartesian rationalism and Pascal's fideism and religion of the heart, inviting us to take another look at an imperfectly understood work. Scholarly treatment of the Demonstratio has generally been cursory or partial because it has been considered less for its own merits than for how it fits into larger research agendas. 2 Some scholars have offered important insights but either do not develop them or do not consider the Demonstratio in a sufficiently broad context. 3 A consensus on the Demonstratio's general thrust thus masks puzzling questions. Everyone acknowledges, for example, that Huet's apologetic responded to Baruch Spinoza's Tractatus theologico-politicus (1670). Like other Christian apologists, he had to maintain "the Mosaic authorship" that was "the supposed guarantee of the truth of the text," 4 devoting "long chapters to the authenticity of the books of the Old Testament." 5 Yet Huet began the Demonstratio with an assault on geometrical knowledge and an epistemological discussion—an odd place to begin refuting the Tractatus. 6 Why? The question deserves a good answer, as Huet was a scholar of international reputation, and his contemporaries judged the Demonstratio an important work. 7More important for us, the evolution of the Demonstratio superbly reflects a vanished intellectual world and scholarly culture and in fact records change [End Page 600] in that world over four decades. Its genesis was the rash promise of a young man inspired by the possibilities of geometry and aspiring to recognition in the Republic of Letters, but before fulfilling that promise Huet developed a lively interest in such diverse fields as biblical criticism and natural philosophy, and he would apply the insights gained there in his apologetic. 8 A central question confronting Huet and his contemporaries was how to achieve certainty in matters religious, historical, and scientific. 9 The Demonstratio reveals Huet's conclusion that certainty in all three was profoundly interrelated. Finally, he composed the Demonstratio during the 1670s, having grasped with much alarm the consequences of the spread of Cartesianism. Thus, the Demonstratio is an intellectual biography not just of the author but of an era.Of course, Huet had to contend with Spinoza's challenge to Mosaic authorship, which fatally undermined the historical certainty of the Bible, but he also wrote... (shrink)

Aristotle’s way of conceiving the relationship between mathematics and other branches of scientific knowledge is completely different from the way a contemporary scientist conceives it. This is one of the causes of the fact that we look at the mathematical passage we find in Aristotle’s works with the wrong expectation. We expect to find more or less stringent proofs, while for the most part Aristotle employs mere analogies. Indeed, this is the primary function of mathematics when employed in a (...) philosophical context: not a demonstrative tool, but a purely analogical model. In the case of the geometrical examples discussed in this paper, the diagrams are not conceived as part of a formalized proof, but as a work in progress. Aristotle is not interested in the final diagram but in the construction viewed in its process of development; namely in the figure a geometer draws, and gradually modifies, when he tries to solve a problem. The way in which the geometer makes use of the elements of his diagram, and the relation between these elements and his inner state of knowledge is the real feature which interests Aristotle. His goal is to use analogy in order to give the reader an idea of the states of mind involved in a more general process of knowing. (shrink)

This paper has two specific goals. The first is to demonstrate that the theorem in MetaphysicsΘ 9, 1051a24-27 is not equiva-lent to Euclid’s Proposition 32 of book I (which contradicts some Aristotelian commentators, such as W. D. Ross, J. L. Heiberg, and T. L. Heith). Agreeing with Henry Mendell’s analysis, I ar-gue that the two theorems are not equivalent, but I offer different reasons for such divergence: I propose a pedagogical-philosoph-ical reason for the Aristotelian theorem being shorter than the Euclidean (...) one (and the previous Aristotelian versions). Aristotle wants to emphasize the deductive procedure as a satisfactory method to discover scientific knowledge. The second objective, opposing some consensus about geometrical deductions/theo-rems in Aristotle, is to briefly propose that the theorem, exactly as we found it in Metaphysics and without any emendation to the text (therefore opposing Henry Mendell’s suggested amend-ments), allows the ancient philosopher to demonstrate that universal mathematical knowledge is in potence in geometrical figures. This tentatively proves that Aristotle emphasizes that geometrical deduction is sufficient to actualize mathematical knowledge. (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)

Since Tversky argued that similarity judgments violate the three metric axioms, asymmetrical similarity judgments have been particularly challenging for standard, geometric models of similarity, such as multidimensional scaling. According to Tversky, asymmetrical similarity judgments are driven by differences in salience or extent of knowledge. However, the notion of salience has been difficult to operationalize, especially for perceptual stimuli for which there are no apparent differences in extent of knowledge. To investigate similarity judgments between perceptual stimuli, across three experiments, (...) we collected data where individuals would rate the similarity of a pair of temporally separated color patches. We identified several violations of symmetry in the empirical results, which the conventional multidimensional scaling model cannot readily capture. Pothos et al. proposed a quantum geometric model of similarity to account for Tversky's findings. In the present work, we extended this model to a more general framework that can be fit to similarity judgments. We fitted several variants of quantum and multidimensional scaling models to the behavioral data and concluded in favor of the quantum approach. Without further modifications of the model, the best-fit quantum model additionally predicted violations of the triangle inequality that we observed in the same data. Overall, by offering a different form of geometric representation, the quantum geometric framework of similarity provides a viable alternative to multidimensional scaling for modeling similarity judgments, while still allowing a convenient, spatial illustration of similarity. (shrink)

It is argued that geometrical intuition, as conceived in Kant, is still crucial to the epistemological foundations of mathematics. For this purpose, I have chosen to target one of the most sympathetic interpreters of Kant's philosophy of mathematics – Michael Friedman – because he has formulated the possible historical limitations of Kant's views most sharply. I claim that there are important insights in Kant's theory that have survived the developments of modern mathematics, and thus, that they are not so (...) intrinsically bound up with the logic and mathematics of Kant's time as Friedman will have it. These insights include the idea that mathematical knowledge relies on the manipulation of objects given in quasi-perceptual intuition, as Charles Parsons has argued, and that pure intuition is a source of knowledge of space itself that cannot be replaced by mere propositional knowledge. In particular, it is pointed out that it is the isomorphism between Kantian intuition and a spatial manifold that underlies both the epistemic intimacy of the most fundamental type of geometrical intuition as well as that of perceptual acquaintance. (shrink)

The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.

The discussion of mathematical knowledge and its relation to the construction of an appropriate diagram in Aristotle's Metaphysics Θ 9. 1051 a21—33 is an important, if compressed, account of Aristotle's most mature thoughts on mathematical knowledge. The discussion of what sort of previous knowledge one must have for understanding a theorem recalls the discussion at An. Post. A 1. 71 a 17–21, where the epistemological point is similar and the examples the same. The first example, that the (...) interior angles of a triangle equal two right angles, appears no less than thirty times in the corpus . The example of the angle inscribed in a semicircle being a right angle also occurs at An. Post. B 11. 94a 27–34, but in a very different context from its two companions. Illustrations of both theorems provided clear stock examples for Aristotle. (shrink)

It is frequently claimed that the human mind is organized in a modular fashion, a hypothesis linked historically, though not inevitably, to the claim that many aspects of the human mind are innately specified. A specific instance of this line of thought is the proposal of an innately specified geometric module for human reorientation. From a massive modularity position, the reorientation module would be one of a large number that organized the mind. From the core knowledge position, the reorientation (...) module is one of five innate and encapsulated modules that can later be supplemented by use of human language. In this paper, we marshall five lines of evidence that cast doubt on the geometric module hypothesis, unfolded in a series of reasons: (1) Language does not play a necessary role in the integration of feature and geometric cues, although it can be helpful. (2) A model of reorientation requires flexibility to explain variable phenomena. (3) Experience matters over short and long periods. (4) Features are used for true reorientation. (5) The nature of geometric information is not as yet clearly specified. In the final section, we review recent theoretical approaches to the known reorientation phenomena. (shrink)

This paper posits Peirce’s logical category of abduction as a necessary component in the learning process. Because of the cardinality of categories, Thirdness always contains in itself the Firstness of abduction. In psychological terms, abduction can be interpreted as intuition or insight. The paper suggests that abduction can be modeled as a vector on a complex plane. Such geometrical interpretation of the triadic sign helps to clarify the paradox of new knowledge that haunted us since Plato first articulated (...) it in his Meno dialogue. Some implications for the actual teaching practice are addressed, and the value of practical knowledge affirmed. (shrink)

Context: Technology has not only changed the way we teach mathematical concepts but also the nature of knowledge, and thus what is possible to learn. While geometric transformations are recognized to be foundational to the formation of students’ geometric conceptions, little research has focused on how these notions can be introduced in elementary schooling. Problem: This project addressed the need for development of students’ reasoning about and with geometric transformations in elementary school. We investigated the nature of students’ understandings (...) of translations, rotations, scaling, and stretching in two dimensions in the context of use of the software application Graphs ’n Glyphs. More specifically, we explored the question “What is the nature of elementary students’ reasoning of geometric transformations when instruction exploits the technological tool Graphs ’n Glyphs?” Method: Using a design research perspective, we present the conduct of a teaching experiment with one pair of fourth-graders, studying translation and rotation. The project investigated how and to what extent activity using Graphs ’n Glyphs can elicit students’ reasoning about geometric transformations, and explored the constraints and affordances of Graphs ’n Glyphs for thinking-in-change about geometric transformations. Results: The students proved adept using the software with carefully designed tasks to explore the behavior of two-dimensional shapes, distinguish among transformations, and develop predictions. In relation to varied conditions of transformations, they formed generalizations about the way a shape behaves, including the use of referent points in predicting outcomes of translations, and recognizing the role of the center of rotation. Implications: The generalizations that the students developed are foundational for developing an understanding of the properties of transformations in the later years of schooling. Dynamic technological approaches to geometry may prove as valuable to elementary students’ understanding of geometry as it is for older students. Constructivist content: This study contributes to ongoing constructivism/constructionism conversations by concentrating on the transformation of ideas when engaging learners in activity through particular contexts and tools. Key Words: Geometry, transformations, constructionist technologies. (shrink)

When non-Euclidean geometry was developed in the nineteenth century, both scientists and philosophers addressed the question as to whether the Kantian theory of space ought to be refurbished or even rejected. The possibility of considering a variety of hypotheses regarding physical space appeared to contradict Kant’s supposition of Euclid’s geometry as a priori knowledge and suggested the view that the geometry of space is a matter for empirical investigation. In this article, I discuss two different attempts to defend the (...) Kantian theory of space against geometrical empiricism. Both Cohen and Riehl defended the apriority of geometrical axioms. Cohen pointed out that knowledge necessarily requires both sensibility and understanding. Therefore, he maintained that space can be proved to be a condition of experience without admitting that some statements about it are intrinsically necessary. By contrast, Riehl emphasized universality and necessity as intrinsic properties of a priori knowledge. He was one of the first philosophers to discuss the space problem in detail and to distinguish between a priori properties of space and empirical properties. Nevertheless, I suggest that Cohen developed a more plausible view because he was not committed to any spatial structure independently of empirical science. (shrink)

Edmund Husserl has argued that we can intuit essences and, moreover, that it is possible to formulate a method for intuiting essences. Husserl calls this method ‘ideation’. In this paper I bring a fresh perspective to bear on these claims by illustrating them in connection with some examples from modern pure geometry. I follow Husserl in describing geometric essences as invariants through different types of free variations and I then link this to the mapping out of geometric invariants in modern (...) mathematics. This view leads naturally to different types of spatial ontologies and it can be used to shed light on Husserl's general claim that there are different ontologies in the eidetic sciences that can be systematically related to one another. The paper is rounded out with a consideration of the role of ideation in the origins of modern geometry, and with a brief discussion of the use of ideation outside of pure geometry. What would be the study that would draw the soul away from the world of becoming to the world of being?… Geometry and arithmetic would be among the studies we are seeking … a philosopher must learn them because he must arise out of the region of generation and lay hold on essence or he can never become a true reckoner … they facilitate the conversion of the soul itself from the world of generation to essence and truth… they are knowledge of that which always is and not of something which at some time comes into being and passes away. (shrink)

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

In ?155 of his New Theory of Vision Berkeley explains that a hypothetical ?unbodied spirit? ?cannot comprehend the manner wherein geometers describe a right line or circle?.1The reason for this, Berkeley continues, is that ?the rule and compass with their use being things of which it is impossible he should have any notion.? This reference to geometrical tools has led virtually all commentators to conclude that at least one reason why the unbodied spirit cannot have knowledge of plane (...) geometry is because it cannot manipulate a ruler or a compass. In this article I will show that such an interpretation is flawed. I will instead argue that Berkeley's understanding of Euclidian geometry was based on Isaac Barrow's account of the foundations of geometry. On this view geometrical objects are conceived in terms of the idealized motion that generates the objects of geometry. Consequently, that what the unbodied spirit cannot do in this context is to form an idea of motion rather than being unable to handle geometrical tools. 1All references to Berkeley are from, A. A. Luce and T. E. Jessop (eds.), The Works of George Berkeley, Bishop of Cloyne (London: Thomas Nelson and Sons, Ltd., 1948) The following abbreviations are used: An Essay Towards A New Theory of Vision, section x = New Theory x; Philosophical Commentaries, entry x = Commentaries x; Part I of A Treatise concerning the Principles of Knowledge, section x = Principles x. All other references to Berkeley's works are of the form The Works of George Berkeley, Bishop of Cloyne, volume x, page y = Works, x, y. (shrink)

WHAT IS PERHAPS the most influential of all theories of knowledge maintains that the content of knowledge and the known are identical in nature. The view goes back to Pythagoras, with his doctrine that reality is at once constituted of and known by means of geometrized numbers. Over the course of time, these numbers have been replaced by other equally startling candidates. The Platonic Idea, the Leibnizian Simple, the Berkeleyan Notion, the Humean Impression, the Peircean Third, and the (...) Whiteheadean Feeling differ from one another in origin, range and career, but like Pythagoras' numbers, all function both as the content of knowledge and the content of the known. Let us term this the "Pure Identity Theory." A "Partial Identity Theory" can be obtained by supposing that the content of knowledge is a fainter version of the content known, or that the content known offers a blurred version of the content of knowledge. In the end, I think, we must hold an Identity Theory. But we must first radically alter the way it has been traditionally viewed and used. (shrink)

This article offers an analysis of the cognitive role of diagrammatic movements in the theater. Based on the recognition of a theatrical work’s inherent ability to provide new insights concerning reality, the article concentrates on the way by which actors’ movements on stage create spatial diagrams that can provide new insights into the spectators’ world. The suggested model of theater’s epistemology results from a combination of Charles S. Peirce’s doctrine of diagrammatic reasoning and David Lewis’s theoretical account of the truth (...) value of counterfactual conditionals. I argue that in several theatrical works – in particular those whose central image is dominated by movements – the relation of what Lewis names “comparative overall similarity” between the fictional and the actual world is based on diagrammatic homology. The cognitive process involved in deciphering them is, hence, based on diagrammatic reasoning. The main emphasis of the analysis is on the previously unnoticed but important cognitive role of observation in the theater: the idea that observation takes an active role in the reasoning process that enables the spectators to form new knowledge about their actual world. Samuel Beckett’s plays Quad and Come and Go serve here as case studies. (shrink)

Does geometry constitues a core set of intuitions present in all humans, regarless of their language or schooling ? We used two non verbal tests to probe the conceptual primitives of geometry in the Munduruku, an isolated Amazonian indigene group. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

Geometrical intuitions spontaneously drive visuo-spatial reasoning in human adults, children and animals. Is their emergence intrinsically linked to visual experience, or does it reflect a core property of cognition shared across sensory modalities? To address this question, we tested the sensitivity of blind-from-birth adults to geometrical-invariants using a haptic deviant-figure detection task. Blind participants spontaneously used many geometric concepts such as parallelism, right angles and geometrical shapes to detect intruders in haptic displays, but experienced difficulties with symmetry (...) and complex spatial transformations. Across items, their performance was highly correlated with that of sighted adults performing the same task in touch (blindfolded) and in vision, as well as with the performances of uneducated preschoolers and Amazonian adults. Our results support the existence of an amodal core-system of geometry that arises independently of visual experience. However, performance at selecting geometric intruders was generally higher in the visual compared to the haptic modality, suggesting that sensory-specific spatial experience may play a role in refining the properties of this core-system of geometry. (shrink)

The most widely used attractive logical account of knowledge uses standard epistemic models, i.e., graphs whose edges are indistinguishability relations for agents. In this paper, we discuss more general topological models for a multi-agent epistemic language, whose main uses so far have been in reasoning about space. We show that this more geometrical perspective affords greater powers of distinction in the study of common knowledge, defining new collective agents, and merging information for groups of agents.

Several studies have emphasized the limits of invariance-based approaches such as Klein’s and Cassirer’s when it comes to account for the shift from the spacetimes of classical mechanics and of special relativity to those of general relativity. Not only is it much more complicated to find such invariants in the case of general relativity, but even if local invariants in Weyl’s fashion are admitted, Cassirer’s attempt at a further generalization of his approach to the spacetime structure of general relativity seems (...) to obscure the fact that the determination of this structure requires a completely different approach that is derived from Riemann rather than Klein. This paper aims to reconsider these issues by drawing attention to the combination of subtractive and additive strategies (more in line with Riemann’s) in Klein’s own considerations about the determination of the structure of spacetime from 1897 to 1910. I will point out that Cassirer relied on Klein’s argument in some central passages from Substance and Function (1910), and elaborated further on his combined approach in Einstein’s Theory of Relativity (1921), also taking into account the application of Riemannian geometry in general relativity. My suggestion is that an appreciation of Cassirer’s continuing commitment to a variety of geometrical traditions may shed light on his particular understanding of a priori elements of knowledge and avoid some of the classical objections to the idea of a relativization of the a priori. (shrink)

The text translated, “Das Wissen von fremden Ichen,” bears particular importance for the early phenomenological movement for two reasons. The first is Lipps’ refutation of the theory that knowledge of other selves arises by way of an inference from analogy. Lipps first developed his account of empathy to explain that we tend to succumb to geometric optical illusions because we project living activity into inanimate objects. In sum, Lipps’ groundbreaking article on The Knowledge of Other Egos deserves as (...) much interest from the philosophical community today as it had in the past. The figure of Lipps the philosopher needs to be taken into consideration anew among other much-celebrated proponents of the phenomenological movement. The supporters of that theory, however, demand that we ought to be able to pick out a justification for the existence of objective reality from the fact of our sensations by way of thought alone. (shrink)

ArgumentThis paper challenges the use of the notion of “culture” to describe a particular organization of mathematical knowledge, shared by a few mathematicians over a short period of time in the second half of the nineteenth century. This knowledge relates to “geometrical equations,” objects that proved crucial for the mechanisms of encounters between equation theory, substitution theory, and geometry at that time, although they were not well-defined mathematical objects. The description of the mathematical collective activities linked to (...) “geometrical equations,” and especially the technical aspects of these activities, is made on the basis of a sociological definition of “culture.” More precisely, after an examination of the social organization of the group of mathematicians, I argue that these activities form an intricate system of patterns, symbols, and values, for which I suggest a characterization as a “cultural system.”. (shrink)

This book concerns Spinoza's theory of knowledge and closely related issues: Spinoza's conceptions of geometrical figure or shape, number, and observational sci.

This book interrogates the ontology of mathematical entities in Spinoza as a basis for addressing a wide range of interpretive issues in Spinoza’s epistemology—from his antiskepticism and philosophy of science to the nature and scope of reason and intuitive knowledge and the intellectual love of God. Going against recent trends in Spinoza scholarship, and drawing on various sources, including Spinoza’s engagements with optical theory and physics, Matthew Homan argues for a realist interpretation of geometrical figures in Spinoza; illustrates (...) their role in a Spinozan hypothetico-deductive scientific method; and develops Spinoza’s mathematical examples to better illuminate the three kinds of knowledge. The result is a portrait of Spinoza’s epistemology as sanguine and distinctive yet at home in the new Cartesian and Galilean scientific-philosophical paradigm. (shrink)

Kant was an idealist. His idealism was in some ways, it is true, less extreme than that of Berkeley. He distinguished his own by calling it ‘transcendental’. It is less extreme than Berkeley's in two ways. First, Kant does not assert that everything which exists is essentially mental, as Berkeley does. Second, those things which he does hold to be essentially mental, he holds to be so in a weaker fashion. Nevertheless he was an idealist; he held that neither intuition (...) nor thought could concern any object that was not essentially related to our minds. Since intuition and thought together provide knowledge, and since we can have no knowledge except through them, it follows that every object of which we have knowledge is essentially within our minds. Moreover, there is at least one respect in which his idealism is more extreme than that of Berkeley. He held that the objects of intuition were essentially within our minds, whereas Berkeley held only that they were essentially within some mind. (shrink)

This introduction to physical science combines a rigorous discussion of scientific principles with sufficient historical background and philosophic interpretation to add a new dimension of interest to the accounts given in more conventional textbooks. It brings out the twofold character of physical science as an expanding body of verifiable knowledge and as an organized human activity whose goals and values are major factors in the revolutionary changes sweeping over the world today.Professor Kemble insists that to understand science one must (...) understand not only what the scientists have discovered, but how the discoveries were made, why the growth of scientific knowledge had to begin slowly, and what it has done to our habits of thought. He has written neither a history of science nor an introduction to the philosophy of science but an introduction to scientific concepts and principles that supplies as much of their historical and philosophical context as limits of space permit.The volume takes up in turn the story of the astronomy of ancient Greece, the Copernican revolution, the idea of the expanding sidereal universe, the rise of Newton's classical mechanics with its many astronomical applications, the concept of energy and its relation to heat, to steam engines, and to thermodynamics. The volume ends with an account of the successes and failures of classical kinetic-molecular theory of heat. (shrink)

Kant was an idealist. His idealism was in some ways, it is true, less extreme than that of Berkeley. He distinguished his own by calling it ‘transcendental’. It is less extreme than Berkeley's in two ways. First, Kant does not assert that everything which exists is essentially mental, as Berkeley does. Second, those things which he does hold to be essentially mental, he holds to be so in a weaker fashion. Nevertheless he was an idealist; he held that neither intuition (...) nor thought could concern any object that was not essentially related to our minds. Since intuition and thought together provide knowledge, and since we can have no knowledge except through them, it follows that every object of which we have knowledge is essentially within our minds. Moreover, there is at least one respect in which his idealism is more extreme than that of Berkeley. He held that the objects of intuition were essentially within our minds, whereas Berkeley held only that they were essentially within some mind. (shrink)

Graphic pattern (e.g. geometric design) and number-based code (e.g. digital sequencing) can store and transmit complex information more efficiently than referential modes of representation. The analysis of the two genres and their relation to one another has not advanced significantly beyond a general classification based on motion-centred geometries of symmetry. This article examines an intriguing example of patchwork coverlets from the maritime societies of Oceania, where information referencing a complex genealogical system is lodged in geometric designs. By drawing attention to (...) the interplay of graphic pattern and number-based code and its role in the knowledge economies of maritime societies, the article offers new insight into possible ways of designing a digital informational surface that captures the behaviour of an operational system, allowing both for differentiation and integration. (shrink)

My goal in this paper is to develop our understanding of the role the imagination plays in Kant’s Critical account of geometry, and I do so by attending to how the imagination factors into the method of reasoning Kant assigns the geometer in the First Critique. Such an approach is not unto itself novel. Recent commentators, such as Friedman (1992) and Young (1992), have taken a careful look at the constructions of the productive imagination in pure intuition and highlighted the (...) importance of the imagination’s activity for securing the universality of geometry knowledge. Specifically, as their respective examinations bring to light, it is only with due attention to the imagination that we can make sense of how a .. (shrink)

At the turn of the seventeenth century, Bruno and Cavalieri independently developed two theories, central to which was the concept of the geometrical indivisible. The introduction of indivisibles had significant implications for geometry – especially in the case of Cavalieri, for whom indivisibles provided a forerunner of the calculus. But how did this event occur? What can we learn from the fact that two theories of indivisibles arose at about the same time? These are the questions addressed in this (...) paper. Relying on the methodology of “historical epistemology”, this paper asserts that the similarities and differences between the theories of Bruno and Cavalieri can be explained in terms of “shared knowledge”. The paper shows that the idea – on which both Bruno and Cavalieri build – that geometrical objects are generated by motion was part of the mathematical culture of the time. Tracing this idea back to its Pythagorean origins thus sheds light on the relationship between motion and continuum in mathematics. (shrink)

This paper joins the conversation on the relationship between Spinoza and George Eliot. After critically examining Atkins’s claim that the novels of George Eliot, as exemplified by Adam Bede, are a presentation of Spinoza’s philosophy stripped of the geometrical method, the paper explores Eliot’s philosophical engagement with Spinoza’s views on sympathy and the imagination. Thus, Eliot is read as a philosopher engaging with the arguments of Spinoza, rather than as someone representing his views in novel form.

The agronomist who wants to study the nutrient and water uptake of roots needs a quantitative three-dimensional dynamic model of the structure of root systems.The model presented takes into account current knowledge about the morphogenesis of root systems. It describes the root system as a set of root axes, characterised by their orders. The morphogenetic properties of root axes differ according to their order. The axes of order 1 are directly inserted on the stem, the axes of order 2 (...) are inserted on axes of order 1, and so on. They tend to be more plagiotropec and to have less vascular bundles as the order increases. (shrink)

It is argued, first, that although Spinoza's early Treatise on the Emendation of the Intellect does show evidence of a foundationalist approach to the justification of knowledge, there are good reasons to think he came to find such an approach unsatisfactory; and second, that the Ethics notion of certainty as adequate knowledge of one's knowledge is a justificational concept which is holistic in that any instance of such certainty depends on knowledge of the entire basic metaphysical (...) system. Finally it is shown that Spinoza's having come to hold a nonlinear view regarding the justificational structure of knowledge explains why he chose to present his philosophy in the geometric form, and why he never succeeded in formulating a method of discovery. (shrink)

The modern reinterpretations of Vico are a good example of the rethinking by historians of one age of the rethinking by historians of previous ages of the original thought of a philosopher. The present volume stresses the unique unity of theory and practice in Vico's thought and dispels some unfounded criticisms, such as his alleged reliance on the geometric method, inconsistencies in his use of the terms "philosophy" and "philology," and the mechanical acceptance of the patterns of development of Greece (...) and Rome as the patterns for all nations. Above all, it is suggested that, although Providence was central to Vico's conception of historical development, his system does not collapse by modern rejection of this force. The principle of the continuity of human nature which lies at the core of Dilthey's insight into the special character of history was stated in its pure form by Vico a century and a half earlier. Vico's ideas were sharpened in his critiques of Grotius, Selden, Pufendorf, and Francis Bacon. While drawing some positive inspiration from each, he found in them the common deficiency of reading their own abstract principles into history. Descartes, however, emerges as Vico's bête noire because of his dismissal of history as incapable of producing certainty. Vico's biting polemic against Descartes pales the criticisms of Arnauld and Gassendi, and, at the same time, illustrates the sweep and power of his own historically-grounded concepts. The author gives major attention to Vico's notion of the limits of knowledge, to his notion of cause, to the historicity of human society and its laws and governments, to the vital role of language, to the distinction between imaginative universals and rational universals, and to the necessary union in his New Science of philosophy and philology. Manson is an unrestrained admirer of Vico and he makes no effort to dilute his enthusiasm by taking critical exception to any fundamental aspect of Vico's epistemology.--H. B. (shrink)

Dans sa violente charge contre la Révolution française, Edmund Burke avait élevé le débat politique à un niveau philosophique. Son argument le plus profond consistait à reprocher aux révolutionnaires de pécher par apriorisme, en cherchant à déduire, comme des géomètres, une nouvelle constitution à partir des principes abstraits énoncés dans la Déclaration des droits de l’homme. Reprise par les disciples allemands de Burke, cette critique de la méthode adoptée par la Constituante tirait des postulats empiristes des Lumières anglo-écossaises toutes les (...) implications conservatrices que recelaient déjà les textes politiques et historiques de Hume. Or c’est d’emblée sur ce terrain spéculatif que se joue, chez Fichte, la défense du droit de révolution. Reprochant à l’empirisme de réduire l’esprit au « mécanisme aveugle de l’association des idées », passivement déterminé par une extériorité en soi, Fichte entend au contraire fonder l’expérience elle-même sur l’activité d’un « Moi-en-soi », absolument indépendant. Conçu comme causalité libre, pouvant donc s’affranchir des « conjonctions coutumières » humiennes pour commencer une nouvelle série dans l’ordre des phénomènes, l’homme, pour Fichte, n’a pas seulement la « faculté de se perfectionner », mais la loi de sa liberté lui en fait même un devoir. Au regard de cet impératif catégorique de perfectionnement, il n’y a pas de Common Law qui tienne. Aussi Fichte ne recule-t-il pas devant la révolution, si les institutions du passé font obstacle à la destination de l’homme. (shrink)

Like most, if not all, of his contemporaries, Spinoza never developed a full-fledged philosophy of mathematics. Still, his numerous remarks about mathematics attest not only to his deep interest in the subject (a point which is also confirmed by the significant presence of mathematical books in his library), but also to his quite elaborate and perhaps unique understanding of the nature of mathematics. At the very center of his thought about mathematics stands a paradox (or, at least, an apparent paradox): (...) mathematics provides Spinoza with an epistemic model. Mathematical knowledge is certain (II/138/9 and II/138/9), clear (IV/261/8), and free from teleological thinking (II/79/33), but the objects of mathematical knowledge – i.e., mathematical entities – are nothing but “auxila imaginationis [aids of the imagination],” (IV/57/16 and II/83/15), entities that are not real and merely assist the imagination in carving the world in manner that is suitable to our limited and distortive cognitive capacities. (shrink)

The connection of Hermann Сohen’s “The Logic of Pure Knowledge” with the revolutionary transformations in physics and mathematics at the end of the 19th century is shown. Сohen criticised Kant’s answer to the question “How is mathematics possible”? If Kant refers to a priori forms of pure intuition, Сohen sees in it a restriction of freedom of mathematical thinking by limits of intuition. It has been shown that Cohen's position is in accordance with the main development of mathematics in (...) the last decades of the 19th century, in particular, with K. Weierstrass’ striving to get rid of geometrical or mechanical images and intuitions in the mathematical analysis. Cohen was also well informed about the latest ideas in physics of his time. They are also discussed in “The Logic of Pure Knowledge.” The revolutionary spirit of physics and mathematics was appealing to Cohen, and he felt a corresponding enthusiasm for it. The ongoing scientific revolution is consonant with Cohen’s assertion that the foundations of science are hypotheses. The purity of pure thinking does not guarantee the correctness of any of its constructions. Each step in the development of science requires a critique of existing notions. The development of knowledge from the naive to the critical one shows a movement from a picture of the world as a set of stable things to a picture of continuous movement and change, where change is more important than what changes. Cohen sees such a development in an evolution of the concept of substance in modern physics and he welcomes the replacement of material substance by energy, seeing this movement as a confirmation of critical idealism. Finally, it is discussed whether we can speak of the actuality of Cohen’s logic of pure knowledge. (shrink)

Basic processes of perception should be cognitively impenetrable so that they are not prey to momentary changes of belief. That said, how does low level vision interact with knowledge to allow recognition? Much more needs to be known about the products of low level vision than that they represent the geometric layout of the world.

The way in which geometrical knowledge has been obtained has always attracted the attention of philosophers. The fact that there is a science that concerns things outside our thinking and that proceeds inferentially appeared striking, and gave rise to specific theories of experience and space. Nonetheless, the geometrical method has not yet been sufficiently investigated. Philosophers who investigate the theory of knowledge discuss the question of whether geometry is an empirical science, but...

The way in which geometrical knowledge has been obtained has always attracted the attention of philosophers. The fact that there is a science that concerns things outside our thinking and that proceeds inferentially appeared striking, and gave rise to specific theories of experience and space. Nonetheless, the geometrical method has not yet been sufficiently investigated. Philosophers who investigate the theory of knowledge discuss the question of whether geometry is an empirical science, but.