This paper aims to show that an examination of some passages in Aristotle’s work can contribute to the resolution of crucial problems related to the interpretation of ancient geometrical analysis. In this context, we will focus in particular on the famous passage of the Posterior Analytics in which Aristotle cryptically refers to the analysis practised by the geometers and we will show the fundamental importance of this passage for a correct understanding of ancient geometrical analysis.

The development of geometrical analysis in the 10th century was partly inspired by the reception of the works of Apollonius, which Arab mathematicians translated as early as the preceding century. Al-Qūhī contributed to this development by writing several collections of problems dealing with Apollonian themes and solved by the method of analysis; however, it seems that they do not all occupy the same place in his work. The author gives here the edition, translation, and mathematical commentary of (...) a short work, entitled The determination of two straight lines from a point along a known angle, which presents the particularity of providing problems as geometrical lemmas to other studies. Indeed, al-Qūhī uses two of these lemmas in more complex constructions which belong to his Treatise on the art of the astrolabe. In this same treatise on the astrolabe, as well as in his Treatise on the perfect compass, this scholar also uses as lemmas several problems solved in another of his works, entitled The generation of points on straight lines according to ratios of which the terms are surfaces. This work is unfortunately lost, and all that remains of it are the traces which subsist in other treatises in which it has been used. This study seems to be necessary if one wishes to understand the organization of the work of al-Qūhī, a mathematician of the first rank who was representative of his time. (shrink)

In this article, we assess the heuristic power of Greek geometrical analysis by trying to reconstruct some analyses of extant propositions of which only the demonstration is found in the text. We have reconstructed the analysis of the trisection of an angle, the property of the tangent to the parabola, to the hyperbola/ellipse, and to the spiral line. In all of these cases, the results and the demonstrations can be found by the analysis alone, without arguments (...) by analogy with other figures or kinematic considerations. We have thus concluded that analysis is a much more powerful tool than is usually assumed to be. We have also examined a part of Apollonius’ Cutting off of a ratio and confirmed that the number of possible cases to be distinguished in analysis tended to increase because the geometric objects and the magnitudes related to them were always considered in the geometric arrangement in which they appear. We suggest that the absence of an algebraic approach that would have detached the magnitudes from their geometrical context may have been one of the major obstacles to further development of Greek geometry. (shrink)

In this analytically oriented work, Peterson articulates and defends five moral principles for addressing ethical issues related to new and existing technologies: the cost-benefit principle, the precautionary principle, the sustainability principle, the autonomy principle, and the fairness principle.

Every linear perspective image has a center of the perspective construction. Only when observed from that location does a 2D image provide the same stimulus as the original 3D scene. Geometric analyses indicate that observing the image from other vantage points should affect the perceived spatial structure of the scene conveyed by the image, involving transformations such as shear, compression, and dilation. Based on previous research, this paper presents a detailed account of these transformations. The analyses are presented in a (...) uniform manner, illustrated with special 3D diagrams, and embedded in a wider framework of related perspective paradigms. Such analyses provide the potential theoretical basis for empirical work on the effects of changes of observer vantage points on the perception of spatial structure in perspective images. (shrink)

Martin Peterson’s The Ethics of Technology: A Geometric Analysis of Five Moral Principles offers a welcome contribution to the ethics of technology, understood by Peterson as a branch of applied ethics that attempts ‘to identify the morally right courses of action when we develop, use, or modify technological artifacts’ (3). He argues that problems within this field are best treated by the use of five domain-specific principles: the Cost-Benefit Principle, the Precautionary Principle, the Sustainability Principle, the Autonomy Principle, and (...) the Fairness Principle. These principles are, in turn, to be understood and applied with reference to the geometric method. This method is perhaps the most interesting and novel part of Peterson’s book, and I’ll devote the bulk of my review to it. (shrink)

This paper examines Aristotle's discussion of the priority of actuality to potentiality in geometry at Metaphysics Θ9, 1051a21–33. Many scholars have assumed what I call the "geometrical construction" interpretation, according to which his point here concerns the relation between an inquirer's thinking and a geometrical figure. In contrast, I defend what I call the "geometrical analysis" interpretation, according to which it concerns the asymmetrical relation between geometrical propositions in which one is proved by means of (...) the other. His argument as so construed is ultimately based on the asymmetrical relation between the corresponding geometrical facts. Then I explore this ontological priority in geometry by drawing attention to a parallel passage, Posterior Analytics II.11, 94a24–35, where Aristotle explains the relation between the same geometrical propositions in connection to material causation. (shrink)

The paper argues that Roger Bacon adhered to a unique form of geometrical atomism, according to which elemental matter can be analysed into cubic (when at rest) or pyramidal (when in motion) portions. Bacon addressed geometrical atomism from the perspective of the Aristotelian review, using his interpretation of Aristotelian principles to render the theory plausible. He was mostly concerned with solving the contradiction between the angular shapes of the portions and the shape of the elemental spheres. His motivation (...) for doing so, I argue, was his conviction in the applicability of geometry onto natural philosophy, and his goal was to render matter capable of geometrical analysis. In this way, he was convinced, geometrical properties can be considered efficient causes of matter’s spatial formation and motion. (shrink)

We present a geometrical analysis of the principles that lay at the basis of Categorial Grammar and of the Lambek Calculus. In Abrusci it is shown that the basic properties known as Residuation laws can be characterized in the framework of Cyclic Multiplicative Linear Logic, a purely non-commutative fragment of Linear Logic. We present a summary of this result and, pursuing this line of investigation, we analyze a well-known set of categorial grammar laws: Monotonicity, Application, Expansion, Type-raising, Composition, (...) Geach laws and Switching laws. (shrink)

Greek ancient and early modern geometry necessarily uses diagrams. Among other things, these enter geometrical analysis. The paper distinguishes two sorts of geometrical analysis and shows that in one of them, dubbed “intra-confgurational” analysis, some diagrams necessarily enter as outcomes of a purely material gesture, namely not as result of a codifed constructive procedure, but as result of a free-hand drawing.

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)

The doctrine of unity of opposites lays in the centre of the debate on Heraclitus’ philosophy. The present article proposes a critical analysis of the mainstream interpretation of geometrical oppositions (fragments DK 22 B 59, B 60 and B 103) as mere examples of different points of view. Instead, we suggest that these fragments are fundamental pieces in Heraclitus cosmology and that they are traces of a circular and archaic paradigm. Indeed, cyclical formulations are spread throughout the fragments (...) and, read in connection with geometrical oppositions, perform a useful model to describe how permanence and change can be complementary events. In this context, circularity can be a key to the comprehension of unity of opposites as a result of interchangeable processes – as day and night, life and death, beginning and end – so that opposites are part of a concrete and vital unity. (shrink)

The object of this paper is to show the predominance of the influence of geometrical ideas in Aristotle's account of first principles in the Posterior Analytics— to show that his analysis of first principles is in its essentials an analysis of the first principles of geometry as he conceived them. My proof of this falls into two parts. I. A consideration of the parallel between Aristotle's and Euclid's account of first principles. II. A comparison between the general (...) movement of thought in the Aristotelian παγωγ and πδειξις and in the δινοια and νησις of the Platonic dialectic. (shrink)

The aim of this paper is to provide a logic-based conceptual analysis of the twin paradox (TwP) theorem within a first-order logic framework. A geometrical characterization of TwP and its variants is given. It is shown that TwP is not logically equivalent to the assumption of the slowing down of moving clocks, and the lack of TwP is not logically equivalent to the Newtonian assumption of absolute time. The logical connection between TwP and a symmetry axiom of special (...) relativity is also studied. (shrink)

This paper is an analysis of the geometrical interpretation of local gauge symmetry for theories of the Yang-Mills type.

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)

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)

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)

This paper examines the second geometrical problem in the Meno. Its purpose is to explore the implication of Cook Wilson’s interpretation, which has been most widely accepted by scholars, in relation to the nature of hypothesis. I argue that (a) the geometrical hypothesis in question is a tentative answer to a more basic problem, which could not be solved by available methods at that time, and that (b) despite the temporary nature of a hypothesis, there is a rational (...) process for formulating it. The paper also contains discussion of the method of analysis, problem reduction and a diorism, which have often been ambiguously explained in relation to the geometrical problem in question. (shrink)

It is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

In an accompanying paper (I), it is shown that the basic equations of the theory of Lorentzian connections with teleparallelism (TP) acquire standard forms of physical field equations upon removal of the constraints represented by the Bianchi identities. A classical physical theory results that supersedes general relativity and Maxwell-Lorentz electrodynamics if the connection is viewed as Finslerian. The theory also encompasses a short-range, strong, classical interaction. It has, however, an open end, since the source side of the torsion field equation (...) is not geometric.In this paper, Kaehler's partial geometrization of the Dirac equation is taken as a starting point for the development of fully geometric Dirac equations via the correspondence principle given in I. For this purpose, Kaehler's calculus (where the spinors are differential forms) is generalized so that it also applies when the torsion is not zero. The point is then made that the forms can take values in tangent Clifford algebras rather than in tensor algebras. The basic “Eigenschaft” of the Kaehler calculus also is examined from the physical perspective of dimensional analysis.Geometric Dirac equations of great structural simplicity are finally inferred from the standard Dirac equation by using the aforementioned correspondence principle. The realm of application of the Dirac theory is thus enriched in principle, though only at an abstract level at this point: the standard spinors, which are scalar-valued forms in the Kaehler version of that theory, become Clifford-valued. In addition, the geometrization of the Dirac equation implies a geometrization of the Dirac current. When this current is replaced in the field equations for the torsion, the theory of Paper I becomes fully geometric. (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 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)

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)

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

Given a quantum system with a very large number of degrees of freedom and a preferred tensor product factorization of the Hilbert space we describe how it can be approximated with a very low-dimensional field theory with geometric degrees of freedom. The geometric approximation procedure consists of three steps. The first step is to construct weighted graphs with vertices representing subsystems and edges representing mutual information between subsystems. The second step is to deform the adjacency matrices of the information graphs (...) to that of a low-dimensional lattice using the graph flow equations introduced in the paper. The third step is to define an emergent metric and to derive an effective description of the metric and possibly other degrees of freedom. To illustrate the procedure we analyze two information graph flows with geometric attractors and metric perturbations obeying a geometric flow equation. Our analysis also suggests a possible approach to quantum gravity in which the geometry emerges directly from a quantum state due to the flow of the information graphs. (shrink)

SummaryIn 1693, Gottfried Wilhelm Leibniz published in the Acta Eruditorum a geometrical proof of the fundamental theorem of the calculus. It is shown that this proof closely resembles Isaac Barrow's proof in Proposition 11, Lecture 10, of his Lectiones Geometricae, published in 1670. This comparison provides evidence that Leibniz gained substantial help from Barrow's book in formulating and presenting his geometrical formulation of this theorem. The analysis herein also supports the work of J. M. Child, who in (...) 1920 studied the early manuscripts of Leibniz and concluded that he had frequently copied his diagrams from Barrow's book, but without acknowledgement. It is also shown that the diagram of Leibniz associated with his 1693 proof has often been reproduced with errors that make some aspects of his text difficult to comprehend. (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)

This paper traces August Comte’s attempts to get clear about the concept of mathematical analysis at various stages in his intellectual development. Comte was especially concerned with distinguishing a method of analysis for the resolution of complex prolems from analysis in the sense of a method of drawing inferences. Geometrical analysis serves as his model for the former. In his attempt to get clear about this notion, he discovers an historical succession of different methods all (...) of which may be labeled “analytic.” In modern terms, Comte reveals how each of these methods of analysis characterizes a research program in mathematics, even showing us how more powerful methods came to supplant less powerful methods of analysis. (shrink)

Late ancient Platonists and Aristotelians describe the method of reasoning to first principles as "analysis." This is a metaphor from geometrical practice. How far back were philosophers taking geometric analysis as a model for philosophy, and what work did they mean this model to do? After giving a logical description of analysis in geometry, and arguing that the standard (not entirely accurate) late ancient logical description of analysis was already familiar in the time of Plato (...) and Aristotle, I argue that Plato, in the second geometrical passage of the "Meno" (86e4-87b2), is taking analysis as a model for one kind of philosophical reasoning, and I explore the advantages and limits of this model for philosophical discovery, and in particular for how first principles can be discovered, without circularity, by argument. (shrink)

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)

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)

Legal reasoning requires identification through search of authoritative legal texts (such as statutes, constitutions, or prior judicial opinions) that apply to a given legal question. In this paper, using a network representation of US Supreme Court opinions that integrates citation connectivity and topical similarity, we model the activity of law search as an organizing principle in the evolution of the corpus of legal texts. The network model and (parametrized) probabilistic search behavior generates a Pagerank-style ranking of the texts that in (...) turn gives rise to a natural geometry of the opinion corpus. This enables us to then measure the ways in which new judicial opinions affect the topography of the network and its future evolution. While we deploy it here on the US Supreme Court opinion corpus, there are obvious extensions to large evolving bodies of legal text (or text corpora in general). The model is a proxy for the way in which new opinions influence the search behavior of litigants and judges and thus affect the law. This type of “legal search effect” is a new legal consequence of research practice that has not been previously identified in jurisprudential thought and has never before been subject to empirical analysis. We quantitatively estimate the extent of this effect and find significant relationships between search-related network structures and propensity of future citation. This finding indicates that “search influence” is a pathway through which judicial opinions can affect future legal development. (shrink)