Equality, similarity and congruence are essential elements of Kant’s theory of geometrical cognition; nevertheless, Kant’s account of them is not well understood. This paper provides historical context for treatments of these geometrical relations, presents Kant’s views on their mathematical definitions, and explains Kant’s theory of their cognition. It also places Kant’s theory within the larger context of his understanding of the quality-quantity distinction. Most importantly, it argues that the relation of equality, in conjunction with the categories of quantity, (...) plays a pivotal and wide-ranging role in Kant’s account of mathematical cognition. (shrink)

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)

Jacques Rancière’s conception of equality as an axiomatic presupposition of the political is important, because it bypasses the tradition which defines equality in terms of Aristotle’s conception of geometric equality. In this paper, I show that Rancière’s theory both espouses a monism, according to which inequality implies equality, and relies on a concept of the free will, which is incompatible with monism. I highlight this tension by bringing Rancière’s theory into conversation with the great monist (...) of the philosophical tradition, Baruch Spinoza. (shrink)

Equality and identity. Bulletin of Symbolic Logic. 19 (2013) 255-6. (Coauthor: Anthony Ramnauth) Also see https://www.academia.edu/s/a6bf02aaab This article uses ‘equals’ [‘is equal to’] and ‘is’ [‘is identical to’, ‘is one and the same as’] as they are used in ordinary exact English. In a logically perfect language the oxymoron ‘the numbers 3 and 2+1 are the same number’ could not be said. Likewise, ‘the number 3 and the number 2+1 are one number’ is just as bad from a logical (...) point of view. In normal English these two sentences are idiomatically taken to express the true proposition that ‘the number 3 is the number 2+1’. Another idiomatic convention that interferes with clarity about equality and identity occurs in discussion of numbers: it is usual to write ‘3 equals 2+1’ when “3 is 2+1” is meant. When ‘3 equals 2+1’ is written there is a suggestion that 3 is not exactly the same number as 2+1 but that they merely have the same value. This becomes clear when we say that two of the sides of a triangle are equal if the two angles they subtend are equal or have the same measure. -/- Acknowledgements: Robert Barnes, Mark Brown, Jack Foran, Ivor Grattan-Guinness, Forest Hansen, David Hitchcock, Spaulding Hoffman, Calvin Jongsma, Justin Legault, Joaquin Miller, Tania Miller, and Wyman Park. -/- ► JOHN CORCORAN AND ANTHONY RAMNAUTH, Equality and identity. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] The two halves of one line are equal but not identical [one and the same]. Otherwise the line would have only one half! Every line equals infinitely many other lines, but no line is [identical to] any other line—taking ‘identical’ strictly here and below. Knowing that two lines equaling a third are equal is useful; the condition “two lines equaling a third” often holds. In fact any two sides of an equilateral triangle is equal to the remaining side! But could knowing that two lines being [identical to] a third are identical be useful? The antecedent condition “two things identical to a third” never holds, nor does the consequent condition “two things being identical”. If two things were identical to a third, they would be the third and thus not be two things but only one. The plural predicate ‘are equal’ as in ‘All diameters of a given circle are equal’ is useful and natural. ‘Are identical’ as in ‘All centers of a given circle are identical’ is awkward or worse; it suggests that a circle has multiple centers. Substituting equals for equals [replacing one of two equals by the other] makes sense. Substituting identicals for identicals is empty—a thing is identical only to itself; substituting one thing for itself leaves that thing alone, does nothing. There are as many types of equality as magnitudes: angles, lines, planes, solids, times, etc. Each admits unit magnitudes. And each such equality analyzes as identity of magnitude: two lines are equal [in length] if the one’s length is identical to the other’s. Tarski [1] hardly mentioned equality-identity distinctions (pp. 54-63). His discussion begins: -/- Among the logical concepts […], the concept of IDENTITY or EQUALITY […] has the greatest importance. -/- Not until page 62 is there an equality-identity distinction. His only “notion of equality”, if such it is, is geometrical congruence—having the same size and shape—an equivalence relation not admitting any unit. Does anyone but Tarski ever say ‘this triangle is equal to that’ to mean that the first is congruent to that? What would motivate him to say such a thing? This lecture treats the history and philosophy of equality-identity distinctions. [1] ALFRED TARSKI, Introduction to Logic, Dover, New York, 1995. [This is expanded from the printed abstract.] . (shrink)

The Fermi energy at 0°K is evaluated for electrons confined to cubical and spherical rigid-walled boxes of equal volume, respectively, in the Sommerfeld approximation. Due primarily to large differences in single-particle degeneracies, Fermi energies compared for equal numbers of particles in these two configurations are found to be unequal. Approximate expressions of the Fermi energy in the large particle-number limit for the spherical case reveal that it agrees in form with the Fermi energy for the cubical configuration. The finite cylindrical (...) box is also examined and it is found that for fixed number of particles and fixed volume, the Fermi energy varies as the two characteristic lengths of the box are changed. Experimental corroboration of the theory is suggested through measurement of the work function for equal-volume micrometallic samples in the different geometries. (shrink)

This article is a study of geometric constructions. We consider, as an illustration, the methods used for dividing the straight line into n equal parts (n-section). Architects and practicioners of classical Europe had at their disposal a broad range of geometric constructions: ancient ones were edited and translated, whereas new solutions were constantly published. The wide variety and reasons for selection of these geometric constructions are puzzling: the most widespread construction was not the simplest one. This article (...) wonders why so many constructions were used to solve the same problem and why tedious methods were used. It is argued that the practical conditions and scale at which the problem was tackled could account for such diversity. (shrink)

This paper analyses the place of sortition in the political philosophy of Jacques Rancière. The idea of sortition is linked to a philosophical reflection on arithmetic equality and geometric equality. Thus, starting from an important work of Cornelius Castoriadis in this sense, we will analyze below the relationship of Rancière’s political philosophy with equality. Finally we will analyze the sortition and the place it occupies in his work. Bernard Manin’s work on ancient democracies and systems of (...) representative government will help us contextualize Rancière’s proposals on the use of random selection in politics. (shrink)

The history and philosophy of science are destined to play a fundamental role in an epoch marked by a major scientific revolution. This ongoing revolution, principally affecting mathematics and physics, entails a profound upheaval of our conception of space, space–time, and, consequently, of natural laws themselves. Briefly, this revolution can be summarized by the following two trends: by the search for a unified theory of the four fundamental forces of nature, which are known, as of now, as gravity, electromagnetism, and (...) strong and weak nuclear forces; by the search for new mathematical concepts capable of elucidating and therefore explaining such a relationship. In fact, the first search is essentially dependent on the second; that is to say, that in order for a new theory of physics to come to light, the development of a deeper geometric theory capable of explaining the structure of space–time on a quantum scale appears to be necessary. On careful consideration, we notice that both of these developments converge in the direction of a unitary and fundamental tendency of modern science—which is the geometrization of theoretical physics and of natural sciences. This new emergent situation carries within it a profound conceptual change, affecting the way in which relations are conceived of, first and foremost, between mathematics and physics. This new paradigm can be summed up by the intimately interdependent points: the immense variety of physical phenomena and of natural forms follows from the equally infinite variety of geometric and topological objects that can be made out in space and from which space is made up; the second point, which ensues from the former one and which is of great historical and epistemological significance, is that mathematics is involved in rather than applied to phenomena. In other words, phenomena are effects that emerge from the geometrical structure of space–time. There is no doubt that this new conception of the relationship between the universe of mathematical ideas and objects and the world of natural phenomena is the true scientific revolution of our century, of great conceptual importance, and consequently, capable of changing our view of science and of nature at one and the same time. It is all at once of a scientific, philosophical and aesthetic order. (shrink)

According to major Latin dictionaries, the word radius is attested as a terminus technicus for the geometrical concept ‘radius’ in Cicero’s Timaeus 17. In this study, however, it is argued that there is good reason to believe that Cicero did not use the word in this sense, but in a metaphorical expression in which radius mainly carries the well-attested sense of ‘rod ’: paribus radiis attingi literally = ‘to be touched by equal rods’, that is to say, ‘to be equidistant’. (...) The first and only convincing attestation of radius in the sense of ‘radius ’ in ancient Latin literature is found in Calcidius’ Commentary on the Timaeus 59. It is suggested that Calcidius was influenced by Cicero’s translation and consequently adopted the word radius; however, its technical sense of ‘radius’ was coined, deliberately or not, by Calcidius himself. As there are no other convincing attestations of this sense of the word in ancient Latin, not even in geometrical texts, it is suggested that the word radius never became a commonly accepted terminus technicus for radius in antiquity. (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)

Objective: Based upon notions to a mental vision of the Vitruvian Man, to determine if any obvious asymmetries exist within Leonardo da Vinci’s timeless schematic—which is famous for its highly symmetrical presentation. Methods: A qualitative analysis performed upon a Vitruvian Man print (taken from the namesake Wikipedia article) to: closely examine if the man’s head is positioned to noticeably tilt toward either direction—left or right—of a dissecting line superimposed for equally splitting (vertically) the circle in the schematic; and, to closely (...) examine the man’s eyes for any artistic asymmetry therein drawn. Results: The man’s head is determined tilting toward his right half/hemisphere of the circle. Also noticeable: the man’s right eye appears being of a much brighter look relative to a darkness observable about his left eye; and, an outline to an inverted equilateral triangle is identifiably shaded surrounding his left eye, whereas a more contrastingly circular shape appears as so shaded around his right eye. Discussion: Once a certain awareness is drawn to such artistic anomalies, they can become so clearly observable that one may wonder if Leonardo da Vinci did as much intentionally. And if so, then why? This paper details a hypothesis necessarily built upon the assumption that Leonardo intentionally embedded such cues within his Vitruvian Man whereby speculations toward a hemispheric brain theory can be established of such cryptic nature, given a then-pronouncedly authoritarian Catholic Church, whereupon he may have figured it possible to receive credit posthumously if ever such notions were proven valid. Incidentally, the concerning analysis as to why the Vitruvian Man’s head tilts in such a fashion aligns with Dr. Iain McGilchrist’s peer-reviewed findings to a hemispheric hypothesis. (shrink)

How does a sense of touch, figuratively and practically, get deployed within equality governance, and to what questions and ways of thinking about the state does this direct us? Taking 2009–2010 as a snap-shot moment in the development of British equality reform—the year leading up to passage of the Equality Act 2010—this article explores the relationship between touch (the haptic) and equality governance from three angles. First, how have governmental bodies used touch language and imagery, including (...) in geometrical representations of disadvantage? Second, what other, more challenging encounters and actions are imaginable; specifically, can touch mobilise the feeling state as a critical form of active citizenship? Third, what re-conceptualisations of the state does the touching, feeling state invoke, and with what effects? Specifically, does conceiving of the state as a multi-identity formation reframe the risks associated with a haptic state, thereby opening up new strategies for political action? (shrink)

Françios Viète was a geometer in search of better techniques for astronomical calculation. Through his theorem on angular sections he found a use for higher-dimensional geometric magnitudes which allowed him to create an algebra for geometry. We show that unlike traditional numerical algebra, the knowns and unknowns in Viète’s logistice speciosa are the relative sizes of non-arithmetized magnitudes in which the “calculations” must respect dimension. Along with this foundational shift Viète adopted a radically new notation based in Greek (...) class='Hi'>geometric equalities. His letters stand for values rather than types, and his given values are undetermined. Where previously algebra was founded in polynomials as aggregations, Viète became the first modern algebraist in working with polynomials built from operations, and the notations reflect these conceptions. Viète’s innovations are situated in the context of sixteenth-century practice, and we examine the interpretation of Jacob Klein, the only historian to have conducted a serious inquiry into the ontology of Viète’s “species”. (shrink)

Techne and Truth. The problem of techne in the dispute between Gorgias and Plato -/- The source of the problem matter of the book is the Plato’s dialogue „Gorgias”. One of the main subjects of the discussion carried out in this multi-aspect work is the issue of the art of rhetoric. In the dialogue the contemporary form of the art of rhetoric, represented by Gorgias, Polos and Callicles, is confronted with Plato’s proposal of rhetoric and concept of art (techne). The (...) ingenuous and dramatic structure of the dialogue composed of three acts, in each of which Socrates’ consecutive interlocutors emerge, is difficult to interpret. It seems, however, that even though the first conversation between Socrates and Gorgias constitutes a small part of the dialogue, it forms the foundations for the discussion. This part, dealing with such important issues as the subject of art, knowledge, power and relation between art and good, is the starting point for the conversations which follow. What do the differences between the Gorgias’ and Plato’s concepts of art consist in? Any attempt to answer the question requires presentation of the background for the discussion taking place in Plato’s work as well as the reconstruction of Gorgias’ vision of the art of rhetoric and art in general, defining the relation between art and knowledge. The latter may be undertaken on the basis of two paraphrases of the treatise “On Non-existence”, two epideictic speeches entitled “Helen” and “Palamedes” and some smaller fragments which have been preserved from Gorgias’ creative output. The treatise “On Non–existence”, very paradoxical in its philosophical meaning, testifies to new, sophistic thinking represented by Gorgias of Leontinoi. The treatise is the polemic and attack on Eleatism based on certain principles and methods elaborated by the Eleatics themselves. It also shows that Gorgias – using traditional Eleatic schemes – is opposed to this tradition. Even though the treatise does not discuss the problem of art directly, it anticipates the considerations from epideictic speeches. “Helen” and “Palamedes” constitute further stages in the development of Gorgias’ thought and concept of art combining typically epideictic elements with some philosophical convictions. Gorgias refers to the Eleatic theme considering the issues of truth, knowledge and belief, yet defining them in the empirical spirit. Solutions in the cognitive sphere determine the horizon of Gorgias’ rhetoric. The word is separated from the existence. Human cognition is limited and therefore the word and visual images affect the belief. This promotes the great importance of rhetoric, painting and sculpture, i.e. the areas which influence in a specific way – creating the “delusion” (ἀπάτη). Through idealization art provides pleasure, which is its objective. Therefore the task of art is to create delusion. The rhetor of Leontinoi discusses the issue of “how?” to use the word ‘rhetoric’. Gorgias is aware that rhetoric may be used to talk about what is unknown and that technical correctness of a rhetoric statement is not identical with the truthfulness of the proclaimed message. Mentioning it, Gorgias shows his respect to the truth as a value which should be respected in speech on the one hand, and on the other emphasizes that the truth in itself has no persuasive power. For this reason it is not enough to proclaim the truth, but, to convince the listeners, the orator should possess the skill of speaking. What is then the relation between truth and art emerging from epideictic speeches? Gorgias undoubtedly emphasizes the formal aspect of speech, as rhetoric is art, thus an ability based on certain principles. According to Gorgias, the truth is the value which should be respected in speeches. Rhetoric is able to create a fictive world, as a great range of reality remains outside direct human experience. Rhetoric does not oppose the truth but discussing not only what is directly given concerns the area outside the sphere of direct experience. This concept of rhetoric is the basis of the discussion about art carried out in the dialogue “Gorgias”. Even though the first of the discussions in the dialogue – the conversation between Socrates and Gorgias – is directly concerned with rhetoric itself, numerous remarks testify to its much wider character. This is best exemplified by the way of carrying out the conversation by Socrates, who, asking questions about rhetoric practised by Gorgias, throughout the whole conversation compares it with other skills. Additionally, the manner and the structure of the conversation prove that Plato considers the issue of rhetoric from a wider perspective determined by the issue of art. The specific feature of the conversation is ordering it around main issues – such notions as πρᾶγμα, ἐπιστήμη, δύναμις, each of which has its own meaning in Plato. In the first part of Gorgias Plato presents the principles previously occurring in other dialogues in a certain order, thus constructing the structure of the conversation with Gorgias. The notions are not novum in Plato but the fact that he clearly wants to systematise them and present a certain concept of art is a new and significant aspect. This tendency is of paramount importance for the discussion because Gorgias’ answers are evaluated by Plato from the perspective of this clearly set concept consisting of not only established notions but also certain principles. The discussion between Socrates and Gorgias introduces a theme which is crucial for the dialogue: the relation between art and good, i.e. the technical and ethical spheres. The issue is introduced by Gorgias, who mentions that rhetoric may be used for evil purposes. On this basis Socrates proves that if rhetoricians can use rhetoric for evil purposes, they have no knowledge concerning the subject of their art – justice, because one who knows what is just, always acts justly. Thus the relation between the technical and ethical spheres transpires to be the main point of contention. The question is whether a technician following the principles of the art he practices may act against good. This paradoxical statement based on the well-known Socrates’ principle "nemo sua sponte peccat" acquires philosophical foundations in the dialogue. In its remaining two parts Plato presents a justification, placing the principle within the framework of a certain concept of art, as a result of which not only rhetoric but also many other skills are criticised. The notions on which the conversation with Gorgias focuses are more precisely defined in the subsequent parts of the dialogue. Additionally, Plato presents twice the conditions which art should fulfill. They oblige the “technicians” to know the nature and purpose of the art and to know the causes of undertaken activities and to be able to justify them. These conditions determine the relation between the two spheres – technical and ethical. According to Plato, art may not act against good because it should be the objective of its activities. Certain concepts of good and knowledge outlined in the dialogue are the guarantees of “technicality”. Good, which should be respected by every art, is understood objectively and is contrasted with subjective pleasure. Knowledge, which should be the basis of art, opposes experience, which uses observation and recollection. Additionally, the definition of knowledge, significance of order in the world assuming the form of “geometrical equality”, significance of mathematics becoming the ideal of exactness indicate that Plato’s concept of knowledge is formed in Gorgias. These conclusions render it impossible that the results of any activity deserving the name of art should oppose good. And this is Plato’s main reproach of Gorgias’ concept, who wrote about false speeches possessing the power of conviction because they were written according to the principles proof of art or who defined tragedy as “a deceit, where he who cheats is more just than he who does not and the cheated is wiser than he who was not cheated”. Such determination of the relation between the technical and ethical sphere resulted from Gorgias’ concept of the world and fundamental philosophical principles. The whole concept of art described by Plato in the dialogue opposes the view presented by Gorgias. According to Plato, every technical activity has a defined object (ἔργον), is based on knowledge (ἐπιστήμη) and is always aimed at the good of human soul or body. The danger revealed and presented in the dialogue by Plato exists in the very concept of art presented by Gorgias, i.e. in its empirical foundations and the resulting understanding of the relation between art and good. Gorgias’ concept of art based on empirical cognition, which concerns the world of phenomena and rejects the possibility of perfect knowledge, leads, according to Plato, to a dangerous falsification of the relation between art and good. Plato proves that this weakness of Gorgias’ concept is revealed when confronted with these representatives of “the magnificent art” who are not protected against radicalism by the adherence to the moral tradition. The comparison of two, completely contrasting concepts of art based on different philosophical understanding of knowledge and good is decisive for the criticism of rhetoric in Gorgias. The concept presented by Plato criticises contemporary rhetoric and sophistry but determines the direction of research of the real art of rhetoric. (shrink)

This article addresses Rancière’s critique of Aristotle’s political theory as parapolitics in order to show that Aristotle is a resource for developing an inclusionary notion of political community. Rancière argues that Aristotle attempts to cut off politics and merely police (maintain) the community by eliminating the political claim of the poor by including it. I respond to three critiques that Rancière makes of Aristotle: that he ends the political dispute by including the demos in the government; that he includes the (...) free masses only incidentally because, by chance, the rule is political, not based on mastery; and that he attempts to close off politics by determining in advance what speech counts as speech. I argue that Aristotle attempts to institute this contest by accepting the incommensurable claims of arithmetic and geometric equality. Aristotle puts this conflict at the root of political life in such a way that serves Rancière’s larger project of perpetuating politics. (shrink)

The theorem that the arithmetic mean is greater than or equal to the geometric mean is investigated for cardinal and ordinal numbers. It is shown that whereas the theorem of the means can be proved for n pairwise comparable cardinal numbers without the axiom of choice, the inequality a2 + b2 ≥ 2ab is equivalent to the axiom of choice. For ordinal numbers, the inequality α2 + β2 ≥ 2αβ is established and the conditions for equality are derived; (...) stronger inequalities are obtained for finite and infinite sequences of ordinals under suitable monotonicity hypotheses. MSC: 03E10, 04A10, 03E25, 04A25. (shrink)

In EN V.3, Aristotle offers an abstract definition of distributive justice that is agreed to by all, namely that it should be governed by geometrical proportionality: ‘equals should be treated equally, unequals should be treated in proportion to their inequalities’. At the same time, he acknowledges that we need a more substantive definition of the currency of equality, i.e. to tell us who are equal and who unequal at each distribution, since this would be the only way to avoid (...) each group of people to identify its own conception with the universal concept of justice. In Pol. III.12 (1282b18-23) Aristotle is arguing that it is the task of the political philosopher to define what kinds of equality or inequality are relevant in distribution of political power. In order to do so, Aristotle has to answer the question who is a citizen, that is, who should share in the constitution (Pol. III.1 1274b40-41). First he clears away the false conceptions and claims about citizenship and then he offers two criteria for political distribution—political capacity and contribution to the polis—which prima facie seem inconsistent. This paper explores the relation of the two criteria and the fundamental role of political capacity within the context of political distribution; I intend to argue that they are actually two stages in the process of political distribution, where political capacity is indexical prior to the contribution to the polis. It is also suggested that the Aristotelian model is egalitarian, in the sense that all politically capable individuals, regardless of the degree of political capacity they possess, are awarded citizenship. (shrink)

The purported fact that geometric theories formulated in terms of points and geometric theories formulated in terms of lines are “equally correct” is often invoked in arguments for conceptual relativity, in particular by Putnam and Goodman. We discuss a few notions of equivalence between first-order theories, and we then demonstrate a precise sense in which this purported fact is true. We argue, however, that this fact does not undermine metaphysical realism.

One can (for the most part) formulate a model of a classical system in either the Lagrangian or the Hamiltonian framework. Though it is often thought that those two formulations are equivalent in all important ways, this is not true: the underlying geometrical structures one uses to formulate each theory are not isomorphic. This raises the question of whether one of the two is a more natural framework for the representation of classical systems. In the event, the answer is yes: (...) I state and sketch proofs of two technical results—inspired by simple physical arguments about the generic properties of classical systems—to the effect that, in a precise sense, classical systems evince exactly the geometric structure Lagrangian mechanics provides for the representation of systems, and none provided by Hamiltonian. The argument not only clarifies the conceptual structure of the two systems of mechanics, but also their relations to each other and their respective mechanisms for representing physical systems. It also shows why naïvely structural approaches to the representational content of physical theories cannot work. [Lagrange] grasped that he had gained a method of stating dynamical truths in a way, which is perfectly indifferent to the particular methods of measurement employed in fixing the positions of the various parts of the system. Accordingly, he went on to deduce equations of motion, which are equally applicable whatever quantitative measurements have been made, provided that they are adequate to fix positions. The beauty and almost divine simplicity of these equations is such that these formulae are worthy to rank with those mysterious symbols which in ancient times were held directly to indicate the Supreme Reason at the base of all things. (Whitehead [1948], p. 63)1. Introduction2.Classical Systems3. The Possible Interactions of a Classical System and the Structure of Its Space of States4. Classical Systems Are Lagrangian5. Classical Systems Are Not Hamiltonian6. How Lagrangian and Hamiltonian Mechanics Represent Classical Systems7. The Conceptual Structure of Classical Mechanics. (shrink)

Formalizing Euclid’s first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) -/- Euclid [fl. 300 BCE] divides his basic principles into what came to be called ‘postulates’ and ‘axioms’—two words that are synonyms today but which are commonly used to translate Greek words meant by Euclid as contrasting terms. -/- Euclid’s postulates are specifically geometric: they concern geometric magnitudes, shapes, figures, etc.—nothing else. The first: “to draw a line from any point to any point”; the (...) last: the parallel postulate. -/- Euclid’s axioms are general principles of magnitude: they concern geometric magnitudes and magnitudes of other kinds as well even numbers. The first is often translated “Things that equal the same thing equal one another”. -/- There are other differences that are or might become important. -/- Aristotle [fl. 350 BCE] meticulously separated his basic principles [archai, singular archê] according to subject matter: geometrical, arithmetic, astronomical, etc. However, he made no distinction that can be assimilated to Euclid’s postulate/axiom distinction. -/- Today we divide basic principles into non-logical [topic-specific] and logical [topic-neutral] but this too is not the same as Euclid’s. In this regard it is important to be cognizant of the difference between equality and identity—a distinction often crudely ignored by modern logicians. Tarski is a rare exception. The four angles of a rectangle are equal to—not identical to—one another; the size of one angle of a rectangle is identical to the size of any other of its angles. No two angles are identical to each other. -/- The sentence ‘Things that equal the same thing equal one another’ contains no occurrence of the word ‘magnitude’. This paper considers the problem of formalizing the proposition Euclid intended as a principle of magnitudes while being faithful to the logical form and to its information content. (shrink)

That space and time should be integrated into a single entity, spacetime, is the great insight of Einstein's special theory of relativity, and leads us to regard spacetime as a fundamental context in which to make sense of the world around us. But it is not the only one. Causality is equally important and at least as far as the special theory goes, it cannot be subsumed under a fundamentally geometrical form of explanation. In fact, the agent of propagation of (...) causal influence is electromagnetic radiation. In this examination, the authors find support for a rationalist approach to physics, never neglecting experimentation, but rejecting a simple empiricist or positivist view of science. (shrink)

Although our eyes receive incomplete and ambiguous information, our perceptual system is usually able to successfully construct a stable representation of the world. In the case of ambiguous figures, however, perception is unstable, spontaneously alternating between equally possible outcomes. The present study compared EEG responses to ambiguous figures and their unambiguous variants. We found that slight figural changes, which turn ambiguous figures into unambiguous ones, lead to a dramatic difference in an ERP (“event-related potential”) component at around 400 ms. This (...) result was obtained across two different categories of figures, namely the geometric Necker cube stimulus and the semantic Old/Young Woman face stimulus. Our results fit well into the Bayesian inference concept, which models the evaluation of a perceptual interpretation’s reliability for subsequent action planning. This process seems to be unconscious and the late.. (shrink)

A “frame of reference” is a standard relative to which motion and rest may be measured; any set of points or objects that are at rest relative to one another enables us, in principle, to describe the relative motions of bodies. A frame of reference is therefore a purely kinematical device, for the geometrical description of motion without regard to the masses or forces involved. A dynamical account of motion leads to the idea of an “inertial frame,” or a reference (...) frame relative to which motions have distinguished dynamical properties. For that reason an inertial frame has to be understood as a spatial reference frame together with some means of measuring time, so that uniform motions can be distinguished from accelerated motions. The laws of Newtonian dynamics provide a simple definition: an inertial frame is a reference-frame with a time-scale, relative to which the motion of a body not subject to forces is always rectilinear and uniform, accelerations are always proportional to and in the direction of applied forces, and applied forces are always met with equal and opposite reactions. It follows that, in an inertial frame, the center of mass of a system of bodies is always at rest or in uniform motion. It also follows that any other frame of reference moving uniformly relative to an inertial frame is also an inertial frame. For example, in Newtonian celestial mechanics, taking the “fixed stars” as a frame of reference, we can determine an inertial frame whose center is the center of mass of the solar system; relative to this frame, every acceleration of every planet can be accounted for as a gravitational interaction with some other planet in accord with Newton 's laws of motion. (shrink)

Book 1 of Euclid's Elements opens with a set of unproved assumptions: definitions, postulates, and ‘common notions’. The common notions are general rules validating deductions that involve the relations of equality and congruence. The attested postulates are five in number, even if a part of the manuscript tradition adds a sixth, almost surely spurious, that in some manuscripts features as the ninth, and last, common notion. The postulates are called αἰτήματα both in the manuscripts of the Elements and in (...) the ancient exegetic tradition. It is not said, however, that this denomination is original, as it coincides with the nomen rei actae associated with the verbal form introducing the postulates themselves. Since antiquity it has been recognized that the postulates naturally split into two groups of quite different character: the first three are rules licensing basic constructions, the fourth and the fifth are assertions stating properties of particular geometric objects. I transcribe postulates 1–3 in the text established by Heiberg : ᾐτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖνκαὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖνκαὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράϕεσθαι. (shrink)

Sydney Shoemaker argues that time without change is possible, but begs the question by assuming an, in effect, Newtonian absolute time, that 'flows equably' in a region in which there is no change and in one in which there is. An equally possible, relativist, assumption, consistent, it seems, with relativity theory, is that where nothing changes there is no time flow, though there may be elsewhere, where there is change. Such an assumption would require some revision of uncritical common thought (...) about time. Aristotle argues that there is no time without change but that time is not change. His arguments for the latter can be faulted both internally and again in terms of the same relativist assumption. From the Physics we can derive, though Aristotle himself did not, an argument that time is to change as geometrical space is to body: the thing itself in abstraction. (shrink)

For a subset A of a Polish group G, we study the (almost) packing index pack( A) (respectively, Pack( A)) of A, equal to the supremum of cardinalities |S| of subsets $S\subset G$ such that the family of shifts $\{xA\}_{x\in S}$ is (almost) disjoint (in the sense that $|xA\cap yA|<|G|$ for any distinct points $x,y\in S$). Subsets $A\subset G$ with small (almost) packing index are large in a geometric sense. We show that $\pack}(A)\in\mathbb{N}\cup\{\aleph_0,\mathfrak{c}\}$ for any σ-compact subset A of (...) a Polish group. In each nondiscrete Polish Abelian group G we construct two closed subsets $A,B\subset G$ with $\mathrm{pack}(A)=\mathrm{pack}(B)=\mathfrak{c}$ and \mathrm{Pack}(A\cup B)=1 and then apply this result to show that G contains a nowhere dense Haar null subset $C\subset G$ with pack(C)=Pack(C)=κ for any given cardinal number $\kappa\in[4,\mathfrak{c}]$. (shrink)

Prior evidence has shown that children’s understanding of balance proceeds through stages. Children go from a stage where they lack a consistent theory ( No Theory ), to becoming Center Theorists at around age 6 (believing that all objects balance in their geometric center), to Mass Theorists at around age 8, when they begin to consider the distribution of objects’ mass. In this study we adapted prior testing paradigms to examine 5-year-olds’ understanding of balance and compared children’s learning about (...) balance from evidence presented through primary sources (a guided activity) or secondary sources (picture books). Most of the research on young children’s understanding of balance has been conducted using a single object, weighted either proportionally (symmetrical object) or disproportionally (asymmetrical object). In this study, instead of using a single object, 5-year-olds ( N = 102) were shown 4 pairs of objects, two with the same weight and two with different weight. Children were told to place the objects on a beam where they thought they would balance. We found evidence for an intermediate level of understanding. Transition Theorists represent children who have two distinct theories, one for balancing same weight objects, and one for balancing different weight objects, but one of these theories is incorrect. Following the assessment of children’s understanding, we compared their learning about balance from evidence that was either presented through primary sources (a guided activity) or secondary sources (picture books). Children learn equally well from both sources of evidence. Findings are discussed in terms of theoretical and practical implications. (shrink)

We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E. In contrast to the well-known interpretation of higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE, but rather by a suitable complete Heyting algebra H. The canonical map relating H and ΩE both serves to interpret equality and provides a modal operator on H in the form of a comonad. Examples of such structures arise from surjective (...) class='Hi'>geometric morphisms f : F → E, where H = f∗ΩF. The logic differs from non-modal higher-order logic in that the principles of functional and propositional extensionality are not longer valid but may be replaced by modalized versions. The usual Kripke, neighborhood, and sheaf semantics for propositional and first-order modal logic are subsumed by this notion. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:TOWARD A GENERAL THEORY OF FICTION by James D. Parsons When nelson Goodman writes, "All fiction is literal, literary falsehood," he seems to be disregarding at least one noteworthy tradition.1 The tradition I have in mind includes works by Jeremy Bendiam, Hans Vaihinger, Tobias Dantzig, Wallace Stevens, and a host ofother writers in many fields who have been laboring for more man two centuries to clear the ground for (...) a general theory of fiction. Such a general dieory would, if its exponents are right about its relevemce, be situated at the very heart of die law, literature, religion, science, mathematics — in fact, of humem life itself. Fiction dien would be no neirrow litereiry falsehood, but a kind of pressure, eis Wedlace Stevens said, which die imagination exerts "against die pressure of reality," and which "helps us to live our lives,"2 or, in Hans Vaihingens metaphor, "a scaffolding afterwards to be demolished" 3 — that is, elfter it heis served its constructive purpose. A fiction is a vitally useful creation of die mind corresponding to nothing in the real world. Most people eire aware of some of these fictions, from the perfect geometric figures of Euclid to the black box of contemporary axiomatics. There are many odiers. The notion diat all human beings are equal before die law is a fiction that can serve opposing purposes: it can work as an instrument for securing greater justice in the courts and in society, or it can function as ceimouflage, helping to perpetuate em unjust legal system and em oppressive social order — emd probably in any actual situation it serves both purposes at once. In die science of mechanics the idea of a pure vacuum, in which moving objects meet with no resistance, was for some centuries a useful fiction diat made possible the exact mathematiced analysis of motion in memy forms — another useful fiction. The notion of normality in psychology — mat diere is a universal standard of reference for human behavior — is stdl utilized by clinical and laboratory workers in the field, in spite of the metssive sociological emd ethnographic criticisms that have been leveled against it. In science generally, the idea that 92 James D. Parsons93 diere is a community of experts who by dieir combined influence decide what the trudi is in every field is a fiction diat seems to have come to the end of its usefulness, diough some philosophers have written about it as if it were a reality and not a fiction at edl. The fictions I have cited so far are none of them literary, nor être diey fedse in the ordinetry meaning of the word. They are simply not true; they describe nothing mat exists, except in die mind, emd cem be considered false only if one forgets what they eire emd tries, illegitimately, to apply tests of trudi to them in a direct way. Among the fictions mat might be called literary are the fictions derived from mydis. Now, a myth is a story told in answer to a question or a series of questions that has never been asked, eidier about the real world or (most likely) about the perennial performance of a ritual. The mydi is designed to forestall the questioning, because the questions that might be raised would be too embarrassing or because questions about some aspects of reality ceuinot be formulated without distorting our relation to them. These are the aspects of reedity that can be viewed only out of die corner of the eye, that disappear when we focus our gaze upon diem, and mat can be identified only indirectly.4 The more explicitly fictional devices, too, of poetry, drama, and the novel develop out of the attempt to achieve some sort of cognizeuice of the seune aspects of reality that are related in mydis. I do not meetn to imply that literature, of which mydi is a part, is concerned only with such disconcerting or evanescent aspects of reality. Literature, like all the great realms of culture, is concerned with the whole of reality. It has to be, omerwise its more passionate, its more ediereed concerns would have no lodging place, no environment in which to maintain... (shrink)

Historians of modern philosophy have been paying increasing attention to contemporaneous scientific developments. Isaac Newton's Principia is of course crucial to any discussion of the influence of scientific advances on the philosophical currents of the modern period, and two philosophers who have been linked especially closely to Newton are John Locke and Immanuel Kant. My dissertation aims to shed new light on the ties each shared with Newtonian science by treating Newton, Locke, and Kant simultaneously. I adopt Newton's philosophy of (...) geometry as the starting point of investigation, for here I believe we have a constructive means by which to assess Locke and Kant's relationship to Newton, In particular, I defend the thesis that the justification Newton, Locke and Kant offer for applying geometrical principles to nature is central to understanding their respective ties to a Newtonian science characterized by the intermingling of mathematics and experiment, Although little is said by Locke in regard to a mathematical approach to nature, I hope to show that his interpretation of the origins of our geometrical ideas has a close affinity to Newton's own characterization of geometry, leading us to reexamine the extent of Locke's 'Newtonianism.' Kant famously attempts to bridge the gap between geometry and the empirical world by establishing space as a "pure form of intuition." My discussion of Kant's Newtonian approach to nature centers on the imagination, and I argue that the mediating work completed by this faculty in geometrical construction and experience in general is equally important to understanding Kant's application of geometry to the empirical realm, In the end, I hope my treatment of the strategies employed by Newton, Locke, and Kant to account for a mathematical-experimental method of natural philosophy sheds further light on the importance of Newton to the progress of modern philosophy. (shrink)

Elaborating the analogy between the sun and the good, Plato's Socrates tells Glaucon to divide a line αβ into two unequal segments at γ. The result is that αγ represents what is intelligible and γβ what is visible.1 Then Glaucon is to divide each of the two segments by the same ratio as he used in the original division.2 Whatever proportion he used to make the cuts γ, δ, and ε in the divided line, generating its four segments, the geometrical (...) implication is that the two middle segments must be equal in length. As both Nicholas D. Smith and Richard Foley have emphasized, when Socrates reiterates the characteristics of the line at 534a3–5, transposing δγ and γε, there should be no doubt that Plato... (shrink)

A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a “geometric technique” used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman’s theorem on the existence of rank-initial embeddings between countable non-standard models of the (...) fragment \ + \-Separation of \; and Gaifman’s technique of iterated ultrapowers is employed to show that any countable model of \ can be elementarily rank-end-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of “strong rank-cut” is characterized in terms of the theory \, and in terms of fixed-point sets of self-embeddings. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:A Hypothesis Concerning the Character of Islamic ArtAsli GocerWhy Islamic art has the distinctive features it has continues to generate clashing explanations. The Islamic visual treasury has no figural images, for instance, and three-dimensional sculpture or large scale oil painting, but instead contains miniatures, vegetal ornaments, arabesque surface patterns, and complex geometrical designs. To account for the phenomena the following radically opposing theories have been offered: the influence of (...) Judaism, the Quranic prohibition against making images, 1 Islamic resentment of the glory of Byzantine icons, 2 logocentricism of classical Islam, 3 the spiritual dimension of Islam, 4 tacit Islamic assertion of “otherness,” 5 and contingency of history. 6 The merits of these theories have been roundly discussed in the literature, and so I will not repeat the task here. I will insist that none of these theories offer a coherent explanation for all arts, but my presumption is that such an explanation is possible. In nonvisual arts the Muslim prefers simple tunes over that of the symphony, for instance, whirling dervishes over the ballet, and lyric poetry over dramatic literature. In my view what explains the common characteristic of all these arts is a certain kind of Islamic attitude informed by a certain philosophical outlook. Although it would be hasty to suggest the existence of a monolithic [End Page 683] Islamic art, it would be equally rash to deny some striking commonalities among its diverse examples. 7The following observation by a Muslim scholar is notable in this regard: “Whether in the great courtyard of the Delhi Mosque or the Qarawiyyin in Fez, one feels oneself within the same artistic and spiritual universe despite all the local variations in material, structural techniques, and the like.” 8 Some cite geometry as what mediates that feeling to the observer despite the cultural divergences in style. 9 Even those who have given up the search for the universal or the spiritual in Islamic art cannot help but see in it the classical ideal of beauty. 10 It is this classical heritage that in fact holds one of the important keys to the puzzle at hand. In particular I shall claim that Islamic art is the living image of what Plato has in mind for art in a theocratic state. Clearly, to refer to Plato and art in the same breath invites trouble. As is well enough known, giving a general interpretation of Plato’s views on the nature of art is a very complex endeavor. 11 Even in the face of grave difficulties in textual exegesis, however, there is sufficient evidence that Plato has some settled views on art. Those pertinent to our discussion here involve two claims, that beauty and goodness are divine and that all human endeavor (including artistic creation) must imitate the divine. 12 Although Platonic strain in Islam has been noted in other areas, its persistent legacy has been overlooked in art. 13 I shall argue that the Platonic influence on Islamic thought can be traced not only to political philosophy 14 and geometry, 15 as it has been commonly accepted, but also aesthetics. 16 A philosophical explanation of the Islamic attitude toward art may therefore be derivable from the Platonic axiom that god is the paradigm of beauty, which is the cosmological principle of [End Page 684] order and harmony; and as all created things, art, too, must reflect the divine paradigm.Before defending this ahistorical hypothesis on the character of Islamic art so reviled in recent literature, a few caveats are in order. “Islamic art” contains redundant terms. Strictly speaking all art in Islam is “Islamic,” because for the Muslim art is a crucial point of contact with the divine. That art should direct the mind to the divine is also Plato’s view. 17 Second, the categories that have traditionally been used to describe Islamic art such as “aniconic” and “nonrepresentational” are unhelpful for the problem at hand. 18 When it comes to the issue of iconography in Islamic arts, there is no consensus, for Islamicists are notably divided on the semiotic dimension of the Islamic visual language. The question of whether the geometric surface pattern is essentially iconography, for instance, is yet... (shrink)

The Interface Theory of Perception, as stated by D. Hoffman, says that perceptual experiences do not to approximate properties of an “objective” world; instead, they have evolved to provide a simplified, species-specific, user interface to the world. Conscious Realism states that the objective world consists of ‘conscious agents’ and their experiences. Under these two theses, consciousness creates all objects and properties of the physical world: the problem of explaining this process reverses the mind-body problem. In support of the interface theory (...) I propose that our perceptions have evolved, not to report the truth, but to guide adaptive behaviors. Using evolutionary game theory, I state a theorem asserting that perceptual strategies that see the truth will, under natural selection, be driven to extinction by perceptual strategies of equal complexity but tuned instead to fitness. I then give a minimal mathematical definition of the essential elements of a “conscious agent.” Under the conscious realism thesis, this leads to a non-dualistic, dynamical theory of conscious process in which both observer and observed have the same mathematical structure. The dynamics raises the possibility of emergence of combinations of conscious agents, in whose experiences those of the component agents are entangled. In support of conscious realism, I discuss two more theorems showing that a conscious agent can consistently see geometric and probabilistic structures of space that are not necessarily in the world per se but are properties of the conscious agent itself. The world simply has to be amenable to such a construction on the part of the agent; and different agents may construct different structures as seeming to belong to the world. This again supports the idea that any true structure of the world is likely quite different from what we see. I conclude by observing that these theorems suggest the need for a new theory which resolves the reverse mind-body problem, a good candidate for which is conscious agent theory. (shrink)

This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is (...) necessary for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed. (shrink)

We investigate the isochronous bifurcations of the straight-line librating orbit in the Hénon–Heiles and related potentials. With increasing scaled energy e, they form a cascade of pitchfork bifurcations that cumulate at the critical saddle-point energy e=1. The stable and unstable orbits created at these bifurcations appear in two sequences whose self-similar properties possess an analytical scaling behavior. Different from the standard Feigenbaum scenario in area preserving two-dimensional maps, here the scaling constants α and β corresponding to the two spatial directions (...) are identical and equal to the root of the scaling constant δ that describes the geometric progression of bifurcation energies en in the limit n→∞. The value of δ is given analytically in terms of the potential parameters. (shrink)

In the selection of job candidates who have the mental ability to become professional ATCOs, psychometric testing has been a ubiquitous activity in the ATM domain. To contribute to psychometric research in the ATM domain, we investigated the extent to which spatial orientation ability (SOA), as conceptualized in the spatial cognition and navigation literature, predicted air traffic conflict detection performance in a simulated free route airspace (FRA) environment. The implementation of free route airspace (FRA) over the past few years, notably (...) in Europe, have facilitated air traffic services by giving greater flexibility to aviation operators in planning and choosing preferred air routes that can lead to quicker arrivals. FRA offers enhanced system safety and efficiency, but these benefits can be outweighed by the introduction of air traffic conflicts that are geometrically more complex. Such conflicts can arise from increased number and distribution of conflict points, as well as from elevated uncertainty in aircraft maneuvering (for instance, during heading changes). Overall, these issues will make conflict detection more challenging for air traffic controllers (ATCOs). Consequently, there is a need to select ATCOs with suitably high levels of spatial orientation ability (SOA) to ensure flight safety under FRA implementation. In this study, we tested 20 participants who are eligible for ATCO job application, and found that response time-based performance on a newly developed, open access, computerized spatial orientation test (SOT) predicted time to loss of minimum separation (tLMS) performance on an air traffic conflict detection task (AT-CDT) we designed. We found this predictive relationship to be significant to a moderately large extent under scenarios with high air traffic density (raw regression coefficient = 0.58). Moreover, we demonstrated our AT-CDT as a valid test in terms of eliciting well-known mental workload and spatial learning effects. We explained these findings in light of similar or overlapping mental processes that were most likely activated optimally under task conditions featuring approximately equal numbers of outcome-relevant stimuli. We conclude by discussing the further application of the SOT to the selection of prospective ATCOs who can demonstrate high levels of conflict detection performance in FRA during training simulations. (shrink)

Historians of relativity theory have puzzled over the fact that, while Einstein regarded Ernst Mach as his chief philosophical mentor, Mach himself publicly rejected relativity in the preface to Die Prinzipien der physikalischen Optik. This work was first published by Mach's son Ludwig in 1921, five years after Mach's death, but the preface is dated “July 1913”, when Einstein was working on general relativity and believing not only that he had Mach's “friendly interest” and support, but also that his project (...) was the working-out of some of Mach's suggestions. To Einstein, whose sympathy for Mach's overall philosophy of science had already begun to wane by 1921, the posthumous appearance of the preface seemed to underscore the inconsistency between Machian positivism and his own program to construct an abstract and geometrical physics; this interpretation appears in important modern analyses like Blackmore, Holton, and Zahar, and it has frequently served the purposes of the philosophical reaction against logical positivism in general. Now Gereon Wolters' book challenges the usual interpretation with a startling claim: that Ernst Mach never wrote the preface, which in fact is a forgery by his son Ludwig. The words “A Forgery and its Consequences” suggest the sweeping consequences that the preface has had for our understanding of the relation between Mach and Einstein; the point of the book is not only to document the dramatic story of the forgery, but also to defend an equally sweeping reconsideration, indeed a rehabilitation, of Mach's philosophy and its role in the history of relativity. (shrink)

Intuitionistic epistemic logic by Artemov and Protopopescu (Rev Symb Log 9:266–298, 2016) accepts the axiom “if A, then A is known” (written $$A \supset K A$$ ) in terms of the Brouwer–Heyting–Kolmogorov interpretation. There are two variants of intuitionistic epistemic logic: one with the axiom “ $$KA \supset \lnot \lnot A$$ ” and one without it. The former is called $$\textbf{IEL}$$, and the latter is called $$\textbf{IEL}^{-}$$. The aim of this paper is to study first-order expansions (with equality and (...) function symbols) of these two intuitionistic epistemic logics. We define Hilbert systems with additional axioms called geometric axioms and sequent calculi with the corresponding rules to geometric axioms and prove that they are sound and complete for the intended semantics. We also prove the cut-elimination theorems for both sequent calculi. As a consequence, the disjunction property and existence property are established for the sequent calculi without geometric implications. Finally, we establish that our sequent calculi can also be formulated with admissible structural rules (i.e., in terms of a G3-style sequent calculus). (shrink)

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than (...) or equal to two in the sense that it is computably equivalent to Weak Kőnig’s Lemma. While we can present two independent proofs for dimension three and upward that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upward. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes. (shrink)

The proportional weight view in epistemology of disagreement generalizes the equal weight view and proposes that we assign to judgments of different people weights that are proportional to their epistemic qualifications. It is shown that if the resulting degrees of confidence are to constitute a probability function, they must be the weighted arithmetic means of individual degrees of confidence, while if the resulting degrees of confidence are to obey the Bayesian rule of conditionalization, they must be the weighted geometric (...) means of individual degrees of confidence. The double bind entails that the proportional weight view (and its moderate adjustment in favor of one’s own judgment) is inconsistent with Bayesianism. (shrink)

It has been argued that Hume's denial of infinite divisibility entails the falsity of most of the familiar theorems of Euclidean geometry, including the Pythagorean theorem and the bisection theorem. I argue that Hume's thesis that there are indivisibles is not incompatible with the Pythagorean theorem and other central theorems of Euclidean geometry, but only with those theorems that deal with matters of minuteness. The key to understanding Hume's view of geometry is the distinction he draws between a precise and (...) an imprecise standard of equality in extension. Hume's project is different from the attempt made by Berkeley in some of his later writings to save Euclidean geometry. Unlike Berkeley, who interprets the theorems of Euclidean geometry as false albeit useful approximations of geometrical facts, Hume is able to save most of the central theorems as true. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:66. ' HUME ON SPACE AND GEOMETRY * : A REJOINDER TO FLEW ' S 'ONE RESERVATION '.? Flew' s reservation about my assertion that the Enquiry contains no significant revision of the Treatise conception of geometry as a body of necessary and synthetic knowledge, appears to involve two charges. Firstly, he alleges that I dismiss but offer no substantial argument against his own view that the Enquiry restores (...) pure geometry "to its place alongside the other two elements of the trinity ". Secondly, he thinks that I have committed a serious sin of omission in failing to take into consideration a certain passage in the Enquiry (page 31, section 27) which, in his opinion, constitutes "a quite decisive reason" for supposing that Hume abandoned the Treatise position, in that it contains an account of applied mathematics, a subject on which the Treatise is silent. To the charge of omission I plead guilty, and I shall presently seek to remedy this defect and to state why I do not share Flew' s view of the significance of the said passage. To the former charge I am not so ready to plead guilty, and I shall counter-charge that Flew has failed not only to do justice to my proposed reading of Enquiry page 25, section 20, but also to provide clear and convincing support for his own. One difficulty encountered in answering Flew lies in understanding whether or not his claim about the status of geometry as a pure science in the Enquiry is put forward independently of his interpretation of E31. As originally presented in Hume's Philosophy of Belief this appears to be the case, and if so, on what evidence does it rest? 2 In his book, Flew' s statement about Hume replacing pure geometry alongside arithmetic and algebra follows directly upon a paragraph in which he comments that Hume's references to geometrical propositions at E25 signify "an important advance from the position of the Treatise". For whereas the status of perfect precision and certainty was 67. there withheld from geometry (T71), Hume now speaks of the certainty and evidence of the propositions of Euclidean geometry, and declares such propositions to be discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. (E25) Now, I argued at some length in my paper (pp. 21-24) that Hume is prepared to grant scientific status to geometry in the Enquiry because he has come to regard its certainty as that proper to a mathematics of space. He has abandoned the Treatise contrast between an ideal or theoretical standard of equality, viewed therein as necessary to geometry considered as a science proper, and the merely sensible standard of equality upon which geometry as practised depends. It should not be forgotten that Hume recognises two kinds of certainty where "Relations of Ideas" are concerned, intuitive and demonstrative. Since he fails to provide any detail of the place of each of these in mathematics, the door is left open for speculation along the lines followed in pp. 28-29 of my paper. For there is nothing in?25 which implies that the certainty of all geometrical propositions is of the same kind, or that the process of geometrical demonstration is of the same logical nature as demonstrative reasoning in arithmetic and algebra. In his comment on my position Flew makes no reference to my proposed reason why Hume classes geometry as a science in the Enquiry.· Yet, even if this reclassification does mark an advance of a kind, the fact remains that E25 does not furnish incontrovertible evidence for that advance which Flew thinks he finds there. Nor, therefore, is it conclusive against my assertion that Hume in both works regards the propositions of geometry as necessary and synthetic truths. Given that Flew does not wish to deny the inclusion of a synthetic element even in the Enquiry account of mathematics (see H. P. B. pp. 64-66), I find it difficult to see why he supposes that the logical status of the propositions of geometry is different in the Enquiry from that in the Treatise, since in the latter too they are held 68. to be truths which depend... (shrink)

Between ca. 400 and 50 BCE, Babylonian astronomers used mathematical methods for predicting ecliptical positions, times and other phenomena of the moon and the planets. Until recently these methods were thought to be of a purely arithmetic nature. A new interpretation of four Babylonian astronomical procedure texts with geometric computations has challenged this view. On these tablets, Jupiter’s total distance travelled along the ecliptic during a certain interval of time is computed from the area of a trapezoidal figure representing (...) the planet’s changing daily displacement along the ecliptic. Moreover, the time when Jupiter reaches half the total distance is computed by bisecting the trapezoid into two smaller ones of equal area. In the present paper these procedures are traced back to precursors from Old Babylonian mathematics. Some implications of the use of geometric methods by Babylonian astronomers are also explored. (shrink)

Habitat fragmentation produces isolated patches characterized by increased edge effects from an originally continuous habitat. The shapes of these patches often show a high degree of irregularity: their shapes deviate significantly from regular geometrical shapes such as rectangular and elliptical ones. In fractal theory, the geometry of patches created by a common landscape transformation process should be statistically similar, i.e. their fractal dimensions and their form factors should be equal. In this paper, we analyze 49 woodlot fragments (Pinus sylvestris L.) (...) in the Belgian Kempen region to study the direct relationship between a transformation process and the concomitant patch geometry. Although the fractal dimension of the woodlots is scattered (i.e. they are not statistically similar), the perimeter-area relation of the fragments is characterized by a single, ‘dimension-like’ exponent. This exponent suggests a certain shape homogeneity among the patches, which is confirmed by the absence of hierarchical levels associated with sharp increases of the fractal dimension at scale transitions. The interaction of different natural (soil factor, vegetation type) and anthropogenic (afforestation, urbanization) processes during patch development is assumed to have generated this feature. Comparison of the area and perimeter fractal dimension with an ecological index for habitat quality, the interior-to-edge ratio, shows that the fractal dimension is suitable for predicting interior habitat presence, which is more likely for patches with smooth perimeters and compact areas. The ratio of the area to the perimeter fractal dimension confirms this observation, with high values for high interior-to-edge ratios, characteristic for regularly shaped patches. (shrink)

Chapter 1 First Person Access to Mental States. Mind Science and Subjective Qualities -/- Abstract. The philosophy of mind as we know it today starts with Ryle. What defines and at the same time differentiates it from the previous tradition of study on mind is the persuasion that any rigorous approach to mental phenomena must conform to the criteria of scientificity applied by the natural sciences, i.e. its investigations and results must be intersubjectively and publicly controllable. In Ryle’s view, philosophy (...) of mind needs to adopt an antimentalist stance to achieve this aim. Antimentalism not only definitively rejects the idea that mind is a substance separated from the body, it also denies that mental phenomena radically differ from physical phenomena by virtue of several unique features. Most problematically, mental phenomena have a conscious character (mental states are related to specific qualitative feelings) and are accessible only to the first-person (only the subject knows directly what s/he is experiencing inside his/her mind). Ryle takes a strong stance on antimentalism going so far as to maintain that an approach to mind which aims to meet the criteria of scientificity set by the natural sciences must avoid any reference to internal, unobservable mental states. In his view (which is considered a specifically philosophical version of psychological behaviorism and also addresses questions put forward in psychological research), mental states can be redescribed in terms of behavioral dispositions. In this chapter, we address the historical roots of the antimentalist view and analyze its relation to the later tradition of research on mind. We show that, compared to the antimentalist stance, functionalism and cognitivism take a step back when they maintain that direct reference to mental states is necessary since mental states cause and therefore explain human behavior. This step backwards is often interpreted as a return to mentalism. However, this is only partially true. Indeed, we suggest that these later traditions retain one important element of Ryle’s antimentalism, i.e. the idea that mental states must be uniquely identified using external and publicly observable criteria, while excluding any reference to introspection and those qualitative dimensions of a mental state, which are accessible only to the first-person. According to the perspective we put forward, this epistemological stance has continued to influence contemporary research on mind and current philosophical and psychological theories which both tend to exclude the subjective qualities of human experience from their accounts of how the mind works. The issue we raise here is whether this is legitimate or whether subjective qualities do play a role with respect to the way our mind works. The conclusion of this chapter anticipates the argument the book makes in in favor of this latter position, starting from a particular angle, i.e. the problem of how we categorize concepts related to our internal states. -/- Chapter 2 The Misleading Aspects of the Mind/Computer Analogy. The Grounding Problem and the Thorny Issue of Propriosensitive Information -/- Abstract. After the crisis of behaviorism, cognitivism and functionalism became the predominant models in the field of psychology and of philosophy, respectively. Their success is mainly due to the new key they use for interpreting mental processes: the mind/computer analogy. On the basis of this analogy, mental operations are seen as cognitive processes based on computations, i.e. on manipulations of abstract symbols which are in turn understood as informational unities (representations). This chapter identifies two main problems with this model. The first is how these symbols can relate to and communicate with perception and thus allow us to identify and classify what we perceive through the senses. Here we limit ourselves to presenting this issue in relation to the classical symbol grounding problem originally put forward by Harnad on the basis of Searle’s Chinese room argument. An attempt to address the problem raised here will be made in chap. 3. The second point we discuss in relation to the mind/computer analogy concerns the idea of information it fosters. Indeed, following this analogy, information is something available in the external world which can be captured by the senses and transmitted to the central system without being influenced or modified by the procedures of transmission. This perspective does not take into account that – unlike computers – in living beings information is acquired by means of the body. As Ulric Neisser has already pointed out, the body is itself an informational source that provides us with additional sensory experience that influences (modifies or complements) the information extracted from the external world by the senses. To develop this line of analysis and to determine exactly what information is provided by the body and how this might influence cognition, we examine Sherrington’s and Gibson’s positions. Moving on from their views, we qualify bodily information in terms of ‘proprioception’. We use ‘proprioception’ in a broad sense to describe any kind of experience we have of our internal states (including postural information as well as sensations related to the general state of the body and its parts). Following Damasio’s and Craig’s studies, we further elaborate this position, arguing that living beings are equipped with an internal propriosensitive monitoring system which maps all the changes that constantly occur in our body and that give us perceptual (‘proprioceptive’ or propriosensitive) information about what happens inside us. Moreover, relying on Goldie’s and Ratcliffe’s view, we show that emotional information can also be considered as a form of ‘proprioception’ which contributes to determining everything we perceive. This analysis leads us to the second main thesis of this book: ‘proprioception’ is a form of internal perception and it is an essential component of the sensory information we can access and use for all cognitive purposes. -/- Chapter 3 Semantic Competence from the Inside: Conceptual Architecture and Composition -/- Abstract. Concepts are essential constituents of thought: they are the instruments we use to categorize our experience, i.e. to classify things and group them together in homogeneous sets. Here we define concepts as the internal mental information (representations) that allows us, among other things, to master words in natural language. By analyzing the way in which individuals master word meanings we explore a number of hypotheses regarding the nature of concepts. Following Diego Marconi’s research, we differentiate between two kind of abilities that underpin lexical competence – so-called ‘referential’ and ‘inferential competence’ – and we suggest that, in order to support these abilities, concepts must also include two corresponding kinds of information, i.e. inferential and referential information. We point out that the most widely used and acknowledged theories of concepts do not make this distinction, instead broadly characterizing the information used for categorization in terms of propositionally described feature lists. However, we show that while feature lists can explain inferential competence, they do not account for referential competence. To address the issue of referential competence we examine Ray Jackendoff’s hypothesis that to account for the possibility of linking perceptual and conceptual information we need to assume the existence of a (visual) representation that encodes the geometric and topological properties of objects and bridges the gap between the percept and the concept. Furthermore, we analyze the extension of this work by Jesse Prinz who introduced the notion of a proxytype, a perceptual representation of a class of objects that incorporates structural and parametric information related to their appearance. However, as we point out, proxytypes can only explain the relationship between perception and concepts with respect to instances that can be perceived through the senses and that belong to the same class by virtue of their physical similarity. We suggest that this notion be extended to include larger conceptual classes. To accomplish this, we further develop Mark Johnson, George Lakoff and Jean Mandler’s idea of a schematic image and argue that conceptual representations include a perceptual schema. Perceptual schemata are non-linguistic, structured experiential gestalts (patterns or maps) that make use of information taken from all sensory modalities, including body perception. They accomplish a quasi-conceptual function: they allow us to recognize and to classify different instances. In this work, we hypothesize that perceptual schemata are an essential component of concepts, but not identical to them. Instead, we suggest that concepts include both perceptual and propositional information with perceptual schemata providing the ‘perceptual core concept’ that grounds related propositional information. -/- Chapter 4 In the Beginning There Were Categories. The Bodily Origin of Prelinguistic Categorical Organization: The Example of Folkbiological Taxonomies -/- Abstract. Studies of categorization in psychology and the cognitive sciences have made use of the notions of ‘category’ and ‘concept’ without precisely defining what is meant by either; in fact, often these terms have been used as synonyms, making it difficult to address specific issues related to conceptual development. This chapter begins by discussing the definitions of, as well as the distinctions between, ‘categories’ and ‘concepts’ in the classical philosophical tradition (Aristotle, Kant and Husserl). We introduce our view of categories with reference to Husserl. Categorization is defined as the way in which our experience is originally (pre-linguistically) organized in a passive and fully unconscious manner on the basis of universal structuring principles. Conceptualization is explained as a later process in which the earlier categorical macro-classes are further subdivided into more specific and detailed sets; this later process also relies on linguistic learning and exposure to culture. On the basis of Ray Jackendoff, Jean Mandler, George Lakoff and Mark Johnson’s work, we hypothesize that categorization is a two-step process that begins with the formation of homogeneous sets of entities partitioned into regions that describe the ontological boundaries of the objects that humans perceive (categories) and continues at a later stage with the development of more specific classifications (concepts). In this chapter, we mainly address three related issues: Why should we assume that there is categorical organization which precedes the development of a conceptual system? How do categories and concepts relate to each other? Shall we hypothesize that categories are innate or that they are formed before concepts on the basis of information and organizational structure available at a very early developmental stage? We show that categorical partitions are necessary for categorization and, following Mandler, that the general categories we form at an early age do not match our adult superordinate concepts. As for the third issue, we argue that there might be no need to assume that categories are innately present in the human mind, since their formation can be explained – at least in certain cases – by basic mechanisms that work on body (propriosensitive) information. This hypothesis will not be discussed in general, but in relation to a particularly relevant example of categorical partition, i.e. the folk-biological dichotomy between ANIMATE/INANIMATE. This is compared with other dichotomies that derive from it, but are not directly categorical such as LIVING/NON-LIVING and BIOLOGICAL/NON-BIOLOGICAL. -/- Chapter 5 Internal States: From Headache to Anger. Conceptualization and Semantic Mastery -/- Abstract. Here we ask if we can also apply the distinction between referential and inferential competence we introduced in Chapter 3 to words that do not refer to things that are perceived using the external senses, especially to words/concepts that denote bodily experiences (such as pain, thirst, hunger, etc.) or emotions. We introduce and discuss the hypothesis that – even though such words/concepts do not refer to intersubjectively identifiable entities in the external world – they do have a kind of referent that can be accessed via direct perception, more specifically ‘proprioception’, as we have defined it in terms of all propriosensitive information we can consciously access. In the first part of the chapter, we specifically consider terms denoting bodily experiences such as ‘pain’ or ‘hunger’ and argue that their referents are identified and classified from a first-personal point of view on the basis of four main characteristics: their specific intensity, their localization in the body, their co-occurrence with other signals and above all their specific qualitative sensations. Emotions are addressed in the second part of the chapter. We suggest that there is a continuity between bodily experiences and emotions. In particular, we argue for a perceptual theory of emotions in line with that proposed by James and Lange at the end of the 19th century and developed more recently by authors such as Damasio (see also chap. 2§5, §6). The hypothesis we put forward is that the referential information that supports the categorization of emotions and therefore also our mastery of terms referring to emotions consists in the perception (i.e. the ‘proprioception’) of those bodily states and changes in bodily states which constitute our emotional experience. In the context of this discussion we examine some objections to this line of reasoning that arise from a cognitivist perspective and following authors such as Oatley, Johnson-Laird and Frijda, we distinguish between basic emotions that can be identified and classified solely on the basis on how they feel and complex emotions whose identification and classification additionally depends on cognitive factors. To describe how emotions are identified and classified on the basis of how they feel, we rely on Marcel and Lambie’s distinction between an ‘emotion state’ and an ‘emotion experience’. Both notions indicate kinds of feelings that we consciously experience. However, they describe first-order and second order emotion awareness respectively. The emotion state is the feeling we have of the bodily states and changes that occur when we are experiencing an emotion, while the emotion experience is the fully developed and integrated emotion we both experience and are, with reflection, aware of experiencing. On the basis of this differentiation, we also show that the same characteristics that aid in the identification and classification of bodily experiences (specific qualitative sensations; somatic localization; specific intensity; presence/absence of specific concomitant sensations) can also be used for the identification and classification of emotions – at least basic emotions. In the last two sections of the chapter we present some clinical evidence on the semantic competence of people who suffer from Alexithymia and Autism Spectrum Disorder which supports the conclusions of our preceding analyses. -/- Chapter 6 The ‘Proprioceptive’ Component of Abstract Concepts -/- Abstract. In this chapter, we address the issue of whether the mastery of abstract words requires only inferential knowledge and thus, if the concepts that support the mastery of abstract words include only linguistic information. We start by differentiating the notions of ‘abstract’ and ‘general’ which are often erroneously confused. We then identify a strict definition of abstract, as contrasted with ‘concrete’, that applies to words or concepts whose referent cannot be experienced by the senses. We argue that abstract words/concepts would be better described as theoretical, because they are usually conceived as structured sets of inferential knowledge expressed linguistically; that is, as small theories. Pointing out parallels with Carnap’s analysis of this issue in philosophy of science, we hypothesize that words/concepts denoting non-observable entities are not all ‘equally theoretical’, because their link to sensory experience can be stronger or weaker. We revive the distinction, inspired by Quine, between theoretical and intertheoretical concepts/words. This distinction relies on the fact that the former – unlike the latter – retains a strong, although indirect, connection with perception. In Quine’s discussion, perception is understood uniquely in terms of observability, i.e. of external sensory experience. Here we argue, however, that bodily, ‘proprioceptive’ (i.e. propriosensitive) experience can also serve to referentially ground theoretical (i.e. abstract) concepts/words. We frame this issue using the example of the theoretical concept ‘freedom’ and Lakoff’s hypothesis that this concept is developed on the basis of bodily information. We contrast this with the example of ‘democracy’ which more closely resembles an intertheoretical concept/word. Furthermore, we show that one of the classical views put forward in psycholinguistic research to explain how abstract concepts are mentally represented – i.e. Paivio’s Dual Coding Theory – points in the same direction as our analysis. The same is true of Barsalou’s work suggesting that we use internal information to understand at least some abstract words. To sustain this position, we put forward two lines of evidence: the first comes from psycholinguistic studies while the second examines deficits of semantic competence exhibited by people with Autism Spectrum Disorder. On the basis of our analysis, we put forward a classification that distinguishes between different kinds of concreteness and different degrees of abstraction: concepts/words referring to body experiences and basic emotions are described as analogous to concrete concepts/words because they are grounded in perceptual (i.e. propriosensitive) experience, while abstract concepts/words are considered more or less abstract depending on whether they are intratheoretical (and rely entirely on inferential information) or theoretical (and are partially grounded in perceptual – or more often in propriosensitive perceptual – information). In the last section of the chapter we consider two scales that have been used in psycholinguistic research to measure the degree of concreteness vs. abstractness of words and we show that – used conjointly – they can provide a measure of the internal vs. external grounding of specific words. (shrink)

In 1803 Louis Poinsot published a textbook on statics, in which he made clear that the subject dealt not only with forces but also with 'couples' (his word), pairs of coplanar non-collinear forces equal in magnitude and direction but opposite in sense. His innovation was not understood or even welcomed by some contemporary mathematicians. Later he adapted his theory to put forward a new relationship between rectilinear and rotational motion in dynamics; its reception was more positive, although not always appreciative (...) of the generality. After summarising the creation of these two theories and noting their respective receptions, this paper considers his advocacy of spatial and geometrical thinking in mechanics and the fact that, despite its importance, historians of statics who cover his period usually ignore his theory of couples. (shrink)

This dissertation explores several issues related to the CPT theorem. Chapter 2 explores the meaning of spacetime symmetries in general and time reversal in particular. It is proposed that a third conception of time reversal, 'geometric time reversal', is more appropriate for certain theoretical purposes than the existing 'active' and 'passive' conceptions. It is argued that, in the case of classical electromagnetism, a particular nonstandard time reversal operation is at least as defensible as the standard view. This unorthodox time (...) reversal operation is of interest because it is the classical counterpart of a view according to which the so-called 'CPT theorem' of quantum field theory is better called 'PT theorem'; on this view, a puzzle about how an operation as apparently non-spatio-temporal as charge conjugation can be linked to spacetime symmetries in as intimate a way as a CPT theorem would seem to suggest dissolves. In chapter 3, we turn to the question of whether the CPT theorem is an essentially quantum-theoretic result. We state and prove a classical analogue of the CPT theorem for systems of tensor fields. This classical analogue, however, appears not to extend to systems of spinor fields. The intriguing answer to our question thus appears to be that the CPT theorem for spinors is essentially quantum-theoretic, but that the CPT theorem for tensor fields applies equally to the classical and quantum cases. Chapter4 explores a puzzle that arises when one puts the CPT theorem alongside a standard way of understanding spacetime symmetries, according to which spacetime symmetries are to be understood in terms of background spacetime structure. The puzzle is that a 'PT theorem' amounts to a statement that the theory may not make essential use of a preferred direction of time, and this seems odd. We propose a solution to that puzzle for the case of tensor field theories. (shrink)

Defences of Kant's foundations of geometry fall short if they are unable to equally ground Euclidean and non-Euclidean geometries. Thus, Kant's account must be separated from geometrical postulates. I argue that characterizing space as the form of outer intuition must be independent of postulates. Geometrical postulates are then expressions of particular spatializing activities made possible by the a priori intuition of space. While Amit Hagar contends that this is to speak of noumena, I argue that a Kantian account of space (...) as the form of outer attention-directing remains seated in the subject. (shrink)

An apparent paradox is obtained in all previous treatments of the Trouton–Noble experiment; there is a three-dimensional (3D) torque T in an inertial frame S in which a thin parallel-plate capacitor is moving, but there is no 3D torque T′ in S′, the rest frame of the capacitor. Different explanations are offered for the existence of another 3D torque, which is equal in magnitude but of opposite direction giving that the total 3D torque is zero. In this paper, it is (...) considered that 4D geometric quantities and not the usual 3D quantities are well-defined both theoretically and experimentally in the 4D spacetime. In analogy with the decomposition of the electromagnetic field F (bivector) into two 1-vectors E and B we introduce decomposition of the 4D torque N (bivector) into 1-vectors N s , N t . It is shown that in the frame of “fiducial” observers, in which the observers who measure N s and N t are at rest, and in the standard basis, only the spatial components $N_{s}^{i}$ and $N_{t}^{i}$ remain, which can be associated with components of two 3D torques T and T t . In such treatment with 4D geometric quantities the mentioned paradox does not appear. The presented explanation is in complete agreement with the principle of relativity and with the Trouton–Noble experiment without the introduction of any additional torque. (shrink)