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)

Evans’ 1968 ANALOGY system was the first computer model of analogy. This paper demonstrates that the structure mapping model of analogy, when combined with high‐level visual processing and qualitative representations, can solve the same kinds of geometric analogy problems as were solved by ANALOGY. Importantly, the bulk of the computations are not particular to the model of this task but are general purpose: We use our existing sketch understanding system, CogSketch, to compute visual structure that is used by our existing (...) analogical matcher, Structure Mapping Engine (SME). We show how SME can be used to facilitate high‐level visual matching, proposing a role for structural alignment in mental rotation. We show how second‐order analogies over differences computed via analogies between pictures provide a more elegant model of the geometric analogy task. We compare our model against human data on a set of problems, showing that the model aligns well with both the answers chosen by people and the reaction times required to choose the answers. (shrink)

Georges J. D. Moyal - Deux notes sur l' «imparfaite science» du geometre athee - Journal of the History of Philosophy 43:3 Journal of the History of Philosophy 43.3 277-300 Deux notes sur l'« imparfaite science » du géomètre athée Georges J. D. Moyal Deux questions. La Ve Méditation de Descartes vise à démontrer que l'existence d'un Dieu vérace est la condition nécessaire de toute science. En effet, Descartes y écrit ceci : « . . . je remarque que la (...) certitude de toutes les autres choses en dépend si absolument [sc. de l'existence de Dieu], que sans cette connaissance il est impossible de pouvoir jamais rien savoir parfaitement. » Ce qui le mène, bien entendu, d'abord à reconnaître que « . . . si j'ignorais qu'il y eût un Dieu [ . . . ] je n'aurais jamais une vraie et certaine science d'aucune chose que ce soit, mais seulement de vagues et inconstantes opinions » et par là même, à refuser ensuite cette parfaite science au géomètre athée : « Or, qu'un athée puisse connaître clairement que les trois angles d'un triangle sont égaux à deux droits, je ne le nie pas ; mais je maintiens seulement qu'il ne le connaît pas par une vraie et certaine science, parce que toute connaissance qui peut être rendue douteuse ne doit pas être appelée science . . . » Il faut reconnaître qu'il y a à nos yeux, aujourd'hui, quelque chose de surprenant à ce que le géomètre athée se voie ainsi refuser cette parfaite science.. (shrink)

The Geometrical Method The Geometrical Method is the style of proof that was used in Euclid’s proofs in geometry, and that was used in philosophy in Spinoza’s proofs in his Ethics. The term appeared first in 16th century Europe when mathematics was on an upswing due to the new science of mechanics. … Continue reading Geometrical Method →.

The cognitive foundations of geometry have puzzled academics for a long time, and even today are mostly unknown to many scholars, including mathematical cognition researchers. -/- Foundations of Geometric Cognition shows that basic geometric skills are deeply hardwired in the visuospatial cognitive capacities of our brains, namely spatial navigation and object recognition. These capacities, shared with non-human animals and appearing in early stages of the human ontogeny, cannot, however, fully explain a uniquely human form of geometric cognition. In the book, (...) Hohol argues that Euclidean geometry would not be possible without the human capacity to create and use abstract concepts, demonstrating how language and diagrams provide cognitive scaffolding for abstract geometric thinking, within a context of a Euclidean system of thought. -/- Taking an interdisciplinary approach and drawing on research from diverse fields including psychology, cognitive science, and mathematics, this book is a must-read for cognitive psychologists and cognitive scientists of mathematics, alongside anyone interested in mathematical education or the philosophical and historical aspects of geometry. (shrink)

_ Source: _Volume 29, Issue 1, pp 39 - 65 Since Euclid, optics has been considered a geometrical science, which Aristotle defines as a “mixed” mathematical science. Hobbes follows this tradition and clearly places optics among physical sciences. However, modern scholars point to a confusion between geometry and physics and do not seem to agree about the way Hobbes mixes both sciences. In this paper, I return to this alleged confusion and intend to emphasize the peculiarity of (...) Hobbes’s geometrical optics. This paper suggests that Hobbes’s conception of geometrical optics, as a mixed mathematical science, greatly differs from Descartes’s one, mainly because they do not share the same “mechanical conception of nature.” I will argue that Hobbes and Descartes also have in common the quest for a different kind of geometry for their optics, different from that of the Ancients. I will show that this departure is not recent since Hobbes’s approach is already evident in 1636, when he judges the demonstrations of his contemporary friends, Claude Mydorge and Walter Warner. Finally the paper broadly suggests what is noteworthy in Hobbes’s optics, that is, the importance of the idea of force in his mechanics, although he was not able to conceptualize it in other terms than “quickness.”. (shrink)

I draw together some recent literature on the debate between dynamical versus geometrical approaches to spacetime theories, in order to argue that there exist defensible versions of the geometr...

In this paper, based on video recordings of Orientation and Mobility (O&M) lessons for visually-disabled students, I will examine how occasioned maps (Psathas, 1979 ; Garfinkel, 2002 ), drawn in the student’s palm are interactionally traced, felt, and noticed in order to represent the shape of a crossing for all practical purposes. Touching will be examined from the perspective of the live production of "trails" on a specific region of the body, the palm of the hand. We will begin to (...) question how such hand-drawing map episodes occur during O&M courses, stressing how the coparticipants establish a participation framework that facilitates the making of the drawing, hand-map drawings are based on lines that are neither evanescent nor permanent. Their “persistence” is not an intrinsic feature but a systematic multimodal accomplishment. We will show how the drawn lines become depictions of the streets and contribute to producing two contrasting geometrical representations of the layout of a junction. (shrink)

This book gives an analysis of Hertz's posthumously published Principles of Mechanics in its philosophical, physical and mathematical context. In a period of heated debates about the true foundation of physical sciences, Hertz's book was conceived and highly regarded as an original and rigorous foundation for a mechanistic research program. Insisting that a law-like account of nature would require hypothetical unobservables, Hertz viewed physical theories as images of the world rather than the true design behind the phenomena. This paved (...) the way for the modern conception of a model. Rejecting the concept of force as a coherent basic notion of physics he built his mechanics on hidden masses and rigid connections, and formulated it as a new differential geometric language.Recently many philosophers have studied Hertz's image theory and historians of physics have discussed his forceless mechanics. The present book shows how these aspects, as well as the hitherto overlooked mathematical aspects, form an integrated whole which is closely connected to the mechanistic world view of the time and which is a natural continuation of Hertz's earlier research on electromagnetism. Therefore it is also a case study of the strong interactions between philosophy, physics and mathematics. Moreover, the book presents an analysis of the genesis of many of the central elements of Hertz's mechanics based on his manuscripts and drafts. Hertz's research program was cut short by the advent of relativity theory but its image theory influenced many philosophers as well as some physicists and mathematicians and its geometric form had a lasting influence on advanced expositions of mechanics. (shrink)

At the end of the last century Paul Tannery published an article on geometry in eleventh-century Europe, which he began with the following statement:“This is not a chapter in the history of science; it is a study of ignorance, in a period immediately before the introduction into the West of Arab mathematics.”.

This article analyzes Galileo's mathematization of motion, focusing in particular on his use of geometrical diagrams. It argues that Galileo regarded his diagrams of acceleration not just as a complement to his mathematical demonstrations, but as a powerful heuristic tool. Galileo probably abandoned the wrong assumption of the proportionality between the degree of velocity and the space traversed in accelerated motion when he realized that it was impossible, on the basis of that hypothesis, to build a diagram of the (...) law of fall. The article also shows how Galileo's discussion of the paradoxes of infinity in the First Day of the Two New Sciences is meant to provide a visual solution to problems linked to the theory of acceleration presented in Day Three of the work. Finally, it explores the reasons why Cavalieri and Gassendi, although endorsing Galileo's law of free fall, replaced Galileo's diagrams of acceleration with alternative ones. (shrink)

The geometrical algebra hypothesis was once the received interpretation of Greek mathematics. In recent decades, however, it has become anathema to many. I give a critical review of all arguments against it and offer a consistent rebuttal case against the modern consensus. Consequently, I find that the geometrical algebra interpretation should be reinstated as a viable historical hypothesis.

We analyze the geometric foundations of classical Yang-Mills theory by studying the relationships between internal relativity, locality, global/local invariance, and background independence. We argue that internal relativity and background independence are the two independent defining principles of Yang-Mills theory. We show that local gauge invariance -heuristically implemented by means of the gauge argument- is a direct consequence of internal relativity. Finally, we analyze the conceptual meaning of BRST symmetry in terms of the invariance of the gauge fixed theory under general (...) local gauge transformations. (shrink)

We present a gravitational quantum dynamics theory that combines quantum field theory for particle dynamics in space-time with classical Einstein’s general relativity in a non-Riemannian Finsler space. This approach is based on the geometrization of quantum mechanics proposed in Tavernelli and combines quantum and gravitational effects into a global curvature of the Finsler space induced by the quantum potential associated to the matter quantum fields. In order to make this theory compatible with general relativity, the quantum effects are described in (...) the framework of quantum field theory, where a covariant definition of ‘simultaneity’ for many-body systems is introduced through the definition of a suited foliation of space-time. As in Einstein’s gravitation theory, the particle dynamics is finally described by means of a geodesic equation in a curved space-time manifold. (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)

In this paper, we focus on the development of geometric cognition. We argue that to understand how geometric cognition has been constituted, one must appreciate not only individual cognitive factors, such as phylogenetically ancient and ontogenetically early core cognitive systems, but also the social history of the spread and use of cognitive artifacts. In particular, we show that the development of Greek mathematics, enshrined in Euclid’s Elements, was driven by the use of two tightly intertwined cognitive artifacts: the use of (...) lettered diagrams; and the creation of linguistic formulae. Together, these artifacts formed the professional language of geometry. In this respect, the case of Greek geometry clearly shows that explanations of geometric reasoning have to go beyond the confines of methodological individualism to account for how the distributed practice of artifact use has stabilized over time. This practice, as we suggest, has also contributed heavily to the understanding of what mathematical proof is; classically, it has been assumed that proofs are not merely deductively correct but also remain invariant over various individuals sharing the same cognitive practice. Cognitive artifacts in Greek geometry constrained the repertoire of admissible inferential operations, which made these proofs inter-subjectively testable and compelling. By focusing on the cognitive operations on artifacts, we also stress that mental mechanisms that contribute to these operations are still poorly understood, in contrast to those mechanisms which drive symbolic logical inference. (shrink)

Pessimistic meta-induction is a powerful argument against scientific realism, so one of the major roles for advocates of scientific realism will be trying their best to give a sustained response to this argument. On the other hand, it is also alleged that structural realism is the most plausible form of scientific realism; therefore, the plausibility of scientific realism is threatened unless one is given the explicit form of a structural continuity and minimal structural preservation for all our current theories. This (...) essay aims to present what we call expansive structures, which are the structures that can be reconstructed from geometrized Newtonian gravitation and are capable of expanding into general relativity, explicitly. In this way, pessimistic meta-induction will be undermined. (shrink)

The question of the existence of gravitational stress-energy in general relativity has exercised investigators in the field since the inception of the theory. Folklore has it that no adequate definition of a localized gravitational stress-energetic quantity can be given. Most arguments to that effect invoke one version or another of the Principle of Equivalence. I argue that not only are such arguments of necessity vague and hand-waving but, worse, are beside the point and do not address the heart of the (...) issue. Based on a novel analysis of what it may mean for one tensor to depend in the proper way on another, which, en passant, provides a precise characterization of the idea of a "geometric object", I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime. I conclude by showing that my results also imply that, under a few natural conditions, the Einstein field equation is the unique equation relating gravitational phenomena to spatiotemporal structure, and discuss how this relates to the non-localizability of gravitational stress-energy. (shrink)

Much of our information comes to us indirectly, in the form of conclusions others have drawn from evidence they gathered. When we hear these conclusions, how can we modify our own opinions so as to gain the benefit of their evidence? In this paper we study the method known as geometric pooling. We consider two arguments in its favour, raising several objections to one, and proposing an amendment to the other.

This paper is a contribution to our understanding of the constructive nature of Greek geometry. By studying the role of constructive processes in Theodoius’s Spherics, we uncover a difference in the function of constructions and problems in the deductive framework of Greek mathematics. In particular, we show that geometric problems originated in the practical issues involved in actually making diagrams, whereas constructions are abstractions of these processes that are used to introduce objects not given at the outset, so that their (...) properties can be used in the argument. We conclude by discussing, more generally, ancient Greek interests in the practical methods of producing diagrams. (shrink)

The brain, rather than being homogeneous, displays an almost infinite topological genus, since it is punctured with a high number of “cavities”. We might think to the brain as a sponge equipped with countless, uniformly placed, holes. Here we show how these holes, termed topological vortexes, stand for nesting, non-concentric brain signal cycles resulting from the activity of inhibitory neurons. Such inhibitory spike activity is inversely correlated with its counterpart, i.e., the excitatory spike activity propagating throughout the whole brain tissue. (...) We illustrate how Pascal’s triangles and linear and nonlinear arithmetic octahedrons are capable of describing the three-dimensional random walks generated by the inhbitory/excitatory activity of the nervous tissue. In case of nonlinear 3D paths, the trajectories of excitatory spiking oscillations can be depicted as the operation of filling the numbers of octahedrons in the form of “islands of numbers”: this leads to excitatory neuronal assemblies, spaced out by empty areas of inhibitory neuronal assemblies. These mathematical procedures allow us to describe the topology of a brain of infinite genus, to represent inhibitory neurons in terms of Betti numbers and to highlight how spike diffusion in neural tissues is generated by the random activation of tiny groups of excitatory neurons. Our approach suggests the existence of a strong mathematical background subtending the intricate oscillatory activity of the central nervous system. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:61. HUME'S GEOMETRIC "OBJECTS" Arithmetic and algebra allow of precision and certainty. The science of geometry is not likewise a perfect and infallible science. At any rate, this is Hume's teaching in the Treatise. When two numbers are so combin ' d, as that the one has always an unite answering to every unite of the other, we pronounce them equal; and 'tis for want of such a standard (...) of equality in extension, that geometry can scarce be esteem'd a perfect and infallible science. 1 The ideas of, say, equality and right line shade into· inequality and curve. It must not be supposed, however, that the vagueness of these ideas renders all judgments uncertain and fallible. 'Tie evident, that the eye, or rather the mind is often able at one view to determine the proportions of bodies....Such judgments are not only common, but in many cases certain and infallible. (T47) Nevertheless, geometry is a fallible science because its essential ideas are vague. Vagueness, due to lack of boundaries between ideas, is in the ideas themselves. For this reason, 'tis impossible to produce any definition of them, which will fix the precise boundaries betwixt them. (T49) (Yet, some forty pages earlier Hume argued at length that the mind cannot form any notion of quantity or quality without forming a precise notion of the degrees of each [T18]) As far as the Treatise goes, geometry is the science of measurement and as such it must allow for error. Even if precise geometric standards could be had, they would not make the propositions of geometry certain. Thus, we might precisely define a right line by a rule of order of indivisible points. But geometry, construed as the science of measurement, would not be perfected. Nor, if it were (the standard), is there any such firmness in our senses or imagination, as to determine when such an order is violated or preserv ' d. The original standard of a right line is in reality nothing but a certain general appearance. (T52) 62. In the Enquiries, Hume continues to hold that notions essential to geometry, for example, equality, right line, and so on, are sensibles. But they are no longer generic appearances. They are determinate. The great advantage of the mathematical sciences above the moral consists in this, that the ideas of the former, being sensible, are always clear and determinate... even when no definition is employed, the object itself may be presented to the senses, and by that means be steadily and clearly apprehended. 2 In addition, geometry is no longer merely the science of measurement. The truths of Euclid, Hume holds, would remain truths even if there were no figures in nature. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence. (EHU25) Professor Zabeeh, in his commentary on Hume, holds that Hume did not mean to imply that Euclid could dispense with the observation of figures. However, whenever possible Hume should be taken at his word: observation of figures is not a truth condition for propositions of geometry. A distinction must be drawn between discovery conditions and what Hume calls the "truth or evidence" of a geometric proposition. We may suppose that observation of figures is a discovery condition. But it is hardly a truth condition. The certainty, as opposed to the discovery, of geometric propositions does not depend on the existence in nature of any figure. I am mainly concerned, however, with Professor Zabeeh' s account of how, according to Hume, geometric propositions retain their certainty and evidence in the face of recalcitrant experience. Noting that Hume does not "fully explain," he gives the real reason for Hume's belief in the apodeictic certainty of mathematical truths. He appears to assume we never allow such truths to be controverted by empirical evidence. That is to say, if perchance we find, by measurement, that the sum of the angles of a Euclidean triangle 63. does not equal 180 degrees, either we say that we measured wrongly or we say that the triangle we have been measuring is not Euclidean.4 This formulation of Hume's reason avoids... (shrink)

This study analyses the predictions of the General Theory of Relativity (GTR) against a slightly modified version of the standard central mass solution (Schwarzschild solution). It is applied to central gravity in the solar system, the Pioneer spacecraft anomalies (which GTR fails to predict correctly), and planetary orbit distances and times, etc (where GTR is thought consistent.) -/- The modified gravity equation was motivated by a theory originally called ‘TFP’ (Time Flow Physics, 2004). This is now replaced by the ‘Geometric (...) Model’, 2014 [20], which retains the same theory of gravity. This analysis is offered partially as supporting detail for the claim in [20] that the theory is realistic in the solar system and explains the Pioneer anomalies. The overall conclusion is that the model can claim to explain the Pioneer anomalies, contingent on the analysis being independently verified and duplicated of course. -/- However the interest lies beyond testing this theory. To start with, it gives us a realistic scale on which gravity might vary from the accepted theory, remain consistent with most solar-scale astronomical observations. It is found here that the modified gravity equation would appear consistent with GTR for most phenomena, but it would retard the Pioneer spacecraft by about the observed amount (15 seconds or so at time). Hence it is a possible explanation of this anomaly, which as far as I know remains unexplained now for 20 years. -/- It also shows what many philosophers of science have emphasized: the pivotal role of counterfactual reasoning. By putting forward an exact alternative solution, and working through the full explanation, we discover a surprising ‘counterfactual paradox’: the modified theory slightly weakens GTR gravity – and yet the effect is to slow down the Pioneer trajectory, making it appear as if gravity is stronger than GTR. The inference that “there must be some tiny extra force…” (Musser, 1998 [1]) is wrong: there is a second option: “…or there may be a slightly weaker form of gravity than GTR.” . (shrink)

Spherical geometry studies the sphere not simply as a solid object in itself, but chiefly as the spatial context of the elements which interact on it in a complex three-dimensional arrangement. This compels to establish graphical conventions appropriate for rendering on the same plane—the plane of the diagram itself—the spatial arrangement of the objects under consideration. We will investigate such “graphical choices” made in the Theodosius’ Spherics from antiquity to the Renaissance. Rather than undertaking a minute analysis of every particular (...) element or single variant, we will try to uncover the more general message each author attempted to convey through his particular graphical choices. From this analysis, it emerges that the different kinds of representation are not the result of merely formal requirements but mirror substantial geometrical requirements expressing different ways of interpreting the sphere and testify to different ways of reasoning about the elements that interact on it. (shrink)

OElig;he thesis is sustained that the definitions of fundamental geometric entities which open Euclids Elements actually are excerpts from the Definitions by Heron of Alexandria, interpolated in late antiquity into Euclids treatise. As a consequence, one of the main bases of the traditional Platonist interpretation of Euclid is refuted. Arguments about the constructivist nature of Euclids mathematical philosophy are given.

We discuss in this paper the question of the scope of the principle of tolerance about languages promoted in Carnap's The Logical Syntax of Language and the nature of the analogy between it and the rudimentary conventionalism purportedly exhibited in the work of Poincaré and Hilbert. We take it more or less for granted that Poincaré and Hilbert do argue for conventionalism. We begin by sketching Coffa's historical account, which suggests that tolerance be interpreted as a conventionalism that allows us (...) complete freedom to select whatever language we wish—an interpretation that generalizes the conventionalism promoted by Poincaré and Hilbert which allows us complete freedom to select whatever axiom system we wish for geometry. We argue that such an interpretation saddles Carnap with a theory of meaning that has unhappy consequences, a theory we believe he did not hold. We suggest that the principle of linguistic tolerance in fact has a more limited scope; but within that scope the analogy between tolerance and geometric conventionalism is quite tight. (shrink)

In his commentary on Euclid, Proclus says both that the first principle of geometry are self-evident and that they are hypotheses received from the single, highest, unhypothetical science, which is probably dialectic. The implication of this seems to be that a geometer both does and does not know geometrical truths. This dilemma only exists if we assume that Proclus follows Aristotle in his understanding of these terms. This paper shows that this is not the case, and explains what Proclus (...) himself means by definition, hypothesis, axiom, postulate, and the self-evident, and how geometry is a science that receives its principles from dialectic. (shrink)

One cannot expect an exact answer to the question “What are the possible structures of space?”, but rough answers to it impact central debates within philosophy of space and time. Recently Gordon Belot has suggested that a rough answer takes the class of metric spaces to represent the possible structures of space. This answer has intuitive appeal, but I argue, focusing on topological characterizations of dimension, examples of prima facie space-like mathematical spaces that have pathological dimension properties, and endorsing a (...) principle of plenitude for possibility, that we get a different rough answer: either the class of metrizable spaces or the class of separable metrizable spaces are possible structures of space. (shrink)

In view of the progress made in recent decades in the fields of stemmatology and the analysis of geometric diagrams, the present article explores the possibility of establishing the stemma codicum of a handwritten tradition from geometric diagrams alone. This exploratory method is tested on Ibn al-Haytham’s Epistle on the Shape of the Eclipse, because this work has not yet been issued in a critical edition. Separate stemmata were constructed on the basis of the diagrams and the text, and a (...) comparison showed no major differences. The greater reliability of a stemma codicum constructed on the basis of the diagrams rather than the text of a mathematical work is discussed and preliminary conclusions are drawn. (shrink)

The paper addresses the capabilities and limitations of extrafoveal processing during a categorical visual search. Previous research has established that a target could be identified from the very first or without any saccade, suggesting that extrafoveal perception is necessarily involved. However, the limits in complexity defining the processed information are still not clear. We performed four experiments with a gradual increase of stimuli complexity to determine the role of extrafoveal processing in searching for the categorically defined geometric shape. The series (...) of experiments demonstrated a significant role of extrafoveal processing while searching for simple two‐dimensional shapes and its gradual decrease in a condition with more complicated three‐dimensional shapes. The factors of objects’ spatial orientation and distractor homogeneity significantly influenced both reaction time and the number of saccades required to identify a categorically defined target. An analysis of the individual p‐value distributions revealed pronounced individual differences in using extrafoveal analysis and allowed examination of the performance of each particular participant. The condition with the forced prohibition of eye movements enabled us to investigate the efficacy of covert attention in the condition with complicated shapes. Our results indicate that both foveal and extrafoveal processing are simultaneously involved during a categorical search, and the specificity of their interaction is determined by the spatial orientation of objects, type of distractors, the prohibition to use overt attention, and individual characteristics of the participants. (shrink)

In a previous article, the writer explored the geometric foundation of the generally covariant spinor calculus. This geometric reasoning can be extended quite naturally to include the Lie covariant differentiation of spinors. The formulas for the Lie covariant derivatives of spinors, adjoint spinors, and operators in spin space are deduced, and it is observed that the Lie covariant derivative of an operator in spin space must vanish when taken with respect to a Killing vector. The commutator of two Lie covariant (...) derivatives is calculated; it is noted that the result is consistent with the geometric interpretation of the Jacobi identity for vectors. Lie current conservation is seen to spring from the result that the operator of spinor affine covariant differentiation commutes with the operator of spinor Lie covariant differentiation with respect to a Killing vector. It is shown that differentiations of the spinor field defined geometrically are Lorentz-covariant. (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)

Most of what is told in this paper has been told before by the same author, in a number of publications of various kinds, but this is the first time that all this material has been brought together and treated in a uniform way. Smaller errors in the earlier publications are corrected here without comment. It has been known since the 1920s that quadratic equations played a prominent role in Babylonian mathematics. See, most recently, Høyrup (Hist Sci 34:1–32, 1996, and (...) Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. Springer, New York, 2002). What has not been known, however, is how quadratic equations came to play that role, since it is difficult to think of any practical use for quadratic equations in the life and work of a Babylonian scribe. One goal of the present paper is to show how the need to find solutions to quadratic equations actually arose in Mesopotamia not later than in the second half of the third millennium BC, and probably before that in connection with certain geometric division of property problems. This issue was brought up for the first time in Friberg (Cuneiform Digit Lib J 2009:3, 2009). In this connection, it is argued that the tool used for the first exact solution of a quadratic equation was either a clever use of the “conjugate rule” or a “completion of the square,” but that both methods ultimately depend on a certain division of a square, the same in both cases. Another, closely related goal of the paper is to discuss briefly certain of the most impressive achievements of anonymous Babylonian mathematicians in the first half of the second millennium BC, namely recursive geometric algorithms for the solution of various problems related to division of figures, more specifically trapezoidal fields. For an earlier, comprehensive (but less accessible) treatment of these issues, see Friberg (Amazing traces of a Babylonian origin in Greek mathematics. WorldScientific, Singapore 2007b, Ch. 11 and App. 1). (shrink)

This paper describes Alfred Clebsch’s 1871 article that gave a geometrical interpretation of elements of the theory of the general algebraic equation of degree 5. Clebsch’s approach is used here to illuminate the relations between geometry, intuition, figures, and visualization at the time. In this paper, we try to delineate clearly what he perceived as geometric in his approach, and to show that Clebsch’s use of geometrical objects and techniques is not intended to aid visualization matters, but rather (...) is a way of directing algebraic calculations. We also discuss the possible reasons why the article of Clebsch has been eventually completely forgotten by the historiography. (shrink)

the view of spinoza as a scion of the mathematico-mechanistic tradition of Galileo and Descartes, albeit perhaps an idiosyncratic one, has been held by many commentators and might be considered standard.1 Although the standard view has a prima facie solid basis in Spinoza's conception of the physical world as extended, law-bound, and deterministic, it has come under sustained criticism of late. Arguing that, for Spinoza, numbers and figures are mere beings of reason and mathematical conceptions of nature belong to the (...) imagination rather than the intellect, recent commentators have cast doubt on the applicability of mathematics to natural science in Spinoza and challenged the association of him with the Galilean and... (shrink)

Book Reviews Clayton peterson, Dialogue: Canadian Philosophical Review/Revue canadienne de philosophie, FirstView Article.

EDMUND HUSSERL in his early writings on space distinguishes three kinds of problems surrounding the presentation of space: psychological, logical, and metaphysical. By the term "psychology" Husserl means a descriptive and genetic psychology which seeks to characterize the contents and structure of particular experiences and to investigate the genetic relations between different experiences. Included among the genetic questions concerning space is the problem of the origin of the science of space.

In lieu of an abstract, here is a brief excerpt of the content:Isaac Barrow on the Mathematization of Nature: Theological Voluntarism and the Rise of Geometrical OpticsAntoni MaletIntroductionIsaac Newton’s Mathematical Principles of Natural Philosophy embodies a strong program of mathematization that departs both from the mechanical philosophy of Cartesian inspiration and from Boyle’s experimental philosophy. The roots of Newton’s mathematization of nature, this paper aims to demonstrate, are to be found in Isaac Barrow’s (1630–77) philosophy of the mathematical (...) class='Hi'>sciences.Barrow’s attitude towards natural philosophy evolved from his earnest interest in medicine of around 1650, when a young Cambridge graduate, to natural philosophy (apparently under Henry More’s influence); from his thesis on the insufficiency of the Cartesian hypothesis to geometrical optics and the strong program of mathematization of natural philosophy of the middle 1660s; from Lucasian professor of Mathematics to Chaplain of his Majesty and eminent Restoration divine. Contemporary accounts of Barrow’s life suggest that he grew ever more skeptical about the worth of natural philosophy and mathematics. 1 In his last years he became a prolific [End Page 265] author of sermons and theological works. Published shortly after his death by John Tillotson (1630–94), later archbishop of Canterbury, they occupy over two thousand folio pages. Overloaded with involved philosophical arguments, Barrow’s sermons were apparently not very popular, but they were highly regarded by scholars and the Anglican hierarchy. 2 It is on certain of his sermons, as well as on the philosophical discussions contained in the Mathematical Lectures that our account of Barrow’s philosophy of the mathematical sciences will rest. 3 Barrow’s understanding of the mathematical sciences will allow us to discuss together three issues often analyzed independently: the theological background to English natural philosophy, the changing notion of mixed mathematical sciences during the seventeenth century, and finally the philosophical foundations of modern geometrical optics.It has long been recognized that significant relationships exist between theological voluntarism or intellectualism and views on natural philosophy. In particular Robert Boyle’s theological voluntarism is seen as grounding his experimentalist approach to natural philosophy. It is not quite so clear, however, how well theological voluntarism may relate to a strong program of mathematization such as the one embodied in Newton’s Principia Mathematica. In fact it has been suggested that the relationship is a negative one. This is derived from the necessary character of mathematical laws, which would put unwanted restrictions on God’s absolute dominion over nature, and also from the notion that theological intellectualism is conducive to a deductive, a priori science—the paradigm of which is of course geometry. However, Barrow’s theological voluntarism lead him to heighten the role of mathematics within natural philosophy.During the seventeenth century the so-called mixed or subalternate mathematical sciences changed profoundly. In the Enlightenment mixed mathematics—meaning above all rational mechanics—became one of the most prestigious and influential disciplines. The mixed mathematics of the Enlightenment, however, was markedly different from the Aristotelian [End Page 266] mixed mathematical sciences. The differences are noticeable both in the subject matter and in the substitution of mathematical infinitesimal analysis for geometrical synthesis, but also in the way of grounding mathematical theory on empirical evidence. Barrow’s Mathematical Lectures (delivered at Cambridge from 1664 to 1666) offer a fresh insight into the metamorphosis of these sciences just when Newton’s “mathematical principles” were in the making. Not the least interesting feature of Barrow’s discussion is that God’s omnipotence allows an evaluation of the truth of mathematical theories that do not apply to this world. Therefore, Barrow is led to introduce the distinction between the internal consistency, or mathematical truth, of a mathematical theory and its physical truth. This, in turn, leads him to the notion that theories need testing.Barrow on Matter and GodRecent literature has established significant correlations between theological voluntarism and empiricism, as well as between theological intellectualism and rationalism. As E. B. Davis writes, “the Christian doctrine of creation is a dialogue between God’s unconstrained will, which utterly transcends the bounds of human comprehension, and God’s orderly intellect, which serves as the model for the human mind.” Intellectualist theology considers God’s omniscience His... (shrink)

This article analyzes the value of geometric models to understand matter with the examples of the Platonic model for the primary four elements (fire, air, water, and earth) and the models of carbon atomic structures in the new science of crystallography. How the geometry of these models is built in order to discover the properties of matter is explained: movement and stability for the primary elements, and hardness, softness and elasticity for the carbon atoms. These geometric models appear to have (...) a double quality: firstly, they exhibit visually the scientific properties of matter, and secondly they give us the possibility to visualize its whole nature. Geometrical models appear to be the expression of the mind in the understanding of physical matter. (shrink)